naginterfaces.library.mip.iqp_​sparse¶

naginterfaces.library.mip.iqp_sparse(m, iobj, ncolh, a, ha, ka, bl, bu, start, names, crname, ns, xs, intvar, istate, strtgy, leniz, lenz, comm, qphx=None, mdepth=None, monit=None, data=None, io_manager=None)[source]

iqp_sparse obtains integer solutions to sparse linear programming and quadratic programming problems.

Note: this function uses optional algorithmic parameters, see also: iqp_sparse_optfile(), iqp_sparse_optstr().

For full information please refer to the NAG Library document for h02ce

https://support.nag.com/numeric/nl/nagdoc_30.1/flhtml/h/h02cef.html

Parameters
mint

, the number of general linear constraints (or slacks). This is the number of rows in , including the free row (if any; see ).

iobjint

If , row of is a free row containing the nonzero elements of the vector appearing in the linear objective term .

If , there is no free row, i.e., the problem is either an FP problem (in which case must be set to zero), or a QP problem with .

ncolhint

, the number of leading nonzero columns of the Hessian matrix . For FP and LP problems, must be set to zero.

afloat, array-like, shape

On entry: the nonzero elements of , ordered by increasing column index. Note that multiple elements with the same row and column indices are not allowed.

On exit: used as internal workspace prior to being restored and hence is unchanged.

haint, array-like, shape

must contain the row index of the nonzero element stored in , for . Note that the row indices for a column may be supplied in any order.

kaint, array-like, shape

must contain the index in of the start of the th column, for . To specify the th column as empty, set . Note that the first and last elements of must be such that and .

blfloat, array-like, shape

, the lower bounds for all the variables and general constraints, in the following order. The first elements of must contain the bounds on the variables , and the next elements the bounds for the general linear constraints (or slacks ) and the free row (if any). To specify a nonexistent lower bound (i.e., ), set , where is the value of the option ‘Infinite Bound Size’ (). 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 .

bufloat, array-like, shape

On entry: , the upper bounds for all the variables and general constraints, in the following order. The first elements of must contain the bounds on the variables , and the next elements the bounds for the general linear constraints (or 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 .

On exit: used as internal workspace prior to being restored and hence is unchanged.

startstr, length 1

Indicates how a starting basis is to be obtained.

An internal crash procedure will be used to choose an initial basis matrix .

A basis is already defined in (probably from a previous call).

namesstr, length 8, array-like, shape

A set of names associated with the so-called MPSX form of the problem.

Must contain the name for the problem (or be blank).

Must contain the name for the free row (or be blank).

Must contain the name for the constraint right-hand side (or be blank).

Must contain the name for the ranges (or be blank).

Must contain the name for the bounds (or be blank).

(These names are used in the monitoring file output; see Monitoring Information.)

crnamestr, length 8, array-like, shape

The optional column and row names.

If , is not referenced and the printed output will use default names for the columns and rows.

If , the first elements must contain the names for the columns and the next elements must contain the names for the rows.

Note that the name for the free row (if any) must be stored in .

nsint

, the number of superbasics. For QP problems, need not be specified if , but must retain its value from a previous call when . For FP and LP problems, need not be initialized.

xsfloat, array-like, shape

The initial values of the variables and slacks . (See the description for .)

intvarint, array-like, shape

Specifies which components of the solution vector are constrained to be integer. Specifically, if elements of are required to take integer values then , for , where is the integer index such that is integer. If then must be set to to signal the end of the integer variable indices.

The order in which the indices of those components of required to be integer is presented determines the order in which the sub-problems are treated and solved.

As such it can be a powerful tool to assist the function in achieving a solution efficiently.

The general advice is to enter the important integer variables in the model early in ; secondary or less important variables should be entered near the end of the list.

However some experimentation might be required to find the optimal order.

istateint, array-like, shape

If , the first elements of and 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 columns of . Possible values for are as follows:

State of during crash procedure

0 or

Eligible for the basis

2

Ignored

3

Eligible for the basis (given preference over or )

4 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 , and must specify the initial states and values, respectively, of the variables and slacks .

If iqp_sparse has been called previously with the same values of and , already contains satisfactory information.

strtgyint

Defines the branching strategy adopted by the function.

Each sub-problem first explored imposes a tighter upper bound on the component of .

