# NAG Library Routine Document

## 1Purpose

h02daf solves general nonlinear programming problems with integer constraints on some of the variables.

## 2Specification

Fortran Interface
 Subroutine h02daf ( n, a, lda, d, ax, bl, bu, x, c, cjac, acc, opts,
 Integer, Intent (In) :: n, nclin, ncnln, lda, varcon(n+nclin+ncnln), maxit, iopts(*) Integer, Intent (Inout) :: iuser(*), ifail Real (Kind=nag_wp), Intent (In) :: a(lda,*), d(nclin), bl(n), bu(n), acc, opts(*) Real (Kind=nag_wp), Intent (Inout) :: x(n), ruser(*) Real (Kind=nag_wp), Intent (Out) :: ax(nclin), c(ncnln), cjac(ncnln,n), objgrd(n), objmip External :: confun, objfun
#include nagmk26.h
 void h02daf_ (const Integer *n, const Integer *nclin, const Integer *ncnln, const double a[], const Integer *lda, const double d[], double ax[], const double bl[], const double bu[], const Integer varcon[], double x[], void (NAG_CALL *confun)(Integer *mode, const Integer *ncnln, const Integer *n, const Integer varcon[], const double x[], double c[], double cjac[], const Integer *nstate, Integer iuser[], double ruser[]),double c[], double cjac[], void (NAG_CALL *objfun)(Integer *mode, const Integer *n, const Integer varcon[], const double x[], double *objmip, double objgrd[], const Integer *nstate, Integer iuser[], double ruser[]),double objgrd[], const Integer *maxit, const double *acc, double *objmip, const Integer iopts[], const double opts[], Integer iuser[], double ruser[], Integer *ifail)
Before calling h02daf, h02zkf must be called with optstr set to ‘Initialize = h02daf’. Optional parameters may also be specified by calling h02zkf before the call to h02daf.

## 3Description

h02daf solves mixed integer nonlinear programming problems using a modified sequential quadratic programming method. The problem is assumed to be stated in the following general form:
 $minimize x∈ Rnc,Zni fx subject to cjx=0, j=1,2,…,me cjx≥0, j=me+1,me+2,…,m l≤xi≤u, i=1,2,…,n$
with ${n}_{c}$ continuous variables and ${n}_{i}$ binary and integer variables in a total of $n$ variables; ${m}_{e}$ equality constraints in a total of $m$ constraint functions.
Partial derivatives of $f\left(x\right)$ and $c\left(x\right)$ are not required for the ${n}_{i}$ integer variables. Gradients with respect to integer variables are approximated by difference formulae.
No assumptions are made regarding $f\left(x\right)$ except that it is twice continuously differentiable with respect to continuous elements of $x$. It is not assumed that integer variables are relaxable. In other words, problem functions are evaluated only at integer points.
The method seeks to minimize the exact penalty function:
 $Pσx = fx +σ gx ∞$
where $\sigma$ is adapted by the algorithm and $g\left(x\right)$ is given by:
 $gx = cjx, j=1,2,…,me = mincjx,0, j=me+1,me+2,…,m.$
Successive quadratic approximations are applied under the assumption that integer variables have a smooth influence on the model functions, that is function values do not change drastically when incrementing or decrementing an integer value. In practice this requires integer variables to be ordinal not categorical. The algorithm is stabilised by a trust region method including Yuan's second order corrections, see Yuan and Sun (2006). The Hessian of the Lagrangian function is approximated by BFGS (see Section 11.4 in e04ucf/e04uca) updates subject to the continuous and integer variables.
The mixed-integer quadratic programming subproblems of the SQP-trust region method are solved by a branch and cut method with continuous subproblem solutions obtained by the primal-dual method of Goldfarb and Idnani, see Powell (1983). Different strategies are available for selecting a branching variable:
• Maximal fractional branching. Select an integer variable from the relaxed subproblem solution with largest distance from next integer value
• Minimal fractional branching. Select an integer variable from the relaxed subproblem solution with smallest distance from next integer value
and a node from where branching, that is the generation of two new subproblems, begins:
• Best of two. The optimal objective function values of the two child nodes are compared and the node with a lower value is chosen
• Best of all. Select an integer variable from the relaxed subproblem solution with the smallest distance from the next integer value
• Depth first. Select a child node whenever possible.
This implementation is based on the algorithm MISQP as described in Exler et al. (2013).
Linear constraints may optionally be supplied by a matrix $A$ and vector $d$ rather than the constraint functions $c\left(x\right)$ such that
 $Ax=d or ​ Ax≥d .$
Partial derivatives with respect to $x$ of these constraint functions are not requested by h02daf.

