Journal of Industrial and Management Optimization (JIMO)

A smoothing scheme for optimization problems with Max-Min constraints

Pages: 209 - 222, Volume 3, Issue 2, May 2007      doi:10.3934/jimo.2007.3.209

       Abstract        Full Text (179.1K)       Related Articles       

X. X. Huang - School of Management, Fudan University, Shanghai 200433, China (email)
Xiaoqi Yang - Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong, China (email)
K. L. Teo - Department of Mathematics and Statistics, Curtin University of Technology, Perth, Australia (email)

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.

Keywords:  Min-max-min problem, smooth approximation, convergence, optimality condition.
Mathematics Subject Classification:  Primary: 90C30; Secondary: 90C46, 90C47.

Received: August 2006;      Revised: January 2007;      Available Online: April 2007.