NAG CL Interface e04nqc (qpconvex2_sparse_solve)
Note:this function usesoptional parametersto define choices in the problem specification and in the details of the algorithm. If you wish to use default settings for all of the optional parameters, you need only read Sections 1 to 10 of this document. If, however, you wish to reset some or all of the settings please refer to Section 11 for a detailed description of the algorithm, to Section 12 for a detailed description of the specification of the optional parameters and to Section 13 for a detailed description of the monitoring information produced by the function.
e04nqc solves sparse linear programming or convex quadratic programming problems. The initialization function e04npcmust have been called before calling e04nqc.
to obtain the current value of an optional parameter.
3Description
e04nqc is designed to solve large-scale linear or quadratic programming problems of the form:
(1)
where is an -vector of variables, and are constant lower and upper bounds, is an sparse matrix and is a linear or quadratic objective function that may be specified in a variety of ways, depending upon the particular problem being solved. The optional parameter may be used to specify a problem in which is maximized instead of minimized.
Upper and lower bounds are specified for all variables and constraints. This form allows full generality in specifying various types of constraint. In particular, the th constraint may be defined as an equality by setting . If certain bounds are not present, the associated elements of or may be set to special values that are treated as or .
The possible forms for the function are summarised in Table 1. The most general form for is
where is a constant, is a constant -vector and is a constant symmetric matrix with elements . In this form, is a quadratic function of and (1) is known as a quadratic program (QP). e04nqc is suitable for all convex quadratic programs. The defining feature of a convex QP is that the matrix must be positive semidefinite, i.e., it must satisfy for all . If not, is nonconvex and e04nqc will terminate with the error indicator NE_HESS_INDEF. If is nonconvex it may be more appropriate to call e04src instead.
Table 1
Choices for the objective function
Problem type
Objective function
Hessian matrix
FP
Not applicable
LP
QP
Symmetric positive semidefinite
If , then and the problem is known as a linear program (LP). In this case, rather than defining an with zero elements, you can define to have no columns by setting (see Section 5).
If , , and , there is no objective function and the problem is a feasible point problem (FP), which is equivalent to finding a point that satisfies the constraints on . In the situation where no feasible point exists, several options are available for finding a point that minimizes the constraint violations (see the description of the optional parameter ).
e04nqc is suitable for large LPs and QPs in which the matrix is sparse, i.e., when the number of zero elements is sufficiently large that it is worthwhile using algorithms which avoid computations and storage involving zero elements. The matrix is input to e04nqc by means of the three array arguments acol, inda and loca. This allows you to specify the pattern of nonzero elements in .
e04nqc exploits structure in by requiring to be defined implicitly in a function
that computes the product for any given vector . In many cases, the product can be computed very efficiently for any given , e.g., may be a sparse matrix, or a sum of matrices of rank-one.
For problems in which can be treated as a dense matrix, it is usually more efficient to use e04mfc,e04nccore04nfc.
There is considerable flexibility allowed in the definition of in Table 1. The vector defining the linear term can be input in three ways: as a sparse row of ; as an explicit dense vector ; or as both a sparse row and an explicit vector (in which case, will be the sum of two linear terms). When stored in , is the iobjth row of , which is known as the objective row. The objective row must always be a free row of in the sense that its lower and upper bounds must be and . Storing as part of is recommended if is a sparse vector. Storing as an explicit vector is recommended for a sequence of problems, each with a different objective (see arguments c and lenc).
The upper and lower bounds on the elements of are said to define the general constraints of the problem. Internally, e04nqc converts the general constraints to equalities by introducing a set of slack variables , where . For example, the linear constraint is replaced by , together with the bounded slack . The problem defined by (1) can, therefore, be re-written in the following equivalent form:
Since the slack variables are subject to the same upper and lower bounds as the elements of , the bounds on and can simply be thought of as bounds on the combined vector . (In order to indicate their special role in QP problems, the original variables are sometimes known as ‘column variables’, and the slack variables are known as ‘row variables’.)
Each LP or QP problem is solved using a two-phase iterative procedure (in which the general constraints are satisfied throughout): a feasibility phase (Phase 1), in which the sum of infeasibilities with respect to the bounds on and is minimized to find a feasible point that satisfies all constraints within a specified feasibility tolerance; and an optimality phase (Phase 2), in which is minimized (or maximized) by constructing a sequence of iterates that lies within the feasible region.
Phase 1 involves solving a linear program of the form
Phase 1
which is equivalent to minimizing the sum of the constraint violations. If the constraints are feasible (i.e., at least one feasible point exists), eventually a point will be found at which both and are zero. Then the associated value of satisfies the original constraints and is used as the starting point for the Phase 2 iterations for minimizing .
If the constraints are infeasible (i.e., or at the end of Phase 1), no solution exists for (1) and you have the option of either terminating or continuing in so-called elastic mode (see the discussion of the optional parameter ). In elastic mode, a ‘relaxed’ or ‘perturbed’ problem is solved in which is minimized while allowing some of the bounds to become ‘elastic’, i.e., to change from their specified values. Variables subject to elastic bounds are known as elastic variables. An elastic variable is free to violate one or both of its original upper or lower bounds. You are able to assign which bounds will become elastic if elastic mode is ever started (see the argument helast in Section 5).
To make the relaxed problem meaningful, e04nqc minimizes while (in some sense) finding the ‘smallest’ violation of the elastic variables. In the situation where all the variables are elastic, the relaxed problem has the form
Phase 2 ()
,
where is a non-negative argument known as the elastic weight (see the description of the optional parameter ), and is called the composite objective. In the more general situation where only a subset of the bounds are elastic, the 's and 's for the non-elastic bounds are fixed at zero.
The elastic weight can be chosen to make the composite objective behave like the original objective , the sum of infeasibilities, or anything in-between. If , e04nqc will attempt to minimize subject to the (true) upper and lower bounds on the non-elastic variables (and declare the problem infeasible if the non-elastic variables cannot be made feasible).
At the other extreme, choosing sufficiently large will have the effect of minimizing the sum of the violations of the elastic variables subject to the original constraints on the non-elastic variables. Choosing a large value of the elastic weight is useful for defining a ‘least-infeasible’ point for an infeasible problem.
In Phase 1 and elastic mode, all calculations involving and are done implicitly in the sense that an elastic variable is allowed to violate its lower bound (say) and an explicit value of can be recovered as .
A constraint is said to be active or binding at if the associated element of either or is equal to one of its upper or lower bounds. Since an active constraint in has its associated slack variable at a bound, the status of both simple and general upper and lower bounds can be conveniently described in terms of the status of the variables . A variable is said to be nonbasic if it is temporarily fixed at its upper or lower bound. It follows that regarding a general constraint as being active is equivalent to thinking of its associated slack as being nonbasic.
At each iteration of an active-set method, the constraints are (conceptually) partitioned into the form
where consists of the nonbasic elements of and the basis matrix is square and nonsingular. The elements of and are called the basic and superbasic variables respectively; with they are a permutation of the elements of and . At a QP solution, the basic and superbasic variables will lie somewhere between their upper or lower bounds, while the nonbasic variables will be equal to one of their bounds. At each iteration, is regarded as a set of independent variables that are free to move in any desired direction, namely one that will improve the value of the objective function (or sum of infeasibilities). The basic variables are then adjusted in order to ensure that continues to satisfy . The number of superbasic variables ( say), therefore, indicates the number of degrees of freedom remaining after the constraints have been satisfied. In broad terms, is a measure of how nonlinear the problem is. In particular, will always be zero for FP and LP problems.
If it appears that no improvement can be made with the current definition of , and , a nonbasic variable is selected to be added to , and the process is repeated with the value of increased by one. At all stages, if a basic or superbasic variable encounters one of its bounds, the variable is made nonbasic and the value of is decreased by one.
Associated with each of the equality constraints is a dual variable . Similarly, each variable in has an associated reduced gradient (also known as a reduced cost). The reduced gradients for the variables are the quantities , where is the gradient of the QP objective function, and the reduced gradients for the slack variables are the dual variables . The QP subproblem is optimal if for all nonbasic variables at their lower bounds, for all nonbasic variables at their upper bounds and for all superbasic variables. In practice, an approximate QP solution is found by slightly relaxing these conditions on (see the description of the optional parameter ).
The process of computing and comparing reduced gradients is known as pricing (a term first introduced in the context of the simplex method for linear programming). To ‘price’ a nonbasic variable means that the reduced gradient associated with the relevant active upper or lower bound on is computed via the formula , where is the th column of . (The variable selected by such a process and the corresponding value of (i.e., its reduced gradient) are the quantities +SBS and dj in the monitoring file output; see Section 9.1.) If has significantly more columns than rows (i.e., ), pricing can be computationally expensive. In this case, a strategy known as partial pricing can be used to compute and compare only a subset of the s.
e04nqc is based on SQOPT, which is part of the SNOPT package described in Gill et al. (2005a). It uses stable numerical methods throughout and includes a reliable basis package (for maintaining sparse factors of the basis matrix ), a practical anti-degeneracy procedure, efficient handling of linear constraints and bounds on the variables (by an active-set strategy), as well as automatic scaling of the constraints. Further details can be found in Section 11.