## 4References

Exler O, Lehmann T and Schittkowski K (2013) A comparative study of SQP-type algorithms for nonlinear and nonconvex mixed-integer optimization Mathematical Programming Computation 4 383–412
Mann A (1986) GAMS/MINOS: Three examples Department of Operations Research Technical Report Stanford University
Powell M J D (1983) On the quadratic programming algorithm of Goldfarb and Idnani Report DAMTP 1983/Na 19 University of Cambridge, Cambridge
Yuan Y-x and Sun W (2006) Optimization Theory and Methods Springer–Verlag

## 5Arguments

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:     $\mathbf{a}\left({\mathbf{lda}},*\right)$ – Real (Kind=nag_wp) arrayInput
Note: the second dimension of the array a must be at least ${\mathbf{n}}$ if ${\mathbf{nclin}}>0$.
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.
5:     $\mathbf{lda}$ – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which h02daf is called.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nclin}}\right)$.
6:     $\mathbf{d}\left({\mathbf{nclin}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: ${d}_{i}$, the constant for the $i$th linear constraint.
If ${\mathbf{nclin}}=0$, the array d is not referenced.
7:     $\mathbf{ax}\left({\mathbf{nclin}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the final values of the linear constraints $Ax$.
If ${\mathbf{nclin}}=0$, ax is not referenced.
8:     $\mathbf{bl}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput
9:     $\mathbf{bu}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput
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: ${\mathbf{bl}}\left(\mathit{i}\right)\le {\mathbf{bu}}\left(\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
10:   $\mathbf{varcon}\left({\mathbf{n}}+{\mathbf{nclin}}+{\mathbf{ncnln}}\right)$ – Integer arrayInput
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).
${\mathbf{varcon}}\left(j\right)=0$
A continuous variable.
${\mathbf{varcon}}\left(j\right)=1$
A binary variable.
${\mathbf{varcon}}\left(j\right)=2$
An integer variable.
${\mathbf{varcon}}\left(j\right)=3$
An equality constraint.
${\mathbf{varcon}}\left(j\right)=4$
An inequality constraint.
Constraints:
• ${\mathbf{varcon}}\left(\mathit{j}\right)=0$, $1$ or $2$, for $\mathit{j}=1,2,\dots ,{\mathbf{n}}$;
• ${\mathbf{varcon}}\left(\mathit{j}\right)=3$ 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:   $\mathbf{x}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput/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:   $\mathbf{confun}$ – Subroutine, supplied by the NAG Library or the user.External Procedure
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 h02daf and confun may be the dummy routine h02ddm. (h02ddm is included in the NAG Library.) If ${\mathbf{ncnln}}>0$, the first call to confun will occur after the first call to objfun.
The specification of confun is:
Fortran Interface
 Subroutine confun ( mode, n, x, c, cjac,
 Integer, Intent (In) :: ncnln, n, varcon(*), nstate Integer, Intent (Inout) :: mode, iuser(*) Real (Kind=nag_wp), Intent (In) :: x(n) Real (Kind=nag_wp), Intent (Inout) :: cjac(ncnln,n), ruser(*) Real (Kind=nag_wp), Intent (Out) :: c(ncnln)
#include nagmk26.h
 void confun (Integer *mode, const Integer *ncnln, const Integer *n, const Integer varcon[], const double x[], double c[], double cjac[], const Integer *nstate, Integer iuser[], double ruser[])
1:     $\mathbf{mode}$ – IntegerInput/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 h02daf will terminate with ifail set to mode.
2:     $\mathbf{ncnln}$ – IntegerInput
On entry: the dimension of the array c and the first dimension of the array cjac as declared in the (sub)program from which h02daf is called. 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 as declared in the (sub)program from which h02daf is called. $n$, the total number of variables, ${n}_{c}+{n}_{i}$.
4:     $\mathbf{varcon}\left(*\right)$ – Integer arrayInput
On entry: the array varcon as supplied to h02daf.
5:     $\mathbf{x}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: the vector of variables at which the objective function and/or all continuous elements of its gradient are to be evaluated.
