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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_opt_qp_dense_solve (e04nf)

## Purpose

nag_opt_qp_dense_solve (e04nf) solves general quadratic programming problems. It is not intended for large sparse problems.

## Syntax

[istate, x, iter, obj, ax, clamda, user, lwsav, iwsav, rwsav, ifail] = e04nf(a, bl, bu, cvec, h, qphess, istate, x, lwsav, iwsav, rwsav, 'n', n, 'nclin', nclin, 'user', user)
[istate, x, iter, obj, ax, clamda, user, lwsav, iwsav, rwsav, ifail] = nag_opt_qp_dense_solve(a, bl, bu, cvec, h, qphess, istate, x, lwsav, iwsav, rwsav, 'n', n, 'nclin', nclin, 'user', user)
Before calling nag_opt_qp_dense_solve (e04nf), or the option setting function nag_opt_qp_dense_option_string (e04nh), nag_opt_init (e04wb) must be called.

## Description

nag_opt_qp_dense_solve (e04nf) is designed to solve a class of quadratic programming problems that are assumed to be stated in the following general form:
 $minimize x∈Rn fx subject to l≤ x Ax ≤u,$
where $A$ is an ${m}_{L}$ by $n$ matrix and $f\left(x\right)$ may be specified in a variety of ways depending upon the particular problem to be solved. The available forms for $f\left(x\right)$ are listed in Table 1, 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 $f\left(x\right)$ Matrix $H$ FP Not applicable Not applicable LP ${c}^{\mathrm{T}}x$ Not applicable QP1 $\phantom{{c}^{\mathrm{T}}x+}\frac{1}{2}{x}^{\mathrm{T}}Hx$ symmetric QP2 ${c}^{\mathrm{T}}x+\frac{1}{2}{x}^{\mathrm{T}}Hx$ symmetric QP3 $\phantom{{c}^{\mathrm{T}}x+}\frac{1}{2}{x}^{\mathrm{T}}{H}^{\mathrm{T}}Hx$ $m$ by $n$ upper trapezoidal QP4 ${c}^{\mathrm{T}}x+\frac{1}{2}{x}^{\mathrm{T}}{H}^{\mathrm{T}}Hx$ $m$ by $n$ upper trapezoidal
Table 1
There is no restriction on $H$ or ${H}^{\mathrm{T}}H$ apart from symmetry. If the quadratic function is convex, a global minimum is found; otherwise, a local minimum is found. The default problem type is QP2 and other objective functions are selected by using the optional parameter Problem Type. For problems of type FP, the objective function is omitted and the function attempts to find a feasible point for the set of constraints.
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 $l$ or $u$ can be set to special values that will be treated as $-\infty$ or $+\infty$. (See the description of the optional parameter Infinite Bound Size.)
The defining feature of a quadratic function $f\left(x\right)$ is that the second-derivative matrix ${\nabla }^{2}f\left(x\right)$ (the Hessian matrix) is constant. For QP1 and QP2 (the default), ${\nabla }^{2}f\left(x\right)=H$; for QP3 and QP4, ${\nabla }^{2}f\left(x\right)={H}^{\mathrm{T}}H$; and for the LP case, ${\nabla }^{2}f\left(x\right)=0$. If $H$ is positive semidefinite, it is usually more efficient to use nag_opt_lsq_lincon_solve (e04nc). If $H$ is defined as the zero matrix, nag_opt_qp_dense_solve (e04nf) will still attempt to solve the resulting linear programming problem; however, this can be accomplished more efficiently by setting the optional parameter ${\mathbf{Problem Type}}=\mathrm{LP}$, or by using nag_opt_lp_solve (e04mf) instead.
You must supply an initial estimate of the solution.
In the QP case, you may supply $H$ either explicitly as an $m$ by $n$ matrix, or implicitly in a function that computes the product $Hx$ or ${H}^{\mathrm{T}}Hx$ for any given vector $x$.
In general, a successful run of nag_opt_qp_dense_solve (e04nf) 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 ${\nabla }^{2}f\left(x\right)$ is positive definite or positive semidefinite, nag_opt_qp_dense_solve (e04nf) 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 ${\nabla }^{2}f\left(x\right)$ is not positive definite. If ${\nabla }^{2}f\left(x\right)$ 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 $x$).
The method used by nag_opt_qp_dense_solve (e04nf) (see Algorithmic Details) is most efficient when many constraints or bounds are active at the solution.

## 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 (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
Gill P E, Murray W and Wright M H (1981) Practical Optimization Academic Press
Pardalos P M and Schnitger G (1988) Checking local optimality in constrained quadratic programming is NP-hard Operations Research Letters 7 33–35

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{a}\left(\mathit{lda},:\right)$ – double array
The first dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nclin}}\right)$.
The second dimension of the array a must be at least ${\mathbf{n}}$ if ${\mathbf{nclin}}>0$ and at least $1$ if ${\mathbf{nclin}}=0$.
The $\mathit{i}$th row of a must contain the coefficients of the $\mathit{i}$th general linear constraint, for $\mathit{i}=1,2,\dots ,{m}_{L}$.
If ${\mathbf{nclin}}=0$, a is not referenced.
2:     $\mathrm{bl}\left({\mathbf{n}}+{\mathbf{nclin}}\right)$ – double array
3:     $\mathrm{bu}\left({\mathbf{n}}+{\mathbf{nclin}}\right)$ – double array
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 ${m}_{L}$ elements the bounds for the general linear constraints (if any). To specify a nonexistent lower bound (i.e., ${l}_{j}=-\infty$), set ${\mathbf{bl}}\left(j\right)\le -\mathit{bigbnd}$, and to specify a nonexistent upper bound (i.e., ${u}_{j}=+\infty$), set ${\mathbf{bu}}\left(j\right)\ge \mathit{bigbnd}$; the default value of $\mathit{bigbnd}$ is ${10}^{20}$, but this may be changed by the optional parameter Infinite Bound Size. To specify the $j$th constraint as an equality, set ${\mathbf{bl}}\left(j\right)={\mathbf{bu}}\left(j\right)=\beta$, say, where $\left|\beta \right|<\mathit{bigbnd}$.
Constraints:
• ${\mathbf{bl}}\left(\mathit{j}\right)\le {\mathbf{bu}}\left(\mathit{j}\right)$, for $\mathit{j}=1,2,\dots ,{\mathbf{n}}+{\mathbf{nclin}}$;
• if ${\mathbf{bl}}\left(j\right)={\mathbf{bu}}\left(j\right)=\beta$, $\left|\beta \right|<\mathit{bigbnd}$.
4:     $\mathrm{cvec}\left(:\right)$ – double array
The dimension of the array cvec must be at least ${\mathbf{n}}$ if the problem is of type LP, QP2 (the default) or QP4, and at least $1$ otherwise
The coefficients of the explicit linear term of the objective function when the problem is of type LP, QP2 (the default) and QP4.
If the problem is of type FP, QP1, or QP3, cvec is not referenced.
5:     $\mathrm{h}\left(\mathit{ldh},\mathit{sdh}\right)$ – double array
ldh, the first dimension of the array, must satisfy the constraint
• if the problem is of type QP1, QP2 (the default), QP3 or QP4, $\mathit{ldh}\ge {\mathbf{n}}$ or at least the value of the optional parameter Hessian Rows;
• if the problem is of type FP or LP, $\mathit{ldh}\ge 1$.
May be used to store the quadratic term $H$ of the QP objective function if desired. In some cases, you need not use h to store $H$ explicitly (see the specification of function qphess). The elements of h are referenced only by function qphess. The number of rows of $H$ is denoted by $m$, whose default value is $n$. (The optional parameter Hessian Rows may be used to specify a value of $m.)
If the default version of qphess is used and the problem is of type QP1 or QP2 (the default), the first $m$ rows and columns of h must contain the leading $m$ by $m$ rows and columns of the symmetric Hessian matrix $H$. 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.
If the default version of qphess is used and the problem is of type QP3 or QP4, the first $m$ rows of h must contain an $m$ by $n$ upper trapezoidal factor of the symmetric Hessian matrix ${H}^{\mathrm{T}}H$. The factor need not be of full rank, i.e., some of the diagonal elements 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 need not be assigned.
If a non-default version of qphess is supplied, then in some cases it may be desirable to use a one-dimensional array to transmit data to qphess. h is then declared as an ldh by $1$ array, where $\mathit{ldh}\ge {\mathbf{n}}×\left({\mathbf{n}}+1\right)/2$.
In other situations, it may be desirable to compute $Hx$ or ${H}^{\mathrm{T}}Hx$ without accessing h – for example, if $H$ or ${H}^{\mathrm{T}}H$ is sparse or has special structure. The arguments h and ldh may then refer to any convenient array.
If the problem is of type FP or LP, h is not referenced.
6:     $\mathrm{qphess}$ – function handle or string containing name of m-file
In general, you need not provide a version of qphess, because a ‘default’ function with name nag_opt_qp_dense_sample_qphess (e54nfu) is included in the Library. However, the algorithm of nag_opt_qp_dense_solve (e04nf) requires only the product of $H$ or ${H}^{\mathrm{T}}H$ 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 matrices $H$ or ${H}^{\mathrm{T}}H$ explicitly.
qphess is not referenced if the problem is of type FP or LP, in which case qphess may be the function nag_opt_qp_dense_sample_qphess (e54nfu).
[hx, user, iwsav] = qphess(n, jthcol, h, ldh, x, user, iwsav)