4References
Fourer R (1982) Solving staircase linear programs by the simplex method Math. Programming23 274–313
Gill P E and Murray W (1978) Numerically stable methods for quadratic programming Math. Programming14 349–372
Gill P E, Murray W and Saunders M A (1995) User's guide for QPOPT 1.0: a Fortran package for quadratic programming Report SOL 95-4 Department of Operations Research, Stanford University
Gill P E, Murray W and Saunders M A (2005a) Users' guide for SQOPT 7: a Fortran package for large-scale linear and quadratic programming Report NA 05-1 Department of Mathematics, University of California, San Diego https://www.ccom.ucsd.edu/~peg/papers/sqdoc7.pdf
Gill P E, Murray W and Saunders M A (2005b) Users' guide for SNOPT 7.1: a Fortran package for large-scale linear nonlinear programming Report NA 05-2 Department of Mathematics, University of California, San Diego https://www.ccom.ucsd.edu/~peg/papers/sndoc7.pdf
Gill P E, Murray W, Saunders M A and Wright M H (1987) Maintaining factors of a general sparse matrix Linear Algebra and its Applics.88/89 239–270
Gill P E, Murray W, Saunders M A and Wright M H (1989) A practical anti-cycling procedure for linearly constrained optimization Math. Programming45 437–474
Gill P E, Murray W, Saunders M A and Wright M H (1991) Inertia-controlling methods for general quadratic programming SIAM Rev.33 1–36
Hall J A J and McKinnon K I M (1996) The simplest examples where the simplex method cycles and conditions where EXPAND fails to prevent cycling Report MS 96–100 Department of Mathematics and Statistics, University of Edinburgh
5Arguments
The first entries of the arguments bl, bu, hs and x refer to the variables . The last entries refer to the slacks .
1: – Nag_StartInput
On entry: indicates how a starting basis (and certain other items) will be obtained.
Requests that an internal Crash procedure be used to choose an initial basis, unless a Basis file is provided via optional parameters , or .
Is the same as but is more meaningful when a Basis file is given.
Means that a basis is already defined in hs and a start point is already defined in x (probably from an earlier call).
Constraint:
, or .
2: – function, supplied by the userExternal Function
For QP problems, you must supply a version of qphx to compute the matrix product for a given vector . If has rows and columns of zeros, it is most efficient to order so that the nonlinear variables appear first. For example, if and only enters the objective quadratically, then
(2)
In this case, ncolh should be the dimension of , and qphx should compute . For FP and LP problems, qphx will never be called by e04nqc and hence qphx may be specified as NULLFN.
On entry: this is the same argument ncolh as supplied to e04nqc.
2: – const doubleInput
On entry: the first ncolh elements of the vector .
3: – doubleOutput
On exit: the product . If ncolh is less than the input argument n, is really the product in (2).
4: – IntegerInput
On entry: allows you to save computation time if certain data must be read or calculated only once. To preserve this data for a subsequent calculation place it in comm.
There is nothing special about the current call of qphx.
e04nqc is calling qphx for the last time. This argument setting allows you to perform some additional computation on the final solution.
The current is optimal.
The problem appears to be infeasible.
The problem appears to be unbounded.
The iterations limit was reached.
5: – Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to qphx.
user – double *
iuser – Integer *
p – Pointer
The type Pointer will be void *. Before calling e04nqc you may allocate memory and initialize these pointers with various quantities for use by qphx when called from e04nqc (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
Note:qphx should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by e04nqc. If your code inadvertently does return any NaNs or infinities, e04nqc is likely to produce unexpected results.
3: – IntegerInput
On entry: , the number of general linear constraints (or slacks). This is the number of rows in the linear constraint matrix , including the free row (if any; see iobj). Note that must have at least one row. If your problem has no constraints, or only upper or lower bounds on the variables, then you must include a dummy row with sufficiently wide upper and lower bounds (see also acol, inda and loca).
Constraint:
.
4: – IntegerInput
On entry: , the number of variables (excluding slacks). This is the number of columns in the linear constraint matrix .
Constraint:
.
5: – IntegerInput
On entry: the number of nonzero elements in .
Constraint:
.
6: – IntegerInput
On entry: the number of column (i.e., variable) and row names supplied in the array names.
There are no names. Default names will be used in the printed output.
All names must be supplied.
Constraint:
or .
7: – IntegerInput
On entry: the number of elements in the constant objective vector .
If , the first lenc elements of belong to variables corresponding to the constant objective term .
Constraint:
.
8: – IntegerInput
On entry: , the number of leading nonzero columns of the Hessian matrix . For FP and LP problems, ncolh must be set to zero.
The first ncolh elements of belong to variables corresponding to the nonzero block of the QP Hessian.
Constraint:
.
9: – IntegerInput
On entry: if , row iobj of is a free row containing the nonzero elements of the vector appearing in the linear objective term .
If , there is no free row, and the linear objective vector should be supplied in array c.
Constraint:
.
10: – doubleInput
On entry: the constant , to be added to the objective for printing purposes. Typically .
11: – const char *Input
On entry: the name for the problem. It is used in the printed solution and in some functions that output Basis files. Only the first eight characters of prob are significant.
12: – const doubleInput
On entry: the nonzero elements of , ordered by increasing column index. Note that all elements must be assigned a value in the calling program.
13: – const IntegerInput
On entry: must contain the row index of the nonzero element stored in , for . Thus a pair of values contains a matrix element and its corresponding row index.
If , the first lenc elements of acol and inda belong to variables corresponding to the constant objectiver term .
If the problem has a quadratic objective, the first ncolh columns of acol and inda belong to variables corresponding to the nonzero block of the Hessian. Function qphx knows about these variables.
Note that the row indices for a column must lie in the range to m, and may be supplied in any order.
Constraint:
, for .
14: – const IntegerInput
On entry: must contain the value , where is the index in acol and inda of the start of the th column, for . Thus, the entries of column are held in , and their corresponding row indices are in , for , where and . To specify the th column as empty, set . Note that the first and last elements of loca must be and . If your problem has no constraints, or just bounds on the variables, you may include a dummy ‘free’ row with a single (zero) element by setting , , , , and , for . This row is made ‘free’ by setting its bounds to be and , where is the value of the optional parameter .
Constraints:
;
, for ;
;
, for .
15: – const doubleInput
On entry: , the lower bounds for all the variables and general constraints, in the following order. The first n elements of bl must contain the bounds on the variables , and the next m elements the bounds for the general linear constraints (which, equivalently, are the bounds for the slacks, ) and the free row (if any). To fix the th variable, set , say, where . To specify a nonexistent lower bound (i.e., ), set . Here, is the value of the optional parameter . To specify the th constraint as an equality, set , say, where . Note that the lower bound corresponding to the free row must be set to and stored in .
On entry: , the upper bounds for all the variables and general constraints, in the following order. The first n elements of bu must contain the bounds on the variables , and the next m elements the bounds for the general linear constraints (which, equivalently, are the bounds for the slacks, ) and the free row (if any). To specify a nonexistent upper bound (i.e., ), set . Note that the upper bound corresponding to the free row must be set to and stored in .
Constraints:
if , ;
otherwise .
17: – const doubleInput
On entry: contains the explicit objective vector (if any). If , c is not referenced and may be NULL.
18: – const char *Input
On entry: the optional column and row names, respectively.
If , names is not referenced and the printed output will use default names for the columns and rows.
If , the first n elements must contain the names for the columns and the next m elements must contain the names for the rows. Note that the name for the free row (if any) must be stored in .
Note: that only the first eight characters of the strings in names are significant.
19: – const IntegerInput
On entry: defines which variables are to be treated as being elastic in elastic mode. The allowed values of helast are:
Status in elastic mode
Variable is non-elastic and cannot be infeasible
Variable can violate its lower bound
Variable can violate its upper bound
Variable can violate either its lower or upper bound
helast need not be assigned if optional parameter .
Constraint:
if , , for .
20: – IntegerInput/Output
On entry: if or , and a Basis file of some sort is to be input (see the description of the optional parameters , or ), then hs and x need not be set at all.
If or and there is no Basis file, the first n elements of hs and x must specify the initial states and values, respectively, of the variables . (The slacks need not be initialized.) An internal Crash procedure is then used to select an initial basis matrix . The initial basis matrix will be triangular (neglecting certain small elements in each column). It is chosen from various rows and columns of . Possible values for are as follows:
State of during Crash procedure
or
Eligible for the basis
Ignored
Eligible for the basis (given preference over or )
or
Ignored
If nothing special is known about the problem, or there is no wish to provide special information, you may set
and , for . All variables will then be eligible for the initial basis. Less trivially, to say that the th variable will probably be equal to one of its bounds, set and or and as appropriate.
Following the Crash procedure, variables for which are made superbasic. Other variables not selected for the basis are then made nonbasic at the value if , or at the value or closest to .
If , hs and x must specify the initial states and values, respectively, of the variables and slacks . If e04nqc has been called previously with the same values of n and m, hs already contains satisfactory information.
