Unconstrained minimization

In this section, we present the constructors for an objective function, supply a prototype function evaluator, and provide examples for solving the unconstrained minimization problem.

Defining an unconstrained problem

Let's consider the two-dimensional Rosenbrock problem with analytic derivatives.

minimize $100(x_2 - x_{1}^2)^2 + (1 - x_1)^2$

Representing the Rosenbrock problem as an NLF1 requires an user-supplied function to evaluate the problem and construction of an NLF1.

Step 1: Write a function that evaluates the Rosenbrock problem and gradient.

   void rosen(int mode, int n, const ColumnVector& x, double& fx, 
              ColumnVector& g, int& result)
   { // Rosenbrock's function
      double f1, f2, x1, x2;
    
      if (n != 2) return;

      x1 = x(1);
      x2 = x(2);
      f1 = (x2 - x1 * x1);
      f2 = 1. - x1;

      if (mode & NLPFunction) {
          fx  = 100.* f1*f1 + f2*f2;
      }
      if (mode & NLPGradient) {
         g(1) = -400.*f1*x1 - 2.*f2; 
         g(2) = 200.*f1;
      }
      result = NLPFunction & NLPGradient;
   }

Step 2: Create an NLF1 object.

   NLF1 rosen_problem(n,rosen,init_rosen);

Specifying the optimization method

There are several algorithms in OPT++ to solve unconstrained problems. We provide examples of solving the Rosenbrock problem with a conjugate gradient and Quasi-Newton method.

Next Section: Bound-constrained Minimization | Back to Solvers Page

Last revised July 13, 2006

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, last revised August 30, 2006.