Input Parameters

1:     $\mathrm{n}$int64int32nag_int scalar
This is the same argument as supplied to this function. See the description for the top level argument n.
2:     $\mathrm{jthcol}$int64int32nag_int scalar
Specifies whether or not the vector $x$ is a column of the identity matrix.
${\mathbf{jthcol}}=j>0$
The vector $x$ is the $j$th column of the identity matrix, and hence $Hx$ or ${H}^{\mathrm{T}}Hx$ is the $j$th column of $H$ or ${H}^{\mathrm{T}}H$, respectively. This may in some cases require very little computation 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.
${\mathbf{jthcol}}=0$
$x$ has no special form.
3:     $\mathrm{h}\left(\mathit{ldh},\mathit{sdh}\right)$ – double array
This is the same argument as supplied to this function. See the description for the top level argument h.
4:     $\mathrm{ldh}$int64int32nag_int scalar
This is the same argument as supplied to this function. See the description for the top level argument ldh.
5:     $\mathrm{x}\left({\mathbf{n}}\right)$ – double array
The vector $x$.
6:     $\mathrm{user}$ – Any MATLAB object
qphess is called from nag_opt_qp_dense_solve (e04nf) with the object supplied to nag_opt_qp_dense_solve (e04nf).
7:     $\mathrm{iwsav}\left(610\right)$int64int32nag_int array
iwsav contains information that is required by the default function 'e54nfu'.

Output Parameters

1:     $\mathrm{hx}\left({\mathbf{n}}\right)$ – double array
The product $Hx$ if the problem is of type QP1 or QP2 (the default), or the product ${H}^{\mathrm{T}}Hx$ if the problem is of type QP3 or QP4.
2:     $\mathrm{user}$ – Any MATLAB object
3:     $\mathrm{iwsav}\left(610\right)$int64int32nag_int array
7:     $\mathrm{istate}\left({\mathbf{n}}+{\mathbf{nclin}}\right)$int64int32nag_int array
Need not be set if the (default) optional parameter Cold Start is used.
If the optional parameter Warm Start has been chosen, istate specifies the desired status of the constraints at the start of the feasibility phase. More precisely, the first $n$ elements of istate refer to the upper and lower bounds on the variables, and the next ${m}_{L}$ elements refer to the general linear constraints (if any). Possible values for ${\mathbf{istate}}\left(j\right)$ are as follows:
 ${\mathbf{istate}}\left(j\right)$ 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 must not be specified unless ${\mathbf{bl}}\left(j\right)={\mathbf{bu}}\left(j\right)$.
The values $-2$, $-1$ and $4$ are also acceptable but will be reset to zero by the function. If nag_opt_qp_dense_solve (e04nf) has been called previously with the same values of n and nclin, istate already contains satisfactory information. (See also the description of the optional parameter Warm Start.) The function also adjusts (if necessary) the values supplied in x to be consistent with istate.
Constraint: $-2\le {\mathbf{istate}}\left(\mathit{j}\right)\le 4$, for $\mathit{j}=1,2,\dots ,{\mathbf{n}}+{\mathbf{nclin}}$.
8:     $\mathrm{x}\left({\mathbf{n}}\right)$ – double array
An initial estimate of the solution.
9:     $\mathrm{lwsav}\left(120\right)$ – logical array
10:   $\mathrm{iwsav}\left(610\right)$int64int32nag_int array
11:   $\mathrm{rwsav}\left(475\right)$ – double array
The arrays lwsav, iwsav and rwsav must not be altered between calls to any of the functions nag_opt_qp_dense_solve (e04nf), nag_opt_qp_dense_option_string (e04nh).

### Optional Input Parameters

1:     $\mathrm{n}$int64int32nag_int scalar
Default: the dimension of the array x.
$n$, the number of variables.
Constraint: ${\mathbf{n}}>0$.
2:     $\mathrm{nclin}$int64int32nag_int scalar
Default: the first dimension of the array a.
${m}_{L}$, the number of general linear constraints.
Constraint: ${\mathbf{nclin}}\ge 0$.
3:     $\mathrm{user}$ – Any MATLAB object
user is not used by nag_opt_qp_dense_solve (e04nf), but is passed to qphess. Note that for large objects it may be more efficient to use a global variable which is accessible from the m-files than to use user.

