*3. Solve by the Jacobian method:

We can draw some general conclusions from the application of the Jacobian method to the linear programming problem. From the numerical example, the necessary conditions require the independent variables to equal zero. Also, the sufficiency conditions indicate that the Hessian is a diagonal matrix. Thus, all its diagonal elements must be positive for a minimum and negative for a maximum. The observations demonstrate that the necessary condition is equivalent to specifying that only basic (feasible) solutions are needed to locate the optimum solution. In this case the independent variables are equivalent to the nonbasic variables in the linear programming problem. Also, the sufficiency condition demonstrates the strong relationship between the diagonal elements of the Hessian matrix and the optimality indicator Zj - Cj (see Section 7.2) in the simplex method:

682 Chapter 18 Classical Optimization Theory n

Minimize f(X) = 2:x7 i=l subject to n IIxi = C

;=1 where C is a positive constant. Suppose that the right-hand side of the constraint is ~

changed to C + 0, where 0 is a small positive quantity. Find the corresponding change in ~

the optimal value off