e04nfc (Integer n, Integer nclin, const double a[], Integer tda, const double bl[], const double bu[], const double cvec[], const double h[], Integer tdh,

void	(qphess)(Integer n, Integer jthcol, const double h[], Integer tdh, const double x[], double hx[], Nag_Comm comm),

double x[], double *objf, Nag_E04_Opt *options, Nag_Comm *comm, NagError *fail)

The function may be called by the names: e04nfc, nag_opt_qp_dense_solve or nag_opt_qp.

3 Description

e04nfc is designed to solve a class of quadratic programming problems stated in the following general form:

\underset{x \in R^{n}}{minimize} f (x) subject to l \leq \{\begin{matrix} x \\ A x \end{matrix}\} \leq u,

where

A

is an

m_{lin}

n

matrix and

f (x)

may be specified in a variety of ways depending upon the particular problem to be solved. The available forms for

f (x)

are listed in Table 1 below, in which the prefixes FP, LP and QP stand for ‘feasible point’, ‘linear programming’ and ‘quadratic programming’ respectively and

c

is an

n

element vector.

Problem Type	$f (x)$	Matrix $H$
FP	Not applicable	Not applicable
LP	$c^{T} x$	Not applicable
QP1	$\frac{1}{2} x^{T} H x$	symmetric
QP2	$c^{T} x + \frac{1}{2} x^{T} H x$	symmetric
QP3	$\frac{1}{2} x^{T} H^{T} H x$	$m$ by $n$ upper trapezoidal
QP4	$c^{T} x + \frac{1}{2} x^{T} H^{T} H x$	$m$ by $n$ upper trapezoidal

Table 1

For problems of type FP a feasible point with respect to a set of linear inequality constraints is sought. The default problem type is QP2, other objective functions are selected by using the optional parameter

options . prob

The constraints involving

A

are called the general constraints. Note that upper and lower bounds are specified for all the variables and for all the general constraints. An equality constraint can be specified by setting

l_{i} = u_{i}

. If certain bounds are not present, the associated elements of

l

u

can be set to special values that will be treated as

- \infty

+ \infty

. (See the description of the optional parameter

options . inf_bound

The defining feature of a quadratic function

f (x)

is that the second-derivative matrix

\nabla^{2} f (x)

(the Hessian matrix) is constant. For the LP case,

\nabla^{2} f (x) = 0

; for QP1 and QP2,

\nabla^{2} f (x) = H

; and for QP3 and QP4,

\nabla^{2} f (x) = H^{T} H

. If

H

is defined as the zero matrix, e04nfc will solve the resulting linear programming problem; however, this can be accomplished more efficiently by setting the optional parameter

options . prob = Nag_LP

, or by using e04mfc.

You must supply an initial estimate of the solution.

In the QP case, you may supply

H

either explicitly as an

m

n

matrix, or implicitly in a C function that computes the product

H x

for any given vector

x

. An example of such a function is included in Section 10. There is no restriction on

H

apart from symmetry. In general, a successful run of e04nfc will indicate one of three situations: (i) a minimizer has been found; (ii) the algorithm has terminated at a so-called dead-point; or (iii) the problem has no bounded solution. If a minimizer is found, and

H

is positive definite or positive semidefinite, e04nfc will obtain a global minimizer; otherwise, the solution will be a local minimizer (which may or may not be a global minimizer). A dead-point is a point at which the necessary conditions for optimality are satisfied but the sufficient conditions are not. At such a point, a feasible direction of decrease may or may not exist, so that the point is not necessarily a local solution of the problem. Verification of optimality in such instances requires further information, and is in general an NP-hard problem (see Pardalos and Schnitger (1988)). Termination at a dead-point can occur only if

H

is not positive definite. If

H

is positive semidefinite, the dead-point will be a weak minimizer (i.e., with a unique optimal objective value, but an infinite set of optimal

x

Details about the algorithm are described in Section 11, but it is not necessary to read this more advanced section before using e04nfc.

4 References

Bunch J R and Kaufman L C (1980) A computational method for the indefinite quadratic programming problem Linear Algebra and its Applications 34 341–370

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 (1984) Procedures for optimization problems with a mixture of bounds and general linear constraints ACM Trans. Math. Software 10 282–298

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

Pardalos P M and Schnitger G (1988) Checking local optimality in constrained quadratic programming is NP-hard Operations Research Letters 7 33–35

5 Arguments

1: $n$ – Integer Input

On entry:

n

, the number of variables.

Constraint:

n > 0

2: $nclin$ – Integer Input

On entry:

m_{lin}

, the number of general linear constraints.

Constraint:

nclin \geq 0

3: $a [nclin \times tda]$ – const double Input

Note: the

(i, j)

th element of the matrix

A

is stored in

a [(i - 1) \times tda + j - 1]

On entry: the

i

th row of a must contain the coefficients of the

i

th general linear constraint (the

i

th row of

A

), for

i = 1, 2, \dots, m_{lin}

. If

nclin = 0

, the array a is not referenced.

4: $tda$ – Integer Input

On entry: the stride separating matrix column elements in the array a.

Constraint: if

nclin > 0

tda \geq n

5: $bl [n + nclin]$ – const double Input

6: $bu [n + nclin]$ – const double Input

On entry: bl must contain the lower bounds and bu the upper bounds, for all the constraints in the following order. The first

n

elements of each array must contain the bounds on the variables, and the next

m_{lin}

elements the bounds for the general linear constraints (if any). To specify a nonexistent lower bound (i.e.,

l_{j} = - \infty

), set

bl [j] \leq - options . inf_bound

, and to specify a nonexistent upper bound (i.e.,

u_{j} = + \infty

), set

bu [j] \geq options . inf_bound

;

options . inf_bound

is the optional parameter, whose default value is

10^{20}

. To specify the

j

th constraint as an equality, set

bl [j] = bu [j] = β

, say, where

|β| < options . inf_bound

Constraints:

$bl [j] \leq bu [j]$ , for $j = 0, 1, \dots, n + nclin - 1$ ;
if $bl [j] = bu [j] = β$ , $|β| < options . inf_bound$ .

7: $cvec [n]$ – const double Input

On entry: the coefficients of the explicit linear term of the objective function when the problem is of type

options . prob = Nag_LP

Nag_QP2

Nag_QP4

. The default problem type is

options . prob = Nag_QP2

corresponding to QP2 described in Section 3; other problem types can be specified using the optional parameter

options . prob

If the problem is of type

options . prob = Nag_FP

Nag_QP1

Nag_QP3

, cvec is not referenced and therefore a NULL pointer may be given.

8: $h [n \times tdh]$ – const double Input

On entry: h may be used to store the quadratic term

H

of the QP objective function if desired. The elements of h are accessed only by the function qphess; thus h is not accessed if the problem is of type

options . prob = Nag_FP

Nag_LP

. The number of rows of

H

is denoted by

m

, its default value is equal to

n

. (The optional parameter

options . hrows

may be used to specify a value of

m < n

If the problem is of type

options . prob = Nag_QP1

Nag_QP2

, the first

m

rows and columns of h must contain the leading

m

m

rows and columns of the symmetric Hessian matrix. Only the diagonal and upper triangular elements of the leading

m

rows and columns of h are referenced. The remaining elements need not be assigned.

For problems

options . prob = Nag_QP3

Nag_QP4

, the first

m

rows of h must contain an

m

n

upper trapezoidal factor of the Hessian matrix. The factor need not be of full rank, i.e., some of the diagonals may be zero. However, as a general rule, the larger the dimension of the leading nonsingular sub-matrix of

H

, the fewer iterations will be required. Elements outside the upper trapezoidal part of the first

m

rows of

H

are assumed to be zero and need not be assigned.

In some cases, you need not use h to store

H

explicitly (see the specification of function qphess).

9: $tdh$ – Integer Input

On entry: the stride separating matrix column elements in the array h.

Constraint:

tdh \geq n

or at least the value of the optional parameter

options . hrows

if it is set.

10: $qphess$ – function, supplied by the user External Function

In general, you need not provide a version of qphess, because a ‘default’ function is included in the NAG Library. If the default function is required then the NAG defined null void function pointer, NULLFN, should be supplied in the call to e04nfc. The algorithm of e04nfc requires only the product of

H

and a vector

x

; and in some cases you may obtain increased efficiency by providing a version of qphess that avoids the need to define the elements of the matrix

H

explicitly.

qphess is not referenced if the problem is of type

options . prob = Nag_FP

Nag_LP

, in which case qphess should be replaced by NULLFN.