### Output Parameters

1:     $\mathrm{istate}\left({\mathbf{n}}+{\mathbf{nclin}}\right)$int64int32nag_int array
The status of the constraints in the working set at the point returned in x. The significance of each possible value of ${\mathbf{istate}}\left(j\right)$ is as follows:
 ${\mathbf{istate}}\left(j\right)$ 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. $\phantom{-}0$ The constraint is satisfied to within the feasibility tolerance, but is not in the working set. $\phantom{-}1$ This inequality constraint is included in the working set at its lower bound. $\phantom{-}2$ This inequality constraint is included in the working set at its upper bound. $\phantom{-}3$ This constraint is included in the working set as an equality. This value of istate can occur only when ${\mathbf{bl}}\left(j\right)={\mathbf{bu}}\left(j\right)$. $\phantom{-}4$ This corresponds to optimality being declared with ${\mathbf{x}}\left(j\right)$ being temporarily fixed at its current value. This value of istate can occur only when ${\mathbf{ifail}}={\mathbf{1}}$ on exit.
2:     $\mathrm{x}\left({\mathbf{n}}\right)$ – double array
The point at which nag_opt_qp_dense_solve (e04nf) terminated. If ${\mathbf{ifail}}={\mathbf{0}}$, ${\mathbf{1}}$ or ${\mathbf{4}}$, x contains an estimate of the solution.
3:     $\mathrm{iter}$int64int32nag_int scalar
The total number of iterations performed.
4:     $\mathrm{obj}$ – double scalar
The value of the objective function at $x$ if $x$ is feasible, or the sum of infeasibilities at $x$ otherwise. If the problem is of type FP and $x$ is feasible, obj is set to zero.
5:     $\mathrm{ax}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nclin}}\right)\right)$ – double array
The final values of the linear constraints $Ax$.
If ${\mathbf{nclin}}=0$, ax is not referenced.
6:     $\mathrm{clamda}\left({\mathbf{n}}+{\mathbf{nclin}}\right)$ – double array
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 ${m}_{L}$ elements contain the multipliers for the general linear constraints (if any). If ${\mathbf{istate}}\left(j\right)=0$ (i.e., constraint $j$ is not in the working set), ${\mathbf{clamda}}\left(j\right)$ is zero. If $x$ is optimal, ${\mathbf{clamda}}\left(j\right)$ should be non-negative if ${\mathbf{istate}}\left(j\right)=1$, non-positive if ${\mathbf{istate}}\left(j\right)=2$ and zero if ${\mathbf{istate}}\left(j\right)=4$.
7:     $\mathrm{user}$ – Any MATLAB object
8:     $\mathrm{lwsav}\left(120\right)$ – logical array
9:     $\mathrm{iwsav}\left(610\right)$int64int32nag_int array
10:   $\mathrm{rwsav}\left(475\right)$ – double array
11:   $\mathrm{ifail}$int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see Error Indicators and Warnings).
nag_opt_qp_dense_solve (e04nf) returns with ${\mathbf{ifail}}={\mathbf{0}}$ if $x$ is a strong local minimizer, i.e., the reduced gradient (Norm Gz; see Printed output) is negligible, the Lagrange multipliers (Lagr Mult; see Printed output) are optimal and ${H}_{R}$ (the reduced Hessian of $f\left(x\right)$; see Definition of Search Direction) is positive semidefinite.

## Error Indicators and Warnings

Note: nag_opt_qp_dense_solve (e04nf) may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

W  ${\mathbf{ifail}}=1$
The iterations were terminated at a dead-point. The necessary conditions for optimality are 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 ${\nabla }^{2}f\left(x\right)$ is not positive definite, $x$ is not necessarily a local solution of the problem and verification of optimality requires further information. If ${\nabla }^{2}f\left(x\right)$ is positive semidefinite or the problem is of type LP, $x$ gives the global minimum value of the objective function, but the final $x$ is not unique.
W  ${\mathbf{ifail}}=2$
The solution appears to be unbounded, i.e., the objective function is not bounded below in the feasible region. This value of ifail occurs if a step larger than Infinite Step Size ($\text{default value}={10}^{20}$) would have to be taken in order to continue the algorithm, or the next step would result in an element of $x$ having magnitude larger than Infinite Bound Size ($\text{default value}={10}^{20}$).
W  ${\mathbf{ifail}}=3$
No feasible point was found, i.e., 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, when the feasibility tolerance for each violated constraint exceeds its Slack (see Printed output) at the final point. The modified problem (with an altered feasibility tolerance) may then be solved using a Warm Start. You should check that there are no constraint redundancies. If the data for the constraints are accurate only to the absolute precision $\sigma$, you should ensure that the value of the optional parameter Feasibility Tolerance ($\text{default value}=\sqrt{\epsilon }$, where $\epsilon$ is the machine precision) is greater than $\sigma$. For example, if all elements of $A$ are of order unity and are accurate only to three decimal places, the Feasibility Tolerance should be at least ${10}^{-3}$.
${\mathbf{ifail}}=4$
The limiting number of iterations was reached before normal termination occurred.
The values of the optional parameters Feasibility Phase Iteration Limit ($\text{default value}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(50,5\left(n+{m}_{L}\right)\right)$) and Optimality Phase Iteration Limit ($\text{default value}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(50,5\left(n+{m}_{L}\right)\right)$) may be too small. If the method appears to be making progress (e.g., the objective function is being satisfactorily reduced), either increase the iterations limit and rerun nag_opt_qp_dense_solve (e04nf) or, alternatively, rerun nag_opt_qp_dense_solve (e04nf) using the Warm Start facility to specify the initial working set.
${\mathbf{ifail}}=5$
The reduced Hessian exceeds its assigned dimension. The algorithm needed to expand the reduced Hessian when it was already at its maximum dimension, as specified by the optional parameter Maximum Degrees of Freedom ($\text{default value}=n$).
The value of the optional parameter Maximum Degrees of Freedom is too small. Rerun nag_opt_qp_dense_solve (e04nf) with a larger value (possibly using the Warm Start facility to specify the initial working set).
${\mathbf{ifail}}=6$
An input argument is invalid.
${\mathbf{ifail}}=7$
The designated problem type was not FP, LP, QP1, QP2, QP3 or QP4. Rerun nag_opt_qp_dense_solve (e04nf) with the optional parameter Problem Type set to one of these values.
$\mathbf{\text{Overflow}}$
If the printed output before the overflow error contains a warning about serious ill-conditioning in the working set when adding the $j$th constraint, it may be possible to avoid the difficulty by increasing the magnitude of the Feasibility Tolerance ($\text{default value}=\sqrt{\epsilon }$, where $\epsilon$ is the machine precision) and rerunning the program. If the message recurs even after this change, the offending linearly dependent constraint (with index ‘$j$’) must be removed from the problem.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

## Accuracy

nag_opt_qp_dense_solve (e04nf) implements a numerically stable active set strategy and returns solutions that are as accurate as the condition of the problem warrants on the machine.

This section contains some comments on scaling and a description of the printed output.

### Scaling

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. (1981) for further information and advice.

