A smoothing scheme for optimization problems with Max-Min constraints

X. X. Huang; Xiaoqi Yang; K. L. Teo

doi:10.3934/jimo.2007.3.209

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2007, Volume 3, Issue 2: 209-222. Doi: 10.3934/jimo.2007.3.209

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A smoothing scheme for optimization problems with Max-Min constraints

1.
School of Management, Fudan University, Shanghai 200433
2.
Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong
3.
Department of Mathematics and Statistics, Curtin University of Technology, Perth

Received: July 31, 2006

Revised: December 31, 2006

Published: March 2007

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Abstract

In this paper, we apply a smoothing approach to a minimization problem with a max-min constraint (i.e., a min-max-min problem). More specifically, we first rewrite the min-max-min problem as an optimization problem with several min-constraints and then approximate each min-constraint function by a smooth function. As a result, the original min-max-min optimization problem can be solved by solving a sequence of smooth optimization problems. We investigate the relationship between the global optimal value and optimal solutions of the original min-max-min optimization problem and that of the approximate smooth problem. Under some conditions, we show that the limit points of the first-order (second-order) stationary points of the smooth optimization problems are first-order (second-order) stationary points of the original min-max-min optimization problem.

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Mathematics Subject Classification: Primary: 90C30; Secondary: 90C46, 90C47.

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