The specification of qphess is:

void	qphess (Integer n, Integer jthcol, const double h[], Integer tdh, const double x[], double hx[], Nag_Comm *comm)

1: $n$ – Integer Input

On entry:

n

, the number of variables.

2: $jthcol$ – Integer Input

On entry: jthcol specifies whether or not the vector

x

is a column of the identity matrix.

$jthcol = j > 0$: The vector $x$ is the $j$ th column of the identity matrix, and hence $H x$ is the $j$ th column of $H$ , which can sometimes be computed very efficiently and qphess may be coded to take advantage of this. However special code is not necessary because $x$ is always stored explicitly in the array x.
$jthcol = 0$: $x$ has no special form.

3: $h [n \times tdh]$ – const double Input

On entry: the matrix

H

of the QP objective function. The matrix element

H_{i j}

is stored in

h [(i - 1) \times tdh + j - 1]

, for

i = 1, 2, \dots, n

and

j = 1, 2, \dots, n

. In some situations, it may be desirable to compute

H x

without accessing h – for example, if

H

is sparse or has special structure. (This is illustrated in the function qphess1 in Section 10.) The arguments h and tdh may then refer to any convenient array.

4: $tdh$ – Integer Input

On entry: the stride separating matrix column elements in the array h.

5: $x [n]$ – const double Input

On entry: the vector

x

6: $hx [n]$ – double Output

On exit: the product

H x

7: $comm$ – Nag_Comm *

Pointer to structure of type Nag_Comm; the following members are relevant to qphess.

flag – IntegerInput/Output: On entry: $comm \to flag$ contains a non-negative number.

On exit: if qphess resets $comm \to flag$ to some negative number e04nfc will terminate immediately with the error indicator NE_USER_STOP. If fail is supplied to e04nfc, $fail . errnum$ will be set to your setting of $comm \to flag$ .

first – Nag_BooleanInput: On entry: will be set to Nag_TRUE on the first call to qphess and Nag_FALSE for all subsequent calls.

nf – IntegerInput: On entry: the number of calls made to qphess including the current one.

user – double *
iuser – Integer *
p – Pointer: The type Pointer will be void * with a C compiler that defines void * and char * otherwise. Before calling e04nfc you may allocate memory to these pointers and they may be initialized with various quantities for use by qphess when called from e04nfc.

Note: qphess should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by e04nfc. If your code inadvertently does return any NaNs or infinities, e04nfc is likely to produce unexpected results.

Note: qphess should be tested separately before being used in conjunction with e04nfc. The input arrays h and x must not be changed within qphess.

11: $x [n]$ – double Input/Output

On entry: an initial estimate of the solution.

On exit: the point at which e04nfc terminated. If

fail . code = NE_NOERROR

, NW_DEAD_POINT, NW_SOLN_NOT_UNIQUE or NW_NOT_FEASIBLE, x contains an estimate of the solution.

12: $objf$ – double * Output

On exit: the value of the objective function at

x

x

is feasible, or the sum of infeasibilities at

x

otherwise. If the problem is of type

options . prob = Nag_FP

and

x

is feasible, objf is set to zero.

13: $options$ – Nag_E04_Opt * Input/Output

On entry/exit: a pointer to a structure of type Nag_E04_Opt whose members are optional parameters for e04nfc. These structure members offer the means of adjusting some of the argument values of the algorithm and on output will supply further details of the results. A description of the members of options is given in Section 12. Some of the results returned in options can be used by e04nfc to perform a ‘warm start’ if it is re-entered (see the optional argument

options . start

If any of these optional parameters are required then the structure options should be declared and initialized by a call to e04xxc and supplied as an argument to e04nfc. However, if the optional parameters are not required the NAG defined null pointer, E04_DEFAULT, can be used in the function call.

14: $comm$ – Nag_Comm * Input/Output

Note: comm is a NAG defined type (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).

On entry/exit: structure containing pointers for user communication with user-supplied functions; see the description of qphess for details. If you do not need to make use of this communication feature the null pointer NAGCOMM_NULL may be used in the call to e04nfc; comm will then be declared internally for use in calls to user-supplied functions.

15: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

If one of NE_USER_STOP, NE_2_INT_ARG_LT, NE_OPT_NOT_INIT, NE_BAD_PARAM, NE_INVALID_INT_RANGE_1, NE_INVALID_INT_RANGE_2, NE_INVALID_REAL_RANGE_FF, NE_INVALID_REAL_RANGE_F, NE_CVEC_NULL, NE_H_NULL, NE_WARM_START, NE_BOUND, NE_BOUND_LCON, NE_STATE_VAL and NE_ALLOC_FAIL occurs, no values will have been assigned to objf, or to $options . ax$ and $options . lambda$ . x and $options . state$ will be unchanged.
NE_2_INT_ARG_LT: On entry, $tda = 〈value〉$ while $n = 〈value〉$ . These arguments must satisfy $tda \geq n$ .

On entry, $tdh = 〈value〉$ while $n = 〈value〉$ . These arguments must satisfy $tdh \geq n$ .

On entry, $tdh = 〈value〉$ while $options . hrows = 〈value〉$ . These arguments must satisfy $tdh \geq options . hrows$ .
NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_BAD_PARAM: On entry, argument $options . print_level$ had an illegal value.

On entry, argument $options . prob$ had an illegal value.

On entry, argument $options . start$ had an illegal value.
NE_BOUND: The lower bound for variable $〈value〉$ (array element $bl [〈value〉]$ ) is greater than the upper bound.
NE_BOUND_LCON: The lower bound for linear constraint $〈value〉$ (array element $bl [〈value〉]$ ) is greater than the upper bound.
NE_CVEC_NULL: $options . prob = 〈value〉$ but argument $cvec =$ NULL.
NE_H_NULL: $options . prob = 〈value〉$ , qphess is NULL but argument h is also NULL. If the default function for qphess is to be used for this problem then an array must be supplied in argument h.
NE_HESS_TOO_BIG: Reduced Hessian exceeds assigned dimension. $options . max_df = 〈value〉$ .
The algorithm needed to expand the reduced Hessian when it was already at its maximum dimension, as specified by the optional parameter $options . max_df$ .
The value of the argument $options . max_df$ is too small. Rerun e04nfc with a larger value (possibly using the $options . start = Nag_Warm$ facility to specify the initial working set).
NE_INT_ARG_LT: On entry, $n = 〈value〉$ .
Constraint: $n \geq 1$ .

On entry, $nclin = 〈value〉$ .
Constraint: $nclin \geq 0$ .
NE_INVALID_INT_RANGE_1: Value $〈value〉$ given to $options . fcheck$ not valid. Correct range is $options . fcheck \geq 1$ .

Value $〈value〉$ given to $options . fmax_iter$ not valid. Correct range is $options . fmax_iter \geq 0$ .

Value $〈value〉$ given to $options . hrows$ not valid. Correct range is $n \geq options . hrows \geq 0$ .

Value $〈value〉$ given to $options . max_df$ not valid. Correct range is $n \geq options . max_df \geq 1$ .

Value $〈value〉$ given to $options . max_iter$ not valid. Correct range is $options . max_iter \geq 0$ .
NE_INVALID_INT_RANGE_2: Value $〈value〉$ given to $options . reset_ftol$ not valid. Correct range is $0 < options . reset_ftol < 10000000$ .
NE_INVALID_REAL_RANGE_F: Value $〈value〉$ given to $options . ftol$ not valid. Correct range is $options . ftol > 0.0$ .

Value $〈value〉$ given to $options . inf_bound$ not valid. Correct range is $options . inf_bound > 0.0$ .

Value $〈value〉$ given to $options . inf_step$ not valid. Correct range is $options . inf_step > 0.0$ .
NE_INVALID_REAL_RANGE_FF: Value $〈value〉$ given to $options . crash_tol$ not valid. Correct range is $0.0 \leq options . crash_tol \leq 1.0$ .

