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Implementing user functions returning their own derivatives

Description
Demonstrates how black box functions that ara capable to calculate their own partial derivatives may be optimized using Xpress NonLinear

Further explanation of this example: 'Xpress NonLinear Reference Manual'

Source Files
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ComplexUserFunctions.c

/* * * * * * * * * * * * * * * * * * * * * * * * * */
/*                                                 */
/* Copyright (C) 2015-2021 Fair Isaac Corporation  */
/*                                                 */
/* * * * * * * * * * * * * * * * * * * * * * * * * */

/*
SLP example demonstrating

1, how to add user functions implemented locally to an SLP problem,
2, tracking how function evaluations happen
3, demonstrating how to define user functions that return derivastives.
4, demonstrating how to solve a problem iteration by iteration
5, demonstrating how to write out the linearization of each iteration for analysis

Solve

Miminze t                               (opt = 27)
s.t.
t = y ^ 2 + z - v + w ^ 2
x ^ 2 + t ^ 2 <= 730                  (= 27*27+1)
and
1 <= x <= 2
2 <= y <= 3
3 <= z <= 4
4 <= v <= 5
5 <= w <= 6

(c) 2021 Fair Isaac Corporation

*/

#include <stdlib.h>
#include <stdio.h>
#include <stdarg.h>
#include <signal.h>
#include <string.h>
#include <math.h>
#include <float.h>

#include <xprs.h>

/* Perform the Xpress specified function call and check its return value. */
#define CHECK_XPRSCALL(call)                             \
do {                                                   \
int result_ = call;                                  \
if ( result_ != 0 ) {                                \
fprintf(stderr, "Line %d: Failed call to %s.\n", \
__LINE__, #call);                          \
goto returnWithError;                              \
}                                                    \
} while (0)

void XPRS_CC Message(XPRSprob my_prob, void* my_object,
const char* msg, int len, int msgtype)
{
switch (msgtype)
{
case 4:  /* error */
case 3:  /* warning */
case 2:  /* dialogue */
case 1:  /* information */
printf("%s\n", msg);
break;
default: /* exiting - buffers need flushing */
fflush(stdout);
break;
}
}

// user function implementing
// y^2 + z - v + w^2
// with derivatives
int XPRS_CC MyObj(double* InputValues, double* Deltas, double* Evaluation, double* Partials, void* Context)
{
// the input values
double y = 0;
double z = 0;
double v = 0;
double w = 0;
// value of the function
double Value = 0;
// the return array counter
int nNeedDeltas = 0;

// set up local variables
y = InputValues[0];
z = InputValues[1];
v = InputValues[2];
w = InputValues[3];

// calculate the function value
*Evaluation = y * y + z - v + w * w;

// report calculation
printf("MyObj(%f,%f,%f,%f) = %f\n", y, z, v, w, Value);

if (Deltas && (Deltas[0] != 0.0 || Deltas[1] != 0.0 || Deltas[2] != 0.0 || Deltas[3] != 0.0)) {
nNeedDeltas = 1;
}

// return deltas (y^2 + z - v + w^2)
// derivatives are expected to follow only for those for which the Delta array holds a nonzero value
if (nNeedDeltas) {
if (Deltas[0]) Partials[0] = 2 * y;
if (Deltas[1]) Partials[1] = 1;
if (Deltas[2]) Partials[2] = -1;
if (Deltas[3]) Partials[3] = 2 * w;
// report derivatives
printf("Diff_MyObj(%f,%f,%f,%f)) = %f,%f,%f,%f\n", y, z, v, w, 2 * y, 1., -1., 2 * w);
}

// return value indicates if the function call was sucessfull
return 0;
}

// user function implementing
// x ^ 2 + t ^ 2
// derivatives will be calculated using tangential approximation, note the perturbed calls in the output
double XPRS_CC Ball(double* InputValues, void* context)
{
// input values
double x = 0;
double t = 0;
// value of the function
double Value = 0;
// the return array counter
int nRes = 0;

x = InputValues[0];
t = InputValues[1];

Value = x * x + t * t;

// report calculation
printf("Ball(%f,%f) = %f\n", x, t, Value);

