NAG Library Function Document

nag_opt_lin_lsq (e04ncc)

1  Purpose

2  Specification

3  Description

4  References

5  Arguments

1:    $\mathbf{m}$ – IntegerInput

On entry: $m$, the number of rows in the matrix $H$. If the problem is of type FP or LP, m is not referenced and is assumed to be zero. The default type is LS1; other problem types can be specified using the optional argument $\mathbf{options}\mathbf{.}\mathbf{prob}$, see Section 12.2.

If the problem is of type QP, m will usually be $n$, the number of variables. However, a value of m less than $n$ is appropriate for problem type 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 QP cases, $m$ should not be greater than $m$; if it is, the last $\left(m-n\right)$ rows of $H$ are ignored.

If the problem is a least squares problem (in particular, the default type LS1), m is also the dimension of the array b. Note that all possibilities ($m<n$, $m=n$ and $m>n$) are allowed in this case.

Constraint: $\mathbf{m}>0$ if problem is not FP or LP.

2:    $\mathbf{n}$ – IntegerInput

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

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

3:    $\mathbf{nclin}$ – IntegerInput

On entry: $n_L$, the number of general linear constraints.

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

4:    $\mathbf{a}\left[\mathbf{nclin}\times \mathbf{tda}\right]$ – const doubleInput

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

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 $\mathit{i}=1,2,\dots ,n_L$. If $\mathbf{nclin}=0$ then the array a is not referenced.

5:    $\mathbf{tda}$ – IntegerInput

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

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

6:    $\mathbf{bl}\left[\mathbf{n}+\mathbf{nclin}\right]$ – const doubleInput

7:    $\mathbf{bu}\left[\mathbf{n}+\mathbf{nclin}\right]$ – const doubleInput

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 $n_L$ elements the bounds for the general linear constraints (if any). To specify a nonexistent lower bound (i.e., $l_j=-\infty$), set $\mathbf{bl}\left[j-1\right]\le -\mathbf{options}\mathbf{.}\mathbf{inf\_bound}$, and to specify a nonexistent upper bound (i.e., $u_j=+\infty$), set $\mathbf{bu}\left[j-1\right]\ge \mathbf{options}\mathbf{.}\mathbf{inf\_bound}$, where $\mathbf{options}\mathbf{.}\mathbf{inf\_bound}$ is one of the optional arguments (default value ${10}^{20}$ (see Section 12.2). To specify the $j$th constraint as an equality, set $\mathbf{bl}\left[j-1\right]=\mathbf{bu}\left[j-1\right]=\beta$, 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 $\mathbf{bl}\left[\mathit{j}\right]=\mathbf{bu}\left[\mathit{j}\right]=\beta$, $\left|\beta \right|<\mathbf{options}\mathbf{.}\mathbf{inf\_bound}$.

8:    $\mathbf{cvec}\left[\mathbf{n}\right]$ – const doubleInput

On entry: the coefficients of the explicit linear term of the objective function when the problem is of type LP, QP2, QP4, LS2 or LS4.

If the problem is of type FP, QP1, QP3, LS1 (the default) or LS3, cvec is not referenced and may be NULL.

9:    $\mathbf{b}\left[\mathbf{m}\right]$ – doubleInput/Output

On entry: the $m$ elements of the vector of observations.

On exit: the transformed residual vector of equation (10).

b is referenced only in the case of least squares problem types (in particular, the default type LS1. For other problem types, b is not referenced and may be NULL.

10:  $\mathbf{h}\left[\mathbf{m}\times \mathbf{tdh}\right]$ – doubleInput/Output

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

On entry: the array h must contain the matrix $H$ as specified in Table 1 (see Section 3).

For problems QP1 and QP2, the first $m$ rows and columns of h must contain the leading $m$ by $m$ 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 are assumed to be zero and need not be assigned.

For problems QP3, QP4, LS3 and LS4, the first $m$ rows of h must contain an $m$ by $n$ upper trapezoidal factor of either the Hessian or the least squares matrix, ordered according to the array kx (see below). 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.

