American Institute of Mathematical Sciences

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

A smoothing scheme for optimization problems with Max-Min constraints

X. X. Huang 1, , Xiaoqi Yang 2, and  K. L. Teo 3,

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.
Keywords: convergence, smooth approximation, Min-max-min problem, optimality condition..
Mathematics Subject Classification: Primary: 90C30; Secondary: 90C46, 90C4.
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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