Constraints:
if or ,
, for ;
if ,
, for .
On exit: the final states of the variables and slacks . The significance of each possible value of is as follows:
State of variable
Normal value of
Nonbasic
Nonbasic
Superbasic
Between and
Basic
Between and
If , basic and superbasic variables may be outside their bounds by as much as the value of the optional parameter . Note that unless the optional parameter is specified, the optional parameter applies to the variables of the scaled problem. In this case, the variables of the original problem may be as much as outside their bounds, but this is unlikely unless the problem is very badly scaled.
Very occasionally some nonbasic variables may be outside their bounds by as much as the optional parameter , and there may be some nonbasic variables for which lies strictly between its bounds.
If , some basic and superbasic variables may be outside their bounds by an arbitrary amount (bounded by sinf if ).
21: – doubleInput/Output
On entry: the initial values of the variables , and, if , the slacks , i.e., . (See the description for argument hs.)
On exit: the final values of the variables and slacks .
22: – doubleOutput
On exit: contains the dual variables (a set of Lagrange multipliers (shadow prices) for the general constraints).
23: – doubleOutput
On exit: contains the reduced costs, . The vector is the gradient of the objective if x is feasible; otherwise, it is the gradient of the Phase 1 objective. In the former case, , for , hence .
24: – Integer *Input/Output
On entry: , the number of superbasics. For QP problems, ns need not be specified if , but must retain its value from a previous call when . For FP and LP problems, ns need not be initialized.
On exit: the final number of superbasics. This will be zero for FP and LP problems.
25: – Integer *Output
On exit: the number of infeasibilities.
26: – double *Output
On exit: the sum of the scaled infeasibilities. This will be zero if , and is most meaningful when .
27: – double *Output
On exit: the value of the objective function.
If , obj includes the quadratic objective term (if any).
If , obj is just the linear objective term (if any).
Note that obj does not include contributions from the constant term objadd or the objective row, if any.
28: – Nag_E04State *Communication Structure
state contains internal information required for functions in this suite. It must not be modified in any way.
29: – Nag_Comm *
The NAG communication argument (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
30: – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
e04nqc returns with NE_NOERROR if the reduced gradient (rgNorm; see Section 9.1) is negligible, the Lagrange multipliers (Lagr Mult; see Section 9.1) are optimal, satisfies the constraints to the accuracy requested by the value of the optional parameter and the reduced Hessian factor (see Section 11.2) is nonsingular.
6Error Indicators and Warnings
NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
Internal memory allocation failed when attempting to obtain workspace sizes , and . Please contact NAG.
NE_ALLOC_INSUFFICIENT
Internal memory allocation was insufficient. Please contact NAG.
NE_ARRAY_INPUT
On entry, , , .
Constraint: or .
On entry, row index in is outside the range to .
NE_BAD_PARAM
Basis file dimensions do not match this problem.
On entry, argument had an illegal value.
NE_BASIS_FAILURE
An error has occurred in the basis package, perhaps indicating incorrect setup of arrays inda and loca. Set the optional parameter and examine the output carefully for further information.
NE_BASIS_ILL_COND
Numerical difficulties have been encountered and no further progress can be made.
Numerical error in trying to satisfy the general constraints. The basis is very ill-conditioned.
An factorization of the basis has just been obtained and used to recompute the basic variables , given the present values of the superbasic and nonbasic variables. However, a row check has revealed that the resulting solution does not satisfy the current constraints sufficiently well.
This probably means that the current basis is very ill-conditioned. Request the if there are any linear constraints and variables.
For certain highly structured basis matrices (notably those with band structure), a systematic growth may occur in the factor . Consult the description of Umax, Umin and Growth in Section 13, and set the optional parameter to (or possibly even smaller, but not less than ).
NE_BASIS_SINGULAR
The basis is singular after several attempts to factorize it (and add slacks where necessary).
Either the problem is badly scaled or the value of the optional parameter is too large.
NE_E04NPC_NOT_INIT
The initialization function e04npc has not been called.
An indefinite matrix was detected during the computation of the reduced Hessian factor (see Section 11.2). This may be caused by being indefinite. Check also that qphx has been coded correctly and that all relevant elements of have been assigned their correct values. If qphx is coded correctly and is positive semidefinite, the failure may be caused by ill conditioning. Try reducing the values of the optional parameters and . If there are very large values in , check the scaling of the variables and constraints.
NE_HESS_TOO_BIG
The value of the optional parameter is too small.
The current set of basic and superbasic variables have been optimized as much as possible and a pricing operation is necessary to continue, but there are already superbasics (and no room for any more).
In general, raise the by a reasonable amount, bearing in mind the storage needed for reduced Hessian (see Section 11.2). (The will also increase to unless specified otherwise, and the associated storage will be about words.) In some cases you may have to set to conserve storage, but beware that the rate of convergence will probably fall off severely.
NE_INT
On entry, .
Constraint: .
On entry, .
Constraint: .
NE_INT_2
On entry, and .
Constraint: .
On entry, and .
Constraint: .
On entry, and .
Constraint: .
On entry, ne is not equal to the number of nonzeros in acol. , nonzeros in .
NE_INT_3
On entry, , and .
Constraint: .
On entry, , and .
Constraint: or .
On entry, , and .
Constraint: .
On entry, , and .
Constraint: or .
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
An unexpected error has occurred. Set the optional parameter and examine the output carefully for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_NOT_REQUIRED_ACC
The requested accuracy could not be achieved.
NE_REAL_2
On entry, bounds bl and bu for are equal and infinite: and .
On entry, bounds bl and bu for are equal and infinite. and .
On entry, bounds for are inconsistent. and .
NE_UNBOUNDED
The problem appears to be unbounded. The constraint violation limit has been reached.
The problem appears to be unbounded. The objective function is unbounded.
The problem is unbounded (or badly scaled). For a minimization problem, the objective function is not bounded below in the feasible region.
For linear problems, unboundedness is detected by the simplex method when a nonbasic variable can be increased or decreased by an arbitrary amount without causing a basic variable to violate a bound. Consider adding an upper or lower bound to the variable. Also, examine the constraints that have nonzeros in the associated column, to see if they have been formulated as intended.
Very rarely, the scaling of the problem could be so poor that numerical error will give an erroneous indication of unboundedness. Consider using the optional parameter .
NW_NOT_FEASIBLE
The linear constraints appear to be infeasible.
The problem appears to be infeasible. Infeasibilites have been minimized.
The problem appears to be infeasible. Nonlinear infeasibilites have been minimized.
The problem appears to be infeasible. The linear equality constraints could not be satisfied.
The problem is infeasible. The general constraints cannot all be satisfied simultaneously to within the value of the optional parameter .
Feasibility is measured with respect to the upper and lower bounds on the variables and slacks. The message tells us that among all the points satisfying the general constraints , there is apparently no point that satisfies the bounds on and . Violations as small as the are ignored, but at least one component of or violates a bound by more than the tolerance.
Note: although the objective function is the sum of infeasibilities (when ), this sum will not necessarily have been minimized when .
If , e04nqc will optimize the QP objective and the sum of infeasibilities, suitably weighted using the optional parameter . The function will tend to determine a ‘good’ infeasible point if the elastic weight is sufficiently large.
NW_SOLN_NOT_UNIQUE
Weak solution found – the solution is not unique.
NW_TOO_MANY_ITER
Iteration limit reached.
Major iteration limit reached.
Too many iterations. The value of the optional parameter is too small.
The Iterations limit was exceeded before the required solution could be found. Check the iteration log to be sure that progress was being made. If so, restart the run using a Basis file that was saved at the end of the run.
7Accuracy
e04nqc implements a numerically stable active-set strategy and returns solutions that are as accurate as the condition of the problem warrants on the machine.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
e04nqc makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
9Further Comments
This section contains a description of the printed output.
9.1Description of the Printed Output
If , one line of information is output to the every th iteration, where is the specified . A heading is printed before the first such line following a basis factorization. The heading contains the items described below. In this description, a pricing operation is defined to be the process by which one or more nonbasic variables are selected to become superbasic (in addition to those already in the superbasic set). The variable selected will be denoted by jq. If the problem is purely linear, variable jq will usually become basic immediately (unless it should happen to reach its opposite bound and return to the nonbasic set).
If optional parameter is in effect, variable jq is selected from or , the ppth segments of the constraint matrix .
Label
Description
Itn
is the iteration count.
pp
is the partial-price indicator. The variable selected by the last pricing operation came from the ppth partition of and . Note that pp is reset to zero whenever the basis is refactorized.
dj
is the value of the reduced gradient (or reduced cost) for the variable selected by the pricing operation at the start of the current iteration.
Algebraically, dj is , for , where is the gradient of the current objective function, is the vector of dual variables, and is the th column of the constraint matrix .
Note that dj is the norm of the reduced-gradient vector at the start of the iteration, just after the pricing operation.
+SBS
is the variable jq selected by the pricing operation to be added to the superbasic set.
-SBS
is the variable chosen to leave the superbasic set. It has become basic if the entry under -B is nonzero, otherwise it becomes nonbasic.
-BS
is the variable removed from the basis to become nonbasic.