If a constrained least squares problem contains a very large number of observations, storage limitations may prevent storage of the entire least squares matrix. In such cases, you should transform the original $H$ into a triangular matrix before the call to nag_opt_lin_lsq (e04ncc) and solve as type LS3 or LS4.

On exit: by default, h contains the upper triangular Cholesky factor $R$ of equation (8), with columns ordered as indicated by kx (see below). If the optional argument $\mathbf{options}\mathbf{.}\mathbf{hessian}=\mathrm{Nag\_TRUE}$ (see Section 12.2), and the problem is one of the LS or QP types, h contains the upper triangular Cholesky factor of the Hessian matrix ${\nabla }^{2}F$, with columns ordered as indicated by kx (see below). In either case, this matrix may be used to obtain the variance-covariance matrix or to recover the upper triangular factor of the original least squares matrix.

If the problem is of type FP or LP, h is not referenced and may be NULL.

11:  $\mathbf{tdh}$ – IntegerInput

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

Constraint: $\mathbf{tdh}\ge \mathbf{n}$.

12:  $\mathbf{kx}\left[\mathbf{n}\right]$ – IntegerInput/Output

On entry: for problems of type QP3, QP4, LS3 or LS4 the array kx must specify the order of the columns of the matrix $H$ with respect to the ordering of x. Thus if column $j$ of $H$ is the column associated with the variable $x_i$ then $\mathbf{kx}\left[j-1\right]=i$.

If the problem is of any other type then the array kx need not be initialized.

Constraints:

• $1\le \mathbf{kx}\left[\mathit{i}\right]\le \mathbf{n}$, for $\mathit{i}=0,1,\dots ,\mathbf{n}-1$;
• if $i\ne j$, $\mathbf{kx}\left[i\right]\ne \mathbf{kx}\left[j\right]$.

On exit: defines the order of the columns of $H$ with respect to the ordering of x, as described above.

13:  $\mathbf{x}\left[\mathbf{n}\right]$ – doubleInput/Output

On entry: an initial estimate of the solution.

On exit: the point at which nag_opt_lin_lsq (e04ncc) terminated. If $\mathbf{fail}\mathbf{.}\mathbf{code}=\mathrm{NE\_NOERROR}$, NW_SOLN_NOT_UNIQUE or NW_NOT_FEASIBLE, x contains an estimate of the solution.

14:  $\mathbf{objf}$ – 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 FP and $x$ is feasible, objf is set to zero.

15:  $\mathbf{options}$ – Nag_E04_Opt *Input/Output

On entry/exit: a pointer to a structure of type Nag_E04_Opt whose members are optional arguments for nag_opt_lin_lsq (e04ncc). 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 below in Section 12. Some of the results returned in options can be used by nag_opt_lin_lsq (e04ncc) to perform a ‘warm start’ (see the member $\mathbf{options}\mathbf{.}\mathbf{start}$ in Section 12.2).

If any of these optional arguments are required then the structure options should be declared and initialized by a call to nag_opt_init (e04xxc) and supplied as an argument to nag_opt_lin_lsq (e04ncc). However, if the optional arguments are not required the NAG defined null pointer, E04_DEFAULT, can be used in the function call.

16:  $\mathbf{comm}$ – Nag_Comm *Input/Output

Note: comm is a NAG defined type (see Section 3.2.1.1 in the Essential Introduction).

On entry/exit: structure containing pointers for communication with an optional user-defined printing function; see Section 12.3.1 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 nag_opt_lin_lsq (e04ncc); comm will then be declared internally for use in calls to user-supplied functions.

17:  $\mathbf{fail}$ – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

5.1  Description of Printed Output

6  Error Indicators and Warnings

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 $\mathbf{tdh}\ge \mathbf{n}$.

NE_ALLOC_FAIL

Dynamic memory allocation failed.

NE_ARRAY_CONS

The contents of array kx are not valid. Constraint: must contain a permutation of integers $1,2,\dots ,\mathbf{n}$.

