Numerical experiments with continuous minimax
In this chapter, we consider the implementation of continuous minimax algorithms. The first is the gradient-based algorithm due to Kiwiel (1987), discussed in Section 2. 4, and the second is the quasi-Newton algorithm in Chapter 4, and a simplified variant derived from this algorithm.
An important complication of continuous minimax is the presence of multiple maximizers. The contribution of these maximizers to the definition of the direction of search of an algorithm is often difficult to evaluate. We consider several strategies suggested by these algorithms for this purpose. The utility of adopting a simplified direction is evaluated. In addition, the general performance of the algorithms are considered in terms of termination criteria, accuracy of the solution and convergence rates.
We observe a superior performance of the full quasi-Newton algorithm and promising results obtained by adopting the variant with a simplified search direction, not requiring convex combination of subgradients.
As in Chapters 3 and 4, we consider the continuous minimax problem, constrained in y, but unconstrained in xand express it as where Φ(x) is the max-function The vector x is defined on the n-dimensional Euclidian space, Y is a convex compact set, f(x, y) is continuous in x, y and twice continuously differentiable in x. In this chapter, we discuss numerical experience using two versions of a quasi-Newton algorithm and Kiwiel's (1987) algorithm for continuous mini