Each sub-problem first explored imposes a tighter lower bound on the component of .

Each branch explored imposes a tighter upper bound on the component of if its fractional part is less than , otherwise it imposes a tighter lower bound.

Random choice is made between first exploring a tighter lower bound or a tighter upper bound sub-problem.

lenizint

The dimension of the internal workspace array .

lenzint

The dimension of the internal workspace array .

commdict, communication object, modified in place

Communication structure.

This argument must have been initialized by a prior call to opt.nlp1_init with .

qphxNone or callable hx = qphx(nstate, x, data=None), optional

Note: if this argument is None then a NAG-supplied facility will be used.

For QP problems, you must supply a version of to compute the matrix product .

If has rows and columns consisting entirely of zeros, it is most efficient to order the variables so that

where the nonlinear variables appear first as shown.

For LP problems, will never be called by iqp_sparse.

Parameters
nstateint

If , then iqp_sparse is calling for the first time on a sub-problem. This argument setting allows you to save computation time if certain data must be read or calculated only once.

If , then iqp_sparse is calling for the last time.

This argument setting allows you to perform some additional computation on the final sub-problem solution.

In general, the last call to is made with (see Exceptions).

Otherwise, .

xfloat, ndarray, shape

The first elements of the vector .

dataarbitrary, optional, modifiable in place

User-communication data for callback functions.

Returns
hxfloat, array-like, shape

The product .

mdepthNone or int, optional

Note: if this argument is None then a default value will be used, determined as follows: .

Specifies the maximum depth the tree of sub-problems may be developed.

monitNone or callable (bstval, halt) = monit(intfnd, nodes, depth, obj, x, bstval, bstsol, bl, bu, halt, data=None), optional

Note: if this argument is None then a NAG-supplied facility will be used.

To provide feed-back on the progress of the branch and bound process.

Additionally provides, via its argument , the ability to terminate the process. (You might choose to do this when an integer solution is found, rather than search for a better solution.) If you do not require any intermediate output then may be None.

Parameters
intfndint

Contains the number of integer solutions obtained so far.

nodesint

Contains the number of nodes (sub-problems) solved so far.

depthint

Contains the depth reached in the tree of problems.

objfloat

Contains the solution value to the sub-problem at this node.

xfloat, ndarray, shape

Contains the solution vector to the sub-problem at this node.

bstvalfloat

Contains the value of the objective function corresponding to the best integer solution obtained so far. If no integer solution has been found contains the largest machine representable number (see machine.real_largest).

bstsolfloat, ndarray, shape

Contains the value of the best integer solution obtained so far.

blfloat, ndarray, shape

Contains the current lower bounds on the variables at this point.

bufloat, ndarray, shape

Contains the current upper bounds on the variables at this point.

haltbool

Will have the value .

dataarbitrary, optional, modifiable in place

User-communication data for callback functions.

Returns
bstvalfloat

May be set to a cut-off value, if you are a sophisticated user, as follows. Before an integer solution has been found will be set by iqp_sparse to the largest machine representable number (see machine.real_largest). If you know that the solution being sought is a much smaller number, then may be set to this number as a cut-off value (see Notes). Beware of setting too small, since then no integer solutions will be discovered. Also, on entry to , should be set using a statement of the form

if intfnd == 0: bstval = cut_off_value


Setting in this way will not prevent the normal operation of the algorithm when subsequent integer solutions are found. It would be a grievous mistake to unconditionally set and if you have any doubts whatsoever about the correct use of this argument then you are strongly recommended to leave it unchanged.

haltbool

If is set to , opt.qpconvex1_sparse_solve will be brought to a halt with exit . This facility may be useful if you are content with any integer solution, or with any integer solution that fits certain criteria. Under these circumstances setting can save considerable unnecessary computation.

dataarbitrary, optional

User-communication data for callback functions.

io_managerFileObjManager, optional

Manager for I/O in this routine.