NE_B_NULL

$\mathbf{options}\mathbf{.}\mathbf{prob}=〈\mathit{\text{value}}〉$ but argument $\mathbf{b}=$ NULL.

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_CYCLING

The algorithm could be cycling, since a total of 50 changes were made to the working set without altering $x$. Check the detailed iteration printout for a repeated pattern of constraint deletions and additions.

If a sequence of constraint changes is being repeated, the iterates are probably cycling. (nag_opt_lin_lsq (e04ncc) does not contain a method that is guaranteed to avoid cycling; such a method would be combinatorial in nature.) Cycling may occur in two circumstances: at a constrained stationary point where there are some small or zero Lagrange multipliers; or at a point (usually a vertex) where the constraints that are satisfied exactly are nearly linearly dependent. In the latter case, you have the option of identifying the offending dependent constraints and removing them from the problem, or restarting the run with a larger value of the optional argument $\mathbf{options}\mathbf{.}\mathbf{ftol}$ (default value $=\sqrt{\epsilon }$, where $\epsilon$ is the machine precision; see Section 12.2). If this error exit occurs but no suspicious pattern of constraint changes can be observed, it may be worthwhile to restart with the final $x$ (with optional argument $\mathbf{options}\mathbf{.}\mathbf{start}=\mathrm{Nag\_Cold}$ or $\mathrm{Nag\_Warm}$).

NE_H_NULL_QP

$\mathbf{options}\mathbf{.}\mathbf{prob}=〈\mathit{\text{value}}〉$ but argument $\mathbf{h}=$ NULL. This problem type requires an array to be supplied in argument h.

NE_INT_ARG_LT

On entry, $\mathbf{m}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{m}\ge 1$.

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_INTERNAL_ERROR

An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.

NE_INVALID_INT_RANGE_1

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

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

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

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

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

NE_INVALID_REAL_RANGE_F

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

NE_INVALID_REAL_RANGE_FF

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

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 error indicator implies that a step as large as optional argument $\mathbf{options}\mathbf{.}\mathbf{inf\_step}$ (default value ${10}^{20}$; see Section 12.2) would have to be taken in order to continue the algorithm. This situation can occur only when $H$ is singular, there is an explicit linear term, and at least one variable has no upper or lower bound.

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_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 (see Section 5.1) at the final point. The modified problem (with an altered value of the optional feasibility tolerance, $\mathbf{options}\mathbf{.}\mathbf{ftol}$) may then be solved using optional argument $\mathbf{options}\mathbf{.}\mathbf{start}=\mathrm{Nag\_Warm}$ (see Section 12.2). 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 $\mathbf{options}\mathbf{.}\mathbf{ftol}$ is greater than $\sigma$. For example, if all elements of $A$ are of order unity and are accurate only to three decimal places, $\mathbf{options}\mathbf{.}\mathbf{ftol}$ should be at least ${10}^{-3}$.

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 argument $\mathbf{options}\mathbf{.}\mathbf{ftol}$ 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 point in x is a weak local minimum, i.e., the projected gradient is negligible, the Lagrange multipliers are optimal, but either $R_z$ (see Section 11.3) is singular or there is a small multiplier. This means that $x$ is not unique.

NW_TOO_MANY_ITER

The maximum number of iterations, $〈\mathit{\text{value}}〉$, have been performed.

