NAG Library Function Document

nag_mip_sqp (h02dac)

1  Purpose

2  Specification

3  Description

4  References

5  Arguments

1:    $\mathbf{n}$ – IntegerInput

On entry: $n$, the total number of variables, $n_c+n_i$.

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

2:    $\mathbf{nclin}$ – IntegerInput

On entry: $n_l$, the number of general linear constraints defined by $A$ and $d$.

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

3:    $\mathbf{ncnln}$ – IntegerInput

On entry: $n_N$, the number of constraints supplied by $c\left(x\right)$.

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

4:    $\mathbf{a}\left[\mathit{dim}\right]$ – const doubleInput

Note: the dimension, dim, of the array a must be at least $\mathbf{n}$ when $\mathbf{nclin}>0$.

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

On entry: the $\mathit{i}$th row of a must contain the coefficients of the $\mathit{i}$th general linear constraint, for $\mathit{i}=1,2,\dots ,n_l$. Any equality constraints must be specified first.

If $\mathbf{nclin}=0$, the array a is not referenced and may be NULL.

5:    $\mathbf{pda}$ – IntegerInput

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

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

6:    $\mathbf{d}\left[\mathbf{nclin}\right]$ – const doubleInput

On entry: $d_i$, the constant for the $i$th linear constraint.

If $\mathbf{nclin}=0$, the array d is not referenced and may be NULL.

7:    $\mathbf{ax}\left[\mathbf{nclin}\right]$ – doubleOutput

On exit: the final values of the linear constraints $Ax$.

If $\mathbf{nclin}=0$, ax is not referenced and may be NULL.

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

9:    $\mathbf{bu}\left[\mathbf{n}\right]$ – const doubleInput

On entry: $\mathbf{bl}$ must contain the lower bounds, $l_i$, and $\mathbf{bu}$ the upper bounds, $u_i$, for the variables; bounds on integer variables are rounded, bounds on binary variables need not be supplied.

Constraint: $\mathbf{bl}\left[\mathit{i}-1\right]\le \mathbf{bu}\left[\mathit{i}-1\right]$, for $\mathit{i}=1,2,\dots ,\mathbf{n}$.

10:  $\mathbf{varcon}\left[\mathbf{n}+\mathbf{nclin}+\mathbf{ncnln}\right]$ – const IntegerInput

On entry: varcon indicates the nature of each variable and constraint in the problem. The first $n$ elements of the array must describe the nature of the variables, the next $n_L$ elements the nature of the general linear constraints (if any) and the next $n_N$ elements the general constraints (if any).

$\mathbf{varcon}\left[j-1\right]=0$

A continuous variable.

$\mathbf{varcon}\left[j-1\right]=1$

A binary variable.

$\mathbf{varcon}\left[j-1\right]=2$

An integer variable.

$\mathbf{varcon}\left[j-1\right]=3$

An equality constraint.

$\mathbf{varcon}\left[j-1\right]=4$

An inequality constraint.

Constraints:

• $\mathbf{varcon}\left[\mathit{j}-1\right]=0$, $1$ or $2$, for $\mathit{j}=1,2,\dots ,\mathbf{n}$;
• $\mathbf{varcon}\left[\mathit{j}-1\right]=3$ or $4$, for $\mathit{j}=\mathbf{n}+1,\dots ,\mathbf{n}+\mathbf{nclin}+\mathbf{ncnln}$;
• At least one variable must be either binary or integer;
• Any equality constraints must precede any inequality constraints.

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

On entry: an initial estimate of the solution, which need not be feasible. Values corresponding to integer variables are rounded; if an initial value less than half is supplied for a binary variable the value zero is used, otherwise the value one is used.

On exit: the final estimate of the solution.

12:  $\mathbf{confun}$ – function, supplied by the userExternal Function

confun must calculate the constraint functions supplied by $c\left(x\right)$ and their Jacobian at $x$. If all constraints are supplied by $A$ and $d$ (i.e., $\mathbf{ncnln}=0$), confun will never be called by nag_mip_sqp (h02dac) and the NAG defined null void function pointer, NULLFN, may be supplied in the call instead. If $\mathbf{ncnln}>0$, the first call to confun will occur after the first call to objfun.

The specification of confun is:

void  confun (Integer *mode, Integer ncnln, Integer n, const Integer varcon[], const double x[], double c[], double cjac[], Integer nstate, Nag_Comm *comm)

1:    $\mathbf{mode}$ – Integer *Input/Output

On entry: indicates which values must be assigned during each call of objfun. Only the following values need be assigned:

$\mathbf{mode}=0$

Elements of c containing continuous variables.

