NAG CL Interface
h02dac (sqp)

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1 Purpose

h02dac solves general nonlinear programming problems with integer constraints on some of the variables.

2 Specification

#include <nag.h>
void  h02dac (Integer n, Integer nclin, Integer ncnln, const double a[], Integer pda, const double d[], double ax[], const double bl[], const double bu[], const Integer varcon[], double x[],
void (*confun)(Integer *mode, Integer ncnln, Integer n, const Integer varcon[], const double x[], double c[], double cjac[], Integer nstate, Nag_Comm *comm),
double c[], double cjac[],
void (*objfun)(Integer *mode, Integer n, const Integer varcon[], const double x[], double *objmip, double objgrd[], Integer nstate, Nag_Comm *comm),
double objgrd[], Integer maxit, double acc, double *objmip, const Integer iopts[], const double opts[], Nag_Comm *comm, NagError *fail)
The function may be called by the names: h02dac or nag_mip_sqp.
Before calling h02dac, h02zkc must be called with optstr set to ‘Initialize = h02dac’. Optional parameters may also be specified by calling h02zkc before the call to h02dac.

3 Description

h02dac 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} f(x) subject to cj(x)=0, j=1,2,,me cj(x)0, j=me+1,me+2,,m lxiu, i=1,2,,n  
with nc continuous variables and ni binary and integer variables in a total of n variables; me equality constraints in a total of m constraint functions.
Partial derivatives of f(x) and c(x) are not required for the ni integer variables. Gradients with respect to integer variables are approximated by difference formulae.
No assumptions are made regarding f(x) 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:
Pσ(x) = f(x) +σ g(x)  
where σ is adapted by the algorithm and g(x) is given by:
g(x) = cj(x), j=1,2,,me = min(cj(x),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 e04ucc) 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: and a node from where branching, that is the generation of two new subproblems, begins:
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(x) such that
Ax=d   or   ​ Axd .  
Partial derivatives with respect to x of these constraint functions are not requested by h02dac.