Returns
nsint

The final number of superbasics. This will be zero for FP and LP problems.

xsfloat, ndarray, shape

contains the final value of , for .

istateint, ndarray, shape

The final states of the variables and slacks from the solution of the last sub-problem tackled. 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 (see Further Comments), basic and superbasic variables may be outside their bounds by as much as the value of the option ‘Feasibility Tolerance’ (, where is the machine precision). Note that unless the option () is specified, the ‘Feasibility Tolerance’ 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 ‘Feasibility Tolerance’, 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 (see Further Comments) if ).

minizint

The minimum value of required to start solving the problem. If = 14, iqp_sparse may be called again with suitably larger than . (The bigger the better, since it is not certain how much workspace the basis factors need.)

minzint

The minimum value of required to start solving the problem. If = 15, iqp_sparse may be called again with suitably larger than . (The bigger the better, since it is not certain how much workspace the basis factors need.)

objfloat

The value of the objective function.

If , includes the quadratic objective term (if any).

If , is just the linear objective term (if any).

For FP problems, is set to zero.

clamdafloat, ndarray, shape

A set of Lagrange-multipliers for the bounds on the variables and the general constraints. More precisely, the first elements contain the multipliers (reduced costs) for the bounds on the variables, and the next elements contain the multipliers (shadow prices) for the general linear constraints.

Other Parameters
‘Check Frequency’int

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 default value is used. If , the value is used and effectively no checks are made.

‘Crash Option’int

Default

Note that this option does not apply when (see Parameters).

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 are initially eligible for the basis, and how many times the crash procedure is called. If , the all-slack basis is chosen. If , the crash procedure is called once (looking for a triangular basis in all rows and columns of the linear constraint matrix ). If , the crash procedure is called twice (looking at any equality constraints first followed by any inequality constraints). If or , the default value is used.

If or , 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.

‘Crash Tolerance’float

Default

This value allows the crash procedure to ignore certain ‘small’ nonzero elements in the constraint matrix while searching for a triangular basis. For each column of , if is the largest element in the column, other nonzeros in that column are ignored if they are less than (or equal to) .

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 with more column variables and fewer (arbitrary) slacks. A feasible solution may be reached earlier for some problems. If or , the default value is used.

‘Defaults’valueless

This special keyword may be used to reset all options to their default values.

‘Expand Frequency’int

Default

This option is part of an anti-cycling procedure (see Miscellaneous) designed to allow progress even on highly degenerate problems.

For LP 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 option ‘Feasibility Tolerance’ is . Over a period of iterations, the feasibility tolerance actually used by iqp_sparse (i.e., the working feasibility tolerance) increases from to (in steps of ).

For QP problems, the same procedure is used for iterations in which there is only one superbasic variable. (Cycling can only occur when the current solution is at a vertex of the feasible region.) Thus, zero steps are allowed if there is more than one superbasic variable, but otherwise positive steps are enforced.

Increasing the value of helps reduce the number of slightly infeasible nonbasic basic 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 option ‘Pivot Tolerance’).

If , the default value is used. If , the value is used and effectively no anti-cycling procedure is invoked.

‘Factorization Frequency’int

Default

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. 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 option ‘Check Frequency’ to ensure that the linear constraints are satisfied. If necessary, the basis will be refactorized before the limit of updates is reached. If , the default value is used.

‘Feasibility Tolerance’float

Default

If , defines the maximum acceptable absolute violation in each constraint at a ‘feasible’ point (including slack variables). For example, if the variables and the coefficients in the linear constraints are of order unity, and the latter are correct to about five decimal digits, it would be appropriate to specify as . If , the default value is used.

iqp_sparse 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 the function does not attempt to find the minimum value of Sinf.

If the constraints and variables have been scaled (see the description of the option ‘Scale Option’), then feasibility is defined in terms of the scaled problem (since it is more likely to be meaningful).

‘Infinite Bound Size’float

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.

‘Infinite Step Size’float

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.

‘Iteration Limit’int

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, but no iterations will be performed. If , the default value is used.

‘Iters’int

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, but no iterations will be performed. If , the default value is used.

‘Itns’int

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, but no iterations will be performed. If , the default value is used.

‘List’valueless

Option ‘List’ enables printing of each option specification as it is supplied. ‘Nolist’ suppresses this printing.

‘Nolist’valueless

Default

Option ‘List’ enables printing of each option specification as it is supplied. ‘Nolist’ suppresses this printing.

‘LU Factor Tolerance’float

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. For large and relatively dense problems, setting and to (say) may give a marked improvement in sparsity without impairing stability to a serious degree. Note that for band matrices it may be necessary to set in the range in order to achieve stability. If or , the default value is used.

