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Maximizing the area of a polygon

Description
Demonstrates how to solve a nonlinear problem in Java

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

Polygon_java.zip[download all files]

Source Files
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Polygon.java[download]





Polygon.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.MapFunction;
import com.dashoptimization.XPRSprob.MapFunctor;

import java.util.ArrayList;


/** Code example how to build nonlinear formulae from strings and token sequences.
<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 Fair Isaac Corporation
</pre>

   Polygon example: maximise the area of an N sided polygon

   *** Demonstrating how to build nonlinear formulae from strings and token sequences ***

<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 Polygon {

    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));

            // 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_IFUN);            // SIN
                val.add(XPRSconstants.IFUN_SIN);
                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 * COS ( THETA%d - THETA%d )",
                        i, j, i, j, j, i));
                }
            }

            IntHolder solvestatus = new IntHolder();
            IntHolder solstatus = new IntHolder();
            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]]);

        }
    }
}

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