HESSIAN Statement :: SAS/OR(R) 12.1 User's Guide: Mathematical Programming Legacy Procedures

HESSIAN Statement

HESSIAN variables ;

The HESSIAN statement defines the Hessian matrix containing the second-order derivatives of the objective function with respect to $x_1,\ldots ,x_ n$ . For more information, see the section Derivatives.

If the DIAHES option is not specified, the HESSIAN statement lists variable names which correspond to the elements $G_{j,k}, \ j \geq k,$ 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

$G = \left[ \begin{array}{ccc} G1 & G2 & G4 \\ G2 & G3 & G5 \\ G4 & G5 & G6 \\ \end{array} \right] = \left[ \begin{array}{ccc} \partial ^2 f / \partial x^2_1 & \partial ^2 f / \partial x_1 \partial x_2 & \partial ^2 f / \partial x_1 \partial x_3 \\ \partial ^2 f / \partial x_2 \partial x_1 & \partial ^2 f / \partial x^2_2 & \partial ^2 f / \partial x_2 \partial x_3 \\ \partial ^2 f / \partial x_3 \partial x_1 & \partial ^2 f / \partial x_3 \partial x_2 & \partial ^2 f / \partial x^2_3 \end{array} \right]$

If the DIAHES option is specified, only the diagonal elements must be listed in the HESSIAN statement. The rows and columns of the Hessian matrix must correspond to the order of the 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;