$\mathbf{mode}=1$

Elements of cjac containing continuous variables.

On exit: may be set to a negative value if you wish to terminate the solution to the current problem, and in this case nag_mip_sqp (h02dac) will terminate with fail set to mode.

2:    $\mathbf{ncnln}$ – IntegerInput

On entry: the dimension of the array c and the first dimension of the array cjac. The number of constraints supplied by $c\left(x\right)$, $n_N$.

3:    $\mathbf{n}$ – IntegerInput

On entry: the second dimension of the array cjac. $n$, the total number of variables, $n_c+n_i$.

4:    $\mathbf{varcon}\left[\mathbf{n}+\mathbf{nclin}+\mathbf{ncnln}\right]$ – const IntegerInput

Note: the dimension, dim, of the array varcon is $\mathbf{n}+\mathbf{nclin}+\mathbf{ncnln}$.

On entry: the array varcon as supplied to nag_mip_sqp (h02dac).

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

On entry: the vector of variables at which the objective function and/or all continuous elements of its gradient are to be evaluated.

6:    $\mathbf{c}\left[\mathbf{ncnln}\right]$ – doubleOutput

On exit: must contain ncnln constraint values, with the value of the $j$th constraint $c_j\left(x\right)$ in $\mathbf{c}\left[j-1\right]$.

7:    $\mathbf{cjac}\left[\mathbf{ncnln}\times \mathbf{n}\right]$ – doubleInput/Output

Note: the derivative of the $i$th constraint with respect to the $j$th variable, $\frac{\partial c_i}{\partial x_{\mathrm{j}}}$, is stored in $\mathbf{cjac}[\left(j-1\right)\times \mathbf{ncnln}+i-1]$.

On entry: continuous elements of cjac are set to the value of NaN.

On exit: the $i$th row of cjac must contain elements of $\frac{\partial c_i}{\partial x_{\mathrm{j}}}$ for each continuous variable $x_j$.

8:    $\mathbf{nstate}$ – IntegerInput

On entry: if $\mathbf{nstate}=1$, nag_mip_sqp (h02dac) is calling confun for the first time. This argument setting allows you to save computation time if certain data must be read or calculated only once.

9:    $\mathbf{comm}$ – Nag_Comm *

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

user – double *

iuser – Integer *

p – Pointer 

The type Pointer will be void *. Before calling nag_mip_sqp (h02dac) you may allocate memory and initialize these pointers with various quantities for use by confun when called from nag_mip_sqp (h02dac) (see Section 2.3.1.1 in How to Use the NAG Library and its Documentation).

13:  $\mathbf{c}\left[\mathbf{ncnln}\right]$ – doubleOutput

On exit: if $\mathbf{ncnln}>0$, $\mathbf{c}\left[\mathit{j}-1\right]$ contains the value of the $\mathit{j}$th constraint function $c_{\mathit{j}}\left(x\right)$ at the final iterate, for $\mathit{j}=1,2,\dots ,\mathbf{ncnln}$.

If $\mathbf{ncnln}=0$, the array c is not referenced and may be NULL.

14:  $\mathbf{cjac}\left[\mathbf{ncnln}\times \mathbf{n}\right]$ – doubleOutput

Note: the derivative of the $i$th constraint with respect to the $j$th variable, $\frac{\partial c_i}{\partial x_{\mathrm{j}}}$, is stored in $\mathbf{cjac}[\left(j-1\right)\times \mathbf{ncnln}+i-1]$.

On exit: if $\mathbf{ncnln}>0$, cjac contains the Jacobian matrix of the constraint functions at the final iterate, i.e., $\mathbf{cjac}\left[\left(\mathit{j}-1\right)\times \mathbf{ncnln}+\mathit{i}-1\right]$ contains the partial derivative of the $\mathit{i}$th constraint function with respect to the $\mathit{j}$th variable, for $\mathit{i}=1,2,\dots ,\mathbf{ncnln}$ and $\mathit{j}=1,2,\dots ,\mathbf{n}$. (See the discussion of argument cjac under confun.)

If $\mathbf{ncnln}=0$, the array cjac is not referenced and may be NULL.

15:  $\mathbf{objfun}$ – function, supplied by the userExternal Function

objfun must calculate the objective function $f\left(x\right)$ and its gradient for a specified $n$-element vector $x$.

The specification of objfun is:

void  objfun (Integer *mode, Integer n, const Integer varcon[], const double x[], double *objmip, double objgrd[], Integer nstate, Nag_Comm *comm)

1:    $\mathbf{mode}$ – Integer *Input/Output

On entry: indicates which values must be assigned during each call of objfun. Only the following values need be assigned:

$\mathbf{mode}=0$

The objective function value, objmip.