4 References

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

5 Arguments

1: n Integer Input
On entry: n, the total number of variables, nc+ni.
Constraint: n>0.
2: nclin Integer Input
On entry: nl, the number of general linear constraints defined by A and d.
Constraint: nclin0.
3: ncnln Integer Input
On entry: nN, the number of constraints supplied by c(x).
Constraint: ncnln0.
4: a[dim] const double Input
Note: the dimension, dim, of the array a must be at least n when nclin>0.
The (i,j)th element of the matrix A is stored in a[(j-1)×pda+i-1].
On entry: the ith row of a must contain the coefficients of the ith general linear constraint, for i=1,2,,nl. Any equality constraints must be specified first.
If nclin=0, the array a is not referenced and may be NULL.
5: pda Integer Input
On entry: the stride separating matrix row elements in the array a.
Constraint: pdanclin.
6: d[nclin] const double Input
On entry: di, the constant for the ith linear constraint.
If nclin=0, the array d is not referenced and may be NULL.
7: ax[nclin] double Output
On exit: the final residuals of the linear constraints Ax-d.
If nclin=0, ax is not referenced and may be NULL.
8: bl[n] const double Input
9: bu[n] const double Input
On entry: bl must contain the lower bounds, li, and bu the upper bounds, ui, for the variables; bounds on integer variables are rounded, bounds on binary variables need not be supplied.
Constraint: bl[i-1]bu[i-1], for i=1,2,,n.
10: varcon[n+nclin+ncnln] const Integer Input
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 nL elements the nature of the general linear constraints (if any) and the next nN elements the general constraints (if any).
varcon[j-1]=0
A continuous variable.
varcon[j-1]=1
A binary variable.
varcon[j-1]=2
An integer variable.
varcon[j-1]=3
An equality constraint.
varcon[j-1]=4
An inequality constraint.
Constraints:
  • varcon[j-1]=0, 1 or 2, for j=1,2,,n;
  • varcon[j-1]=3 or 4, for j=n+1,,n+nclin+ncnln;
  • At least one variable must be either binary or integer;
  • Any equality constraints must precede any inequality constraints.
11: x[n] double Input/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: confun function, supplied by the user External Function
confun must calculate the constraint functions supplied by c(x) and their Jacobian at x. If all constraints are supplied by A and d (i.e., ncnln=0), confun will never be called by h02dac and If 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: mode Integer * Input/Output
On entry: indicates which values must be assigned during each call of objfun. Only the following values need be assigned:
mode=0
Elements of c containing continuous variables.
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 h02dac will terminate with fail set to mode.
2: ncnln Integer Input
On entry: the dimension of the array c and the first dimension of the array cjac. The number of constraints supplied by c(x), nN.
3: n Integer Input
On entry: the second dimension of the array cjac. n, the total number of variables, nc+ni.
4: varcon[n+nclin+ncnln] const Integer Input
On entry: the array varcon as supplied to h02dac.
5: x[n] const double Input
On entry: the vector of variables at which the objective function and/or all continuous elements of its gradient are to be evaluated.
6: c[ncnln] double Output
On exit: must contain ncnln constraint values, with the value of the jth constraint cj(x) in c[j-1].
7: cjac[ncnln×n] double Input/Output
Note: the derivative of the ith constraint with respect to the jth variable, ci xj , is stored in cjac[(j-1)×ncnln+i-1].
On entry: continuous elements of cjac are set to the value of NaN.
On exit: the ith row of cjac must contain elements of ci xj for each continuous variable xj.
8: nstate Integer Input
On entry: if nstate=1, 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: comm Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to confun.
userdouble *
iuserInteger *
pPointer 
The type Pointer will be void *. Before calling h02dac you may allocate memory and initialize these pointers with various quantities for use by confun when called from h02dac (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
Note: confun should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by h02dac. If your code inadvertently does return any NaNs or infinities, h02dac is likely to produce unexpected results.
13: c[ncnln] double Output
On exit: if ncnln>0, c[j-1] contains the value of the jth constraint function cj(x) at the final iterate, for j=1,2,,ncnln.
If ncnln=0, the array c is not referenced and may be NULL.
14: cjac[ncnln×n] double Output
Note: the derivative of the ith constraint with respect to the jth variable, ci xj , is stored in cjac[(j-1)×ncnln+i-1].
On exit: if ncnln>0, cjac contains the Jacobian matrix of the constraint functions at the final iterate, i.e., cjac[(j-1)×ncnln+i-1] contains the partial derivative of the ith constraint function with respect to the jth variable, for i=1,2,,ncnln and j=1,2,,n. (See the discussion of argument cjac under confun.)
If ncnln=0, the array cjac is not referenced and may be NULL.
15: objfun function, supplied by the user External Function
objfun must calculate the objective function f(x) 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: mode Integer * Input/Output
On entry: indicates which values must be assigned during each call of objfun. Only the following values need be assigned:
mode=0
The objective function value, objmip.
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 h02dac will terminate with fail set to mode.
2: n Integer Input
On entry: n, the total number of variables, nc+ni.
3: varcon[n+nclin+ncnln] const Integer Input
On entry: the array varcon as supplied to h02dac.
4: x[n] const double Input
On entry: the vector of variables at which the objective function and/or all continuous elements of its gradient are to be evaluated.
5: objmip double * Output
On exit: must be set to the objective function value, f.
6: objgrd[n] double Input/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 mode=1, with objgrd[j-1] containing the partial derivative of f with respect to continuous variable xj; otherwise objgrd is not referenced.
7: nstate Integer Input
On entry: if nstate=1, 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: comm Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to objfun.
userdouble *
iuserInteger *
pPointer 
The type Pointer will be void *. Before calling h02dac you may allocate memory and initialize these pointers with various quantities for use by objfun when called from h02dac (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
Note: objfun should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by h02dac. If your code inadvertently does return any NaNs or infinities, h02dac is likely to produce unexpected results.
16: objgrd[n] double Output
On exit: the objective function gradient at the solution.
17: maxit Integer Input
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: maxit=500.
18: acc double 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 ε (ε being the machine precision as given by X02AJC), the below suggested value is used.
Suggested value: acc=0.0001.
19: objmip double * Output
On exit: with fail.code= NE_NOERROR, objmip contains the value of the objective function for the MINLP solution.
20: iopts[dim] const Integer Communication Array
Note: the dimension, 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 h02zkc.
21: opts[dim] const double Communication Array
Note: the dimension, 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 h02zkc.
On entry: the real option array as returned by h02zkc.
22: comm Nag_Comm *
The NAG communication argument (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
23: 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

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_BOUND
On entry, bl[value]>bu[value].
Constraint: bl[i-1]bu[i-1], for i=1,2,,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 σ is relatively low value, try a higher one, for example 1020.
Optional parameter Penalty=value.
NE_INITIALIZATION
On entry, the optional parameter arrays iopts and opts have either not been initialized or been corrupted.
NE_INT
On entry, n=value.
Constraint: n>0.
On entry, nclin=value.
Constraint: nclin0.
On entry, ncnln=value.
Constraint: ncnln0.
NE_INT_3
On entry, pda=value and nclin=value.
Constraint: pdanclin.
NE_INT_ARRAY_CONS
On entry, varcon[value]=value.
Constraint: varcon[i-1]=0, 1 or 2, for i=1,2,,n.
On entry, varcon[value]=value.
Constraint: varcon[i-1]=3 or 4, for i=n+1,,n+nclin+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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_NUM_DIFFICULTIES
The optimization failed due to numerical difficulties. Set optional parameter 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 Feasible Steps=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 value.
NE_ZERO_COEFF
Termination with zero integer trust region for integer variables; try a different starting value.
Optional parameter Integer Trust Radius=value.
NE_ZERO_VARS
On entry, there are no binary or integer variables.
NW_TOO_MANY_ITER
On entry, maxit=value. Exceeded the maximum number of iterations.