// for functions returning a single value, the returm value is the calculated value of the function
return Value;
}

int main(int argc, char** argv)
{

XPRSprob xprob = NULL;
int ReturnValue = 0;

char buffer[1000];
int nReturn, loop;

int idBall, idMyObj;

char* sProblem = NULL;        /* Problem name */

/* Initialise Optimizer */
nReturn = XPRSinit(NULL);
CHECK_XPRSCALL( XPRSgetbanner(buffer); printf("%s\n", buffer) );

/* Initialise problem object */
CHECK_XPRSCALL(XPRScreateprob(&xprob));

// set messaginf callback

// use XSLP

/*  Miminze t
s.t.
t = y ^ 2 + z - v + w ^ 2
x ^ 2 + t ^ 2 <= 730                  (27*27+1)
and
1 <= x <= 2
2 <= y <= 3
3 <= z <= 4
4 <= v <= 5
5 <= w <= 6

Optimum should be 27 with y=2, z=3, v=5, w=5 (forced by objective) and x being 1 (forced by the second constrained) */

{
/* Linear data */
int nRow = 2;                  /* Number of rows */
char sRowType[] = { 'E','L' };          /* Row types */
double dRHS[] = { 0,730 };            /* RHS values */
double* dRange = NULL;               /* Range values - none in this example */
char sRowName[] = "obj\0ball";        /* Row names */

int nCol = 7;                  /* Number of columns */
double dObj[] = { 1,0,0,0,0,0,0 };      /* Objective function coefficients */
double dLowerBd[] = { 0,1,2,3,4,5,1 };      /* Lower bounds on the columns */
double dUpperBd[] = { XPRS_PLUSINFINITY,2,3,4,5,6,1 };    /* Upper bounds - note macro for infinity */
char sColName[] = "t\0x\0y\0z\0v\0w\0="; /* Column names */

/* Note the existence of the "=" column fixed to 1 and used to add nonlinear terms that are not coefficients to the problem */
/* Future releases of XSLP (XSLP with optimizer version 21.00 on onwards) will not require this to be done manually */

/* Matrix data */
int nColStart[] = { 0,1,1,1,1,1,1,1,1 };  /* Start positions of the columns in dMatElem
- note there are nCol+1 entries, the last
indicating where the next column would
start if there was one */
int* nColElem = NULL;               /* Number of elements in each column - not
needed thanks to the last (optional) entry
in nColStart */
int nRowInd[] = { 0 };   /* Row indices for the matrix elements */
double dMatElem[] = { -1 };  /* Matrix elements - arranged by column */

/* Load the matrix into Optimizer */
CHECK_XPRSCALL(XPRSloadlp(xprob, "UserFunctionExample_tracking", nCol, nRow, sRowType, dRHS, dRange, dObj, nColStart, nColElem, nRowInd, dMatElem, dLowerBd, dUpperBd));
}

CHECK_XPRSCALL(XPRSnlpadduserfunction(xprob, "MyObj", XPRS_USERFUNCTION_VECMAPDELTA, 4, 1, 0, (XPRSfunctionptr)MyObj, NULL, &idMyObj));
CHECK_XPRSCALL(XPRSnlpadduserfunction(xprob, "Ball", XPRS_USERFUNCTION_VECMAP, 2, 1, 0, (XPRSfunctionptr)Ball, NULL, &idBall));

// add the nonlinear functions to the slp problem
{
int Col;
int nToken = 0;
int nCoef = 0;
int RowIndex[10];
int Type[100];
double Value[100];
int FormulaStart[10];

// MyObj(y,z,v,w) = y^2 + z - v + w^2 into row 0
RowIndex[nCoef] = 0;
FormulaStart[nCoef++] = nToken;

