NAG CL Interface
e04nfc (qp_​dense_​solve)

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

1: $\mathbf{n}$ – Integer Input

On entry: $n$, the number of variables.

Constraint: $\mathbf{n}>0$.

2: $\mathbf{nclin}$ – Integer Input

On entry: $m_{\mathrm{lin}}$, the number of general linear constraints.

Constraint: $\mathbf{nclin}\ge 0$.

3: $\mathbf{a}\left[\mathbf{nclin}\times \mathbf{tda}\right]$ – const double Input

Note: the $(i,j)$th element of the matrix $A$ is stored in $\mathbf{a}\left[(i-1)\times \mathbf{tda}+j-1\right]$.

On entry: the $\mathit{i}$th row of a must contain the coefficients of the $\mathit{i}$th general linear constraint (the $\mathit{i}$th row of $A$), for $\mathit{i}=1,2,\dots ,m_{\mathrm{lin}}$. If $\mathbf{nclin}=0$, the array a is not referenced.

4: $\mathbf{tda}$ – Integer Input

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

Constraint: if $\mathbf{nclin}>0$, $\mathbf{tda}\ge \mathbf{n}$

5: $\mathbf{bl}\left[\mathbf{n}+\mathbf{nclin}\right]$ – const double Input

6: $\mathbf{bu}\left[\mathbf{n}+\mathbf{nclin}\right]$ – 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 $m_{\mathrm{lin}}$ elements the bounds for the general linear constraints (if any). To specify a nonexistent lower bound (i.e., $l_j=-\infty$), set $\mathbf{bl}\left[j\right]\le -\mathbf{options}\mathbf{.}\mathbf{inf\_bound}$, and to specify a nonexistent upper bound (i.e., $u_j=+\infty$), set $\mathbf{bu}\left[j\right]\ge \mathbf{options}\mathbf{.}\mathbf{inf\_bound}$; $\mathbf{options}\mathbf{.}\mathbf{inf\_bound}$ is the optional parameter, whose default value is ${10}^{20}$. To specify the $j$th constraint as an equality, set $\mathbf{bl}\left[j\right]=\mathbf{bu}\left[j\right]=\beta$, say, where $\left|\beta \right|<\mathbf{options}\mathbf{.}\mathbf{inf\_bound}$.

Constraints:

• $\mathbf{bl}\left[\mathit{j}\right]\le \mathbf{bu}\left[\mathit{j}\right]$, for $\mathit{j}=0,1,\dots ,\mathbf{n}+\mathbf{nclin}-1$;
• if $\mathbf{bl}\left[j\right]=\mathbf{bu}\left[j\right]=\beta$, $\left|\beta \right|<\mathbf{options}\mathbf{.}\mathbf{inf\_bound}$.

7: $\mathbf{cvec}\left[\mathbf{n}\right]$ – const double Input

On entry: the coefficients of the explicit linear term of the objective function when the problem is of type $\mathbf{options}\mathbf{.}\mathbf{prob}=\mathrm{Nag\_LP}$, $\mathrm{Nag\_QP2}$ or $\mathrm{Nag\_QP4}$. The default problem type is $\mathbf{options}\mathbf{.}\mathbf{prob}=\mathrm{Nag\_QP2}$ corresponding to QP2 described in Section 3; other problem types can be specified using the optional parameter $\mathbf{options}\mathbf{.}\mathbf{prob}$.

If the problem is of type $\mathbf{options}\mathbf{.}\mathbf{prob}=\mathrm{Nag\_FP}$, $\mathrm{Nag\_QP1}$ or $\mathrm{Nag\_QP3}$, cvec is not referenced and, therefore, a NULL pointer may be given.

8: $\mathbf{h}\left[\mathbf{n}\times \mathbf{tdh}\right]$ – 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 $\mathbf{options}\mathbf{.}\mathbf{prob}=\mathrm{Nag\_FP}$ or $\mathrm{Nag\_LP}$. The number of rows of $H$ is denoted by $m$, its default value is equal to $n$. (The optional parameter $\mathbf{options}\mathbf{.}\mathbf{hrows}$ may be used to specify a value of $m<n$.)

If the problem is of type $\mathbf{options}\mathbf{.}\mathbf{prob}=\mathrm{Nag\_QP1}$ or $\mathrm{Nag\_QP2}$, the first $m$ rows and columns of h must contain the leading $m\times m$ 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 $\mathbf{options}\mathbf{.}\mathbf{prob}=\mathrm{Nag\_QP3}$ or $\mathrm{Nag\_QP4}$, the first $m$ rows of h must contain an $m\times n$ 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: $\mathbf{tdh}$ – Integer Input

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

Constraint: $\mathbf{tdh}\ge \mathbf{n}$ or at least the value of the optional parameter $\mathbf{options}\mathbf{.}\mathbf{hrows}$ if it is set.