### Description of the Printed Output

The following line of summary output ($\text{}<80$ characters) 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 is the iteration count. Step is the step taken along the computed search direction. If a constraint is added during the current iteration, Step will be the step to the nearest constraint. When the problem is of type LP, the step can be greater than one during the optimality phase. Ninf is the number of violated constraints (infeasibilities). This will be zero during the optimality phase. Sinf/Objective is 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, Objective is the value of the objective function of (1). 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 nonincreasing. 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. Norm Gz is $‖{Z}_{R}^{\mathrm{T}}{g}_{\mathrm{FR}}‖$, 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. (See Definition of Search Direction and Main Iteration.)
The final printout includes a listing of the status of every variable and constraint.
The following describes the printout for each variable. A full stop (.) is printed for any numerical value that is zero.
Varbl gives the name (V) and index $\mathit{j}$, for $\mathit{j}=1,2,\dots ,n$, 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.
A key is sometimes printed before State.
 A Alternative optimum possible. The variable is active at one of its bounds, but its Lagrange multiplier is essentially zero. This means that if the variable were allowed to start moving away from its bound then there would be no change to the objective function. The values of the other free variables might change, giving a genuine alternative solution. However, if there are any degenerate variables (labelled D), the actual change might prove to be zero, since one of them could encounter a bound immediately. In either case the values of the Lagrange multipliers might also change. D Degenerate. The variable is free, but it is equal to (or very close to) one of its bounds. I Infeasible. The variable is currently violating one of its bounds by more than the Feasibility Tolerance.
Value is the value of the variable at the final iteration.
Lower Bound is the lower bound specified for the variable. None indicates that ${\mathbf{bl}}\left(j\right)\le -\mathit{bigbnd}$.
Upper Bound is the upper bound specified for the variable. None indicates that ${\mathbf{bu}}\left(j\right)\ge \mathit{bigbnd}$.
Lagr Mult is the Lagrange multiplier for the associated bound. This will be zero if State is FR unless ${\mathbf{bl}}\left(j\right)\le -\mathit{bigbnd}$ and ${\mathbf{bu}}\left(j\right)\ge \mathit{bigbnd}$, in which case the entry will be blank. If $x$ is optimal, the multiplier should be non-negative if State is LL and non-positive if State is UL.
Slack is the difference between the variable Value and the nearer of its (finite) bounds ${\mathbf{bl}}\left(j\right)$ and ${\mathbf{bu}}\left(j\right)$. A blank entry indicates that the associated variable is not bounded (i.e., ${\mathbf{bl}}\left(j\right)\le -\mathit{bigbnd}$ and ${\mathbf{bu}}\left(j\right)\ge \mathit{bigbnd}$).
The meaning of the printout for general constraints is the same as that given above for variables, with ‘variable’ replaced by ‘constraint’, ${\mathbf{bl}}\left(j\right)$ and ${\mathbf{bu}}\left(j\right)$ are replaced by ${\mathbf{bl}}\left(n+j\right)$ and ${\mathbf{bu}}\left(n+j\right)$ respectively, and with the following change in the heading:
 L Con gives the name (L) and index $\mathit{j}$, for $\mathit{j}=1,2,\dots ,{n}_{L}$, of the linear constraint.
Note that movement off a constraint (as opposed to a variable moving away from its bound) can be interpreted as allowing the entry in the Slack column to become positive.
Numerical values are output with a fixed number of digits; they are not guaranteed to be accurate to this precision.

## Example

This example minimizes the quadratic function $f\left(x\right)={c}^{\mathrm{T}}x+\frac{1}{2}{x}^{\mathrm{T}}Hx$, where
 $c=-0.02,-0.2,-0.2,-0.2,-0.2,0.04,0.04T$
 $H= 2 0 0 0 0 -0 -0 0 2 0 0 0 -0 -0 0 0 2 2 0 -0 -0 0 0 2 2 0 -0 -0 0 0 0 0 2 -0 -0 0 0 0 0 0 -2 -2 0 0 0 0 0 -2 -2$
subject to the bounds
 $-0.01≤x1≤0.01 -0.10≤x2≤0.15 -0.01≤x3≤0.03 -0.04≤x4≤0.02 -0.10≤x5≤0.05 -0.01≤x6≤0.00 -0.01≤x7≤0.00$
and to the general constraints
 $x1 + x2 + x3 + x4 + x5 + x6 + x7 = -0.13 0.15x1 + 0.04x2 + 0.02x3 + 0.04x4 + 0.02x5 + 0.01x6 + 0.03x7 ≤ -0.0049 0.03x1 + 0.05x2 + 0.08x3 + 0.02x4 + 0.06x5 + 0.01x6 ≤ -0.0064 00.02x1 + 0.04x2 + 0.01x3 + 0.02x4 + 0.02x5 ≤ -0.0037 0.02x1 + 0.03x2 + 0.01x5 ≤ -0.0012 -0.0992 ≤ 0.70x1 + 0.75x2 + 0.80x3 + 0.75x4 + 0.80x5 + 0.97x6 -0.003 ≤ 0.02x1 + 0.06x2 + 0.08x3 + 0.12x4 + 0.02x5 + 0.01x6 + 0.97x7 ≤ -0.002$
The initial point, which is infeasible, is
 $x0 = -0.01,-0.03,0.0,-0.01,-0.1,0.02,0.01T .$
The optimal solution (to five figures) is
 $x*=-0.01,-0.069865,0.018259,-0.24261,-0.62006,0.013805,0.0040665T.$
One bound constraint and four general constraints are active at the solution.
```function e04nf_example

fprintf('e04nf example results\n\n');

% Problem specification
n     = 7;
m     = 7;
nclin = 7;
a     = [ 1,     1,       1,       1,       1,      1,       1;
0.15,  0.04,    0.02,    0.04,    0.02,   0.01,    0.03;
0.03,  0.05,    0.08,    0.02,    0.06,   0.01,    0;
0.02,  0.04,    0.01,    0.02,    0.02,   0.00,    0;
0.02,  0.03,    0.00,    0.00,    0.01,   0.00,    0;
0.70,  0.75,    0.80,    0.75,    0.80,   0.97,    0;
0.02,  0.06,    0.08,    0.12,    0.02,   0.01,    0.97];
bl    = [-0.01; -0.10;   -0.01;   -0.04;   -0.10;  -0.01;   -0.01;...
-0.13; -1e25;   -1e25;   -1e25;   -1e25;  -0.0992; -0.003];
bu    = [ 0.01;  0.15;    0.03;    0.02;    0.05;   1e25;    1e25;...
-0.13; -0.0049; -0.0064; -0.0037; -0.0012; 1e25;    0.002];
cvec  = [-0.02; -0.2;    -0.2;    -0.2;    -0.2;    0.04;    0.04];
x     = [-0.01; -0.03;    0;      -0.01;   -0.1;    0.02;    0.01];

h     = zeros(m,n);
h(1,  1)   = 2;
h(2,  2)   = 2;
h(3:4,3:4) = 2;
h(5,  5)   = 2;
h(6:7,6:7) = -2;

% Initialize and set options
istate     = zeros(14, 1, 'int64');
[cwsav,lwsav,iwsav,rwsav,ifail] = e04wb('e04nf');
[lwsav, iwsav, rwsav, inform] = e04nh(...
'NoList', lwsav, iwsav, rwsav);
[lwsav, iwsav, rwsav, inform] = e04nh(...
'Print Level = 0', lwsav, iwsav, rwsav);

% Solve
[istate, x, iter, obj, ax, clamda, user, lwsav, iwsav, rwsav, ifail] = ...
e04nf(...
a, bl, bu, cvec, h, @qphess, istate, x, lwsav, iwsav, rwsav);
fprintf('Minimum value     :  %9.4f\n\n',obj);
fprintf('Found after %3d iterations at x:\n   ',iter);
fprintf(' %9.4f',x);
fprintf('\nLinear contrained values Ax:\n   ');
fprintf(' %9.4f',ax);
fprintf('\n');

function [hx, user, iwsav] = qphess(n, jthcol, h, ldh, x, user, iwsav)

if (iwsav(365) == 3 || iwsav(365) == 4)
% Problem type QP1 and QP2
hx = h*x;
elseif(iwsav(365) == 5 || iwsav(365) == 6)
hx = transpose(h)*h*x;
else
hx = zeros(n,1);
end
```
```e04nf example results

Minimum value     :     0.0370

Found after   7 iterations at x:
-0.0100   -0.0699    0.0183   -0.0243   -0.0620    0.0138    0.0041
Linear contrained values Ax:
-0.1300   -0.0059   -0.0064   -0.0045   -0.0029   -0.0992   -0.0030
```
Note: the remainder of this document is intended for more advanced users. Algorithmic Details contains a detailed description of the algorithm which may be needed in order to understand Optional Parameters and Description of Monitoring Information. Optional Parameters describes the optional parameters which may be set by calls to nag_opt_qp_dense_option_string (e04nh). Description of Monitoring Information describes the quantities which can be requested to monitor the course of the computation.