$\mathbf{mode}=1$

The continuous elements of objgrd.

On exit: may be set to a negative value if you wish to terminate the solution to the current problem, and in this case nag_mip_sqp (h02dac) will terminate with fail set to mode.

2:    $\mathbf{n}$ – IntegerInput

On entry: $n$, the total number of variables, $n_c+n_i$.

3:    $\mathbf{varcon}\left[\mathbf{n}+\mathbf{nclin}+\mathbf{ncnln}\right]$ – const IntegerInput

Note: the dimension, dim, of the array varcon is $\mathbf{n}+\mathbf{nclin}+\mathbf{ncnln}$.

On entry: the array varcon as supplied to nag_mip_sqp (h02dac).

4:    $\mathbf{x}\left[\mathbf{n}\right]$ – const doubleInput

On entry: the vector of variables at which the objective function and/or all continuous elements of its gradient are to be evaluated.

5:    $\mathbf{objmip}$ – double *Output

On exit: must be set to the objective function value, $f$, if $\mathbf{mode}=0$; otherwise objmip is not referenced.

6:    $\mathbf{objgrd}\left[\mathbf{n}\right]$ – doubleInput/Output

On entry: continuous elements of objgrd are set to the value of NaN.

On exit: must contain the gradient vector of the objective function if $\mathbf{mode}=1$, with $\mathbf{objgrd}\left[j-1\right]$ containing the partial derivative of $f$ with respect to continuous variable $x_j$; otherwise objgrd is not referenced.

7:    $\mathbf{nstate}$ – IntegerInput

On entry: if $\mathbf{nstate}=1$, nag_mip_sqp (h02dac) is calling objfun for the first time. This argument setting allows you to save computation time if certain data must be read or calculated only once.

8:    $\mathbf{comm}$ – Nag_Comm *

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

user – double *

iuser – Integer *

p – Pointer 

The type Pointer will be void *. Before calling nag_mip_sqp (h02dac) you may allocate memory and initialize these pointers with various quantities for use by objfun when called from nag_mip_sqp (h02dac) (see Section 2.3.1.1 in How to Use the NAG Library and its Documentation).

16:  $\mathbf{objgrd}\left[\mathbf{n}\right]$ – doubleOutput

On exit: the objective function gradient at the solution.

17:  $\mathbf{maxit}$ – IntegerInput

On entry: the maximum number of iterations within which to find a solution. If maxit is less than or equal to zero, the suggested value below is used.

Suggested value: $\mathbf{maxit}=500$.

18:  $\mathbf{acc}$ – doubleInput

On entry: the requested accuracy for determining feasible points during iterations and for halting the method when the predicted improvement in objective function is less than acc. If acc is less than or equal to $\epsilon$ ($\epsilon$ being the machine precision as given by nag_machine_precision (X02AJC)), the below suggested value is used.

Suggested value: $\mathbf{acc}=0.0001$.

19:  $\mathbf{objmip}$ – double *Output

On exit: with $\mathbf{fail}\mathbf{.}\mathbf{code}=$ NE_NOERROR, objmip contains the value of the objective function for the MINLP solution.

20:  $\mathbf{iopts}\left[\mathit{dim}\right]$ – const IntegerCommunication Array

Note: the dimension, $\mathit{dim}$, of this array is dictated by the requirements of associated functions that must have been previously called. This array MUST be the same array passed as argument iopts in the previous call to nag_mip_opt_set (h02zkc).

21:  $\mathbf{opts}\left[\mathit{dim}\right]$ – const doubleCommunication Array

Note: the dimension, $\mathit{dim}$, of this array is dictated by the requirements of associated functions that must have been previously called. This array MUST be the same array passed as argument opts in the previous call to nag_mip_opt_set (h02zkc).

On entry: the real option array as returned by nag_mip_opt_set (h02zkc).

22:  $\mathbf{comm}$ – Nag_Comm *

The NAG communication argument (see Section 2.3.1.1 in How to Use the NAG Library and its Documentation).

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

The NAG error argument (see Section 2.7 in How to Use the NAG Library and its Documentation).

6  Error Indicators and Warnings

NE_ALLOC_FAIL

Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.

NE_BAD_PARAM

On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.

NE_BOUND

On entry, $\mathbf{bl}\left[〈\mathit{\text{value}}〉\right]>\mathbf{bu}\left[〈\mathit{\text{value}}〉\right]$.
Constraint: $\mathbf{bl}\left[\mathit{i}-1\right]\le \mathbf{bu}\left[\mathit{i}-1\right]$, for $\mathit{i}=1,2,\dots ,\mathbf{n}$.