The limiting number of iterations (determined by the optional arguments $\mathbf{options}\mathbf{.}\mathbf{max\_iter}$ and $\mathbf{options}\mathbf{.}\mathbf{fmax\_iter}$, see Section 12.2) was reached before normal termination occurred. If the method appears to be making progress (e.g., the objective function is being satisfactorily reduced), either increase the iteration limits or, alternatively, rerun nag_opt_lin_lsq (e04ncc) using the optional argument $\mathbf{options}\mathbf{.}\mathbf{start}=\mathrm{Nag\_Warm}$ to specify the initial working set. If the iteration limit is already large, but some of the constraints could be nearly linearly dependent, check the extended iteration printout (see Section 12.3) for a repeated pattern of constraints entering and leaving the working set. (Near-dependencies are often indicated by wide variations in size in the diagonal elements of the matrix $T$ (see Section 11.2), which will be printed if optional argument $\mathbf{options}\mathbf{.}\mathbf{print\_level}=\mathrm{Nag\_Soln\_Iter\_Full}$ (default value $\mathbf{options}\mathbf{.}\mathbf{print\_level}=\mathrm{Nag\_Soln\_Iter}$; see Section 12.2.) In this case, the algorithm could be cycling (see the comments below for $\mathbf{fail}\mathbf{.}\mathbf{code}=\mathbf{NE\_CYCLING}$).

7  Accuracy

8  Parallelism and Performance

9  Further Comments

9.1  Termination Criteria

9.2  Scaling

10  Example

10.1  Program Text

Program Text (e04ncce.c)

10.2  Program Data

Program Data (e04ncce.d)

10.3  Program Results

Program Results (e04ncce.r)

11  Further Description

11.1  Overview

11.2  Definition of the Search Direction

11.3  The Main Iteration

12  Optional Arguments

12.1  Optional Argument Checklist and Default Values

12.2  Description of the Optional Arguments

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)

n – Integer

The number of variables.

nclin – Integer

The number of linear constraints.

iter – Integer

The iteration count.

jdel – Integer

Index of constraint deleted from the working set.

jadd – Integer

Index of constraint added to the working set.

step – double

The step taken along the computed search direction.

ninf – Integer

The number of violated constraints (infeasibilities).

f – double

The current value of the objective function if $\mathbf{st}\mathbf{\to }\mathbf{ninf}=0$; otherwise, $\mathbf{st}\mathbf{\to }\mathbf{f}$ is a weighted sum of the magnitudes of constraint violations.

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 (see Section 11.3).

nrank – Integer

The rank of the upper triangular matrix $R$ (see Section 11.3).

nrz – Integer

Number of columns of $Z_1$ (see Section 11.2).

norm_gz – double

Euclidean norm of the reduced gradient, $‖Z_1^{\mathrm{T}}{g}_{\mathrm{FR}}‖$ (see Section 11.3).

norm_gf – double

Euclidean norm of the gradient function with respect to the free variables.

cond_t – double

A lower bound on the condition number of the working set.

cond_r – double

A lower bound on the condition number of the triangular factor $R_1$ (see Section 11.3).

x – double *

The components $\mathbf{st}\mathbf{\to }\mathbf{x}\left[\mathit{j}-1\right]$ of the current point $x$, for $\mathit{j}=1,2,\dots ,\mathbf{st}\mathbf{\to }\mathbf{n}$.

ax – double *

If $\mathbf{st}\mathbf{\to }\mathbf{nclin}>0$, the $\mathbf{st}\mathbf{\to }\mathbf{nclin}$ components of the linear constraints $Ax$.

state – Integer *

$\mathbf{options}\mathbf{.}\mathbf{state}$ contains the status of the $\mathbf{st}\mathbf{\to }\mathbf{n}$ variables and $\mathbf{st}\mathbf{\to }\mathbf{nclin}$ general linear constraints. See Section 12.2 for a description of the possible status values.

diagt – double *

If $\mathbf{st}\mathbf{\to }\mathbf{lin}>0$, the $\mathbf{st}\mathbf{\to }\mathbf{lin}$ elements in the diagonal of the matrix $T$.

diagr – double *

If $\mathbf{st}\mathbf{\to }\mathbf{nrank}>0$, the first $\mathbf{st}\mathbf{\to }\mathbf{nrank}$ elements of the diagonal of the upper triangular matrix $R$.

bnd – Integer

The number of bound constraints in the working set.

kx – Integer *

bclambda – double *

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

lin – Integer

The number of linear constraints in the working set.