6:     $\mathbf{c}\left({\mathbf{ncnln}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: must contain ncnln constraint values, with the value of the $j$th constraint ${c}_{j}\left(x\right)$ in ${\mathbf{c}}\left(j\right)$.
7:     $\mathbf{cjac}\left({\mathbf{ncnln}},{\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput/Output
Note: the derivative of the $i$th constraint with respect to the $j$th variable, $\frac{\partial {c}_{i}}{\partial {x}_{j}}$, is stored in ${\mathbf{cjac}}\left(i,j\right)$.
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 $\frac{\partial {c}_{i}}{\partial {x}_{j}}$ for each continuous variable ${x}_{j}$.
8:     $\mathbf{nstate}$ – IntegerInput
On entry: if ${\mathbf{nstate}}=1$, h02daf 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:     $\mathbf{iuser}\left(*\right)$ – Integer arrayUser Workspace
10:   $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) arrayUser Workspace
confun must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which h02daf is called. Arguments denoted as Input must not be changed by this procedure.
Note: confun should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by h02daf. If your code inadvertently does return any NaNs or infinities, h02daf is likely to produce unexpected results.
13:   $\mathbf{c}\left({\mathbf{ncnln}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: if ${\mathbf{ncnln}}>0$, ${\mathbf{c}}\left(\mathit{j}\right)$ 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.
14:   $\mathbf{cjac}\left({\mathbf{ncnln}},{\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
Note: the derivative of the $i$th constraint with respect to the $j$th variable, $\frac{\partial {c}_{i}}{\partial {x}_{j}}$, is stored in ${\mathbf{cjac}}\left(i,j\right)$.
On exit: if ${\mathbf{ncnln}}>0$, cjac contains the Jacobian matrix of the constraint functions at the final iterate, i.e., ${\mathbf{cjac}}\left(\mathit{i},\mathit{j}\right)$ 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.
15:   $\mathbf{objfun}$ – Subroutine, supplied by the user.External Procedure
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:
Fortran Interface
 Subroutine objfun ( mode, n, x,
 Integer, Intent (In) :: n, varcon(*), nstate Integer, Intent (Inout) :: mode, iuser(*) Real (Kind=nag_wp), Intent (In) :: x(n) Real (Kind=nag_wp), Intent (Inout) :: objgrd(n), ruser(*) Real (Kind=nag_wp), Intent (Out) :: objmip
#include nagmk26.h
 void objfun (Integer *mode, const Integer *n, const Integer varcon[], const double x[], double *objmip, double objgrd[], const Integer *nstate, Integer iuser[], double ruser[])
1:     $\mathbf{mode}$ – IntegerInput/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 h02daf will terminate with ifail set to mode.
2:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the total number of variables, ${n}_{c}+{n}_{i}$.
3:     $\mathbf{varcon}\left(*\right)$ – Integer arrayInput
On entry: the array varcon as supplied to h02daf.
4:     $\mathbf{x}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: the vector of variables at which the objective function and/or all continuous elements of its gradient are to be evaluated.
5:     $\mathbf{objmip}$ – Real (Kind=nag_wp)Output
On exit: must be set to the objective function value, $f$, if ${\mathbf{mode}}=0$; otherwise objmip is not referenced.
6:     $\mathbf{objgrd}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput/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 ${\mathbf{objgrd}}\left(j\right)$ 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 ${\mathbf{nstate}}=1$, h02daf 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:     $\mathbf{iuser}\left(*\right)$ – Integer arrayUser Workspace
9:     $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) arrayUser Workspace
objfun is called with the arguments iuser and ruser as supplied to h02daf. You should use the arrays iuser and ruser to supply information to objfun.
objfun must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which h02daf is called. Arguments denoted as Input must not be changed by this procedure.
Note: objfun should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by h02daf. If your code inadvertently does return any NaNs or infinities, h02daf is likely to produce unexpected results.