NE_CONSTRAINT

On entry, linear equality constraints do not precede linear inequality constraints.

On entry, nonlinear equality constraints do not precede nonlinear inequality constraints.

NE_DERIV_ERRORS

One or more constraint gradients appear to be incorrect.

One or more objective gradients appear to be incorrect.

NE_INFEASIBLE

Termination at an infeasible iterate; if the problem is feasible, try a different starting value.

NE_INFINITE

Penalty parameter tends to infinity in an underlying mixed-integer quadratic program; the problem may be infeasible. If $\sigma$ is relatively low value, try a higher one, for example ${10}^{20}$.
Optional parameter $\mathbf{Penalty}=〈\mathit{\text{value}}〉$.

NE_INITIALIZATION

On entry, the optional parameter arrays iopts and opts have either not been initialized or been corrupted.

NE_INT

On entry, $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{n}>0$.

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

On entry, $\mathbf{ncnln}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{ncnln}\ge 0$.

NE_INT_3

On entry, $\mathbf{pda}=〈\mathit{\text{value}}〉$ and $\mathbf{nclin}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{pda}\ge \mathbf{nclin}$.

NE_INT_ARRAY_CONS

On entry, $\mathbf{varcon}\left[〈\mathit{\text{value}}〉\right]=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{varcon}\left[\mathit{i}-1\right]=0$, $1$ or $2$, for $\mathit{i}=1,2,\dots ,\mathbf{n}$.

On entry, $\mathbf{varcon}\left[〈\mathit{\text{value}}〉\right]=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{varcon}\left[\mathit{i}-1\right]=3$ or $4$, for $\mathit{i}=\mathbf{n}+1,\dots ,\mathbf{n}+\mathbf{nclin}+\mathbf{ncnln}$.

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.

An unexpected error has been triggered by this function. Please contact NAG.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.

NE_NO_LICENCE

Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.

NE_NUM_DIFFICULTIES

The optimization failed due to numerical difficulties. Set optional parameter $\mathbf{Print\; Level}=3$ for more information.

NE_TOO_MANY

More than the maximum number of feasible steps without improvement in the objective function. If the maximum number of feasible steps is small, say less than $5$, try increasing it. Optional parameter $\mathbf{Feasible\; Steps}=〈\mathit{\text{value}}〉$.

NE_USER_NAN

The supplied confun returned a NaN value.

The supplied objfun returned a NaN value.

NE_USER_STOP

The optimization halted because you set mode negative in objfun or mode negative in confun, to $〈\mathit{\text{value}}〉$.

NE_ZERO_COEFF

Termination with zero integer trust region for integer variables; try a different starting value.
Optional parameter $\mathbf{Integer\; Trust\; Radius}=〈\mathit{\text{value}}〉$.

NE_ZERO_VARS

On entry, there are no binary or integer variables.

NW_TOO_MANY_ITER

On entry, $\mathbf{maxit}=〈\mathit{\text{value}}〉$. Exceeded the maximum number of iterations.

7  Accuracy

8  Parallelism and Performance

9  Further Comments

10  Example

10.1  Program Text

Program Text (h02dace.c)

10.2  Program Data

10.3  Program Results

Program Results (h02dace.r)

11  Optional Parameters

• Branch Bound Steps
• Branching Rule
• Check Gradients
• Continuous Trust Radius
• Descent
• Descent Factor
• Feasible Steps
• Improved Bounds
• Integer Trust Radius
• Maximum Restarts
• Minor Print Level
• Modify Hessian
• Node Selection
• Non Monotone
• Objective Scale Bound
• Penalty
• Penalty Factor
• Print Level
• QP Accuracy
• Scale Continuous Variables
• Scale Objective Function
• Warm Starts

11.1  Description of the Optional Parameters

• the keywords;
• a parameter value, where the letters $a$, $i$ and $r$ denote options that take character, integer and real values respectively.

Call H02ZKF('Check Gradients = Yes', iopts, liopts, opts, lopts, ifail)

$\mathbf{Branching\; Rule}=\mathrm{Maximum}$

Maximum fractional branching.

$\mathbf{Branching\; Rule}=\mathrm{Minimum}$

Minimum fractional branching.

$\mathbf{Node\; Selection}=\mathrm{Best\; of\; all}$

Large tree search; this method is the slowest as it solves all subproblem QPs independently.

$\mathbf{Node\; Selection}=\mathrm{Best\; of\; two}$

Uses warm starts and less memory.

$\mathbf{Node\; Selection}=\mathrm{Depth\; first}$

Uses more warm starts. If warm starts are applied, they can speed up the solution of mixed integer subproblems significantly when solving almost identical QPs.