This example is almost identical to LPex1.java using only the function populatebyrow to create the model. Also, this function differs only in the creation of the objective from its LPex1.java counterpart. Here the objective function is created and added to the model like this:

 // Q = 0.5 ( 33*x0*x0 + 22*x1*x1 + 11*x2*x2 - 12*x0*x1 - 23*x1*x2 )
      IloNumExpr x00 = model.prod( 33, x[0], x[0]);
      IloNumExpr x11 = model.prod( 22, x[1], x[1]);
      IloNumExpr x22 = model.prod( 11, x[2], x[2]);
      IloNumExpr x01 = model.prod(-12, x[0], x[1]);
      IloNumExpr x12 = model.prod(-23, x[1], x[2]);
      IloNumExpr Q   = model.prod(0.5, model.sum(x00, x11, x22, x01, x12));

      double[] objvals = {1.0, 2.0, 3.0};
      model.add(model.maximize(model.diff(model.scalProd(x, objvals), Q)));

A quadratic objective may be built with square, prod or sum methods. Note that inclusion of IloPiecewiseLinear will change the model from a QP to a MIQP.

Complete Program: QPex1.java

// --------------------------------------------------------------------------
// File: examples/src/QPex1.java
// Version 9.0    
// --------------------------------------------------------------------------
//  Copyright (C) 2001-2003 by ILOG.
//  All Rights Reserved.
//  Permission is expressly granted to use this example in the
//  course of developing applications that use ILOG products.
// --------------------------------------------------------------------------
//
// QPex1.java - Entering and optimizing a QP problem

import ilog.concert.*;
import ilog.cplex.*;


public class QPex1 {
   public static void main(String[] args) {
      try {
         IloCplex cplex = new IloCplex();
         IloLPMatrix lp = populateByRow(cplex);

//#ifdef FULLTEST
         cplex.setParam(IloCplex.IntParam.Threads,
                        cplex.getMax(IloCplex.IntParam.Threads));
//#endif
       
         if ( cplex.solve() ) {
            double[] x     = cplex.getValues(lp);
            double[] dj    = cplex.getReducedCosts(lp);
            double[] pi    = cplex.getDuals(lp);
            double[] slack = cplex.getSlacks(lp);
          
            System.out.println("Solution status = " + cplex.getStatus());
            System.out.println("Solution value  = " + cplex.getObjValue());
          
            int ncols = lp.getNcols();
            for (int j = 0; j < ncols; ++j) {
               System.out.println("Column: " + j +
                                  " Value = " + x[j] +
                                  " Reduced cost = " + dj[j]);
            }
          
            int nrows = lp.getNrows();
            for (int i = 0; i < nrows; ++i) {
               System.out.println("Row   : " + i +
                                  " Slack = " + slack[i] +
                                  " Pi = " + pi[i]);
            }
          
            cplex.exportModel("qpex1.lp");
         }
         cplex.end();
      }
      catch (IloException e) {
         System.err.println("Concert exception '" + e + "' caught");
      }
   }

   static IloLPMatrix populateByRow(IloMPModeler model) throws IloException {
      IloLPMatrix lp = model.addLPMatrix();
    
      double[]    lb = {0.0, 0.0, 0.0};
      double[]    ub = {40.0, Double.MAX_VALUE, Double.MAX_VALUE};
      IloNumVar[] x  = model.numVarArray(model.columnArray(lp, 3), lb, ub);
    
      double[]   lhs = {-Double.MAX_VALUE, -Double.MAX_VALUE};
      double[]   rhs = {20.0, 30.0};
      double[][] val = { {-1.0,  1.0,  1.0},
                         { 1.0, -3.0,  1.0} };
      int[][]    ind = { {0, 1, 2},
                         {0, 1, 2} };
      lp.addRows(lhs, rhs, ind, val);
    
      // Q = 0.5 ( 33*x0*x0 + 22*x1*x1 + 11*x2*x2 - 12*x0*x1 - 23*x1*x2 )
      IloNumExpr x00 = model.prod( 33, x[0], x[0]);
      IloNumExpr x11 = model.prod( 22, x[1], x[1]);
      IloNumExpr x22 = model.prod( 11, x[2], x[2]);
      IloNumExpr x01 = model.prod(-12, x[0], x[1]);
      IloNumExpr x12 = model.prod(-23, x[1], x[2]);
      IloNumExpr Q   = model.prod(0.5, model.sum(x00, x11, x22, x01, x12));
    
      double[] objvals = {1.0, 2.0, 3.0};
      model.add(model.maximize(model.diff(model.scalProd(x, objvals), Q)));
    
      return (lp);
   }
}