JACOBIAN Statement

The NLP Procedure

JACOBIAN Statement

JACOBIAN variables ;

The JACOBIAN statement defines the JACOBIAN matrix J for a system of objective functions. For more information, see the section "Derivatives".

The JACOBIAN statement lists m*n variable names that correspond to the elements J_i,j, i = 1, ... ,m, j = 1, ... ,n, of the Jacobian matrix listed by rows.

For example, the statements

  lsq f1-f3;
  decvar x1 x2;
  jacobian j1-j6;

correspond to the Jacobian matrix

$J = [ J1 & J2 \ J3 & J4 \ J5 & J6 \ ] = [ \partial f_1/ \partial x_1 & \partial... ... \partial x_2 \ \partial f_3/ \partial x_1 & \partial f_3/ \partial x_2 \ ] .$

The m rows of the Jacobian matrix must correspond to the order of the m function names listed in the MIN, MAX, or LSQ statement. The n columns of the Jacobian matrix must correspond to the order of the n decision variables listed in the DECVAR statement. To specify the values of nonzero derivatives, the variables specified in the JACOBIAN statement have to be defined in the left-hand side of algebraic expressions in programming statements.

For example, consider the Rosenbrock Function:

       proc nlp tech=levmar;
         array j[2,2] j1-j4;
         lsq f1 f2;
         decvar x1 x2;
         jacobian j1-j4;

         f1 = 10 * (x2 - x1 * x1);
         f2 = 1 - x1;

         j[1,1] = -20 * x1;
         j[1,2] = 10;
         j[2,1] = -1;
         j[2,2] = 0;   /* is not needed */
       run;

The JACOBIAN statement is useful only if more than one objective function is given in the MIN, MAX, or LSQ statement, or if a DATA= input data set specifies more than one function. If the MIN, MAX, or LSQ statement contains only one objective function and no DATA= input data set is used, the JACOBIAN and GRADIENT statements are equivalent. In the case of least-squares minimization, the crossproduct Jacobian is used as an approximative Hessian matrix.

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