Step
is the value of the step length taken along the current search direction . The variables have just been changed to . If a variable is made superbasic during the current iteration (i.e., +SBS is positive), Step will be the step to the nearest bound. During the optimality phase, the step can be greater than unity only if the reduced Hessian is not positive definite.
Pivot
is the th element of a vector satisfying whenever (the th column of the constraint matrix replaces the th column of the basis matrix . Wherever possible, Step is chosen so as to avoid extremely small values of Pivot (since they may cause the basis to be nearly singular). In extreme cases, it may be necessary to increase the value of the optional parameter to exclude very small elements of from consideration during the computation of Step.
nInf
is the number of violated constraints (infeasibilities) before the present iteration. This number will not increase unless iterations are in elastic mode.
sInf
is the sum of infeasibilities before the present iteration. It will usually decrease at each nonzero step, but if nInf decreases by or more, sInf may occasionally increase. However, in elastic mode it will decrease monotonically.
Objective
is the value of the current objective function after the present iteration. Note, if is , the heading is Composite Obj.
L+U
L is the number of nonzeros in the basis factor . Immediately after a basis factorization , L contains lenL (see Section 13). Further nonzeros are added to L when various columns of are later replaced. (Thus, L increases monotonically.) U is the number of nonzeros in the basis factor . Immediately after a basis factorization , U contains lenU (see Section 13). As columns of are replaced, the matrix is maintained explicitly (in sparse form). The value of U may fluctuate up or down; in general, it will tend to increase.
ncp
is the number of compressions required to recover workspace in the data structure for . This includes the number of compressions needed during the previous basis factorization. Normally, ncp should increase very slowly.
The following will be output if the problem is QP or if the superbasic set is non-empty.
Label
Description
rgNorm
is the largest reduced-gradient among the superbasic variables after the current iteration. During the optimality phase, this will be approximately zero after a unit step.
nS
is the current number of superbasic variables.
condHz
is a lower bound on the condition number of the reduced Hessian (see Section 11.2). The larger this number, the more difficult the problem. Attention should be given to the scaling of the variables and the constraints to guard against high values of condHz.
10Example
This example minimizes the quadratic function , where
subject to the bounds
and to the linear constraints
The initial point, which is infeasible, is
The optimal solution (to five figures) is
One bound constraint and four linear constraints are active at the solution. Note that the Hessian matrix is positive semidefinite.
Note: the remainder of this document is intended for more advanced users. Section 11 contains a detailed description of the algorithm which may be needed in order to understand Sections 12 and 13. Section 12 describes the optional parameters which may be set by calls to e04nrc,e04nsc,e04ntcand/ore04nuc. Section 13 describes the quantities which can be requested to monitor the course of the computation.
11Algorithmic Details
This section contains a detailed description of the method used by e04nqc.
11.1Overview
e04nqc is based on an inertia-controlling method that maintains a Cholesky factorization of the reduced Hessian (see below). The method is similar to that of Gill and Murray (1978), and is described in detail by Gill et al. (1991). Here we briefly summarise the main features of the method. Where possible, explicit reference is made to the names of variables that are arguments of the function or appear in the printed output.
The method used has two distinct phases: finding an initial feasible point by minimizing the sum of infeasibilities (the feasibility phase), and minimizing the quadratic objective function within the feasible region (the optimality phase). The computations in both phases are performed by the same functions. The two-phase nature of the algorithm is reflected by changing the function being minimized from the sum of infeasibilities (the printed quantity sInf; see Section 9.1) to the quadratic objective function (the printed quantity Objective; see Section 9.1).
In general, an iterative process is required to solve a quadratic program. Given an iterate in both the original variables and the slack variables , a new iterate is defined by
(3)
where the step length
is a non-negative scalar (the printed quantity Step; see Section 13), and is called the search direction. (For simplicity, we shall consider a typical iteration and avoid reference to the index of the iteration.) Once an iterate is feasible (i.e., satisfies the constraints), all subsequent iterates remain feasible.
11.2Definition of the Working Set and Search Direction
At each iterate , a working set of constraints is defined to be a linearly independent subset of the constraints that are satisfied ‘exactly’ (to within the value of the optional parameter ). The working set is the current prediction of the constraints that hold with equality at a solution of the LP or QP problem. Let denote the number of constraints in the working set (including bounds), and let denote the associated working set matrix consisting of the gradients of the working set constraints.
The search direction is defined so that constraints in the working set remain unaltered for any value of the step length. It follows that must satisfy the identity
(4)
This characterisation allows to be computed using any full-rank matrix that spans the null space of . (Thus, and .) The null space matrix is defined from a sparse factorization of part of (see (7) and (8)). The direction will satisfy (4) if
(5)
where is any -vector.
The working set contains the constraints and a subset of the upper and lower bounds on the variables . Since the gradient of a bound constraint or is a vector of all zeros except for in position , it follows that the working set matrix contains the rows of and the unit rows associated with the upper and lower bounds in the working set.
The working set matrix can be represented in terms of a certain column partition of the matrix by (conceptually) partitioning the constraints so that
(6)
where is a square nonsingular basis and , and are the basic, superbasic and nonbasic variables respectively. The nonbasic variables are equal to their upper or lower bounds at , and the superbasic variables are independent variables that are chosen to improve the value of the current objective function. The number of superbasic variables is (the printed quantity nS; see Section 9.1). Given values of and , the basic variables are adjusted so that satisfies (6).
If is a permutation matrix such that , then satisfies
(7)
where is the identity matrix with the same number of columns as .
The null space matrix is defined from a sparse factorization of part of . In particular, is maintained in ‘reduced gradient’ form, using the LUSOL package (see Gill et al. (1991)) to maintain sparse factors of the basis matrix as the partition changes. Given the permutation , the null space basis is given by
(8)
This matrix is used only as an operator, i.e., it is never computed explicitly. Products of the form and are obtained by solving with or . This choice of implies that , the number of ‘degrees of freedom’ at , is the same as , the number of superbasic variables.
Let and denote the reduced gradient and reduced Hessian of the objective function:
(9)
where is the objective gradient at . Roughly speaking, and describe the first and second derivatives of an -dimensional unconstrained problem for the calculation of . (The condition estimator of is the quantity condHz in the monitoring file output; see Section 9.1.)
At each iteration, an upper triangular factor is available such that . Normally, is computed from at the start of the optimality phase and then updated as the QP working set changes. For efficiency, the dimension of should not be excessive (say, ). This is guaranteed if the number of nonlinear variables is ‘moderate’.
If the QP problem contains linear variables, is positive semidefinite and may be singular with at least one zero diagonal element. However, an inertia-controlling strategy is used to ensure that only the last diagonal element of can be zero. (See Gill et al. (1991) for a discussion of a similar strategy for indefinite quadratic programming.)
If the initial is singular, enough variables are fixed at their current value to give a nonsingular . This is equivalent to including temporary bound constraints in the working set. Thereafter, can become singular only when a constraint is deleted from the working set (in which case no further constraints are deleted until becomes nonsingular).
11.3Main Iteration
If the reduced gradient is zero, is a constrained stationary point on the working set. During the feasibility phase, the reduced gradient will usually be zero only at a vertex (although it may be zero elsewhere in the presence of constraint dependencies). During the optimality phase, a zero reduced gradient implies that minimizes the quadratic objective function when the constraints in the working set are treated as equalities. At a constrained stationary point, Lagrange multipliers are defined from the equations
(10)
A Lagrange multiplier, , corresponding to an inequality constraint in the working set is said to be optimal if when the associated constraint is at its upper bound, or if when the associated constraint is at its lower bound, where depends on the value of the optional parameter . If a multiplier is nonoptimal, the objective function (either the true objective or the sum of infeasibilities) can be reduced by continuing the minimization with the corresponding constraint excluded from the working set. (This step is sometimes referred to as ‘deleting’ a constraint from the working set.) If optimal multipliers occur during the feasibility phase but the sum of infeasibilities is nonzero, there is no feasible point and the function terminates immediately with NE_NOT_REQUIRED_ACC.
The special form (7) of the working set allows the multiplier vector , the solution of (10), to be written in terms of the vector
(11)
where satisfies the equations , and denotes the basic elements of . The elements of are the Lagrange multipliers associated with the equality constraints . The vector of nonbasic elements of consists of the Lagrange multipliers associated with the upper and lower bound constraints in the working set. The vector of superbasic elements of is the reduced gradient in (9). The vector of basic elements of is zero, by construction. (The Euclidean norm of and the final values of , and are the quantities rgNorm, Reduced Gradnt, Obj Gradient and Dual Activity in the monitoring file output; see Section 13.)
If the reduced gradient is not zero, Lagrange multipliers need not be computed and the search direction is given by (see (8) and (12)). The step length is chosen to maintain feasibility with respect to the satisfied constraints.
There are two possible choices for , depending on whether or not is singular. If is nonsingular, is nonsingular and in (5) is computed from the equations
(12)
where is the reduced gradient at . In this case, is the minimizer of the objective function subject to the working set constraints being treated as equalities. If is feasible, is defined to be unity. In this case, the reduced gradient at will be zero, and Lagrange multipliers are computed at the next iteration. Otherwise, is set to , the step to the ‘nearest’ constraint along . This constraint is then added to the working set at the next iteration.