16:   $\mathbf{objgrd}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
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:   $\mathbf{acc}$ – Real (Kind=nag_wp)Input
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 x02ajf), the below suggested value is used.
Suggested value: ${\mathbf{acc}}=0.0001$.
19:   $\mathbf{objmip}$ – Real (Kind=nag_wp)Output
On exit: with ${\mathbf{ifail}}={\mathbf{0}}$, objmip contains the value of the objective function for the MINLP solution.
20:   $\mathbf{iopts}\left(*\right)$ – Integer arrayCommunication Array
Note: the dimension 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 h02zkf.
21:   $\mathbf{opts}\left(*\right)$ – Real (Kind=nag_wp) arrayCommunication Array
Note: the dimension 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 h02zkf.
On entry: the real option array as returned by h02zkf.
22:   $\mathbf{iuser}\left(*\right)$ – Integer arrayUser Workspace
23:   $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) arrayUser Workspace
iuser and ruser are not used by h02daf, but are passed directly to confun and objfun and may be used to pass information to these routines.
24:   $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$. When the value $-\mathbf{1}\text{​ or ​}\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}>0$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{nclin}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{nclin}}\ge 0$.
${\mathbf{ifail}}=3$
On entry, ${\mathbf{ncnln}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ncnln}}\ge 0$.
${\mathbf{ifail}}=4$
On entry, ${\mathbf{lda}}=〈\mathit{\text{value}}〉$ and ${\mathbf{nclin}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{lda}}\ge {\mathbf{nclin}}$.
${\mathbf{ifail}}=5$
On entry, ${\mathbf{bl}}\left(〈\mathit{\text{value}}〉\right)>{\mathbf{bu}}\left(〈\mathit{\text{value}}〉\right)$.
Constraint: ${\mathbf{bl}}\left(\mathit{i}\right)\le {\mathbf{bu}}\left(\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
${\mathbf{ifail}}=6$
On entry, ${\mathbf{varcon}}\left(〈\mathit{\text{value}}〉\right)=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{varcon}}\left(\mathit{i}\right)=0$, $1$ or $2$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
${\mathbf{ifail}}=7$
On entry, ${\mathbf{varcon}}\left(〈\mathit{\text{value}}〉\right)=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{varcon}}\left(\mathit{i}\right)=3$ or $4$, for $\mathit{i}={\mathbf{n}}+1,\dots ,{\mathbf{n}}+{\mathbf{nclin}}+{\mathbf{ncnln}}$.
${\mathbf{ifail}}=8$
The supplied objfun returned a NaN value.
${\mathbf{ifail}}=9$
The supplied confun returned a NaN value.
${\mathbf{ifail}}=10$
On entry, the optional parameter arrays iopts and opts have either not been initialized or been corrupted.
${\mathbf{ifail}}=11$
On entry, there are no binary or integer variables.
${\mathbf{ifail}}=12$
On entry, linear equality constraints do not precede linear inequality constraints.
${\mathbf{ifail}}=13$
On entry, nonlinear equality constraints do not precede nonlinear inequality constraints.
${\mathbf{ifail}}=81$
One or more objective gradients appear to be incorrect.
${\mathbf{ifail}}=91$
One or more constraint gradients appear to be incorrect.
${\mathbf{ifail}}=1001$
On entry, ${\mathbf{maxit}}=〈\mathit{\text{value}}〉$. Exceeded the maximum number of iterations.
${\mathbf{ifail}}=1002$
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 ${\mathbf{Feasible Steps}}=〈\mathit{\text{value}}〉$.
${\mathbf{ifail}}=1003$
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 ${\mathbf{Penalty}}=〈\mathit{\text{value}}〉$.
${\mathbf{ifail}}=1004$
Termination at an infeasible iterate; if the problem is feasible, try a different starting value.
${\mathbf{ifail}}=1005$
Termination with zero integer trust region for integer variables; try a different starting value.
Optional parameter ${\mathbf{Integer Trust Radius}}=〈\mathit{\text{value}}〉$.
${\mathbf{ifail}}=1008$
The optimization failed due to numerical difficulties. Set optional parameter ${\mathbf{Print Level}}=3$ for more information.
${\mathbf{ifail}}<0$
The optimization halted because you set mode negative in objfun or mode negative in confun, to $〈\mathit{\text{value}}〉$.
${\mathbf{ifail}}=-99$
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

