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Maximizing the area of a polygon using a 'mapdelta' userfunction

Description
Demonstrates how to solve a nonlinear problem in Java

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

 Polygon_java_mapdelta.zip [download all files]

Source Files
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 PolygonMapDelta.java [download]

PolygonMapDelta.java

import com.dashoptimization.DefaultMessageListener;
import com.dashoptimization.IntHolder;
import com.dashoptimization.XPRSconstants;
import com.dashoptimization.XPRSenumerations.ObjSense;
import com.dashoptimization.XPRSprob;
import com.dashoptimization.XPRSprob.MapDeltaFunction;
import com.dashoptimization.XPRSprob.MapDeltaFunctor;

import java.util.ArrayList;

/** Code example that uses a user function of type "mapdelta".
<pre>
Xpress Optimizer Examples
=========================

Maximize the area of polygon of N vertices and diameter of 1
The position of vertices is indicated as (rho,theta) coordinates
where rho denotes the distance to the base point
(vertex with number N) and theta the angle from the x-axis.

(c) 2021-2024 Fair Isaac Corporation
</pre>

Polygon example: maximise the area of an N sided polygon

*** Demonstrating using a simple map (R-&gt;R) userfunction ***

<pre>
Variables:

rho   : 0..N-1   ! Distance of vertex from the base point
theta : 0..N-1   ! Angle from x-axis

Objective:
(sum (i in 1..N-2) (rho(i)*rho(i-1)*sin(theta(i)-theta(i-1)))) * 0.5

Constraints:
Vertices in increasing degree order:
theta(i) >= theta(i-1) +.0001  : i = 1..N-2
Boundary conditions:
theta(N-1) <= Pi
0.1 <= rho(i) <= 1   : i = 0..N-2
Third side of all triangles <= 1
rho(i)^2 + rho(j)^2 - rho(i)*rho(j)*2*cos(theta(j)-theta(i)) <= 1    : i in 0..N-3, j in i..N-2
</pre>
*/

public final class PolygonMapDelta {

/** User function that maps a double to a double.
* This just forwards to <code>sin()</code>.
* In addition to the function evaluated at the given point we must
* also provide the value of the derivative evaluated at the given value.
*/
private static final MapDeltaFunctor mySin = new MapDeltaFunctor() {
@Override
public double map(double value, double delta, double[] partial) {
if (delta != 0.0)
partial[0] = Math.cos(value);
return Math.sin(value);
}
};
/** User function that maps a double to a double and also evaluates
* the derivative.
* This illustrates another way to set a user function: by direct
* reference to a function.
*/
private static double myCos(double value, double delta, double[] partial) {
if (delta != 0.0)
partial[0] = -Math.sin(value);
return Math.cos(value);
}

public static void main(String[] args) {
try (XPRSprob prob = new XPRSprob(null)) {
prob.addMessageListener(new DefaultMessageListener());

// Number of sides of the Polygon
int nSide = 5;

// Theta
int[] theta = prob.varArray('C', nSide - 1, 0.0, Math.PI,
i -> String.format("THETA%d", i + 1));
// Rho
int[] rho = prob.varArray('C', nSide - 1, 0.01, 1.0,
i -> String.format("RHO%d", i + 1));

// Add the user functions
MapDeltaFunction sin = prob.nlpAddUserFunction("mySin", 0, mySin);
MapDeltaFunction cos = prob.nlpAddUserFunction("myCos", 0, PolygonMapDelta::myCos);

// Objective function. We build the objective function as
// a formula in infix notation. See below for submitting a
// formula as string.
// Tokens are always integers, while values may be integers
// (for example operator or delimiter constants) or double
// values (actual numbers). That is why the val list has
// elements of type Number.
ArrayList<Integer> tok = new ArrayList<Integer>();
ArrayList<Number> val = new ArrayList<Number>();
for (int i = 1; i < nSide - 1; ++i) {
if ( tok.size() > 0 ) {
tok.add(XPRSconstants.TOK_OP);
val.add(XPRSconstants.OP_PLUS);
}
tok.add(XPRSconstants.TOK_COL);             // RHO(i)
val.add(rho[i]);
tok.add(XPRSconstants.TOK_OP);              // *
val.add(XPRSconstants.OP_MULTIPLY);
tok.add(XPRSconstants.TOK_COL);             // RHO(i-1)
val.add(rho[i-1]);
tok.add(XPRSconstants.TOK_OP);              // *
val.add(XPRSconstants.OP_MULTIPLY);
tok.add(XPRSconstants.TOK_FUN);             // mySin
val.add(sin.getId());
tok.add(XPRSconstants.TOK_LB);              // (
val.add(XPRSconstants.TOK_LB);
tok.add(XPRSconstants.TOK_COL);             // THETA(i)
val.add(theta[i]);
tok.add(XPRSconstants.TOK_OP);              // -
val.add(XPRSconstants.OP_MINUS);
tok.add(XPRSconstants.TOK_COL);             // THETA(i-1)
val.add(theta[i-1]);
tok.add(XPRSconstants.TOK_RB);              // )
val.add(XPRSconstants.TOK_RB);
tok.add(XPRSconstants.TOK_OP);              // *
val.add(XPRSconstants.OP_MULTIPLY);
tok.add(XPRSconstants.TOK_CON);             // 0.5
val.add(0.5);
}
tok.add(XPRSconstants.TOK_EOF);
val.add(0.0);
// Since nonlinear objectives cannot be directly expressed in Xpress, we maximize a free
// variable objx and constrain this variable to be equal to the nonlinear objective.
int objx = prob.addCol(1.0, XPRSconstants.MINUSINFINITY, XPRSconstants.PLUSINFINITY);
int objeq = prob.addRow(new int[]{objx}, new double[]{-1.0}, 'E', 0.0);
prob.nlpChgFormula(objeq, 0,
tok.stream().mapToInt(i -> i).toArray(),
val.stream().mapToDouble(d -> d.doubleValue()).toArray());
prob.chgObjSense(ObjSense.MAXIMIZE);

// Vertices in increasing degree order
for (int i = 0; i < nSide - 2; ++i)
prob.addRow(new int[]{ theta[i], theta[i+1] },
new double[]{ -1.0, 1.0 },
'G',
0.001);

// Third side of all triangles <= 1
for (int i = 1; i < nSide - 1; i++) {
for (int j = i + 1; j < nSide; j++) {
int row = prob.addRow(new int[0], new double[0], 'L', 1.0);
prob.nlpChgFormulaStr(row,
String.format("RHO%d ^ 2 + RHO%d ^ 2 - RHO%d * RHO%d * 2 * myCos ( THETA%d - THETA%d )",
i, j, i, j, j, i));
}
}

//prob.writeProb("map.lp", "l");
IntHolder solvestatus = new IntHolder();
IntHolder solstatus = new IntHolder();
//solve problem to local optimality
prob.controls().setNLPSolver(XPRSconstants.NLPSOLVER_LOCAL);
prob.optimize("s", solvestatus, solstatus);
System.out.printf("solvestatus: %s%n", solvestatus);
System.out.printf("solstatus:   %s%n", solstatus);
System.out.printf("Solution value: %f%n", prob.attributes().getObjVal());

double[] x = new double[prob.attributes().getCols()];
prob.getSolution(null, x, 0, x.length - 1);
for (int i = 0; i < rho.length; ++i)
System.out.printf("RHO%d = %f%n", i + 1, x[rho[i]]);
for (int i = 0; i < theta.length; ++i)
System.out.printf("THETA%d = %f%n", i + 1, x[theta[i]]);

}
}
}