We present illustrations of the performance of minimax from an algorithmic point of view. The quasi-Newton algorithm and Kiwiel's algorithm have been implemented as discussed in Chapter 5. Because both algorithms require a maximization problem to be solved in Step 1, they are both relatively computationally more expensive compared to most nonlinear programming algorithms. For the maximization subproblem, we used the comprehensive NAG54 optimization routine E04VDF which can handle both linear and nonlinear constraints of a nonlinear programming problem. All the illustrations were solved by using both algorithms. Kiwiel's algorithm has been implemented as a check to the solutions found by the quasiNewton algorithm. In the implementation of Kiwiel's algorithm, we set the linear approximation parameter m to a constant for all examples: m = 20e2°−3.
The stopping criterion for both algorithms is the condition that the approximate directional derivative is sufficiently close to zero, that is,
The values of the stopping parameter ε used in the examples are within the range [1.0e−6, l.Oe−14]. The values reported here are for the quasi-Newton and Kiwiel's algorithms, as in the chapter containing numerical results.____________________