Type[nToken] = XPRS_TOK_FUN; Value[nToken++] = idMyObj;
Type[nToken] = XPRS_TOK_LB;  Value[nToken++] = 0;
CHECK_XPRSCALL(XPRSgetindex(xprob, 2, "y", &Col)); Type[nToken] = XPRS_TOK_COL; Value[nToken++] = Col;
Type[nToken] = XPRS_TOK_DEL;  Value[nToken++] = XPRS_DEL_COMMA;
CHECK_XPRSCALL(XPRSgetindex(xprob, 2, "z", &Col)); Type[nToken] = XPRS_TOK_COL; Value[nToken++] = Col;
Type[nToken] = XPRS_TOK_DEL;  Value[nToken++] = XPRS_DEL_COMMA;
CHECK_XPRSCALL(XPRSgetindex(xprob, 2, "v", &Col)); Type[nToken] = XPRS_TOK_COL; Value[nToken++] = Col;
Type[nToken] = XPRS_TOK_DEL;  Value[nToken++] = XPRS_DEL_COMMA;
CHECK_XPRSCALL(XPRSgetindex(xprob, 2, "w", &Col)); Type[nToken] = XPRS_TOK_COL; Value[nToken++] = Col;
Type[nToken] = XPRS_TOK_RB;  Value[nToken++] = 0;
Type[nToken] = XPRS_TOK_EOF; Value[nToken++] = 0;

// BALL(x,t) = ( x ^ 2 + t ^ 2 ) into row 1
RowIndex[nCoef] = 1;
FormulaStart[nCoef++] = nToken;

Type[nToken] = XPRS_TOK_FUN; Value[nToken++] = 2;
Type[nToken] = XPRS_TOK_LB;  Value[nToken++] = 0;
CHECK_XPRSCALL(XPRSgetindex(xprob, 2, "x", &Col)); Type[nToken] = XPRS_TOK_COL; Value[nToken++] = Col;
Type[nToken] = XPRS_TOK_DEL;  Value[nToken++] = XPRS_DEL_COMMA;
CHECK_XPRSCALL(XPRSgetindex(xprob, 2, "t", &Col)); Type[nToken] = XPRS_TOK_COL; Value[nToken++] = Col;
Type[nToken] = XPRS_TOK_RB;  Value[nToken++] = 0;
Type[nToken] = XPRS_TOK_EOF; Value[nToken++] = 0;
FormulaStart[nCoef] = nToken;
CHECK_XPRSCALL(XPRSnlpaddformulas(xprob, nCoef, RowIndex, FormulaStart, 0, Type, Value));
}

// Solve problem
CHECK_XPRSCALL(XPRSnlpoptimize(xprob, "")); // 'c' as in continue

// retrive and report the optimal solution
{
int nCol, Col, status;
double* dsol;
CHECK_XPRSCALL(XPRSgetintattrib(xprob, XPRS_COLS, &nCol));
dsol = (double*)malloc(nCol * sizeof(double));
CHECK_XPRSCALL(XPRSgetsolution(xprob, &status, dsol, 0, nCol - 1));
printf("Solution:\n");
CHECK_XPRSCALL(XPRSgetindex(xprob, 2, "x", &Col); printf("  x = %f\n", dsol[Col]));
CHECK_XPRSCALL(XPRSgetindex(xprob, 2, "y", &Col); printf("  y = %f\n", dsol[Col]));
CHECK_XPRSCALL( XPRSgetindex(xprob, 2, "z", &Col); printf("  z = %f\n", dsol[Col]));
CHECK_XPRSCALL(XPRSgetindex(xprob, 2, "v", &Col); printf("  v = %f\n", dsol[Col]));
CHECK_XPRSCALL(XPRSgetindex(xprob, 2, "w", &Col); printf("  w = %f\n", dsol[Col]));
}

goto NormalReturn;
returnWithError:
printf("\nError %d\n", ReturnValue);
ReturnValue = -1;
NormalReturn:

// Retrieve error from Xpress
if (ReturnValue) {
fprintf(stderr, "An error was detected during execution.\n");
if (xprob) {
int errorcode;
char errorMessage[512];
XPRSgetintattrib(xprob, XPRS_ERRORCODE, &errorcode);
XPRSgetlasterror(xprob, errorMessage);
fprintf(stderr, "Optimizer returned error code '%i' with message:\n%s\n", errorcode, errorMessage);
}
}

/* Destroy problem object and free environment */
XPRSdestroyprob(xprob);
XPRSfree();
}