Not applicable.

## 8Parallelism and Performance

h02daf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

None.

## 10Example

Select a portfolio of at most $p$ assets from $n$ available with expected return $\rho$, is fully invested and that minimizes
 $xTΣx subject to rTx = ρ ∑ i=1 n xi = 1 xi ≤ yi ∑ i=1 n yi ≤ p xi ≥ 0 yi = 0 or 1$
where
• $x$ is a vector of proportions of selected assets
• $y$ is an indicator variable that describes if an asset is in or out
• $r$ is a vector of mean returns
• $\Sigma$ is the covariance matrix of returns.
This example is taken from Mann (1986) with
 $r = 89127 Σ = 43-10 3610 -11100 0000 p = 3 ρ = 10.$
Linear constraints are supplied through both $A$ and $d$, and confun.

### 10.1Program Text

Program Text (h02dafe.f90)

None.

### 10.3Program Results

Program Results (h02dafe.r)

## 11Optional Parameters

This section can be skipped if you wish to use the default values for all optional parameters, otherwise, the following is a list of the optional parameters available and a full description of each optional parameter is provided in Section 11.1.

### 11.1Description 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;
• a parameter value, where the letters $a$, $i$ and $r$ denote options that take character, integer and real values respectively.
All options accept the value $\mathrm{DEFAULT}$ in order to return single options to their default states.
Keywords and character values are case insensitive, however they must be separated by at least one space.
h02zkf can be called to supply options, one call being necessary for each optional parameter. For example,
```Call h02zkf('Check Gradients = Yes', iopts, liopts, opts, lopts, ifail)
```
h02zkf should be consulted for a full description of the method of supplying optional parameters.
For h02daf the maximum length of the argument cvalue used by h02zlf is $12$.
 Branch Bound Steps $i$ Default$\text{}=500$
Maximum number of branch-and-bound steps for solving the mixed integer quadratic problems.
Constraint: ${\mathbf{Branch Bound Steps}}>1$.
 Branching Rule $a$ Default$\text{}=\mathrm{Maximum}$
Branching rule for branch and bound search.
${\mathbf{Branching Rule}}=\mathrm{Maximum}$
Maximum fractional branching.
${\mathbf{Branching Rule}}=\mathrm{Minimum}$
Minimum fractional branching.
 Check Gradients $a$ Default$\text{}=\mathrm{No}$
Perform an internal check of supplied objective and constraint gradients. It is advisable to set ${\mathbf{Check Gradients}}=\mathrm{Yes}$ during code development to avoid difficulties associated with incorrect user-supplied gradients.
 Continuous Trust Radius $r$ Default$\text{}=10.0$
Initial continuous trust region radius, ${\Delta }_{0}^{c}$; the initial trial step $d\in {R}^{{n}_{c}}$ for the SQP approximation must satisfy ${‖d‖}_{\infty }\le {\Delta }_{0}^{c}$.
Constraint: ${\mathbf{Continuous Trust Radius}}>0.0$.
 Descent $r$ Default$\text{}=0.05$
Initial descent parameter, $\delta$, larger values of $\delta$ allow penalty optional parameter $\sigma$ to increase faster which can lead to faster convergence.
Constraint: $0.0<{\mathbf{Descent}}<1.0$.
 Descent Factor $r$ Default$\text{}=0.1$
Factor for decreasing the internal descent parameter, $\delta$, between iterations.
Constraint: $0.0<{\mathbf{Descent Factor}}<1.0$.
 Feasible Steps $i$ Default$\text{}=10$