10: $\mathbf{qphess}$ – 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 $H$ 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 matrix $H$ or $H^{\mathrm{T}}H$ explicitly.

qphess is not referenced if the problem is of type $\mathbf{options}\mathbf{.}\mathbf{prob}=\mathrm{Nag\_FP}$ or $\mathrm{Nag\_LP}$, in which case qphess should be replaced by NULLFN.

The specification of qphess is:

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

1: $\mathbf{n}$ – Integer Input

On entry: $n$, the number of variables.

2: $\mathbf{jthcol}$ – Integer Input

On entry: jthcol 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 $Hx$ 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.

$\mathbf{jthcol}=0$

$x$ has no special form.

3: $\mathbf{h}\left[\mathbf{n}\times \mathbf{tdh}\right]$ – const double Input

On entry: the matrix $H$ of the QP objective function. The matrix element $H_{\mathit{i}\mathit{j}}$ is stored in $\mathbf{h}\left[(\mathit{i}-1)\times \mathbf{tdh}+\mathit{j}-1\right]$, for $\mathit{i}=1,2,\dots ,n$ and $\mathit{j}=1,2,\dots ,n$. In some situations, it may be desirable to compute $Hx$ 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: $\mathbf{tdh}$ – Integer Input

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

5: $\mathbf{x}\left[\mathbf{n}\right]$ – const double Input

On entry: the vector $x$.

6: $\mathbf{hx}\left[\mathbf{n}\right]$ – double Output

On exit: the product $Hx$.

7: $\mathbf{comm}$ – Nag_Comm *

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

flag – IntegerInput/Output

On entry: $\mathbf{comm}\mathbf{\to }\mathbf{flag}$ contains a non-negative number.

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

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: $\mathbf{x}\left[\mathbf{n}\right]$ – double Input/Output

On entry: an initial estimate of the solution.

On exit: the point at which e04nfc terminated. If $\mathbf{fail}\mathbf{.}\mathbf{code}=\mathrm{NE\_NOERROR}$, NW_DEAD_POINT, NW_SOLN_NOT_UNIQUE or NW_NOT_FEASIBLE, x contains an estimate of the solution.

12: $\mathbf{objf}$ – double * Output

On exit: 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 $\mathbf{options}\mathbf{.}\mathbf{prob}=\mathrm{Nag\_FP}$ and $x$ is feasible, objf is set to zero.

13: $\mathbf{options}$ – 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 $\mathbf{options}\mathbf{.}\mathbf{start}$).

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: $\mathbf{comm}$ – 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: $\mathbf{fail}$ – 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 $\mathbf{options}\mathbf{.}\mathbf{ax}$ and $\mathbf{options}\mathbf{.}\mathbf{lambda}$. x and $\mathbf{options}\mathbf{.}\mathbf{state}$ will be unchanged.

NE_2_INT_ARG_LT

On entry, $\mathbf{tda}=⟨\mathit{\text{value}}⟩$ while $\mathbf{n}=⟨\mathit{\text{value}}⟩$. These arguments must satisfy $\mathbf{tda}\ge \mathbf{n}$.

On entry, $\mathbf{tdh}=⟨\mathit{\text{value}}⟩$ while $\mathbf{n}=⟨\mathit{\text{value}}⟩$. These arguments must satisfy $\mathbf{tdh}\ge \mathbf{n}$.

On entry, $\mathbf{tdh}=⟨\mathit{\text{value}}⟩$ while $\mathbf{options}\mathbf{.}\mathbf{hrows}=⟨\mathit{\text{value}}⟩$. These arguments must satisfy $\mathbf{tdh}\ge \mathbf{options}\mathbf{.}\mathbf{hrows}$.

NE_ALLOC_FAIL

Dynamic memory allocation failed.

NE_BAD_PARAM

On entry, argument $\mathbf{options}\mathbf{.}\mathbf{print\_level}$ had an illegal value.

On entry, argument $\mathbf{options}\mathbf{.}\mathbf{prob}$ had an illegal value.

On entry, argument $\mathbf{options}\mathbf{.}\mathbf{start}$ had an illegal value.

NE_BOUND

The lower bound for variable $⟨\mathit{\text{value}}⟩$ (array element $\mathbf{bl}\left[⟨\mathit{\text{value}}⟩\right]$) is greater than the upper bound.

NE_BOUND_LCON

The lower bound for linear constraint $⟨\mathit{\text{value}}⟩$ (array element $\mathbf{bl}\left[⟨\mathit{\text{value}}⟩\right]$) is greater than the upper bound.

NE_CVEC_NULL

$\mathbf{options}\mathbf{.}\mathbf{prob}=⟨\mathit{\text{value}}⟩$ but argument $\mathbf{cvec}=$ NULL.

