An interior point algorithm for continuous minimax: implementation and computation

Authors:
B. Rustem;S. Zakovic;P. Parpas
Affiliations:
Department of Computing, Imperial College, London, UK;Department of Computing, Imperial College, London, UK;Department of Computing, Imperial College, London, UK
Venue:
Optimization Methods & Software
Year:
2008

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Abstract

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 [B. Rustem, S. Zakovic, and P. Parpas, Convergence of an interior point algorithm for continuous minimax, J. Optim. Theory Appl. (2007), in press]. 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.