Maximum number of feasible steps without improvements, where feasibility is measured by ${‖g\left(x\right)‖}_{\infty }\le \sqrt{{\mathbf{acc}}}$.
Constraint: ${\mathbf{Feasible Steps}}>1$.
 Improved Bounds $a$ Default$\text{}=\mathrm{No}$
Calculate improved bounds in case of ‘Best of all’ node selection strategy.
 Integer Trust Radius $r$ Default$\text{}=10.0$
Initial integer trust region radius, ${\Delta }_{0}^{i}$; the initial trial step $e\in {R}^{{n}_{i}}$ for the SQP approximation must satisfy ${‖e‖}_{\infty }\le {\Delta }_{0}^{i}$.
Constraint: ${\mathbf{Integer Trust Radius}}\ge 1.0$.
 Maximum Restarts $i$ Default$\text{}=2$
Maximum number of restarts that allow the mixed integer SQP algorithm to return to a better solution. Setting a value higher than the default might lead to better results at the expense of function evaluations.
Constraint: $0<{\mathbf{Maximum Restarts}}\le 15$.
 Minor Print Level $i$ Default$\text{}=0$
Print level of the subproblem solver. Active only if ${\mathbf{Print Level}}\ne 0$.
Constraint: $0<{\mathbf{Minor Print Level}}<4$.
 Modify Hessian $a$ Default$\text{}=\mathrm{Yes}$
Modify the Hessian approximation in an attempt to get more accurate search directions. Calculation time is increased when the number of integer variables is large.
 Node Selection $a$ Default$\text{}=\mathrm{Depth First}$
Node selection strategy for branch and bound.
${\mathbf{Node Selection}}=\mathrm{Best of all}$
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.
 Non Monotone $i$ Default$\text{}=10$
Maximum number of successive iterations considered for the non-monotone trust region algorithm, allowing the penalty function to increase.
Constraint: $0<{\mathbf{Non Monotone}}<100$.
 Objective Scale Bound $r$ Default$\text{}=1.0$
When ${\mathbf{Scale Objective Function}}>0$ internally scale absolute function values greater than $1.0$ or Objective Scale Bound.
Constraint: ${\mathbf{Objective Scale Bound}}>0.0$.
 Penalty $r$ Default$\text{}=1000.0$
Initial penalty optional parameter, $\sigma$.
Constraint: ${\mathbf{Penalty}}\ge 0.0$.
 Penalty Factor $r$ Default$\text{}=10.0$
Factor for increasing penalty optional parameter $\sigma$ when the trust regions cannot be enlarged at a trial step.
Constraint: ${\mathbf{Penalty Factor}}>1.0$.
 Print Level $i$ Default$\text{}=0$
Specifies the desired output level of printing.
${\mathbf{Print Level}}=\mathrm{0}$
No output.
${\mathbf{Print Level}}=\mathrm{1}$
Final convergence analysis.
${\mathbf{Print Level}}=\mathrm{2}$
One line of intermediate results per iteration.
${\mathbf{Print Level}}=\mathrm{3}$
Detailed information printed per iteration.
 QP Accuracy $r$ Default$\text{}=\text{1.0E−10}$
Termination tolerance of the relaxed quadratic program subproblems.
Constraint: ${\mathbf{QP Accuracy}}>0.0$.
 Scale Continuous Variables $a$ Default$\text{}=\mathrm{Yes}$
Internally scale continuous variables values.
 Scale Objective Function $i$ Default$\text{}=1$
Internally scale objective function values.
${\mathbf{Scale Objective Function}}=\mathrm{0}$
No scaling.
${\mathbf{Scale Objective Function}}=\mathrm{1}$
Scale absolute values greater than Objective Scale Bound.
 Warm Starts $i$ Default$\text{}=100$
Maximum number of warm starts within the mixed integer QP solver, see Node Selection.
Constraint: ${\mathbf{Warm Starts}}\ge 0$.
© The Numerical Algorithms Group Ltd, Oxford, UK. 2017