First-Order Algorithms for Generalized Semi-Infinite Min-Max Problems

  • Authors:

  • Elijah Polak;Liqun Qi;Defeng Sun

  • Affiliations:

  • Department of Electrical Engineering and Computer Sciences, University of California at Berkeley, Berkeley, CA 94720, USA. polak@optimum.eecs.berkeley.edu;School of Mathematics, the University of New South Wales, Sydney 2052, Australia. l.qi@unsw.edu.au;School of Mathematics, the University of New South Wales, Sydney 2052, Australia. sun@maths.unsw.edu.au

  • Venue:
  • Computational Optimization and Applications - Special issue on computational optimization—a tribute to Olvi Mangasarian, part II
  • Year:
  • 1999

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Abstract

We present a first-order algorithm for solving semi-infinitegeneralized min-max problems which consist of minimizing a function f^0(x) = F(ψ^1(x), …, ψ^m(x)), where F is a smooth functionand each ψ^i is the maximum of an infinite number of smooth functions.In Section 3.3 of [17] Polak finds a methodology for solving infinitedimensional problems by expanding them into an infinite sequence ofconsistent finite dimensional approximating problems, and then using amaster algorithm that selects an appropriate subsequence of theseproblems and applies a number of iterations of a finite dimensionaloptimization algorithm to each of these problems, sequentially. Ouralgorithm was constructed within this framework; it calls an algorithmby Kiwiel as a subroutine. The number of iterations of the Kiwielalgorithm to be applied to the approximating problems is determined bya test that ensures that the overall scheme retains the rate ofconvergence of the Kiwiel algorithm.Under reasonable assumptions we show that all the accumulation pointsof sequences constructed by our algorithm are stationary, and, under anadditional strong convexity assumption, that the Kiwiel algorithmconverges at least linearly, and that our algorithm also converges atleast linearly, with the same rate constant bounds as Kiwiel‘s.