PROC NLP: HESSIAN Statement

HESSIAN Statement

HESSIAN variables ;

The HESSIAN statement defines the Hessian matrix $\text{[math]}$ containing the second-order derivatives of the objective function $\text{[math]}$ with respect to $\text{[math]}$ . For more information, see the section Derivatives.

If the DIAHES option is not specified, the HESSIAN statement lists $\text{[math]}$ variable names which correspond to the elements $\text{[math]}$ of the lower triangle of the symmetric Hessian matrix listed by rows. For example, the statements

      min f;
      decvar x1 - x3;
      hessian g1-g6;

correspond to the Hessian matrix

$\text{[math]}$

If the DIAHES option is specified, only the $\text{[math]}$ diagonal elements must be listed in the HESSIAN statement. The $\text{[math]}$ rows and $\text{[math]}$ columns of the Hessian matrix $\text{[math]}$ must correspond to the order of the $\text{[math]}$ parameter names listed in the DECVAR statement. To specify the values of nonzero derivatives, the variables specified in the HESSIAN statement must be defined on the left-hand side of algebraic expressions in the programming statements. For example, consider the Rosenbrock function:

proc nlp tech=nrridg;
   min f;
   decvar x1 x2;
   gradient g1 g2;
   hessian h1-h3;
   
   f1 = 10 * (x2 - x1 * x1);
   f2 = 1 - x1;
   
   f = .5 * (f1 * f1 + f2 * f2);
   
   g1 = -200 * x1 * (x2 - x1 * x1) - (1 - x1);
   g2 = 100 * (x2 - x1 * x1);
   
   h1 = -200 * (x2 - 3 * x1 * x1) + 1;
   h2 = -200 * x1;
   h3 = 100;
run;