## Algorithmic Details

This section contains a detailed description of the method used by nag_opt_qp_dense_solve (e04nf).

### Overview

nag_opt_qp_dense_solve (e04nf) 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 nag_opt_qp_dense_solve (e04nf) or appear in the printed output. nag_opt_qp_dense_solve (e04nf) has two phases:
 (i) finding an initial feasible point by minimizing the sum of infeasibilities (the feasibility phase), and (ii) 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 ${m}_{L}\le n$. Once any iterate is feasible, all subsequent iterates remain feasible.
nag_opt_qp_dense_solve (e04nf) 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 (e.g., nag_opt_nlp1_rcomm (e04uf) or nag_opt_nlp2_solve (e04wd)). 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 Warm 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 $\stackrel{-}{x}$ is defined by
 $x-=x+αp$ (1)
where the step length $\alpha$ 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 Feasibility Tolerance). The working set is the current prediction of the constraints that hold with equality at the 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 element 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 elements of the search direction are zero, the columns of $A$ corresponding to fixed variables may be ignored.
Let ${m}_{\mathrm{W}}$ denote the number of general constraints in the working set and let ${n}_{\mathrm{FX}}$ denote the number of variables fixed at one of their bounds (${m}_{\mathrm{W}}$ and ${n}_{\mathrm{FX}}$ are the quantities Lin and Bnd in the monitoring file output from nag_opt_qp_dense_solve (e04nf); see Description of Monitoring Information). Similarly, let ${n}_{\mathrm{FR}}$ (${n}_{\mathrm{FR}}=n-{n}_{\mathrm{FX}}$) denote the number of free variables. At every iteration, the variables are reordered so that the last ${n}_{\mathrm{FX}}$ variables are fixed, with all other relevant vectors and matrices ordered accordingly.

### Definition of Search Direction

Let ${A}_{\mathrm{FR}}$ denote the ${m}_{\mathrm{W}}$ by ${n}_{\mathrm{FR}}$ sub-matrix of general constraints in the working set corresponding to the free variables and let ${p}_{\mathrm{FR}}$ 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 $p$ if
 $AFRpFR=0.$ (2)
In order to compute ${p}_{\mathrm{FR}}$, the $TQ$ factorization of ${A}_{\mathrm{FR}}$ is used:
 $AFRQFR=0 T,$ (3)
where $T$ is a nonsingular ${m}_{\mathrm{W}}$ by ${m}_{\mathrm{W}}$ upper triangular matrix (i.e., ${t}_{ij}=0$ if $i>j$), and the nonsingular ${n}_{\mathrm{FR}}$ by ${n}_{\mathrm{FR}}$ matrix ${Q}_{\mathrm{FR}}$ is the product of orthogonal transformations (see Gill et al. (1984)). If the columns of ${Q}_{\mathrm{FR}}$ are partitioned so that
 $QFR=Z Y,$
where $Y$ is ${n}_{\mathrm{FR}}$ by ${m}_{\mathrm{W}}$, then the ${n}_{Z}$ $\left({n}_{Z}={n}_{\mathrm{FR}}-{m}_{\mathrm{W}}\right)$ columns of $Z$ form a basis for the null space of ${A}_{\mathrm{FR}}$. Let ${n}_{R}$ be an integer such that $0\le {n}_{R}\le {n}_{Z}$, 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 Zr in the monitoring output from nag_opt_qp_dense_solve (e04nf). In many cases, ${Z}_{R}$ will include all the columns of $Z$.) The direction ${p}_{\mathrm{FR}}$ will satisfy (2) if
 $pFR=ZRpR,$ (4)
where ${p}_{R}$ is any ${n}_{R}$-vector.
Let $Q$ denote the $n$ by $n$ matrix
 $Q= QFR IFX ,$
where ${I}_{\mathrm{FX}}$ is the identity matrix of order ${n}_{\mathrm{FX}}$. Let ${H}_{Q}$ and ${g}_{Q}$ denote the $n$ by $n$ transformed Hessian and transformed gradient
 $HQ=QTHQ and gQ=QTc+Hx$
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 $f\left(x\right)$, 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, ${H}_{R}={R}^{\mathrm{T}}R$, where $R$ is the upper triangular Cholesky factor of ${H}_{R}$. If ${H}_{R}$ is not positive definite, ${H}_{R}={R}^{\mathrm{T}}DR$, where $D=\mathrm{diag}\left(1,1,\dots ,1,\mu \right)$, with $\mu \le 0$.
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
 $RpR=γeR$ (5)
where $\gamma$ 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
 $pRT HR pR < 0 and gRT pR ≤ 0 ,$
which allow the objective function to be reduced by any positive step of the form $x+\alpha p$.