Value $〈value〉$ given to $options . rank_tol$ not valid. Correct range is $0.0 \leq options . rank_tol < 1.0$ .
NE_NOT_APPEND_FILE: Cannot open file $〈string〉$ for appending.
NE_NOT_CLOSE_FILE: Cannot close file $〈string〉$ .
NE_OPT_NOT_INIT: Options structure not initialized.
NE_STATE_VAL: $options . state [〈value〉]$ is out of range. $options . state [〈value〉] = 〈value〉$ .
NE_UNBOUNDED: Solution appears to be unbounded.
This value of fail implies that a step as large as $options . inf_step$ would have to be taken in order to continue the algorithm. This situation can occur only when $H$ is not positive definite and at least one variable has no upper or lower bound.
NE_USER_STOP: User requested termination, user flag value $= 〈value〉$ .
This exit occurs if you set $comm \to flag$ to a negative value in qphess. If fail is supplied the value of $fail . errnum$ will be the same as your setting of $comm \to flag$ .
NE_WARM_START: $options . start = Nag_Warm$ but pointer $options . state =$ NULL.
NE_WRITE_ERROR: Error occurred when writing to file $〈string〉$ .
NW_DEAD_POINT: Iterations terminated at a dead point (check the optimality conditions).
The necessary conditions for optimality have been satisfied but the sufficient conditions are not. (The reduced gradient is negligible, the Lagrange multipliers are optimal, but $H_{r}$ is singular or there are some very small multipliers.) If $H$ is not positive definite, $x$ is not necessarily a local solution of the problem and verification of optimality requires further information.
NW_NOT_FEASIBLE: No feasible point was found for the linear constraints.
It was not possible to satisfy all the constraints to within the feasibility tolerance. In this case, the constraint violations at the final $x$ will reveal a value of the tolerance for which a feasible point will exist – for example, if the feasibility tolerance for each violated constraint exceeds its Residual at the final point. You should check that there are no constraint redundancies. If the data for the constraints are accurate only to the absolute precision $σ$ , you should ensure that the value of the optional parameter $options . ftol$ is greater than $σ$ . For example, if all elements of $A$ are of order unity and are accurate only to three decimal places, the optional parameter $options . ftol$ should be at least $10^{- 3}$ .
NW_OVERFLOW_WARN: Serious ill conditioning in the working set after adding constraint $〈value〉$ . Overflow may occur in subsequent iterations.
If overflow occurs preceded by this warning then serious ill conditioning has probably occurred in the working set when adding a constraint. It may be possible to avoid the difficulty by increasing the magnitude of the optional parameter $options . ftol$ and re-running the program. If the message recurs even after this change, the offending linearly dependent constraint $j$ must be removed from the problem.
NW_SOLN_NOT_UNIQUE: Optimal solution is not unique.
The necessary conditions for optimality have been satisfied but the sufficient conditions are not. (The reduced gradient is negligible, the Lagrange multipliers are optimal, but $H_{r}$ is singular or there are some very small multipliers.) If $H$ is positive semidefinite, $x$ gives the global minimum value of the objective function, but the final $x$ is not unique.
NW_TOO_MANY_ITER: The maximum number of iterations, $〈value〉$ , have been performed.
The value of the optional parameter $options . max_iter$ may be too small. If the method appears to be making progress (e.g., the objective function is being satisfactorily reduced), increase the value of $options . max_iter$ and rerun e04nfc (possibly using the $options . start = Nag_Warm$ facility to specify the initial working set).

7 Accuracy

e04nfc implements a numerically stable active set strategy and returns solutions that are as accurate as the condition of the problem warrants on the machine.

8 Parallelism and Performance

e04nfc is not threaded in any implementation.

9 Further Comments

Sensible scaling of the problem is likely to reduce the number of iterations required and make the problem less sensitive to perturbations in the data, thus improving the condition of the problem. In the absence of better information it is usually sensible to make the Euclidean lengths of each constraint of comparable magnitude. See the E04 Chapter Introduction and Gill et al. (1986) for further information and advice.

10 Example

To minimize the quadratic function

f (x) = c^{T} x + \frac{1}{2} x^{T} H x

, where

c = {(- 0.02, - 0.2, - 0.2, - 0.2, - 0.2, 0.04, 0.04)}^{T}

H = (\begin{matrix} 2 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 2 & 2 & 0 & 0 & 0 \\ 0 & 0 & 2 & 2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 2 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & - 2 & - 2 \\ 0 & 0 & 0 & 0 & 0 & - 2 & - 2 \end{matrix})

subject to the bounds

\begin{matrix} - 0.01 \leq x_{1} \leq 0.01 \\ - 0.10 \leq x_{2} \leq 0.15 \\ - 0.01 \leq x_{3} \leq 0.03 \\ - 0.04 \leq x_{4} \leq 0.02 \\ - 0.10 \leq x_{5} \leq 0.05 \\ - 0.01 \leq x_{6} \\ - 0.01 \leq x_{7} \end{matrix}

and the general constraints

\begin{array}{r} x_{1} + & x_{2} + & x_{3} + & x_{4} + & x_{5} + & x_{6} + & x_{7} = & - 0.13 \\ 0.15 x_{1} + & 0.04 x_{2} + & 0.02 x_{3} + & 0.04 x_{4} + & 0.02 x_{5} + & 0.01 x_{6} + & 0.03 x_{7} \leq & - 0.0049 \\ 0.03 x_{1} + & 0.05 x_{2} + & 0.08 x_{3} + & 0.02 x_{4} + & 0.06 x_{5} + & 0.01 x_{6} & \leq & - 0.0064 \\ 0.02 x_{1} + & 0.04 x_{2} + & 0.01 x_{3} + & 0.02 x_{4} + & 0.02 x_{5} & \leq & - 0.0037 \\ 0.02 x_{1} + & 0.03 x_{2} & + & 0.01 x_{5} & \leq & - 0.0012 \\ - 0.0992 \leq 0.70 x_{1} + & 0.75 x_{2} + & 0.80 x_{3} + & 0.75 x_{4} + & 0.80 x_{5} + & 0.97 x_{6} & \leq & - 0.0020 \\ - 0.003 \leq 0.02 x_{1} + & 0.06 x_{2} + & 0.08 x_{3} + & 0.12 x_{4} + & 0.02 x_{5} + & 0.01 x_{6} + & 0.97 x_{7} \leq & 0.002 \end{array}

The initial point, which is infeasible, is

x_{0} = {(- 0.01, - 0.03, 0.0, - 0.01, - 0.1, 0.02, 0.01)}^{T} .

The computed solution (to five figures) is

x^{*} = {(- 0.01, - 0.069865, 0.018259, - 0.024261, - 0.062006, 0.0138054, 0.0040665)}^{T} .

One bound constraint and four general constraints are active at the solution.

This example shows the use of certain optional parameters. Option values are assigned directly within the program text and by reading values from a data file. The options structure is declared and initialized by e04xxc. Values are then assigned directly to

options . outfile

and

options . inf_bound

and two further options are read from the data file by use of e04xyc. e04nfc is then called to solve the problem using the function qphess1, with the Hessian implicit, for argument qphess. On successful return two further options are set, selecting a warm start and a reduced level of printout, and the problem is solved again using the function qphess2. In this case the Hessian is defined explicitly. Finally the memory freeing function e04xzc is used to free the memory assigned to the pointers in the options structure. You must not use the standard C function free() for this purpose.

11 Further Description

This section gives a detailed description of the algorithm used in e04nfc. This, and possibly the next section, Section 12, may be omitted if the more sophisticated features of the algorithm and software are not currently of interest.

11.1 Overview

e04nfc is based on an inertia-controlling method that maintains a Cholesky factorization of the reduced Hessian (see below). The method is based on 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 e04nfc or appear in the printed output. e04nfc has two 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 to the quadratic objective function. The feasibility phase does not perform the standard simplex method (i.e., it does not necessarily find a vertex), except in the LP case when

m_{lin} \leq n

. Once any iterate is feasible, all subsequent iterates remain feasible.

e04nfc has been designed to be efficient when used to solve a sequence of related problems – for example, within a sequential quadratic programming method for nonlinearly constrained optimization. In particular, you may specify an initial working set (the indices of the constraints believed to be satisfied exactly at the solution); see the discussion of the optional parameter

options . start

In general, an iterative process is required to solve a quadratic program. (For simplicity, we shall always consider a typical iteration and avoid reference to the index of the iteration.) Each new iterate

\bar{x}

is defined by

\bar{x} = x + α p,

(1)

where the steplength

α

is a non-negative scalar, and

p

is called the search direction.

At each point

x

, a working set of constraints is defined to be a linearly independent subset of the constraints that are satisfied ‘exactly’ (to within the tolerance defined by the optional parameter

options . ftol

). The working set is the current prediction of the constraints that hold with equality at a solution of a linearly constrained QP problem. The search direction is constructed so that the constraints in the working set remain unaltered for any value of the step length. For a bound constraint in the working set, this property is achieved by setting the corresponding component of the search direction to zero. Thus, the associated variable is fixed, and specification of the working set induces a partition of

x

into fixed and free variables. During a given iteration, the fixed variables are effectively removed from the problem; since the relevant components of the search direction are zero, the columns of

A

corresponding to fixed variables may be ignored.

