An interior point algorithm for continuous minimax: implementation and computation

In this paper we propose an algorithm for the constrained continuous minimax problem. The algorithm, motivation and numerical experience are reported in this paper. Theoretical properties, and the convergence of the proposed method are discussed in a separate paper[31]. The algorithm uses quasi–Newton search direction, based on sub–gradient information, conditional on maximizers. The initial problem is transformed to an equivalent equality constrained problem, where the logarithmic barrier function is used to ensure feasibility. In the case of multiple maximizers, the algorithm adopts semi–infinite programming iterations towards epi–convergence. Satisfaction of the equality constraints is ensured by an adaptive quadratic penalty function. The algorithm is augmented by a discrete minimax procedure to compute the semi-infinite programming steps and ensure overall progress when required by the adaptive penalty procedure. Progress towards the solution is maintained using merit functions. Computational results are included to illustrate the efficient performance of the algorithm. 1 Formulation of the Problem Consider the following problem: min x max y∈Y {f(x, y) | g(x) = 0, x ≥ 0} , (1) where Y is a compact subset of R, x ∈ R, f(x, y) is continuous in x and y, twice continuously differentiable in x, and g : R → R is continuous and twice differentiable in x. We also denote the feasible region with Xf = {x ∈ R| g(x) = 0, x ≥ 0}. When the maximizer y is defined on a discrete set, (1) becomes a discrete minimax problem and algorithms for solving such problems have been considered by a number of authors, including Womersley and Fletcher [34], Polak, Mayne and Higgins [22] and Rustem and Nguyen [29], Obasanjo and Rustem [19]. Financial support of the EPSRC grants GR/R51377/01 and EP/C513584/1 are gratefully acknowledged. The authors are also grateful to Professor Elijah Polak for valuable suggestions and advice on this paper. Any remaining errors are the responsibility of the authors. Corresponding author. E-mail: [email protected], tel: +44-20-7594-8345. Department of Computing, Imperial College, London SW7 2AZ