If is singular, then must also be singular, and an inertia-controlling strategy is used to ensure that only the last diagonal element of is zero. (See Gill et al. (1991) for a discussion of a similar strategy for indefinite quadratic programming.) In this case, satisfies
(13)
which allows the objective function to be reduced by any step of the form , where . The vector is a direction of unbounded descent for the QP problem in the sense that the QP objective is linear and decreases without bound along . If no finite step of the form (where ) reaches a constraint not in the working set, the QP problem is unbounded and the function terminates immediately with NE_UNBOUNDED. Otherwise, is defined as the maximum feasible step along and a constraint active at is added to the working set for the next iteration.
e04nqc makes explicit allowance for infeasible constraints. Infeasible linear constraints are detected first by solving a problem of the form
(14)
where . This is equivalent to minimizing the sum of the general linear constraint violations subject to the simple bounds. (In the linear programming literature, the approach is often called elastic programming.)
11.4Miscellaneous
If the basis matrix is not chosen carefully, the condition of the null space matrix in (8) could be arbitrarily high. To guard against this, the function implements a ‘basis repair’ feature in which the LUSOL package (see Gill et al. (1991)) is used to compute the rectangular factorization
(15)
returning just the permutation that makes unit lower triangular. The pivot tolerance is set to require , and the permutation is used to define in (7). It can be shown that is likely to be little more than unity. Hence, should be well-conditioned regardless of the condition of
. This feature is applied at the beginning of the optimality phase if a potential ordering is known.
The EXPAND procedure (see Gill et al. (1989)) is used to reduce the possibility of cycling at a point where the active constraints are nearly linearly dependent. Although there is no absolute guarantee that cycling will not occur, the probability of cycling is extremely small (see Hall and McKinnon (1996)). The main feature of EXPAND is that the feasibility tolerance is increased at the start of every iteration. This allows a positive step to be taken at every iteration, perhaps at the expense of violating the bounds on by a small amount.
Suppose that the value of the optional parameter is . Over a period of iterations (where is the value of the optional parameter ), the feasibility tolerance actually used by the function (i.e., the working feasibility tolerance) increases from to (in steps of ).
At certain stages the following ‘resetting procedure’ is used to remove small constraint infeasibilities. First, all nonbasic variables are moved exactly onto their bounds. A count is kept of the number of nontrivial adjustments made. If the count is nonzero, the basic variables are recomputed. Finally, the working feasibility tolerance is reinitialized to .
If a problem requires more than iterations, the resetting procedure is invoked and a new cycle of iterations is started. (The decision to resume the feasibility phase or optimality phase is based on comparing any constraint infeasibilities with .)
The resetting procedure is also invoked when the function reaches an apparently optimal, infeasible or unbounded solution, unless this situation has already occurred twice. If any nontrivial adjustments are made, iterations are continued.
The EXPAND procedure not only allows a positive step to be taken at every iteration, but also provides a potential choice of constraints to be added to the working set. All constraints at a distance (where ) along from the current point are then viewed as acceptable candidates for inclusion in the working set. The constraint whose normal makes the largest angle with the search direction is added to the working set. This strategy helps keep the basis matrix well-conditioned.
12Optional Parameters
Several optional parameters in e04nqc define choices in the problem specification or the algorithm logic. In order to reduce the number of formal arguments of e04nqc these optional parameters have associated default values that are appropriate for most problems. Therefore, you need only specify those optional parameters whose values are to be different from their default values.
The remainder of this section can be skipped if you wish to use the default values for all optional parameters.
The following is a list of the optional parameters available. A full description of each optional parameter is provided in Section 12.1.
Optional parameters may be specified by calling one, or any, of the functions e04nrc,e04nsc,e04ntcande04nuc before a call to e04nqc, but after a call to e04npc.
e04nrc reads options from an external options file, with Begin and End as the first and last lines respectively and each intermediate line defining a single optional parameter. For example,
Begin
Print Level = 5
End
The call
e04nrc (ioptns, &state, &fail);
can then be used to read the file on
descriptor ioptns. NE_NOERROR
on successful exit. e04nrc should be consulted for a full description of this method of supplying optional parameters.
e04nsc,e04ntcore04nuc can be called to supply options directly, one call being necessary for each optional parameter. e04nsc,e04ntcore04nuc should be consulted for a full description of this method of supplying optional parameters.
All optional parameters not specified by you are set to their default values. Optional parameters specified by you are unaltered by e04nqc (unless they define invalid values) and so remain in effect for subsequent calls unless altered by you.
12.1Description of the Optional Parameters
For each option, we give a summary line, a description of the optional parameter and details of constraints.
The summary line contains:
the keywords;
a parameter value,
where the letters , and denote options that take character, integer and real values respectively;
the default value is used whenever the condition is satisfied and where the symbol is a generic notation for machine precision (see X02AJC);
The variable holds the value of .
Keywords and character values are case and white space insensitive.
Optional parameters used to specify files (e.g., optional parameters and ) have type Nag_FileID (see Section 3.1.1 in the Introduction to the NAG Library CL Interface). This ID value must either be set to (the default value) in which case there will be no output, or will be as returned by a call of x04acc.
Check Frequency
Default
Every th iteration after the most recent basis factorization, a numerical test is made to see if the current solution satisfies the linear constraints . If the largest element of the residual vector is judged to be too large, the current basis is refactorized and the basic variables recomputed to satisfy the constraints more accurately. If , the value is used and effectively no checks are made.
is useful for debugging purposes, but otherwise this option should not be needed.
Crash Option
Default
Crash Tolerance
Default
Note that these options do not apply when (see Section 5).
If , an internal Crash procedure is used to select an initial basis from various rows and columns of the constraint matrix . The value of determines which rows and columns of are initially eligible for the basis, and how many times the Crash procedure is called. Columns of are used to pad the basis where necessary.
Meaning
The initial basis contains only slack variables: .
The Crash procedure is called once, looking for a triangular basis in all rows and columns of the matrix .
The Crash procedure is called once, looking for a triangular basis in rows.
The Crash procedure is called twice, treating linear equalities and linear inequalities separately.
If , certain slacks on inequality rows are selected for the basis first. (If , numerical values are used to exclude slacks that are close to a bound.) The Crash procedure then makes several passes through the columns of , searching for a basis matrix that is essentially triangular. A column is assigned to ‘pivot’ on a particular row if the column contains a suitably large element in a row that has not yet been assigned. (The pivot elements ultimately form the diagonals of the triangular basis.) For remaining unassigned rows, slack variables are inserted to complete the basis.
The allows the Crash procedure to ignore certain ‘small’ nonzero elements in each column of . If is the largest element in column , other nonzeros in the column are ignored if . (To be meaningful, should be in the range .)
When , the basis obtained by the Crash procedure may not be strictly triangular, but it is likely to be nonsingular and almost triangular. The intention is to obtain a starting basis containing more columns of and fewer (arbitrary) slacks. A feasible solution may be reached sooner on some problems.
For example, suppose the first columns of form the matrix shown under ; i.e., a tridiagonal matrix with entries , , . To help the Crash procedure choose all columns for the initial basis, we would specify a of for some value of .
Defaults
This special keyword may be used to reset all optional parameters to their default values.
Dump File
Default
Load File
Default
(See Section 3.1.1 in the Introduction to the NAG Library CL Interface for further information on NAG data types.)
Optional parameters and are similar to optional parameters and , but they record solution information in a manner that is more direct and more easily modified. A full description of information recorded in optional parameters and is given in Gill et al. (2005a).
If , the last solution obtained will be output to the file .
If , the containing basis information will be read.
The file will usually have been output previously as a . The file will not be accessed if optional parameters or are specified.
Elastic Mode
Default
This argument determines if (and when) elastic mode is to be started. Three elastic modes are available as follows:
Meaning
Elastic mode is never invoked. e04nqc will terminate as soon as infeasibility is detected. There may be other points with significantly smaller sums of infeasibilities.
Elastic mode is invoked only if the constraints are found to be infeasible (the default). If the constraints are infeasible, continue in elastic mode with the composite objective determined by the values of the optional parameters and .
The iterations start and remain in elastic mode. This option allows you to minimize the composite objective function directly without first performing Phase 1 iterations.
The success of this option will depend critically on your choice of . If is sufficiently large and the constraints are feasible, the minimizer of the composite objective and the solution of the original problem are identical. However, if the is not sufficiently large, the minimizer of the composite function may be infeasible, even if a feasible point exists.
Elastic Objective
Default
This determines the form of the composite objective in Phase 2 (). Three types of composite objectives are available.
Meaning
Include only the true objective in the composite objective. This option sets in the composite objective and allows e04nqc to ignore the elastic bounds and find a solution that minimizes subject to the non-elastic constraints. This option is useful if there are some ‘soft’ constraints that you would like to ignore if the constraints are infeasible.
Use a composite objective defined with determined by the value of . This value is intended to be used in conjunction with .