Let

m_{w}

denote the number of general constraints in the working set and let

n_{f x}

denote the number of variables fixed at one of their bounds (

m_{w}

and

n_{f x}

are the quantities Lin and Bnd in the printed output from e04nfc). Similarly, let

n_{f r}

(

n_{f r} = n - n_{f x}

) denote the number of free variables. At every iteration, the variables are re-ordered so that the last

n_{f x}

variables are fixed, with all other relevant vectors and matrices ordered accordingly.

11.2 Definition of the Search Direction

Let

A_{f r}

denote the

m_{w}

n_{f r}

sub-matrix of general constraints in the working set corresponding to the free variables, and let

p_{f r}

denote the search direction with respect to the free variables only. The general constraints in the working set will be unaltered by any move along

p

A_{f r} p_{f r} = 0 .

(2)

In order to compute

p_{f r}

, the

T Q

factorization of

A_{f r}

is used:

A_{f r} Q_{f r} = (\begin{matrix} 0 & T \end{matrix}),

(3)

where

T

is a nonsingular

m_{w}

m_{w}

upper triangular matrix (i.e.,

t_{i j} = 0

i > j

), and the nonsingular

n_{f r}

n_{f r}

matrix

Q_{f r}

is the product of orthogonal transformations (see Gill et al. (1984)). If the columns of

Q_{f r}

are partitioned so that

Q_{f r} = (\begin{matrix} Z & Y \end{matrix}),

where

Y

n_{f r} \times m_{w}

, then the

n_{z}

(

n_{z} = n_{f r} - m_{w}

) columns of

Z

form a basis for the null space of

A_{f r}

. Let

n_{r}

be an integer such that

0 \leq n_{r} \leq n_{z}

, and let

Z_{r}

denote a matrix whose

n_{r}

columns are a subset of the columns of

Z

. (The integer

n_{r}

is the quantity Nrz in the printed output from e04nfc. In many cases,

Z_{r}

will include all the columns of

Z

.) The direction

p_{f r}

will satisfy (2) if

p_{f r} = Z_{r} p_{r},

(4)

where

p_{r}

is any

n_{r}

-vector.

Let

Q

denote the

n

n

matrix

Q = (\begin{matrix} Q_{f r} \\ I_{f x} \end{matrix}),

where

I_{f x}

is the identity matrix of order

n_{f x}

. Let

H_{q}

and

g_{q}

denote the

n

n

transformed Hessian and the transformed gradient

H_{q} = Q^{T} H Q and g_{q} = Q^{T} (c + H x)

and let the matrix of first

n_{r}

rows and columns of

H_{q}

be denoted by

H_{r}

and the vector of the first

n_{r}

elements of

g_{q}

be denoted by

g_{r}

. The quantities

H_{r}

and

g_{r}

are known as the reduced Hessian and reduced gradient of

f (x)

, respectively. Roughly speaking,

g_{r}

and

H_{r}

describe the first and second derivatives of an unconstrained problem for the calculation of

p_{r}

At each iteration, a triangular factorization of

H_{r}

is available. If

H_{r}

is positive definite,

H_{r} = R^{T} R

, where

R

is the upper triangular Cholesky factor of

H_{r}

. If

H_{r}

is not positive definite,

H_{r} = R^{T} D R

, where

D = diag (1, 1, \dots, 1, μ)

, with

μ \leq 0

The computation is arranged so that the reduced gradient vector is a multiple of

e_{r}

, a vector of all zeros except in the last (i.e.,

n_{r}

th) position. This allows the vector

p_{r}

in (4) to be computed from a single back-substitution

R p_{r} = γ e_{r},

(5)

where

γ

is a scalar that depends on whether or not the reduced Hessian is positive definite at

x

. In the positive definite case,

x + p

is the minimizer of the objective function subject to the constraints (bounds and general) in the working set treated as equalities. If

H_{r}

is not positive definite,

p_{r}

satisfies the conditions

p_{r}^{T} H_{r} p_{r} < 0 and g_{r}^{T} p_{r} \leq 0,

which allow the objective function to be reduced by any positive step of the form

x + α p

11.3 The Main Iteration

If the reduced gradient is zero,

x

is a constrained stationary point in the subspace defined by

Z

. During the feasibility phase, the reduced gradient will usually be zero only at a vertex (although it may be zero at non-vertices in the presence of constraint dependencies). During the optimality phase, a zero reduced gradient implies that

x

minimizes the quadratic objective when the constraints in the working set are treated as equalities. At a constrained stationary point, Lagrange multipliers

λ_{c}

and

λ_{b}

for the general and bound constraints are defined from the equations

A_{f r}^{T} λ_{c} = g_{f r} and λ_{b} = g_{f x} - A_{f x}^{T} λ_{c} .

(6)

Given a positive constant

δ

of the order of the machine precision, a Lagrange multiplier

λ_{j}

corresponding to an inequality constraint in the working set is said to be optimal if

λ_{j} \leq δ

when the associated constraint is at its upper bound, or if

λ_{j} \geq - δ

when the associated constraint is at its lower bound. If a multiplier is non-optimal, the objective function (either the true objective or the sum of infeasibilities) can be reduced by deleting the corresponding constraint (with index Jdel; see Section 12.3) from the working set.

If optimal multipliers occur during the feasibility phase and the sum of infeasibilities is nonzero, there is no feasible point, and you can force e04nfc to continue until the minimum value of the sum of infeasibilities has been found (see the discussion of the optional parameter

options . min_infeas

in Section 12.2). At this point, the Lagrange multiplier

λ_{j}

corresponding to an inequality constraint in the working set will be such that

- (1 + δ) \leq λ_{j} \leq δ

when the associated constraint is at its upper bound, and

- δ \leq λ_{j} \leq 1 + δ

when the associated constraint is at its lower bound. Lagrange multipliers for equality constraints will satisfy

‖λ_{j}‖ \leq 1 + δ

If the reduced gradient is not zero, Lagrange multipliers need not be computed and the nonzero elements of the search direction

p

are given by

Z_{r} p_{r}

(see (5)). The choice of step length is influenced by the need to maintain feasibility with respect to the satisfied constraints. If

H_{r}

is positive definite and

x + p

is feasible,

α

will be taken as unity. In this case, the reduced gradient at

\bar{x}

will be zero, and Lagrange multipliers are computed. Otherwise,

α

is set to

α_{m}

, the step to the ‘nearest’ constraint (with index Jadd; see Section 12.3), which is added to the working set at the next iteration.

Each change in the working set leads to a simple change to

A_{f r}

: if the status of a general constraint changes, a row of

A_{f r}

is altered; if a bound constraint enters or leaves the working set, a column of

A_{f r}

changes. Explicit representations are recurred of the matrices

T

Q_{f r}

and

R

; and of vectors

Q^{T} g

, and

Q^{T} c

. The triangular factor

R

associated with the reduced Hessian is only updated during the optimality phase.

One of the most important features of e04nfc is its control of the conditioning of the working set, whose nearness to linear dependence is estimated by the ratio of the largest to smallest diagonal elements of the

T Q

factor

T

(the printed value Cond T; see Section 12.3). In constructing the initial working set, constraints are excluded that would result in a large value of Cond T.

e04nfc includes a rigorous procedure that prevents the possibility of cycling at a point where the active constraints are nearly linearly dependent (see Gill et al. (1989)). The main feature of the anti-cycling procedure is that the feasibility tolerance is increased slightly at the start of every iteration. This not only allows a positive step to be taken at every iteration, but also provides, whenever possible, a choice of constraints to be added to the working set. Let

α_{m}

denote the maximum step at which

x + α_{m} p

does not violate any constraint by more than its feasibility tolerance. All constraints at a distance

α

(

α \leq α_{m}

) along

p

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.

11.4 Choosing the Initial Working Set

At the start of the optimality phase, a positive definite

H_{r}

can be defined if enough constraints are included in the initial working set. (The matrix with no rows and columns is positive definite by definition, corresponding to the case when

A_{f r}

contains

n_{f r}

constraints.) The idea is to include as many general constraints as necessary to ensure that the reduced Hessian is positive definite.

Let

H_{z}

denote the matrix of the first

n_{z}

rows and columns of the matrix

H_{q} = Q^{T} H Q

at the beginning of the optimality phase. A partial Cholesky factorization is used to find an upper triangular matrix

R

that is the factor of the largest positive definite leading sub-matrix of

H_{z}

. The use of interchanges during the factorization of

H_{z}

tends to maximize the dimension of

R

. (The condition of

R

may be controlled using the optional parameter

options . rank_tol

.) Let

Z_{r}

denote the columns of

Z

corresponding to

R

, and let

Z

be partitioned as

Z = (Z_{r} Z_{a})

. A working set, for which

Z_{r}

defines the null space, can be obtained by including the rows of

Z_{a}^{T}

as ‘artificial constraints’. Minimization of the objective function then proceeds within the subspace defined by

Z_{r}

, as described in Section 11.2.

