# American Institute of Mathematical Sciences

April  2007, 3(2): 209-222. doi: 10.3934/jimo.2007.3.209

## 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  August 2006 Revised  January 2007 Published  April 2007

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.
Citation: X. X. Huang, Xiaoqi Yang, K. L. Teo. A smoothing scheme for optimization problems with Max-Min constraints. Journal of Industrial and Management Optimization, 2007, 3 (2) : 209-222. doi: 10.3934/jimo.2007.3.209
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