Include only the elastic variables in the composite objective. The elastics are weighted by . This choice minimizes the violations of the elastic variables at the expense of possibly increasing the true objective. This option can be used to find a point that minimizes the sum of the violations of a subset of constraints specified by the input array helast.
Elastic Weight
Default
This defines the value of in the composite objective in Phase 2 ().
At each iteration of elastic mode, the composite objective is defined to be
where for , for , and is the quadratic objective.
Note that the effect of is not disabled once a feasible point is obtained.
Expand Frequency
Default
This option is part of an anti-cycling procedure (see Section 11.4) designed to allow progress even on highly degenerate problems.
The strategy is to force a positive step at every iteration, at the expense of violating the constraints by a small amount. Suppose that the value of the optional parameter is . Over a period of iterations, the feasibility tolerance actually used by e04nqc (i.e., the working feasibility tolerance) increases from to (in steps of ).
Increasing the value of helps reduce the number of slightly infeasible nonbasic variables (most of which are eliminated during the resetting procedure). However, it also diminishes the freedom to choose a large pivot element (see the description of the optional parameter ).
If , the value is used and effectively no anti-cycling procedure is invoked.
Factorization Frequency
Default or
If , at most basis changes will occur between factorizations of the basis matrix.
For LP problems, the basis factors are usually updated at every iteration. Higher values of may be more efficient on problems that are extremely sparse and well scaled.
For QP problems, fewer basis updates will occur as the solution is approached. The number of iterations between basis factorizations will, therefore, increase. During these iterations a test is made regularly according to the value of optional parameter to ensure that the linear constraints are satisfied. Occasionally, the basis will be refactorized before the limit of updates is reached. If , the default value is used.
Feasibility Tolerance
Default
A feasible problem is one in which all variables satisfy their upper and lower bounds to within the absolute tolerance . (This includes slack variables. Hence, the general constraints are also satisfied to within .)
e04nqc attempts to find a feasible solution before optimizing the objective function. If the sum of infeasibilities cannot be reduced to zero, the problem is assumed to be infeasible. Let sInf be the corresponding sum of infeasibilities. If sInf is quite small, it may be appropriate to raise by a factor of or . Otherwise, some error in the data should be suspected.
Note that if sInf is not small and you have not asked e04nqc to minimize the violations of the elastic variables (i.e., you have not specified ), there may be other points that have a significantly smaller sum of infeasibilities. e04nqc will not attempt to find the solution that minimizes the sum unless .
If the constraints and variables have been scaled (see the description of the optional parameter ), then feasibility is defined in terms of the scaled problem (since it is more likely to be meaningful).
Infinite Bound Size
Default
If , defines the ‘infinite’ bound in the definition of the problem constraints. Any upper bound greater than or equal to will be regarded as (and similarly any lower bound less than or equal to will be regarded as ). If , the default value is used.
Iterations Limit
Default
The value of specifies the maximum number of iterations allowed before termination. Setting and means that: the workspace needed to start solving the problem will be computed and printed; and feasibility and optimality will be checked. No iterations will be performed. If , the default value is used.
LU Density Tolerance
Default
LU Singularity Tolerance
Default
The density tolerance is used during factorization of the basis matrix. Columns of and rows of are formed one at a time, and the remaining rows and columns of the basis are altered appropriately. At any stage, if the density of the remaining matrix exceeds , the Markowitz strategy for choosing pivots is terminated. The remaining matrix is factored by a dense procedure. Raising the density tolerance towards may give slightly sparser factors, with a slight increase in factorization time.
If , defines the singularity tolerance used to guard against ill-conditioned basis matrices. After is refactorized, the diagonal elements of are tested as follows. If or , the th column of the basis is replaced by the corresponding slack variable. If , the default value is used.
LU Factor Tolerance
Default
LU Update Tolerance
Default
The values of and affect the stability and sparsity of the basis factorization , during refactorization and updates respectively. The lower triangular matrix is a product of matrices of the form
where the multipliers will satisfy . The default values of and usually strike a good compromise between stability and sparsity. They must satisfy , .
For large and relatively dense problems, or (say) may give a useful improvement in stability without impairing sparsity to a serious degree.
For certain very regular structures (e.g., band matrices) it may be necessary to reduce in order to achieve stability. For example, if the columns of include a sub-matrix of the form
one should set both and to values in the range .
LU Partial Pivoting
Default
LU Complete Pivoting
LU Rook Pivoting
The factorization implements a Markowitz-type search for pivots that locally minimize the fill-in subject to a threshold pivoting stability criterion. The default option is to use threshold partial pivoting. The optional parameters and are more expensive but more stable and better at revealing rank, as long as the is not too large (say ).
Minimize
Default
Maximize
Feasible Point
This option specifies the required direction of the optimization. It applies to both linear and nonlinear terms (if any) in the objective function. Note that if two problems are the same except that one minimizes and the other maximizes , their solutions will be the same but the signs of the dual variables and the reduced gradients (see Section 11.3) will be reversed.
The optional parameter means ‘ignore the objective function, while finding a feasible point for the linear constraints’. It can be used to check that the constraints are feasible without altering the call to e04nqc.
New Basis File
Default
Backup Basis File
Default
Save Frequency
Default
(See Section 3.1.1 in the Introduction to the NAG Library CL Interface for further information on NAG data types.)
Optional parameters and are sometimes referred to as basis maps. They contain the most compact representation of the state of each variable. They are intended for restarting the solution of a problem at a point that was reached by an earlier run. For nontrivial problems, it is advisable to save basis maps at the end of a run, in order to restart the run if necessary.
If , a basis map will be saved on file
every th iteration, where is the .
The first record of the file will contain the word PROCEEDING if the run is still in progress. A basis map will also be saved at the end of a run, with some other word indicating the final solution status.
If ,
is intended as a safeguard against losing the results of a long run. Suppose that a is being saved every () iterations, and that e04nqc is about to save such a basis at iteration . It is conceivable that the run may be interrupted during the next few milliseconds (in the middle of the save). In this case the Basis file will be corrupted and the run will have been essentially wasted.
To eliminate this risk, both a and a may be specified.
The following would be suitable for the above example:
The current basis will then be saved every iterations, first on and then immediately on . If the run is interrupted at iteration during the save on , there will still be a usable basis on (corresponding to iteration ).
Note that a new basis will be saved in at the end of a run if it terminates normally, but it will not be saved in . In the above example, if an optimum solution is found at iteration (or if the iteration limit is ), the final basis on will correspond to iteration , but the last basis saved on will be the one for iteration .
A full description of information recorded in and is given in Gill et al. (2005a).
Nolist
Default
List
Optional parameter enables printing of each optional parameter specification as it is supplied. suppresses this printing.
Old Basis File
Default
(See Section 3.1.1 in the Introduction to the NAG Library CL Interface for further information on NAG data types.)
If , the basis maps information will be obtained from the file associated with ID .
The file will usually have been output previously as a or .
A full description of information recorded in and is given in Gill et al. (2005a).
The file will not be acceptable if the number of rows or columns in the problem has been altered.
Optimality Tolerance
Default
This is used to judge the size of the reduced gradients , where is the th component of the gradient, is the associated column of the constraint matrix , and is the set of dual variables.
By construction, the reduced gradients for basic variables are always zero. The problem will be declared optimal if the reduced gradients for nonbasic variables at their lower or upper bounds satisfy
respectively, and if for superbasic variables.
In the above tests, is a measure of the size of the dual variables. It is included to make the tests independent of a scale factor on the objective function. The quantity actually used is defined by
so that only large scale factors are allowed for.
If the objective is scaled down to be very small, the optimality test reduces to comparing against .
Partial Price
Default or
This option is recommended for large FP or LP problems that have significantly more variables than constraints (i.e., ). It reduces the work required for each pricing operation (i.e., when a nonbasic variable is selected to enter the basis). If , all columns of the constraint matrix are searched. If , and are partitioned to give roughly equal segments , for (modulo ). If the previous pricing search was successful on , the next search begins on the segments and . If a reduced gradient is found that is larger than some dynamic tolerance, the variable with the largest such reduced gradient (of appropriate sign) is selected to enter the basis. If nothing is found, the search continues on the next segments , and so on. If , the default value is used.
Pivot Tolerance
Default
Broadly speaking, the pivot tolerance is used to prevent columns entering the basis if they would cause the basis to become almost singular.
When changes to for some search direction , a ‘ratio test’ determines which component of reaches an upper or lower bound first. The corresponding element of is called the pivot element. Elements of are ignored (and, therefore, cannot be pivot elements) if they are smaller than the pivot tolerance .
It is common for two or more variables to reach a bound at essentially the same time. In such cases, the optional parameter (say ) provides some freedom to maximize the pivot element and thereby improve numerical stability. Excessively small values of should, therefore, not be specified. To a lesser extent, the optional parameter (say ) also provides some freedom to maximize the pivot element. Excessively large values of should, therefore, not be specified.
Print File
Default
(See Section 3.1.1 in the Introduction to the NAG Library CL Interface for further information on NAG data types.)