The artificially augmented working set is given by

{\bar{A}}_{f r} = (\begin{matrix} Z_{a}^{T} \\ A_{f r} \end{matrix}),

(7)

so that

p_{f r}

will satisfy

A_{f r} p_{f r} = 0

and

Z_{a}^{T} p_{f r} = 0

. By definition of the

T Q

factorization,

{\bar{A}}_{f r}

automatically satisfies the following:

{\bar{A}}_{f r} Q_{f r} = (\begin{matrix} Z_{a}^{T} \\ A_{f r} \end{matrix}) Q_{f r} = (\begin{matrix} Z_{a}^{T} \\ A_{f r} \end{matrix}) (\begin{matrix} Z_{r} & Z_{a} Y \end{matrix}) = (\begin{matrix} 0 & \bar{T} \end{matrix}),

where

\bar{T} = (\begin{matrix} I & 0 \\ 0 & T \end{matrix}),

and hence the

T Q

factorization of (7) is available trivially from

T

and

Q_{f r}

without additional expense.

The matrix

Z_{a}

is not kept fixed, since its role is purely to define an appropriate null space; the

T Q

factorization can therefore be updated in the normal fashion as the iterations proceed. No work is required to ‘delete’ the artificial constraints associated with

Z_{a}

when

Z_{r}^{T} g_{f r} = 0

, since this simply involves repartitioning

Q_{f r}

. The ‘artificial’ multiplier vector associated with the rows of

Z_{a}^{T}

is equal to

Z_{a}^{T} g_{f r}

, and the multipliers corresponding to the rows of the ‘true’ working set are the multipliers that would be obtained if the artificial constraints were not present. If an artificial constraint is ‘deleted’ from the working set, an A appears alongside the entry in the Jdel column of the printed output (see Section 12.3).

The number of columns in

Z_{a}

and

Z_{r}

, the Euclidean norm of

Z_{r}^{T} g_{f r}

, and the condition estimator of

R

appear in the printed output as Nart, Nrz, Norm Gz and Cond Rz (see Section 12.3).

Under some circumstances, a different type of artificial constraint is used when solving a linear program. Although the algorithm of e04nfc does not usually perform simplex steps (in the traditional sense), there is one exception: a linear program with fewer general constraints than variables (i.e.,

m_{lin} \leq n

). (Use of the simplex method in this situation leads to savings in storage.) At the starting point, the ‘natural’ working set (the set of constraints exactly or nearly satisfied at the starting point) is augmented with a suitable number of ‘temporary’ bounds, each of which has the effect of temporarily fixing a variable at its current value. In subsequent iterations, a temporary bound is treated as a standard constraint until it is deleted from the working set, in which case it is never added again. If a temporary bound is ‘deleted’ from the working set, an F (for ‘Fixed’) appears alongside the entry in the Jdel column of the printed output (see Section 12.3).

12 Optional Parameters

A number of optional input and output arguments to e04nfc are available through the structure argument options, type Nag_E04_Opt. An argument may be selected by assigning an appropriate value to the relevant structure member; those arguments not selected will be assigned default values. If no use is to be made of any of the optional parameters you should use the NAG defined null pointer, E04_DEFAULT, in place of options when calling e04nfc; the default settings will then be used for all arguments.

Before assigning values to options directly the structure must be initialized by a call to the function e04xxc. Values may then be assigned to the structure members in the normal C manner.

After return from e04nfc, the options structure may only be re-used for future calls of e04nfc if the dimensions of the new problem are the same. Otherwise, the structure must be cleared by a call of e04xzc) and re-initialized by a call of e04xxc before future calls. Failure to do this will result in unpredictable behaviour.

Option settings may also be read from a text file using the function e04xyc in which case initialization of the options structure will be performed automatically if not already done. Any subsequent direct assignment to the options structure must not be preceded by initialization.

If assignment of functions and memory to pointers in the options structure is required, this must be done directly in the calling program; they cannot be assigned using e04xyc.

12.1 Optional Parameter Checklist and Default Values

For easy reference, the following list shows the members of options which are valid for e04nfc together with their default values where relevant. The number

ε

is a generic notation for machine precision (see X02AJC).

Nag_ProblemType prob	Nag_QP2
Nag_Start start	$Nag_Cold$
Boolean list	Nag_TRUE
Nag_PrintType print_level	Nag_Soln_Iter
char outfile[512]	stdout
void (*print_fun)()	NULL
Integer fmax_iter	$\max (50, 5 (n + nclin))$
Integer max_iter	$\max (50, 5 (n + nclin))$
Boolean min_infeas	Nag_FALSE
double crash_tol	0.01
double ftol	$\sqrt{ε}$
double optim_tol	$ε^{0.8}$
Integer reset_ftol	10000
Integer fcheck	50
double inf_bound	$10^{20}$
double inf_step	$\max (options . inf_bound, 10^{20})$
Integer hrows	n
Integer max_df	n
double rank_tol	$100 ε$
Integer *state	size $n + nclin$
double *ax	size nclin
double *lambda	size $n + nclin$
Integer iter
Integer nf

12.2 Description of the Optional Parameters

prob – Nag_ProblemType

Default

= Nag_QP2

On entry: specifies the type of objective function to be minimized during the optimality phase. The following are the six possible values of

options . prob

and the size of the arrays h and cvec that are required to define the objective function:

$Nag_FP$	h and cvec not accessed;
$Nag_LP$	h not accessed, $cvec [n]$ required;
$Nag_QP1$	$h [n \times tdh]$ symmetric, cvec not referenced;
$Nag_QP2$	$h [n \times tdh]$ symmetric, $cvec [n]$ required;
$Nag_QP3$	$h [n \times tdh]$ upper trapezoidal, cvec not referenced;
$Nag_QP4$	$h [n \times tdh]$ upper trapezoidal, $cvec [n]$ required.

H = 0

, i.e., the objective function is purely linear, the efficiency of e04nfc may be increased by specifying

options . prob

Nag_LP

Constraint:

options . prob = Nag_FP

Nag_LP

Nag_QP1

Nag_QP2

Nag_QP3

Nag_QP4

start – Nag_Start

Default

= Nag_Cold

On entry: specifies how the initial working set is chosen. With

options . start = Nag_Cold

, e04nfc chooses the initial working set based on the values of the variables and constraints at the initial point. Broadly speaking, the initial working set will include equality constraints and bounds or inequality constraints that violate or ‘nearly’ satisfy their bounds (to within

options . crash_tol

With

options . start = Nag_Warm

, you must provide a valid definition of every element of the array pointer

options . state

(see below for the definition of this member of options). e04nfc will override your specification of

options . state

if necessary, so that a poor choice of the working set will not cause a fatal error.

Nag_Warm

will be advantageous if a good estimate of the initial working set is available – for example, when e04nfc is called repeatedly to solve related problems.

Constraint:

options . start = Nag_Cold

Nag_Warm

list – Nag_Boolean

Default

= Nag_TRUE

On entry: if

options . list = Nag_TRUE

the argument settings in the call to e04nfc will be printed.

print_level – Nag_PrintType

Default

= Nag_Soln_Iter

On entry: the level of results printout produced by e04nfc. The following values are available:

$Nag_NoPrint$	No output.
$Nag_Soln$	The final solution.
$Nag_Iter$	One line of output for each iteration.
$Nag_Iter_Long$	A longer line of output for each iteration with more information (line exceeds 80 characters).
$Nag_Soln_Iter$	The final solution and one line of output for each iteration.
$Nag_Soln_Iter_Long$	The final solution and one long line of output for each iteration (line exceeds 80 characters).
$Nag_Soln_Iter_Const$	As $Nag_Soln_Iter_Long$ with the Lagrange multipliers, the variables $x$ , the constraint values $A x$ and the constraint status also printed at each iteration.
$Nag_Soln_Iter_Full$	As $Nag_Soln_Iter_Const$ with the diagonal elements of the upper triangular matrix $T$ associated with the $T Q$ factorization (3) of the working set, and the diagonal elements of the upper triangular matrix $R$ printed at each iteration.

Details of each level of results printout are described in Section 12.3.