7 Accuracy

Not applicable.

8 Parallelism and Performance

h02dac 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 function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

None.

10 Example

Select a portfolio of at most p assets from n available with expected return ρ, 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
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.

10.1 Program Text

Program Text (h02dace.c)

10.2 Program Data

None.

10.3 Program Results

Program Results (h02dace.r)

11 Optional 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.

11.1 Description of the Optional Parameters

For each option, we give a summary line, a description of the optional parameter and details of constraints.
The summary line contains:
All options accept the value 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.
h02zkc can be called to supply options, one call being necessary for each optional parameter. For example,
Call h02zkc('Check Gradients = Yes', iopts, liopts, opts, lopts, ifail)
h02zkc should be consulted for a full description of the method of supplying optional parameters.
For h02dac the maximum length of the argument cvalue used by h02zlc is 12.
Branch Bound StepsiDefault =500
Maximum number of branch-and-bound steps for solving the mixed integer quadratic problems.
Constraint: Branch Bound Steps>1.
Branching RuleaDefault =Maximum
Branching rule for branch and bound search.
Branching Rule=Maximum
Maximum fractional branching.
Branching Rule=Minimum
Minimum fractional branching.
Check GradientsaDefault =No
Perform an internal check of supplied objective and constraint gradients. It is advisable to set Check Gradients=Yes during code development to avoid difficulties associated with incorrect user-supplied gradients.
Continuous Trust RadiusrDefault =10.0
Initial continuous trust region radius, Δ0c; the initial trial step dRnc for the SQP approximation must satisfy d Δ0c.
Constraint: Continuous Trust Radius>0.0.
DescentrDefault =0.05
Initial descent parameter, δ, larger values of δ allow penalty optional parameter σ to increase faster which can lead to faster convergence.
Constraint: 0.0<Descent<1.0.
Descent FactorrDefault =0.1
Factor for decreasing the internal descent parameter, δ, between iterations.
Constraint: 0.0<Descent Factor<1.0.
Feasible StepsiDefault =10
Maximum number of feasible steps without improvements, where feasibility is measured by g(x)acc.
Constraint: Feasible Steps>1.
Improved BoundsaDefault =No
Calculate improved bounds in case of ‘Best of all’ node selection strategy.
Integer Trust RadiusrDefault =10.0
Initial integer trust region radius, Δ0i; the initial trial step eRni for the SQP approximation must satisfy eΔ0i.
Constraint: Integer Trust Radius1.0.
Maximum RestartsiDefault =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<Maximum Restarts14.
Minor Print LeveliDefault =0
Print level of the subproblem solver. Active only if Print Level0.
Constraint: 0<Minor Print Level<4.
Modify HessianaDefault =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 SelectionaDefault =Depth First
Node selection strategy for branch and bound.
Node Selection=Best of all
Large tree search; this method is the slowest as it solves all subproblem QPs independently.
Node Selection=Best of two
Uses warm starts and less memory.
Node Selection=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 MonotoneiDefault =10
Maximum number of successive iterations considered for the non-monotone trust region algorithm, allowing the penalty function to increase.
Constraint: 0<Non Monotone<100.
Objective Scale BoundrDefault =1.0
When Scale Objective Function>0 internally scale absolute function values greater than 1.0 or Objective Scale Bound.
Constraint: Objective Scale Bound>0.0.
PenaltyrDefault =1000.0
Initial penalty optional parameter, σ.
Constraint: Penalty0.0.
Penalty FactorrDefault =10.0
Factor for increasing penalty optional parameter σ when the trust regions cannot be enlarged at a trial step.
Constraint: Penalty Factor>1.0.
Print LeveliDefault =0
Specifies the desired output level of printing.
Print Level=0
No output.
Print Level=1
Final convergence analysis.
Print Level=2
One line of intermediate results per iteration.
Print Level=3
Detailed information printed per iteration.
QP AccuracyrDefault =1.0e−10
Termination tolerance of the relaxed quadratic program subproblems.
Constraint: QP Accuracy>0.0.
Scale Continuous VariablesaDefault =Yes
Internally scale continuous variables values.
Scale Objective FunctioniDefault =1
Internally scale objective function values.
Scale Objective Function=0
No scaling.
Scale Objective Function=1
Scale absolute values greater than Objective Scale Bound.
Warm StartsiDefault =100
Maximum number of warm starts within the mixed integer QP solver, see Node Selection.
Constraint: Warm Starts0.