‘LU Update Tolerance’float

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. For large and relatively dense problems, setting and to (say) may give a marked improvement in sparsity without impairing stability to a serious degree. Note that for band matrices it may be necessary to set in the range in order to achieve stability. If or , the default value is used.

‘LU Singularity Tolerance’float

Default

If , defines the singularity tolerance used to guard against ill-conditioned basis matrices. Whenever the basis 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.

‘Minimize’valueless

Default

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 The Main Iteration) will be reversed.

‘Maximize’valueless

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 The Main Iteration) will be reversed.

‘Monitoring File’int

Default

If and , monitoring information produced by iqp_sparse is sent to a file with logical unit number . If and/or , the default value is used and hence no monitoring information is produced.

‘Optimality Tolerance’float

Default

If , is used to judge the size of the reduced gradients . By definition, the reduced gradients for basic variables are always zero. Optimality is declared if the reduced gradients for any nonbasic variables at their lower or upper bounds satisfy , and if for any superbasic variables. If , the default value is used.

‘Partial Price’int

Default

Note that this option does not apply to QP problems.

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 . 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’float

Default

If , is used to prevent columns entering the basis if they would cause the basis to become almost singular. If , the default value is used.

‘Print Level’int

Default

The value of controls the amount of printout produced by iqp_sparse, as indicated below. A detailed description of the printed output is given in Further Comments (summary output at each iteration and the final solution) and Monitoring Information (monitoring information at each iteration). Note that the summary output will not exceed characters per line and that the monitoring information will not exceed characters per line. If , the default value is used. The following printout is sent to the file object associated with the advisory I/O unit (see FileObjManager):

Output

No output.

The final solution only.

One line of summary output for each iteration (no printout of the final solution).

The final solution and one line of summary output for each iteration.

The following printout is sent to the logical unit number defined by the ‘Monitoring File’:

Output

No output.

The final solution only.

One long line of output for each iteration (no printout of the final solution).

The final solution and one long line of output for each iteration.

The final solution, one long line of output for each iteration, matrix statistics (initial status of rows and columns, number of elements, density, biggest and smallest elements, etc.), details of the scale factors resulting from the scaling procedure (if or ), basis factorization statistics and details of the initial basis resulting from the crash procedure (if ; see Parameters).

If and the unit number defined by ‘Monitoring File’ is the advisory unit number, then the summary output is suppressed.

‘Rank Tolerance’float

Default

See above.

‘Scale Option’int

Default

This option enables you to scale the variables and constraints using an iterative procedure due to Fourer (see Hock and Schittkowski (1981)), which attempts to compute row scales and column scales such that the scaled matrix coefficients are as close as possible to unity. This may improve the overall efficiency of the function on some problems. (The lower and upper bounds on the variables and slacks for the scaled problem are redefined as and respectively, where if .)

If , no scaling is performed. If , all rows and columns of the constraint matrix are scaled. If , an additional scaling is performed that may be helpful when the solution is large; it takes into account columns of that are fixed or have positive lower bounds or negative upper bounds. If or , the default value is used.

‘Scale Tolerance’float

Default

Note that this option does not apply when .

If , is used to control the number of scaling passes to be made through the constraint matrix . At least (and at most ) passes will be made. More precisely, let denote the largest column ratio (i.e., in some sense) after the th scaling pass through . The scaling procedure is terminated if for some . Thus, increasing the value of from to (say) will probably increase the number of passes through . If or , the default value is used.

‘Superbasics Limit’int

Default

Note that this option does not apply to FP or LP problems.

The value of specifies ‘how nonlinear’ you expect the QP problem to be. If , the default value is used.

Raises
NagValueError
(errno )

On entry, .

Constraint: , , or .

(errno )

On entry, .

Constraint: .

(errno )

On entry, .

Constraint: .

(errno )

On entry, , .

Constraint: with .

(errno )

On entry, , .

Constraint: with .

(errno )

On entry, , and .

Constraint: or .

(errno )

On entry, .

Constraint: or .

(errno )

On entry, , and .

Constraint: .

(errno )

On entry, and .

Constraint: .

(errno )

On entry, , and .

Constraint: .

(errno )

On entry, .

Constraint: .

(errno )

On entry, , and .

Constraint: .

(errno )

On entry, and .