### 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 ${\lambda }_{C}$ and ${\lambda }_{B}$ for the general and bound constraints are defined from the equations
 $AFRTλC=gFR and λB=gFX-AFXTλC.$ (6)
Given a positive constant $\delta$ of the order of the machine precision, a Lagrange multiplier ${\lambda }_{j}$ corresponding to an inequality constraint in the working set is said to be optimal if ${\lambda }_{j}\le \delta$ when the associated constraint is at its upper bound, or if ${\lambda }_{j}\ge -\delta$ when the associated constraint is at its lower bound. If a multiplier is nonoptimal, the objective function (either the true objective or the sum of infeasibilities) can be reduced by deleting the corresponding constraint (with index Jdel; see Description of Monitoring Information) 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 nag_opt_qp_dense_solve (e04nf) to continue until the minimum value of the sum of infeasibilities has been found; see the discussion of the optional parameter Minimum Sum of Infeasibilities. At such a point, the Lagrange multiplier ${\lambda }_{j}$ corresponding to an inequality constraint in the working set will be such that $-\left(1+\delta \right)\le {\lambda }_{j}\le \delta$ when the associated constraint is at its upper bound, and $-\delta \le {\lambda }_{j}\le \left(1+\delta \right)$ when the associated constraint is at its lower bound. Lagrange multipliers for equality constraints will satisfy $\left|{\lambda }_{j}\right|\le 1+\delta$.
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 (4) and (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, $\alpha$ will be taken as unity. In this case, the reduced gradient at $\stackrel{-}{x}$ will be zero, and Lagrange multipliers are computed. Otherwise, $\alpha$ is set to ${\alpha }_{\mathrm{M}}$, the step to the ‘nearest’ constraint (with index Jadd; see Description of Monitoring Information), which is added to the working set at the next iteration.
Each change in the working set leads to a simple change to ${A}_{\mathrm{FR}}$: if the status of a general constraint changes, a row of ${A}_{\mathrm{FR}}$ is altered; if a bound constraint enters or leaves the working set, a column of ${A}_{\mathrm{FR}}$ changes. Explicit representations are recurred of the matrices $T$, ${Q}_{\mathrm{FR}}$ and $R$; and of vectors ${Q}^{\mathrm{T}}g$, and ${Q}^{\mathrm{T}}c$. The triangular factor $R$ associated with the reduced Hessian is only updated during the optimality phase.
One of the most important features of nag_opt_qp_dense_solve (e04nf) 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 $TQ$ factor $T$ (the printed value Cond T; see Description of Monitoring Information). In constructing the initial working set, constraints are excluded that would result in a large value of Cond T.
nag_opt_qp_dense_solve (e04nf) 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 ${\alpha }_{\mathrm{M}}$ denote the maximum step at which $x+{\alpha }_{\mathrm{M}}p$ does not violate any constraint by more than its feasibility tolerance. All constraints at a distance $\alpha$ ($\alpha \le {\alpha }_{\mathrm{M}}$) 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.

### 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 ${A}_{\mathrm{FR}}$ contains ${n}_{\mathrm{FR}}$ 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 ${H}_{Q}={Q}^{\mathrm{T}}HQ$ 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 Rank Tolerance.) Let ${Z}_{R}$ denote the columns of $Z$ corresponding to $R$, and let $Z$ be partitioned as $Z=\left(\begin{array}{cc}{Z}_{R}& {Z}_{A}\end{array}\right)$. A working set for which ${Z}_{R}$ defines the null space can be obtained by including the rows of ${Z}_{A}^{\mathrm{T}}$ as ‘artificial constraints’. Minimization of the objective function then proceeds within the subspace defined by ${Z}_{R}$, as described in Definition of Search Direction.
The artificially augmented working set is given by
 $A-FR= ZAT AFR ,$ (7)
so that ${p}_{\mathrm{FR}}$ will satisfy ${A}_{\mathrm{FR}}{p}_{\mathrm{FR}}=0$ and ${Z}_{A}^{\mathrm{T}}{p}_{\mathrm{FR}}=0$. By definition of the $TQ$ factorization, ${\stackrel{-}{A}}_{\mathrm{FR}}$ automatically satisfies the following:
 $A-FRQFR= ZAT AFR QFR= ZAT AFR ZR ZA Y = 0 T- ,$
where
 $T-= I 0 0 T ,$
and hence the $TQ$ factorization of (7) is available trivially from $T$ and ${Q}_{\mathrm{FR}}$ without additional expense.
The matrix ${Z}_{A}$ is not kept fixed, since its role is purely to define an appropriate null space; the $TQ$ 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 ${Z}_{R}^{\mathrm{T}}{g}_{\mathrm{FR}}=0$, since this simply involves repartitioning ${Q}_{\mathrm{FR}}$. The ‘artificial’ multiplier vector associated with the rows of ${Z}_{A}^{\mathrm{T}}$ is equal to ${Z}_{A}^{\mathrm{T}}{g}_{\mathrm{FR}}$, 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 monitoring file output (see Description of Monitoring Information).
The number of columns in ${Z}_{A}$ and ${Z}_{R}$, the Euclidean norm of ${Z}_{R}^{\mathrm{T}}{g}_{\mathrm{FR}}$, and the condition estimator of $R$ appear in the monitoring file output as Art, Zr, Norm Gz and Cond Rz respectively (see Description of Monitoring Information).
Under some circumstances, a different type of artificial constraint isused when solving a linear program. Although the algorithm of nag_opt_qp_dense_solve (e04nf) 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., ${m}_{L}\le n$). 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 monitoring file output (see Description of Monitoring Information).

## Optional Parameters

Several optional parameters in nag_opt_qp_dense_solve (e04nf) define choices in the problem specification or the algorithm logic. In order to reduce the number of formal arguments of nag_opt_qp_dense_solve (e04nf) these optional parameters have associated default values that are appropriate for most problems. Therefore, you need only specify those optional parameters whose values are to be different from their default values.
The remainder of this section can be skipped if you wish to use the default values for all optional parameters.
The following is a list of the optional parameters available. A full description of each optional parameter is provided in Description of the s.
Optional parameters may be specified by calling nag_opt_qp_dense_option_string (e04nh) before a call to nag_opt_qp_dense_solve (e04nf).
nag_opt_qp_dense_option_string (e04nh) can be called to supply options directly, one call being necessary for each optional parameter. For example,
```[lwsav, iwsav, rwsav, inform] = e04nh('Print Level = 5', lwsav, iwsav, rwsav);
```
nag_opt_qp_dense_option_string (e04nh) should be consulted for a full description of this method of supplying optional parameters.
All optional parameters not specified by you are set to their default values. Optional parameters specified by you are unaltered by nag_opt_qp_dense_solve (e04nf) (unless they define invalid values) and so remain in effect for subsequent calls unless altered by you.