NE_H_NULL

$\mathbf{options}\mathbf{.}\mathbf{prob}=⟨\mathit{\text{value}}⟩$, 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. $\mathbf{options}\mathbf{.}\mathbf{max\_df}=⟨\mathit{\text{value}}⟩$.
The algorithm needed to expand the reduced Hessian when it was already at its maximum dimension, as specified by the optional parameter $\mathbf{options}\mathbf{.}\mathbf{max\_df}$.
The value of the argument $\mathbf{options}\mathbf{.}\mathbf{max\_df}$ is too small. Rerun e04nfc with a larger value (possibly using the $\mathbf{options}\mathbf{.}\mathbf{start}=\mathrm{Nag\_Warm}$ facility to specify the initial working set).

NE_INT_ARG_LT

On entry, $\mathbf{n}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{n}\ge 1$.

On entry, $\mathbf{nclin}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{nclin}\ge 0$.

NE_INVALID_INT_RANGE_1

Value $⟨\mathit{\text{value}}⟩$ given to $\mathbf{options}\mathbf{.}\mathbf{fcheck}$ not valid. Correct range is $\mathbf{options}\mathbf{.}\mathbf{fcheck}\ge 1$.

Value $⟨\mathit{\text{value}}⟩$ given to $\mathbf{options}\mathbf{.}\mathbf{fmax\_iter}$ not valid. Correct range is $\mathbf{options}\mathbf{.}\mathbf{fmax\_iter}\ge 0$.

Value $⟨\mathit{\text{value}}⟩$ given to $\mathbf{options}\mathbf{.}\mathbf{hrows}$ not valid. Correct range is $\mathbf{n}\ge \mathbf{options}\mathbf{.}\mathbf{hrows}\ge 0$.

Value $⟨\mathit{\text{value}}⟩$ given to $\mathbf{options}\mathbf{.}\mathbf{max\_df}$ not valid. Correct range is $\mathbf{n}\ge \mathbf{options}\mathbf{.}\mathbf{max\_df}\ge 1$.

Value $⟨\mathit{\text{value}}⟩$ given to $\mathbf{options}\mathbf{.}\mathbf{max\_iter}$ not valid. Correct range is $\mathbf{options}\mathbf{.}\mathbf{max\_iter}\ge 0$.

NE_INVALID_INT_RANGE_2

Value $⟨\mathit{\text{value}}⟩$ given to $\mathbf{options}\mathbf{.}\mathbf{reset\_ftol}$ not valid. Correct range is $0<\mathbf{options}\mathbf{.}\mathbf{reset\_ftol}<10000000$.

NE_INVALID_REAL_RANGE_F

Value $⟨\mathit{\text{value}}⟩$ given to $\mathbf{options}\mathbf{.}\mathbf{ftol}$ not valid. Correct range is $\mathbf{options}\mathbf{.}\mathbf{ftol}>0.0$.

Value $⟨\mathit{\text{value}}⟩$ given to $\mathbf{options}\mathbf{.}\mathbf{inf\_bound}$ not valid. Correct range is $\mathbf{options}\mathbf{.}\mathbf{inf\_bound}>0.0$.

Value $⟨\mathit{\text{value}}⟩$ given to $\mathbf{options}\mathbf{.}\mathbf{inf\_step}$ not valid. Correct range is $\mathbf{options}\mathbf{.}\mathbf{inf\_step}>0.0$.

NE_INVALID_REAL_RANGE_FF

Value $⟨\mathit{\text{value}}⟩$ given to $\mathbf{options}\mathbf{.}\mathbf{crash\_tol}$ not valid. Correct range is $0.0\le \mathbf{options}\mathbf{.}\mathbf{crash\_tol}\le 1.0$.

Value $⟨\mathit{\text{value}}⟩$ given to $\mathbf{options}\mathbf{.}\mathbf{rank\_tol}$ not valid. Correct range is $0.0\le \mathbf{options}\mathbf{.}\mathbf{rank\_tol}<1.0$.

NE_NOT_APPEND_FILE

Cannot open file $⟨\mathit{string}⟩$ for appending.

NE_NOT_CLOSE_FILE

Cannot close file $⟨\mathit{string}⟩$.

NE_OPT_NOT_INIT

Options structure not initialized.

NE_STATE_VAL

$\mathbf{options}\mathbf{.}\mathbf{state}\left[⟨\mathit{\text{value}}⟩\right]$ is out of range. $\mathbf{options}\mathbf{.}\mathbf{state}\left[⟨\mathit{\text{value}}⟩\right]=⟨\mathit{\text{value}}⟩$.

NE_UNBOUNDED

Solution appears to be unbounded.
This value of fail implies that a step as large as $\mathbf{options}\mathbf{.}\mathbf{inf\_step}$ 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 $=⟨\mathit{\text{value}}⟩$.
This exit occurs if you set $\mathbf{comm}\mathbf{\to }\mathbf{flag}$ to a negative value in qphess. If fail is supplied the value of $\mathbf{fail}\mathbf{.}\mathbf{errnum}$ will be the same as your setting of $\mathbf{comm}\mathbf{\to }\mathbf{flag}$.