If , the following information is output to
during the solution of each problem:
–a listing of the optional parameters;
–some statistics about the problem;
–the amount of storage available for the factorization of the basis matrix;
–notes about the initial basis resulting from a Crash procedure or a Basis file;
–the iteration log;
–basis factorization statistics;
–the exit fail condition and some statistics about the solution obtained;
–the printed solution, if requested.
The last four items are described in Sections 9 and 13. Further brief output may be directed to the .
Print Frequency
Default
If , one line of the iteration log will be printed every th iteration. A value such as is suggested for those interested only in the final solution. If , the value of is used and effectively no checks are made.
Print Level
Default
This controls the amount of printing produced by e04nqc as follows.
Meaning
0
No output except error messages. If you want to suppress all output, set .
The set of selected options, problem statistics, summary of the scaling procedure, information about the initial basis resulting from a Crash or a Basis file, a single line of output at each iteration (controlled by the optional parameter ), and the exit condition with a summary of the final solution.
Basis factorization statistics.
Punch File
Default
Insert File
Default
(See Section 3.1.1 in the Introduction to the NAG Library CL Interface for further information on NAG data types.)
These files provide compatibility with commercial mathematical programming systems. The from a previous run may be used as an for a later run on the same problem. A full description of information recorded in and is given in Gill et al. (2005a).
If , the final solution obtained will be output to file .
For linear programs, this format is compatible with various commercial systems.
If ,
the containing basis information will be read. The file will usually have been output previously as a . The file will not be accessed if is specified.
QPSolver Cholesky
Default
QPSolver CG
QPSolver QN
Specifies the active-set algorithm used to solve the quadratic program in Phase 2 (). holds the full Cholesky factor of the reduced Hessian . As the QP iterations proceed, the dimension of changes with the number of superbasic variables. If the number of superbasic variables needs to increase beyond the value of , the reduced Hessian cannot be stored and the solver switches to . The Cholesky solver is reactivated if the number of superbasics stabilizes at a value less than .
solves the QP using a quasi-Newton method. In this case, is the factor of a quasi-Newton approximate Hessian.
uses an active-set method similar to , but uses the conjugate-gradient method to solve all systems involving the reduced Hessian.
The Cholesky QP solver is the most robust, but may require a significant amount of computation if there are many superbasics.
The quasi-Newton QP solver does not require computation of the exact at the start of Phase 2 (). It may be appropriate when the number of superbasics is large but relatively few iterations are needed to reach a solution (e.g., if e04nqc is called with a Warm Start).
The conjugate-gradient QP solver is appropriate for problems with many degrees of freedom (say, more than superbasics).
Reduced Hessian Dimension
Default
This specifies that an triangular matrix (to define the reduced Hessian according to ). is to be available for use by the Cholesky QP solver.
Scale Option
Default
Scale Tolerance
Default
Scale Print
Three scale options are available as follows:
Meaning
0
No scaling. This is recommended if it is known that and the constraint matrix never have very large elements (say, larger than ).
1
The constraints and variables are scaled by an iterative procedure that attempts to make the matrix coefficients as close as possible to (see Fourer (1982)). This will sometimes improve the performance of the solution procedures.
2
The constraints and variables are scaled by the iterative procedure. Also, a certain additional scaling is performed that may be helpful if the right-hand side or the solution is large. This takes into account columns of that are fixed or have positive lower bounds or negative upper bounds.
Optional parameter affects how many passes might be needed through the constraint matrix. On each pass, the scaling procedure computes the ratio of the largest and smallest nonzero coefficients in each column:
If is less than times its previous value, another scaling pass is performed to adjust the row and column scales. Raising from to (say) usually increases the number of scaling passes through . At most passes are made. The value of should lie in the range .
causes the row scales and column scales to be printed to , if has been specified. The scaled matrix coefficients are , and the scaled bounds on the variables and slacks are , , where if .
Solution Yes
Default
Solution No
This option determines if the final obtained solution is to be output to the
. Note that the option operates independently.
Solution File
Default
(See Section 3.1.1 in the Introduction to the NAG Library CL Interface for further information on NAG data types.)
If , the final solution will be output to file
(whether optimal or not).
To see more significant digits in the printed solution, it will sometimes be useful to make
.
Summary File
Default
Summary Frequency
Default
(See Section 3.1.1 in the Introduction to the NAG Library CL Interface for further information on NAG data types.)
If , a brief log will be output to file , including one line of information every th iteration.
In an interactive environment, it is useful to direct this output to the terminal, to allow a run to be monitored online. (If something looks wrong, the run can be manually terminated.) Further details are given in Section 13. If , the value of is used and effectively no checks are made.
Superbasics Limit
Default
This places a limit on the storage allocated for superbasic variables. Ideally, should be set slightly larger than the ‘number of degrees of freedom’ expected at an optimal solution.
For linear programs, an optimum is normally a basic solution with no degrees of freedom. (The number of variables lying strictly between their bounds is no more than , the number of general constraints.) The default value of is, therefore, .
For quadratic problems, the number of degrees of freedom is often called the ‘number of independent variables’. Normally, need not be greater than , where is the number of leading nonzero columns of . For many problems, may be considerably smaller than . This will save storage if is very large.
Suppress Parameters
Normally e04nqc prints the options file as it is being read, and then prints a complete list of the available keywords and their final values. The optional parameter tells e04nqc not to print the full list.
System Information No
Default
System Information Yes
This option prints additional information on the progress of major and minor iterations, and Crash statistics. See Section 13.
Timing Level
Default
If , some timing information will be output to the Print file, if .
Unbounded Step Size
Default
If , specifies the magnitude of the change in variables that will be considered a step to an unbounded solution. (Note that an unbounded solution can occur only when the Hessian is not positive definite.) If the change in during an iteration would exceed the value of , the objective function is considered to be unbounded below in the feasible region. If , the default value is used. See for the definition of .
13Description of Monitoring Information
This section describes the intermediate printout and final printout which constitutes the monitoring information produced by e04nqc. (See also the description of the optional parameters and .) You can control the level of printed output.
13.1Crash Statistics
When , and has been specified, the following lines of intermediate printout (less than characters) are produced on the unit number specified by optional parameter whenever (see Section 5). They refer to the number of columns selected by the Crash procedure during each of several passes through , whilst searching for a triangular basis matrix.
Label
Description
Slacks
is the number of slacks selected initially.
Free cols
is the number of free columns in the basis, including those whose bounds are rather far apart.
Preferred
is the number of ‘preferred’ columns in the basis (i.e., for some ). It will be a subset of the columns for which was specified.
Unit
is the number of unit columns in the basis.
Double
is the number of double columns in the basis.
Triangle
is the number of triangular columns in the basis.
Pad
is the number of slacks used to pad the basis (to make it a nonsingular triangle).
13.2Basis Factorization Statistics
When and , the first seven items of intermediate printout in the list below are produced on the unit number specified by optional parameter whenever the matrix or is factorized. Gaussian elimination is used to compute an factorization of or , where is a lower triangular matrix and is an upper triangular matrix for some permutation matrices and . The factorization is stabilized in the manner described under the optional parameter . In addition, if has been specified, the entries from Elems onwards are also output.
Label
Description
Factor
the number of factorizations since the start of the run.
Demand
a code giving the reason for the present factorization, as follows:
Code
Meaning
0
First factorization.
1
The number of updates reached the .
2
The nonzeros in the updated factors have increased significantly.
7
Not enough storage to update factors.
10
Row residuals are too large (see the description of the optional parameter ).
11
Ill-conditioning has caused inconsistent results.
Itn
is the current minor iteration number.
Nonlin
is the number of nonlinear variables in the current basis .
Linear
is the number of linear variables in .
Slacks
is the number of slack variables in .
B, BR, BS or BTfactorize
is the type of factorization.
B
periodic factorization of the basis .
BR
more careful rank-revealing factorization of using threshold rook pivoting. This occurs mainly at the start, if the first basis factors seem singular or ill-conditioned. Followed by a normal B factorize.
BS
is factorized to choose a well-conditioned from the current . Followed by a normal B factorize.
BT
same as BS except the current is tried first and accepted if it appears to be not much more ill-conditioned than after the previous BS factorize.
m
is the number of rows in or .
n
is the number of columns in or . Preceded by ‘=’ or ‘>’ respectively.
Elems
is the number of nonzero elements in or .
Amax
is the largest nonzero in or .
Density
is the percentage nonzero density of or .
Merit/MerRP/MerCP
Merit is the average Markowitz merit count for the elements chosen to be the diagonals of . Each merit count is defined to be where and are the number of nonzeros in the column and row containing the element at the time it is selected to be the next diagonal. Merit is the average of n such quantities. It gives an indication of how much work was required to preserve sparsity during the factorization. If or has been selected, this heading is changed to MerCP, respectively MerRP.
lenL
is the number of nonzeros in .
L+U
is the number of nonzeros representing the basis factors and . Immediately after a basis factorization , this is lenL+lenU, the number of subdiagonal elements in the columns of a lower triangular matrix and the number of diagonal and superdiagonal elements in the rows of an upper-triangular matrix. Further nonzeros are added to L when various columns of are later replaced. As columns of are replaced, the matrix is maintained explicitly (in sparse form). The value of L will steadily increase, whereas the value of U may fluctuate up or down. Thus the value of L+U may fluctuate up or down (in general, it will tend to increase).