### Description of the Optional Parameters

For each option, we give a summary line, a description of the optional parameter and details of constraints.
The summary line contains:
• the keywords, where the minimum abbreviation of each keyword is underlined (if no characters of an optional qualifier are underlined, the qualifier may be omitted);
• a parameter value, where the letters $a$, $i$ and $r$ denote options that take character, integer and real values respectively;
• the default value, where the symbol $\epsilon$ is a generic notation for machine precision (see nag_machine_precision (x02aj)).
Keywords and character values are case and white space insensitive.
Check Frequency  $r$
Default $\text{}=50$
Every $i$th iteration, 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 refactorized and the variables are recomputed to satisfy the constraints more accurately. If $i\le 0$, the default value is used.
Cold Start
Default
Warm Start
This option specifies how the initial working set is chosen. With a Cold Start, nag_opt_qp_dense_solve (e04nf) 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 Crash Tolerance).
With a Warm Start, you must provide a valid definition of every element of the array istate. nag_opt_qp_dense_solve (e04nf) will override your specification of istate if necessary, so that a poor choice of the working set will not cause a fatal error. For instance, any elements of istate which are set to $-2$, $-1\text{​ or ​}4$ will be reset to zero, as will any elements which are set to $3$ when the corresponding elements of bl and bu are not equal. A warm start will be advantageous if a good estimate of the initial working set is available – for example, when nag_opt_qp_dense_solve (e04nf) is called repeatedly to solve related problems.
Crash Tolerance  $r$
Default $\text{}=0.01$
This value is used in conjunction with the optional parameter Cold Start (the default value) when nag_opt_qp_dense_solve (e04nf) selects an initial working set. If $0\le r\le 1$, the initial working set will include (if possible) bounds or general inequality constraints that lie within $r$ of their bounds. In particular, a constraint of the form ${a}_{j}^{\mathrm{T}}x\ge l$ will be included in the initial working set if $\left|{a}_{j}^{\mathrm{T}}x-l\right|\le r\left(1+\left|l\right|\right)$. If $r<0$ or $r>1$, the default value is used.
Defaults
This special keyword may be used to reset all optional parameters to their default values.
Expand Frequency  $i$
Default $\text{}=5$
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 Feasibility Tolerance is $\delta$. Over a period of $i$ iterations, the feasibility tolerance actually used by nag_opt_qp_dense_solve (e04nf) (i.e., the working feasibility tolerance) increases from $0.5\delta$ to $\delta$ (in steps of $0.5\delta /i$).
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 working feasibility tolerance is reinitialized to $0.5\delta$.
If a problem requires more than $i$ iterations, the resetting procedure is invoked and a new cycle of $i$ iterations is started with $i$ incremented by $10$. (The decision to resume the feasibility phase or optimality phase is based on comparing any constraint infeasibilities with $\delta$.)
The resetting procedure is also invoked when nag_opt_qp_dense_solve (e04nf) reaches an apparently optimal, infeasible or unbounded solution, unless this situation has already occurred twice. If any nontrivial adjustments are made, iterations are continued.
If $i\le 0$, the default value is used. If $i\ge 9999999$, no anti-cycling procedure is invoked.
Feasibility Phase Iteration Limit  ${i}_{1}$
Default $\text{}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(50,5\left(n+{m}_{L}\right)\right)$
Optimality Phase Iteration Limit  ${i}_{2}$
Default $\text{}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(50,5\left(n+{m}_{L}\right)\right)$
For problems of type FP, the scalar ${i}_{1}$ specifies the maximum number of iterations allowed before temination. Setting ${i}_{1}=0$ and ${\mathbf{Print Level}}>0$ means that the workspace needed will be computed and printed, but no iterations will be performed.
For problems of type LP, the maximum number of iterations allowed before temination is taken as $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({i}_{1},{i}_{2}\right)$. Setting ${i}_{1}=0$, ${i}_{2}=0$ and ${\mathbf{Print Level}}>0$ means that the workspace needed will be computed and printed, but no iterations will be performed.
For problems of type QP, the scalars ${i}_{1}$ and ${i}_{2}$ specify the maximum number of iterations allowed in the feasibility and optimality phases. Optimality Phase Iteration Limit is equivalent to Iteration Limit. Setting ${i}_{1}=0$ and ${\mathbf{Print Level}}>0$ means that the workspace needed will be computed and printed, but no iterations will be performed.
If ${i}_{1}<0$ or ${i}_{2}<0$, the default value is used.
Feasibility Tolerance  $r$
Default $\text{}=\sqrt{\epsilon }$
If $r\ge \epsilon$, $r$ 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 $r$ as ${10}^{-6}$. If $0\le r<\epsilon$, the default value is used.
nag_opt_qp_dense_solve (e04nf) attempts to find a feasible solution before optimizing the objective function. If the sum of infeasibilities cannot be reduced to zero, the optional parameter Minimum Sum of Infeasibilities 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 $r$ 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 $r$.
Hessian Rows  $i$
Default $\text{}=n$
Note that this option does not apply to problems of type FP or LP.
This specifies $m$, the number of rows of the Hessian matrix $H$. The default value of $m$ is $n$, the number of variables of the problem.
If the problem is of type QP then $m$ will usually be $n$, the number of variables. However, a value of $m$ less than $n$ is appropriate for QP3 or QP4 if $H$ is an upper trapezoidal matrix with $m$ rows. Similarly, $m$ may be used to define the dimension of a leading block of nonzeros in the Hessian matrices of QP1 or QP2. In this case the last $n-m$ rows and columns of $H$ are assumed to be zero. In the QP case $m$ should not be greater than $n$; if it is, the last $m-n$ rows of $H$ are ignored.
If $i<0$ or $i>n$, the default value is used.
Infinite Bound Size  $r$
Default $\text{}={10}^{20}$
If $r>0$, $r$ defines the ‘infinite’ bound $\mathit{bigbnd}$ in the definition of the problem constraints. Any upper bound greater than or equal to $\mathit{bigbnd}$ will be regarded as $+\infty$ (and similarly any lower bound less than or equal to $-\mathit{bigbnd}$ will be regarded as $-\infty$). If $r<0$, the default value is used.
Infinite Step Size  $r$
Default $\text{}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(\mathit{bigbnd},{10}^{20}\right)$
If $r>0$, $r$ 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 $r$ then the objective function is considered to be unbounded below in the feasible region. If $r\le 0$, the default value is used.
Iteration Limit  $i$
Default $\text{}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(50,5\left(n+{m}_{L}\right)\right)$
Iters
Itns
See optional parameter Feasibility Phase Iteration Limit.
List
Nolist
Default for nag_opt_qp_dense_solve (e04nf) $\text{}={\mathbf{Nolist}}$
Normally each optional parameter specification is printed as it is supplied. Optional parameter Nolist may be used to suppress the printing and optional parameter List may be used to restore printing.
Maximum Degrees of Freedom  $i$
Default $\text{}=n$
Note that this option does not apply to problems of type FP or LP.
This places a limit on the storage allocated for the triangular factor $R$ of the reduced Hessian ${H}_{R}$. Ideally, $i$ should be set slightly larger than the value of ${n}_{R}$ expected at the solution. It need not be larger than ${m}_{{\mathbf{n}}}+1$, where ${m}_{{\mathbf{n}}}$ is the number of variables that appear nonlinearly in the quadratic objective function. For many problems it can be much smaller than ${m}_{{\mathbf{n}}}$.
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 of $i$ is therefore the number of variables $n$. If Hessian Rows $m$ is specified, the default value of $i$ is the same number, $m$.
Minimum Sum of Infeasibilities  $a$
Default $=\mathrm{NO}$
If no feasible point exists for the constraints then this option is used to control whether or not nag_opt_qp_dense_solve (e04nf) will calculate a point that minimizes the constraint violations. If ${\mathbf{Minimum Sum of Infeasibilities}}=\mathrm{NO}$, nag_opt_qp_dense_solve (e04nf) will terminate as soon as it is evident that no feasible point exists for the constraints. The final point will generally not be the point at which the sum of infeasibilities is minimized. If ${\mathbf{Minimum Sum of Infeasibilities}}=\mathrm{YES}$, nag_opt_qp_dense_solve (e04nf) will continue until the sum of infeasibilities is minimized.
Monitoring File  $i$
Default $\text{}=-1$
If $i>6$ and ${\mathbf{Print Level}}\ge 5$, monitoring information produced by nag_opt_qp_dense_solve (e04nf) at every iteration is sent to a file with logical unit number $i$.
If $i<0$ and/or ${\mathbf{Print Level}}<5$, no monitoring information is produced.
Optimality Tolerance  $r$
Default $\text{}={\epsilon }^{0.5}$
If $r\ge \epsilon$, $r$ defines the tolerance used to determine if the bounds and general constraints have the right ‘sign’ for the solution to be judged to be optimal.
If $0\le r<\epsilon$, the default value is used.
Print Level  $i$
The value of $i$ controls the amount of printout produced by nag_opt_qp_dense_solve (e04nf), as indicated below. A detailed description of the printed output is given in Printed output (summary output at each iteration and the final solution) and Description of Monitoring Information (monitoring information at each iteration). If $i<0$, the default value is used.
The following printout is sent to the current advisory message unit (as defined by nag_file_set_unit_advisory (x04ab)):
 $i$ Output $\phantom{\ge 0}0$ No output. $\phantom{\ge 0}1$ The final solution only. $\phantom{\ge 0}5$ One line of summary output ($\text{}<80$ characters; see Printed output) for each iteration (no printout of the final solution). $\text{}\ge 10$ 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 optional parameter Monitoring File:
 $i$ Output $\text{}<5$ No output. $\text{}\ge 5$ One long line of output ($\text{}>80$ characters; see Description of Monitoring Information) for each iteration (no printout of the final solution). $\text{}\ge 20$ At each iteration: the Lagrange multipliers, the variables $x$, the constraint values $Ax$ and the constraint status (see istate). $\text{}\ge 30$ At each iteration: the diagonal elements of the upper triangular matrix $T$ associated with the $TQ$ factorization (3) (see Definition of Search Direction) of the working set and the diagonal elements of the upper triangular matrix $R$.
If ${\mathbf{Print Level}}\ge 5$ and the unit number defined by the Monitoring File is the same as that defined by nag_file_set_unit_advisory (x04ab) then the summary output is suppressed.
Problem Type  $a$
Default $=$ QP2
This option specifies the type of objective function to be minimized during the optimality phase. The following are the five optional keywords and the dimensions of the arrays that must be specified in order to define the objective function:
 LP h not referenced, ${\mathbf{cvec}}\left({\mathbf{n}}\right)$ required; QP1 ${\mathbf{h}}\left(\mathit{ldh},*\right)$ symmetric, cvec not referenced; QP2 ${\mathbf{h}}\left(\mathit{ldh},*\right)$ symmetric, ${\mathbf{cvec}}\left({\mathbf{n}}\right)$ required; QP3 ${\mathbf{h}}\left(\mathit{ldh},*\right)$ upper trapezoidal, cvec not referenced; QP4 ${\mathbf{h}}\left(\mathit{ldh},*\right)$ upper trapezoidal, ${\mathbf{cvec}}\left({\mathbf{n}}\right)$ required.
For problems of type FP the objective function is omitted and neither h nor cvec are referenced.
The following keywords are also acceptable. The minimum abbreviation of each keyword is underlined.
 $a$ Option Quadratic QP2 Linear LP Feasible FP
In addition, the keyword QP is equivalent to the default option QP2.
If $H=0$ (i.e., the objective function is purely linear), the efficiency of nag_opt_qp_dense_solve (e04nf) may be increased by specifying $a$ as LP.
Rank Tolerance  $r$
Default $\text{}=100\epsilon$
Note that this option does not apply to problems of type FP or LP.
This optional parameter enables you to control the condition number of the triangular factor $R$ (see Algorithmic Details). If ${\rho }_{i}$ denotes the function ${\rho }_{i}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left\{\left|{R}_{11}\right|,\left|{R}_{22}\right|,\dots ,\left|{R}_{ii}\right|\right\}$, the dimension of $R$ is defined to be smallest index $i$ such that $\left|{R}_{i+1,i+1}\right|\le \sqrt{r}\left|{\rho }_{i+1}\right|$. If $r\le 0$, the default value is used.