kactive – Integer *

lambda – double *

Indices of the linear constraints in the working set, with associated multipliers. $\mathbf{st}\mathbf{\to }\mathbf{kactive}\left[\mathit{i}\right]$ 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 $\mathit{i}=0,1,\dots ,\mathbf{st}\mathbf{\to }\mathbf{lin}-1$.

nart – Integer

The number of artificial constraints in the working set (see Section 11.3).

gq – double *

$\mathbf{st}\mathbf{\to }\mathbf{gq}\left[\mathit{i}\right]$, for $\mathit{i}=0,1,\dots ,\mathbf{st}\mathbf{\to }\mathbf{nart}-1$, hold the multipliers for the artificial constraints.

n – Integer

The number of variables.

nclin – Integer

The number of linear constraints.

iter – Integer

The iteration count.

x – double *

The components $\mathbf{st}\mathbf{\to }\mathbf{x}\left[\mathit{j}-1\right]$ of the final point $x$, for $\mathit{j}=1,2,\dots ,\mathbf{st}\mathbf{\to }\mathbf{n}$.

feasible – Nag_Boolean

Will be Nag_TRUE if the final point is feasible.

f – double

The final value of the objective function if $\mathbf{st}\mathbf{\to }\mathbf{feasible}$ is Nag_TRUE; otherwise, the sum of infeasibilities. If the problem is of type FP and $x$ is feasible then $\mathbf{st}\mathbf{\to }\mathbf{f}$ is set to zero.

ax – double *

If $\mathbf{st}\mathbf{\to }\mathbf{nclin}>0$, the $\mathbf{st}\mathbf{\to }\mathbf{nclin}$ components of the final linear constraint activities, $Ax$.

state – Integer *

Contains the final status of the $\mathbf{st}\mathbf{\to }\mathbf{n}$ variables and $\mathbf{st}\mathbf{\to }\mathbf{nclin}$ general linear constraints. See Section 12.2 for a description of the possible status values.

lambda – double *

Contains the $\mathbf{st}\mathbf{\to }\mathbf{n}+\mathbf{st}\mathbf{\to }\mathbf{nclin}$ final values of the Lagrange multipliers.

bl – double *

Contains the $\mathbf{st}\mathbf{\to }\mathbf{n}+\mathbf{st}\mathbf{\to }\mathbf{nclin}$ lower bounds.

bu – double *

Contains the $\mathbf{st}\mathbf{\to }\mathbf{n}+\mathbf{st}\mathbf{\to }\mathbf{nclin}$ upper bounds.

endstate – Nag_EndState 

The state of termination of nag_opt_lin_lsq (e04ncc). Possible values of $\mathbf{st}\mathbf{\to }\mathbf{endstate}$ and their correspondence to the exit value of fail are:

Value of $\mathbf{st}\mathbf{\to }\mathbf{endstate}$	Value of fail
$\mathrm{Nag\_Feasible}$ or $\mathrm{Nag\_Optimal}$	NE_NOERROR
$\mathrm{Nag\_Weakmin}$	NW_SOLN_NOT_UNIQUE
$\mathrm{Nag\_Unbounded}$	NE_UNBOUNDED
$\mathrm{Nag\_Infeasible}$	NW_NOT_FEASIBLE
Nag_Too_Many_Iter	NW_TOO_MANY_ITER
$\mathrm{Nag\_Cycling}$	NE_CYCLING

it_prt – Nag_Boolean

Will be Nag_TRUE when the print function is called with the result of the current iteration.

sol_prt – Nag_Boolean

Will be Nag_TRUE when the print function is called with the final result.

new_lm – Nag_Boolean

Will be Nag_TRUE when the Lagrange multipliers have been updated.

user – double *

iuser – Integer *

p – Pointer 

Pointers for communication of user information. If used they must be allocated memory either before entry to nag_opt_lin_lsq (e04ncc) or during a call to $\mathbf{options}\mathbf{.}\mathbf{print\_fun}$. The type Pointer will be void * with a C compiler that defines void * and char * otherwise.