Constraint:

options . print_level = Nag_NoPrint

Nag_Soln

Nag_Iter

Nag_Soln_Iter

Nag_Iter_Long

Nag_Soln_Iter_Long

Nag_Soln_Iter_Const

Nag_Soln_Iter_Full

outfile – const char[512]

Default

= stdout

On entry: the name of the file to which results should be printed. If

options . outfile [0] =' \0'

then the stdout stream is used.

print_fun – pointer to function

Default

=

NULL

On entry: printing function defined by you; the prototype of

options . print_fun

void (*print_fun)(const Nag_Search_State *st, Nag_Comm *comm);

See Section 12.3.1 below for further details.

fmax_iter – Integer

Default

= \max (50, 5 (n + nclin))

max_iter – Integer

Default

= \max (50, 5 (n + nclin))

On entry:

options . fmax_iter

specifies the maximum number of iterations allowed in the feasibility phase.

options . max_iter

specifies the maximum number of iterations permitted in the optimality phase.

If you wish to check that a call to e04nfc is correct before attempting to solve the problem in full then

options . fmax_iter

may be set to

0

. No iterations will then be performed but the initialization stages prior to the first iteration will be processed and a listing of argument settings output, if

options . list = Nag_TRUE

(the default setting).

Constraint:

options . fmax_iter \geq 0

and

options . max_iter \geq 0

min_infeas – Nag_Boolean

Default

= Nag_FALSE

On entry:

options . min_infeas

specifies whether e04nfc should minimize the sum of infeasibilities if no feasible point exists for the constraints.

$options . min_infeas = Nag_FALSE$: e04nfc will terminate as soon as it is evident that the problem is infeasible, in which case the final point will generally not be the point at which the sum of infeasibilities is minimized.
$options . min_infeas = Nag_TRUE$: e04nfc will continue until the sum of infeasibilities is minimized.

crash_tol – double

Default

= 0.01

On entry:

options . crash_tol

is used in conjunction with the optional parameter

options . start

when

options . start

has the default setting, i.e.,

options . start = Nag_Cold

, e04nfc selects an initial working set. The initial working set will include bounds or general inequality constraints that lie within

options . crash_tol

of their bounds. In particular, a constraint of the form

a_{j}^{T} x \geq l

will be included in the initial working set if

|a_{j}^{T} x - l| \leq options . crash_tol \times (1 + |l|)

Constraint:

0.0 \leq options . crash_tol \leq 1.0

ftol – double

Default

= \sqrt{ε}

On entry:

options . ftol

defines the maximum acceptable absolute violation in each constraint at a ‘feasible’ point. For example, if the variables and the coefficients in the general constraints are of order unity, and the latter are correct to about 6 decimal digits, it would be appropriate to specify

options . ftol

10^{- 6}

e04nfc attempts to find a feasible solution before optimizing the objective function. If the sum of infeasibilities cannot be reduced to zero,

options . min_infeas

can be used to find the minimum value of the sum. Let Sinf be the corresponding sum of infeasibilities. If Sinf is quite small, it may be appropriate to raise

options . ftol

by a factor of 10 or

100

. Otherwise, some error in the data should be suspected.

Note that a ‘feasible solution’ is a solution that satisfies the current constraints to within the tolerance

options . ftol

Constraint:

options . ftol > 0.0

optim_tol – double

Default

= ε^{0.8}

On entry:

options . optim_tol

defines the tolerance used to determine whether the bounds and generated constraints have the correct sign for the solution to be judged optimal.

reset_ftol – Integer

Default

= 5

On entry: this option is part of an anti-cycling procedure designed to guarantee 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

options . ftol

δ

. Over a period of

options . reset_ftol

iterations, the feasibility tolerance actually used by e04nfc increases from

0.5 δ

δ

(in steps of

0.5 δ / options . reset_ftol

At certain stages the following ‘resetting procedure’ is used to remove constraint infeasibilities. First, all variables whose upper or lower bounds are in the working set are moved exactly onto their bounds. A count is kept of the number of nontrivial adjustments made. If the count is positive, iterative refinement is used to give variables that satisfy the working set to (essentially) machine precision. Finally, the current feasibility tolerance is reinitialized to

0.5 δ

If a problem requires more than

options . reset_ftol

iterations, the resetting procedure is invoked and a new cycle of

options . reset_ftol

iterations is started with

options . reset_ftol

incremented by

10

. (The decision to resume the feasibility phase or optimality phase is based on comparing any constraint infeasibilities with

δ

The resetting procedure is also invoked when e04nfc reaches an apparently optimal, infeasible or unbounded solution, unless this situation has already occurred twice. If any nontrivial adjustments are made, iterations are continued.

Constraint:

0 < options . reset_ftol < 10000000

fcheck – Integer

Default

= 50

On entry: every

options . fcheck

iterations, a numerical test is made to see if the current solution

x

satisfies the constraints in the working set. If the largest residual of the constraints in the working set is judged to be too large, the current working set is re-factorized and the variables are recomputed to satisfy the constraints more accurately.

Constraint:

options . fcheck \geq 1

inf_bound – double

Default

= 10^{20}

On entry:

options . inf_bound

defines the ‘infinite’ bound in the definition of the problem constraints. Any upper bound greater than or equal to

options . inf_bound

will be regarded as

+ \infty

(and similarly for a lower bound less than or equal to

- options . inf_bound

Constraint:

options . inf_bound > 0.0

inf_step – double

Default

= \max (options . inf_bound, 10^{20})

On entry:

options . inf_step

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

x

during an iteration would exceed the value of

options . inf_step

, the objective function is considered to be unbounded below in the feasible region.

Constraint:

options . inf_step > 0.0

hrows – Integer

Default

= n

On entry: specifies

m

, the number of rows of the quadratic term

H

of the QP objective function. The default value of

options . hrows

n

, the number of variables of the problem, except that if the problem is specified as type

options . prob = Nag_FP

Nag_LP

, the default value of

options . hrows

is zero.

If the problem is of type QP,

options . hrows

will usually be

n

, the number of variables. However, a value of

options . hrows

less than

n

is appropriate for

options . prob = Nag_QP3

Nag_QP4

H

is an upper trapezoidal matrix with

m

rows. Similarly,

options . hrows

may be used to define the dimension of a leading block of nonzeros in the Hessian matrices of

options . prob = Nag_QP1

Nag_QP2

, in which case the last

n - m

rows and columns of

H

are assumed to be zero.

Constraint:

0 \leq options . hrows \leq n

max_df – Integer

Default

= n

On entry: places a limit on the storage allocated for the triangular factor

R

of the reduced Hessian

H_{r}

. Ideally,

options . max_df

should be set slightly larger than the value of

n_{r}

expected at the solution. It need not be larger than

m_{n} + 1

, where

m_{n}

is the number of variables that appear nonlinearly in the quadratic objective function. For many problems it can be much smaller than

m_{n}

For quadratic problems, a minimizer may lie on any number of constraints, so that

n_{r}

may vary between

1

and

n

. The default value is therefore normally n but if the optional parameter

options . hrows

is specified then the default value of

options . max_df

is set to the value in

options . hrows

Constraint:

1 \leq options . max_df \leq n

rank_tol – double

Default

= 100 ε

On entry:

options . rank_tol

enables you to control the condition number of the triangular factor

R

(see Section 11). If

ρ_{i}

denotes the function

ρ_{i} = \max \{|R_{11}|, |R_{22}|, \dots, |R_{i i}|\}

, the dimension of

R

is defined to be smallest index

i

such that

|R_{i + 1, i + 1}| \leq options . rank_tol \times |ρ_{i + 1}|

Constraint:

0.0 \leq options . rank_tol < 1.0

state – Integer *

Default memory

= n + nclin

On entry:

options . state

need not be set if the default option of

options . start = Nag_Cold

is used as

n + nclin

values of memory will be automatically allocated by e04nfc.

If the option

options . start = Nag_Warm

has been chosen,

options . state

must point to a minimum of

n + nclin

elements of memory. This memory will already be available if the options structure has been used in a previous call to e04nfc from the calling program, using the same values of n and nclin and

options . start = Nag_Cold

. If a previous call has not been made you must be allocate sufficient memory to

options . state

When a warm start is chosen

options . state

should specify the desired status of the constraints at the start of the feasibility phase. More precisely, the first

n

elements of

options . state

refer to the upper and lower bounds on the variables, and the next

m_{lin}

elements refer to the general linear constraints (if any). Possible values for

options . state [j]

are as follows:

$options . state [j]$	Meaning
0	The corresponding constraint should not be in the initial working set.
1	The constraint should be in the initial working set at its lower bound.
2	The constraint should be in the initial working set at its upper bound.
3	The constraint should be in the initial working set as an equality. This value should only be specified if $bl [j] = bu [j]$ . The values $1$ , 2 or 3 all have the same effect when $bl [j] = bu [j]$ .