## Description of Monitoring Information

This section describes the long line of output ($\text{}>80$ characters) which forms part of the monitoring information produced by nag_opt_qp_dense_solve (e04nf). (See also the description of the optional parameters Monitoring File and Print Level.) You can control the level of printed output.
To aid interpretation of the printed results the following convention is used for numbering the constraints: indices $1$ through $n$ refer to the bounds on the variables and indices $n+1$ through $n+{m}_{L}$ 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 ${\mathbf{Print Level}}\ge 5$ and ${\mathbf{Monitoring File}}\ge 0$, the following line of output is produced at every iteration on the unit number specified by the Monitoring File. In all cases the values of the quantities printed are those in effect on completion of the given iteration.
 Itn is the iteration count. Jdel is the index of the constraint deleted from the working set. If Jdel is zero, no constraint was deleted. Jadd is the index of the constraint added to the working set. If Jadd is zero, no constraint was added. Step is the step taken along the computed search direction. If a constraint is added during the current iteration, Step will be the step to the nearest constraint. When the problem is of type LP, the step can be greater than one during the optimality phase. Ninf is the number of violated constraints (infeasibilities). This will be zero during the optimality phase. Sinf/Objective is 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, Objective is the value of the objective function of (1). 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 nonincreasing. 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 is the number of simple bound constraints in the current working set. Lin is the number of general linear constraints in the current working set. Art is the number of artificial constraints in the working set, i.e., the number of columns of ${Z}_{A}$ (see Choosing the Initial Working Set). Zr is the number of columns of ${Z}_{1}$ (see Definition of Search Direction). Zr is the dimension of the subspace in which the objective function is currently being minimized. The value of Zr is the number of variables minus the number of constraints in the working set; i.e., $\mathtt{Zr}=n-\left(\mathtt{Bnd}+\mathtt{Lin}+\mathtt{Art}\right)$.The value of ${n}_{Z}$, the number of columns of $Z$ (see Definition of Search Direction) can be calculated as ${n}_{Z}=n-\left(\mathtt{Bnd}+\mathtt{Lin}\right)$. A zero value of ${n}_{Z}$ implies that $x$ lies at a vertex of the feasible region. Norm Gz is $‖{Z}_{R}^{\mathrm{T}}{g}_{\mathrm{FR}}‖$, 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. NOpt is the number of nonoptimal 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 is 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 is a lower bound on the condition number of the working set. Cond Rz is a lower bound on the condition number of the triangular factor $R$ (the Cholesky factor of the current reduced Hessian; see Definition of Search Direction). If the problem is specified to be of type LP then Cond Rz is not printed. Rzz is the last diagonal element $\mu$ of the matrix $D$ associated with the ${R}^{\mathrm{T}}DR$ factorization of the reduced Hessian ${H}_{R}$ (see Definition of Search Direction). Rzz is only printed if ${H}_{R}$ is not positive definite (in which case $\mu \ne 1$). 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.