Constraint: .

(errno )

On entry, and .

Constraint: .

(errno )

On entry, , and .

Constraint:

(errno )

On entry, .

Constraint: .

(errno )

On entry, .

Constraint: .

(errno )

No integer solutions found.

(errno )

Need to increase depth, .

(errno )

The problem is unbounded (or badly scaled).

(errno )

The problem is infeasible.

(errno )

Too many iterations. Try increasing Iteration Limit.

(errno )

Reduced Hessian exceeds assigned dimension.

(errno )

Hessian appears to be indefinite.

(errno )

Invalid input argument for internal call to local optimizer.

(errno )

Numerical error trying to satisfy constraints. The basis is very ill-conditioned.

(errno )

Singular basis after attempts to factorize.

(errno )

Unexpected error in NAG function – contact NAG.

Warns
NagAlgorithmicWarning
(errno )

Notes

No equivalent traditional C interface for this routine exists in the NAG Library.

iqp_sparse is designed to obtain integer solutions to a class of quadratic programming problems addressed by opt.qpconvex1_sparse_solve. Specifically it solves the following problem:

where is a set of variables (some of which may be required to be integer), is an matrix and the objective function may be specified in a variety of ways depending upon the particular problem to be solved. The option ‘Maximize’ may be used to specify an alternative problem in which is maximized. The possible forms for are listed in Table [label omitted], in which the prefixes LP and QP stand for ‘linear programming’ and ‘quadratic programming’ respectively, is an -element vector and is the second-derivative matrix (the Hessian matrix).

Problem type

Objective function

Hessian matrix

LP

Not applicable

QP

Symmetric positive semidefinite

For LP and QP problems, the unique global minimum value of is found. For QP problems, you must also provide a function that computes for any given vector . ( need not be stored explicitly.)

(It is not expected that the feasibility problem of opt.qpconvex1_sparse_solve would be relevant here.)

The function employs a ‘Branch and Bound’ technique to enforce the integer constraints. In this technique the problem is first solved without the integer constraints. If a variable is found to be non-integral when it is required to have an integer value then two additional problems are constructed. One bounds the variable above by the nearest integer value below the optimal value previously obtained. The second problem is formed by bounding the variable below by the nearest integer value above the optimal value. This process is continued until an integer solution is found. At this point you may elect to stop, or may continue to search for better integer solutions by examining any other sub-problems that remain to be explained.

In practice the function tries to compute an integer solution as quickly as possible using a depth-first approach, since this helps determine a realistic cut-off value. If we have a cut-off value, say the value of the function at this first integer solution, and any sub-problem, say, has a solution value greater than this cut-off value, then subsequent sub-problems of must have solutions greater than the value of the solution at and, therefore, need not be computed. Thus a knowledge of a good cut-off value can result in fewer sub-problems being solved and thus speed up the operation of the function. (See the description of in Parameters for details of how you can supply your own cut-off value.)

Each sub-problem is solved using opt.qpconvex1_sparse_solve. You are referred to the function document for opt.qpconvex1_sparse_solve for details of the algorithm used.

References

Gill, P E, Hammarling, S, Murray, W, Saunders, M A and Wright, M H, 1986, Users’ guide for LSSOL (Version 1.0), Report SOL 86-1, Department of Operations Research, Stanford University

Gill, P E and Murray, W, 1978, Numerically stable methods for quadratic programming, Math. Programming (14), 349–372

Gill, P E, Murray, W, Saunders, M A and Wright, M H, 1986, Some theoretical properties of an augmented Lagrangian merit function, Report SOL, 86–6R, Department of Operations Research, Stanford University

Gill, P E, Murray, W, Saunders, M A and Wright, M H, 1989, A practical anti-cycling procedure for linearly constrained optimization, Math. Programming (45), 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

Hock, W and Schittkowski, K, 1981, Test Examples for Nonlinear Programming Codes. Lecture Notes in Economics and Mathematical Systems (187), Springer–Verlag

Lawson, C L, Hanson, R J, Kincaid, D R and Krogh, F T, 1979, Basic linear algebra supbrograms for Fortran usage, ACM Trans. Math. Software (5), 308–325

Murtagh, B A and Saunders, M A, 1983, MINOS 5.0 user’s guide, Report SOL 83-20, Department of Operations Research, Stanford University