The values

- 2

- 1

and 4 are also acceptable but will be reset to zero by the function. In particular, if e04nfc has been called previously with the same values of n and nclin,

options . state

already contains satisfactory information. (See also the description of the optional parameter

options . start

.) The function also adjusts (if necessary) the values supplied in x to be consistent with the values supplied in

options . state

On exit: if e04nfc exits with a value of

fail . code = NE_NOERROR

, NW_DEAD_POINT, NW_SOLN_NOT_UNIQUE or NW_NOT_FEASIBLE, the values in

options . state

indicate the status of the constraints in the working set at the solution. Otherwise,

options . state

indicates the composition of the working set at the final iterate. The significance of each possible value of

options . state [j]

is as follows:

$options . state [j]$	Meaning
$- 2$	The constraint violates its lower bound by more than the feasibility tolerance.
$- 1$	The constraint violates its upper bound by more than the feasibility tolerance.
$0$	The constraint is satisfied to within the feasibility tolerance, but is not in the working set.
$1$	This inequality constraint is included in the working set at its lower bound.
$2$	This inequality constraint is included in the working set at its upper bound.
$3$	This constraint is included in the working set as an equality. This value of $options . state$ can occur only when $bl [j] = bu [j]$ .
$4$	This corresponds to optimality being declared with $x [j]$ being temporarily fixed at its current value. This value of $options . state$ can only occur when $fail . code = NW_DEAD_POINT$ or NW_SOLN_NOT_UNIQUE.

ax – double *

Default memory

= nclin

On entry: nclin values of memory will be automatically allocated by e04nfc and this is the recommended method of use of

options . ax

. However you may supply memory from the calling program.

On exit: if

nclin > 0

options . ax

points to the final values of the linear constraints

A x

lambda – double *

Default memory

= n + nclin

On entry:

n + nclin

values of memory will be automatically allocated by e04nfc and this is the recommended method of use of

options . lambda

. However you may supply memory from the calling program.

On exit: the values of the Lagrange multipliers for each constraint with respect to the current working set. The first

n

elements contain the multipliers for the bound constraints on the variables, and the next

m_{lin}

elements contain the multipliers for the general linear constraints (if any). If

options . state [j] = 0

(i.e., constraint

j

is not in the working set),

options . lambda [j]

is zero. If

x

is optimal,

options . lambda [j]

should be non-negative if

options . state [j] = 1

, non-positive if

options . state [j] = 2

and zero if

options . state [j] = 4

iter – Integer

On exit: the total number of iterations performed in the feasibility phase and (if appropriate) the optimality phase.

nf – Integer

On exit: the number of times the product

H x

has been calculated (i.e., number of calls of qphess).

12.3 Description of Printed Output

You can control the level of printed output with the structure members

options . list

and

options . print_level

(see Section 12.2). If

options . list = Nag_TRUE

then the argument values to e04nfc are listed, whereas the printout of results is governed by the value of

options . print_level

. The default of

options . print_level = Nag_Soln_Iter

provides a single line of output at each iteration and the final result. This section describes all of the possible levels of results printout available from e04nfc.

The convention for numbering the constraints in the iteration results is that indices 1 to

n

refer to the bounds on the variables, and indices

n + 1

n + m_{lin}

refer to the general constraints. When the status of a constraint changes, the index of the constraint is printed, along with the designation L (lower bound), U (upper bound), E (equality), F (temporarily fixed variable) or A (artificial constraint).

When

options . print_level = Nag_Iter

Nag_Soln_Iter

the following line of output is produced at every iteration. In all cases, the values of the quantities printed are those in effect on completion of the given iteration.

Itn	the iteration count.
Jdel	the index of the constraint deleted from the working set. If Jdel is zero, no constraint was deleted.
Jadd	the index of the constraint added to the working set. If Jadd is zero, no constraint was added.
Step	the step taken along the computed search direction. If a constraint is added during the current iteration (i.e., Jadd is positive), Step will be the step to the nearest constraint. During the optimality phase, the step can be greater than $1.0$ only if the reduced Hessian is not positive definite.
Ninf	the number of violated constraints (infeasibilities). This will be zero during the optimality phase.
Sinf/Obj	the value of the current objective function. If $x$ is not feasible, Sinf gives a weighted sum of the magnitudes of constraint violations. If $x$ is feasible, Obj is the value of the objective function. The output line for the final iteration of the feasibility phase (i.e., the first iteration for which Ninf is zero) will give the value of the true objective at the first feasible point.
	During the optimality phase, the value of the objective function will be non-increasing. During the feasibility phase, the number of constraint infeasibilities will not increase until either a feasible point is found, or the optimality of the multipliers implies that no feasible point exists. Once optimal multipliers are obtained, the number of infeasibilities can increase, but the sum of infeasibilities will either remain constant or be reduced until the minimum sum of infeasibilities is found.
Bnd	the number of simple bound constraints in the current working set.
Lin	the number of general linear constraints in the current working set.
Nart	the number of artificial constraints in the working set, i.e., the number of columns of $Z_{a}$ (see Section 11). At the start of the optimality phase, Nart provides an estimate of the number of non-positive eigenvalues in the reduced Hessian.
Nrz	the number of columns of $Z_{r}$ (see Section 11). Nrz is the dimension of the subspace in which the objective function is currently being minimized. The value of Nrz is the number of variables minus the number of constraints in the working set; i.e., $Nrz = n - (Bnd + Lin + Nart)$ .
	The value of $n_{z}$ , the number of columns of $Z$ (see Section 11) can be calculated as $n_{z} = n - (Bnd + Lin)$ . A zero value of $n_{z}$ implies that $x$ lies at a vertex of the feasible region.
Norm Gz	$‖Z_{r}^{T} g_{f r}‖$ , the Euclidean norm of the reduced gradient with respect to $Z_{r}$ . During the optimality phase, this norm will be approximately zero after a unit step.

options . print_level = Nag_Iter_Long

Nag_Soln_Iter_Long

Nag_Soln_Iter_Const

Nag_Soln_Iter_Full

the line of printout is extended to give the following information. (Note this longer line extends over more than 80 characters.)

NOpt	the number of non-optimal Lagrange multipliers at the current point. NOpt is not printed if the current $x$ is infeasible or no multipliers have been calculated. At a minimizer, NOpt will be zero.
Min LM	the value of the Lagrange multiplier associated with the deleted constraint. If Min LM is negative, a lower bound constraint has been deleted; if Min LM is positive, an upper bound constraint has been deleted. If no multipliers are calculated during a given iteration, Min LM will be zero.
Cond T	a lower bound on the condition number of the working set.
Cond Rz	a lower bound on the condition number of the triangular factor $R$ (the Cholesky factor of the current reduced Hessian). If the problem is specified to be of type $options . prob = Nag_LP$ , Cond Rz is not printed.
Rzz	the last diagonal element $μ$ of the matrix $D$ associated with the $R^{T} D R$ factorization of the reduced Hessian $H_{r}$ (see Section 11.2). Rzz is only printed if $H_{r}$ is not positive definite (in which case $μ \neq 1$ ). If the printed value of Rzz is small in absolute value, then $H_{r}$ is approximately singular. A negative value of Rzz implies that the objective function has negative curvature on the current working set.

When

options . print_level = Nag_Soln_Iter_Const

Nag_Soln_Iter_Full

more detailed results are given at each iteration. For the setting

options . print_level = Nag_Soln_Iter_Const

additional values output are:

Value of x	the value of $x$ currently held in x.
State	the current value of $options . state$ associated with $x$ .
Value of Ax	the value of $A x$ currently held in $options . ax$ .
State	the current value of $options . state$ associated with $A x$ .

Also printed are the Lagrange Multipliers for the bound constraints, linear constraints and artificial constraints.

options . print_level = Nag_Soln_Iter_Full

then the diagonal of

T

and

Z_{r}

are also output at each iteration.

When

options . print_level = Nag_Soln

Nag_Soln_Iter

Nag_Soln_Iter_Const

Nag_Soln_Iter_Full

the final printout from e04nfc includes a listing of the status of every variable and constraint. The following describes the printout for each variable.