Cmpressns
is the number of times the data structure holding the partially factored matrix needed to be compressed to recover unused storage. Ideally this number should be zero. If it is more than or , the amount of workspace available to e04nqc should be increased for efficiency.
Incres
is the percentage increase in the number of nonzeros in and relative to the number of nonzeros in or .
Utri
is the number of triangular rows of or at the top of .
lenU
the number of nonzeros in , including its diagonals.
Ltol
is the largest subdiagonal element allowed in . This is the specified or a smaller value that is currently being used for greater stability.
Umax
the maximum nonzero element in .
Ugrwth
is the ratio , which ideally should not be substantially larger than or . If it is orders of magnitude larger, it may be advisable to reduce the to , , or , say (but bigger than ).
As long as Lmax is not large (say or less), gives an estimate of the condition number . If this is extremely large, the basis is nearly singular. Slacks are used to replace suspect columns of and the modified basis is refactored.
Ltri
is the number of triangular columns of or at the left of .
dense1
is the number of columns remaining when the density of the basis matrix being factorized reached .
Lmax
is the actual maximum subdiagonal element in (bounded by Ltol).
Akmax
is the largest nonzero generated at any stage of the factorization. (Values much larger than Amax indicate instability.) Akmax is not printed if is selected.
Agrwth
is the ratio . Values much larger than (say) indicate instability. Agrwth is not printed if is selected.
bump
is the size of the block to be factorized nontrivially after the triangular rows and columns of or have been removed.
dense2
is the number of columns remaining when the density of the basis matrix being factorized reached . (The Markowitz pivot strategy searches fewer columns at that stage.)
DUmax
is the largest diagonal of .
DUmin
is the smallest diagonal of .
condU
the ratio , which estimates the condition number of (and of if Ltol is less than , say).
13.3Basis Map
When and , the following lines of intermediate printout (less than characters) are produced on the unit number specified by optional parameter . They refer to the elements of the
names
array (see Section 5).
Label
Description
Name
gives the name for the problem (blank if problem unnamed).
Infeasibilities
gives the number of infeasibilities. Printed only if the final point is infeasible.
Objective Value
gives the objective value at the final point (or the value of the sum of infeasibilities). Printed only if the final point is feasible.
Status
gives the exit status for the problem (i.e., Optimal soln, Weak soln, Unbounded, Infeasible, Excess itns, Error condn or Feasble soln) followed by details of the direction of the optimization (i.e., (Min) or (Max)).
Iteration
gives the iteration number when the file was created.
Superbasics
gives the number of superbasic variables.
Objective
gives the name of the free row for the problem (blank if objective unnamed).
RHS
gives the name of the constraint right-hand side for the problem (blank if objective unnamed).
Ranges
gives the name of the ranges for the problem (blank if objective unnamed).
Bounds
gives the name of the bounds for the problem (blank if objective unnamed).
13.4Solution Output
At the end of a run, the final solution will be output to the Print file. Some header information appears first to identify the problem and the final state of the optimization procedure. A ROWS section and a COLUMNS section then follow, giving one line of information for each row and column.
13.4.1The ROWS section
General constraints take the form . The th constraint is, therefore, of the form
where is the th row of .
Internally, the constraints take the form , where is the set of slack variables (which happen to satisfy the bounds ). For the th constraint, the slack variable is directly available, and it is sometimes convenient to refer to its state. It should satisfy . A fullstop (.) is printed for any numerical value that is exactly zero.
Label
Description
Number
is the value of . (This is used internally to refer to in the intermediate output.)
Row
gives the name of .
State
the state of (the state of relative to the bounds and ). The various states possible are as follows:
LL
is nonbasic at its lower limit, .
UL
is nonbasic at its upper limit, .
EQ
is nonbasic and fixed at the value .
FR
is nonbasic and currently zero, even though it is free to take any value between its bounds and .
BS
is basic.
SBS
is superbasic.
A key is sometimes printed before State.
Note that unless the optional parameter is specified, the tests for assigning a key are applied to the variables of the scaled problem.
A
Alternative optimum possible. The variable is nonbasic, but its reduced gradient is essentially zero. This means that if the variable were allowed to start moving away from its bound, there would be no change in the value of the objective function. The values of the other free variables might change, giving a genuine alternative solution. However, if there are any degenerate variables (labelled D), the actual change might prove to be zero, since one of them could encounter a bound immediately. In either case, the values of the Lagrange multipliers might also change.
D
Degenerate. The variable is basic or superbasic, but it is equal (or very close) to one of its bounds.
I
Infeasible. The variable is basic or superbasic and is currently violating one of its bounds by more than the value of the .
N
Not precisely optimal. If the slack is superbasic, the dual variable is not sufficiently small, as measured by the . If the slack is nonbasic, is not sufficiently positive or negative. If a loose has been used, or if iterations were terminated before optimality, this key might be helpful in deciding whether or not to restart the run.
Activity
is the value of at the final iterate.
Slack Activity
is the value by which the row differs from its nearest bound. (For the free row (if any), it is set to Activity.)
Lower Limit
is , the lower bound specified for the variable . None indicates that .
Upper Limit
is , the upper bound specified for the variable . None indicates that .
Dual Activity
is the value of the dual variable (the Lagrange multiplier for ; see Section 11.3). For FP problems, is set to zero.
i
gives the index of the th row.
13.4.2The COLUMNS Section
Let the th component of be the variable and assume that it satisfies the bounds . A fullstop (.) is printed for any numerical value that is exactly zero.
Label
Description
Number
is the column number . (This is used internally to refer to in the intermediate output.)
Column
gives the name of .
State
the state of relative to the bounds and . The various states possible are as follows:
LL
is nonbasic at its lower limit, .
UL
is nonbasic at its upper limit, .
EQ
is nonbasic and fixed at the value .
FR
is nonbasic and currently zero, even though it is free to take any value between its bounds and .
BS
is basic.
SBS
is superbasic.
A key is sometimes printed before State.
Note that unless the optional parameter is specified, the tests for assigning a key are applied to the variables of the scaled problem.
A
Alternative optimum possible. The variable is nonbasic, but its reduced gradient is essentially zero. This means that if the variable were allowed to start moving away from its bound, there would be no change in the value of the objective function. The values of the other free variables might change, giving a genuine alternative solution. However, if there are any degenerate variables (labelled D), the actual change might prove to be zero, since one of them could encounter a bound immediately. In either case, the values of the Lagrange multipliers might also change.
D
Degenerate. The variable is basic or superbasic, but it is equal (or very close) to one of its bounds.
I
Infeasible. The variable is basic or superbasic and is currently violating one of its bounds by more than the value of the .
N
Not precisely optimal. If the slack is superbasic, the dual variable is not sufficiently small, as measured by the . If the slack is nonbasic, is not sufficiently positive or negative. If a loose has been used, or if iterations were terminated before optimality, this key might be helpful in deciding whether or not to restart the run.
Activity
is the value of at the final iterate.
Obj Gradient
is the value of at the final iterate. For FP problems, is set to zero.
Lower Limit
is the lower bound specified for the variable. None indicates that .
Upper Limit
is the upper bound specified for the variable. None indicates that .
Reduced Gradnt
is the value of at the final iterate (see Section 11.3). For FP problems, is set to zero.
m + j
is the value of .
Note: if two problems are the same except that one minimizes and the other maximizes , their solutions will be the same but the signs of the dual variables and the reduced gradients will be reversed.
13.5The Solution File
If ,
the information contained in a printed solution may also be output to the relevant file (which may be the Print file if so desired). Infinite Upper and Lower limits appear as rather than None.
The maximum line length is characters.
A Solution file is intended to be read from disk by a self-contained program that extracts and saves certain values as required for possible further computation. Typically the first lines would be ignored.
The end of the ROWS section is marked by a line that starts with a and is otherwise blank. If this and the next lines are skipped, the COLUMNS section (see Section 13.4.2) can then be read under the same format.
13.6The Summary File
If , certain brief information will be output to file. A disk file should be used to retain a concise log of each run if desired. (A is more easily perused than the associated ).
The following information is included:
1.The optional parameters supplied via the option setting functions, if any;
2.The Basis file loaded, if any;
3.The status of the solution after each basis factorization (whether feasible; the objective value; the number of function calls so far);
4.The same information every th iteration, where is the specified ;
5.Warnings and error messages;
6.The exit condition and a summary of the final solution.
Item 4 is preceded by a blank line, but item 5 is not.
The meaning of the printout for linear constraints is the same as that given above for variables, with ‘variable’ replaced by ‘constraint’, replaced by , replaced by , and are replaced by and respectively, and with the following change in the heading:
Constrnt
gives the name of the linear constraint.
Note that movement off a constraint (as opposed to a variable moving away from its bound) can be interpreted as allowing the entry in the Residual column to become positive.
Numerical values are output with a fixed number of digits; they are not guaranteed to be accurate to this precision.