NE_WARM_START

$\mathbf{options}\mathbf{.}\mathbf{start}=\mathrm{Nag\_Warm}$ but pointer $\mathbf{options}\mathbf{.}\mathbf{state}=$ NULL.

NE_WRITE_ERROR

Error occurred when writing to file $⟨\mathit{string}⟩$.

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 $\sigma$, you should ensure that the value of the optional parameter $\mathbf{options}\mathbf{.}\mathbf{ftol}$ is greater than $\sigma$. For example, if all elements of $A$ are of order unity and are accurate only to three decimal places, the optional parameter $\mathbf{options}\mathbf{.}\mathbf{ftol}$ should be at least ${10}^{\mathrm{-3}}$.

NW_OVERFLOW_WARN

Serious ill conditioning in the working set after adding constraint $⟨\mathit{\text{value}}⟩$. 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 $\mathbf{options}\mathbf{.}\mathbf{ftol}$ 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, $⟨\mathit{\text{value}}⟩$, have been performed.
The value of the optional parameter $\mathbf{options}\mathbf{.}\mathbf{max\_iter}$ 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 $\mathbf{options}\mathbf{.}\mathbf{max\_iter}$ and rerun e04nfc (possibly using the $\mathbf{options}\mathbf{.}\mathbf{start}=\mathrm{Nag\_Warm}$ facility to specify the initial working set).

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

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

12.3 Description of Printed Output

12.3.1 Output of results via a user-defined printing function

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

first – Nag_Boolean

Nag_TRUE on the first call to $\mathbf{options}\mathbf{.}\mathbf{print\_fun}$.

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, $\Vert Z_r^{\mathrm{T}}{g}_{fr}\Vert$.

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

$\mathbf{options}\mathbf{.}\mathbf{ax}$ points to the nclin memory locations holding the current values $Ax$.

state – Integer

$\mathbf{options}\mathbf{.}\mathbf{state}$ points to the $\mathbf{n}+\mathbf{nclin}$ 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 $\mathbf{st}\mathbf{\to }\mathbf{lin}$ columns. Matrix element $i,j$ is held in $\mathbf{st}\mathbf{\to }\mathbf{t}\left[(i-1)\times \mathbf{st}\mathbf{\to }\mathbf{tdt}+j-1\right]$.

tdt – Integer

The trailing dimension for $\mathbf{st}\mathbf{\to }\mathbf{t}$.

r – double

The upper triangular matrix $R$ with $\mathbf{st}\mathbf{\to }\mathbf{nrz}$ columns. Matrix element $i,j$ is held in $\mathbf{st}\mathbf{\to }\mathbf{r}\left[(i-1)\times \mathbf{st}\mathbf{\to }\mathbf{tdr}+j-1\right]$.

tdr – Integer

The trailing dimension for $\mathbf{st}\mathbf{\to }\mathbf{r}$.

condr – double

A lower bound on the condition number of the triangular factor $R$.

rzz – double

Last diagonal element $\mu$ of the matrix $D$.

kx – Integer

Indices of the bound constraints with associated multipliers. Value of $\mathbf{st}\mathbf{\to }\mathbf{kx}\left[\mathit{i}\right]$ is the index of the constraint with multiplier $\mathbf{st}\mathbf{\to }\mathbf{lambda}\left[\mathit{i}\right]$, for $\mathit{i}=0,1,\dots ,\mathbf{st}\mathbf{\to }\mathbf{bnd}-1$.

kactive – Integer

Indices of the linear constraints with associated multipliers. Value of $\mathbf{st}\mathbf{\to }\mathbf{kactive}\left[\mathit{i}\right]$ is the index of the constraint with multiplier $\mathbf{st}\mathbf{\to }\mathbf{lambda}\left[\mathbf{st}\mathbf{\to }\mathbf{bnd}+\mathit{i}\right]$, for $\mathit{i}=0,1,\dots ,\mathbf{st}\mathbf{\to }\mathbf{lin}-1$.

lambda – double

The multipliers for the constraints in the working set. $\mathbf{options}\mathbf{.}\mathbf{lambda}\left[\mathit{i}\right]$, for $\mathit{i}=0,1,\dots ,\mathbf{st}\mathbf{\to }\mathbf{bnd}-1$, hold the multipliers for the bound constraints while the multipliers for the linear constraints are held at indices $i=\mathbf{st}\mathbf{\to }\mathbf{bnd},\dots ,\mathbf{st}\mathbf{\to }\mathbf{bnd}+\mathbf{st}\mathbf{\to }\mathbf{lin}-1$.