Varbl	gives the name (V) and index $j$ , for $j = 1, 2, \dots, n$ , of the variable.
State	gives the state of the variable (FR if neither bound is in the working set, EQ if a fixed variable, LL if on its lower bound, UL if on its upper bound, TF if temporarily fixed at its current value). If Value lies outside the upper or lower bounds by more than the feasibility tolerance, State will be ++ or -- respectively.
Value	is the value of the variable at the final iteration.
Lower bound	is the lower bound specified for the variable. (None indicates that $bl [j - 1] \leq - options . inf_bound$ .)
Upper bound	is the upper bound specified for the variable. (None indicates that $bu [j - 1] \geq options . inf_bound$ .)
Lagr mult	is the value of the Lagrange multiplier for the associated bound constraint. This will be zero if State is FR. If $x$ is optimal, the multiplier should be non-negative if State is LL, and non-positive if State is UL.
Residual	is the difference between the variable Value and the nearer of its bounds $bl [j - 1]$ and $bu [j - 1]$ .

The meaning of the printout for general constraints is the same as that given above for variables, with ‘variable’ replaced by ‘constraint’, and with the following change in the heading:

LCon	is the name (L) and index $j$ , for $j = 1, 2, \dots, m_{lin}$ , of the constraint.

12.3.1 Output of results via a user-defined printing function

You may also specify your own print function for output of iteration results and the final solution by use of the

options . print_fun

function pointer, which has prototype

void (*print_fun)(const Nag_Search_State *st, Nag_Comm *comm);

The rest of this section can be skipped if you only wish to use the default printing facilities.

When a user-defined function is assigned to

options . print_fun

this will be called in preference to the internal print function of e04nfc. Calls to the user-defined function are again controlled by means of the

options . print_level

member. Information is provided through st and comm, the two structure arguments to

options . print_fun

comm \to it_prt = Nag_TRUE

then the results from the last iteration of e04nfc are set in the following members of st:

first – Nag_Boolean: Nag_TRUE on the first call to $options . print_fun$ .

iter – Integer: The number of iterations performed.

n – Integer: The number of variables.

nclin – Integer: The number of linear constraints.

jdel – Integer: Index of constraint deleted.

jadd – Integer: Index of constraint added.

step – double: The step taken along the current search direction.

ninf – Integer: The number of infeasibilities.

f – double: The value of the current objective function.

bnd – Integer: Number of bound constraints in the working set.

lin – Integer: Number of general linear constraints in the working set.

nart – Integer: Number of artificial constraints in the working set.

nrz – Integer: Number of columns of $Z_{r}$ .

norm_gz – double: Euclidean norm of the reduced gradient, $‖Z_{r}^{T} g_{f r}‖$ .

nopt – Integer: Number of non-optimal Lagrange multipliers.

min_lm – double: Value of the Lagrange multiplier associated with the deleted constraint.

condt – double: A lower bound on the condition number of the working set.

x – double: x points to the n memory locations holding the current point $x$ .

ax – double: $options . ax$ points to the nclin memory locations holding the current values $A x$ .

state – Integer: $options . state$ points to the $n + nclin$ memory locations holding the status of the variables and general linear constraints. See Section 12.2 for a description of the possible status values.

t – double: The upper triangular matrix $T$ with $st \to lin$ columns. Matrix element $i, j$ is held in $st \to t [(i - 1) \times st \to tdt + j - 1]$ .

tdt – Integer: The trailing dimension for $st \to t$ .

st \to rset = Nag_TRUE

then the problem is QP, e04nfc is executing the optimality phase and the following members of st are also set:

r – double: The upper triangular matrix $R$ with $st \to nrz$ columns. Matrix element $i, j$ is held in $st \to r [(i - 1) \times st \to tdr + j - 1]$ .

tdr – Integer: The trailing dimension for $st \to r$ .

condr – double: A lower bound on the condition number of the triangular factor $R$ .

rzz – double: Last diagonal element $μ$ of the matrix $D$ .

comm \to new_lm = Nag_TRUE

then the Lagrange multipliers have been updated and the following members of st are set:

kx – Integer: Indices of the bound constraints with associated multipliers. Value of $st \to kx [i]$ is the index of the constraint with multiplier $st \to lambda [i]$ , for $i = 0, 1, \dots, st \to bnd - 1$ .

kactive – Integer: Indices of the linear constraints with associated multipliers. Value of $st \to kactive [i]$ is the index of the constraint with multiplier $st \to lambda [st \to bnd + i]$ , for $i = 0, 1, \dots, st \to lin - 1$ .

lambda – double: The multipliers for the constraints in the working set. $options . lambda [i]$ , for $i = 0, 1, \dots, st \to bnd - 1$ , hold the multipliers for the bound constraints while the multipliers for the linear constraints are held at indices $i = st \to bnd, \dots, st \to bnd + st \to lin - 1$ .

gq – double: $st \to gq [i]$ , for $i = 0, 1, \dots, st \to nart - 1$ , hold the multipliers for the artificial constraints.

The following members of st are also relevant and apply when

comm \to it_prt

comm \to new_lm

is Nag_TRUE.

refactor – Nag_Boolean: Nag_TRUE if iterative refinement performed. See Section 12.2 and optional parameter $options . reset_ftol$ .

jmax – Integer: If $st \to refactor = Nag_TRUE$ then $st \to jmax$ holds the index of the constraint with the maximum violation.

errmax – double: If $st \to refactor = Nag_TRUE$ then $st \to errmax$ holds the value of the maximum violation.

moved – Nag_Boolean: Nag_TRUE if some variables have been moved to their bounds. See the optional parameter $options . reset_ftol$ .

nmoved – Integer: If $st \to moved = Nag_TRUE$ then $st \to nmoved$ holds the number of variables which were moved to their bounds.

rowerr – Nag_Boolean: Nag_TRUE if some constraints are not satisfied to within $options . ftol$ .

feasible – Nag_Boolean: Nag_TRUE when a feasible point has been found.

comm \to sol_prt = Nag_TRUE

then the final result from e04nfc is available and the following members of st are set:

iter – Integer: The number of iterations performed.

n – Integer: The number of variables.

nclin – Integer: The number of linear constraints.

x – double: x points to the n memory locations holding the final point $x$ .

f – double: The final objective function value or, if $x$ is not feasible, the sum of infeasibilities. If the problem is of type $options . prob = Nag_FP$ and $x$ is feasible then $st \to f$ is set to zero.

ax – double: $st \to ax$ points to the nclin memory locations holding the final values $A x$ .

state – Integer: $st \to state$ points to the $n + nclin$ memory locations holding the final status of the variables and general linear constraints. See Section 12.2 for a description of the possible status values.

lambda – double: $st \to lambda$ points to the $n + nclin$ final values of the Lagrange multipliers.

bl – double: $st \to bl$ points to the $n + nclin$ lower bound values.

bu – double: $st \to bu$ points to the $n + nclin$ upper bound values.

endstate – Nag_EndState

The state of termination of e04nfc. Possible values of

st \to endstate

and their correspondence to the exit value of fail.code are:

Value of $st \to endstate$	Value of fail.code
$Nag_Feasible$ and $Nag_Optimal$	NE_NOERROR
$Nag_Deadpoint$ and $Nag_Weakmin$	If the problem is QP NW_DEAD_POINT otherwise NW_SOLN_NOT_UNIQUE
$Nag_Unbounded$	NE_UNBOUNDED
$Nag_Infeasible$	NW_NOT_FEASIBLE
$Nag_Too_Many_Iter$	NW_TOO_MANY_ITER
$Nag_Hess_Too_Big$	NE_HESS_TOO_BIG

The relevant members of the structure comm are:

it_prt – Nag_Boolean: Will be Nag_TRUE when the print function is called with the result of the current iteration.

sol_prt – Nag_Boolean: Will be Nag_TRUE when the print function is called with the final result.

new_lm – Nag_Boolean: Will be Nag_TRUE when the Lagrange multipliers have been updated.

user – double
iuser – Integer
p – Pointer: Pointers for communication of user information. You must allocate memory either before entry to e04nfc or during a call to qphess or $options . print_fun$ . The type Pointer will be void * with a C compiler that defines void * and char * otherwise.

NAG Library Manual, Mark 27

Interfaces: FL CL AD

NAG CL Interface Introduction

E04 (Opt) Chapter Contents

E04 (Opt) Chapter Introduction

e04nf: FL CL AD

NAG CL Interfacee04nfc (qp_​dense_​solve)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

11 Further Description

11.1 Overview

11.2 Definition of the Search Direction

11.3 The Main Iteration

11.4 Choosing the Initial Working Set

12 Optional Parameters

12.1 Optional Parameter Checklist and Default Values

12.2 Description of the Optional Parameters

12.3 Description of Printed Output

12.3.1 Output of results via a user-defined printing function

NAG CL Interface
e04nfc (qp_dense_solve)