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Journal of Industrial and Management Optimization (JIMO)
 

Doubly nonnegative relaxation method for solving multiple objective quadratic programming problems

Pages: 543 - 556, Volume 10, Issue 2, April 2014      doi:10.3934/jimo.2014.10.543

 
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Yanqin Bai - Department of Mathematics, Shanghai University, Shanghai 200444, China (email)
Chuanhao Guo - Department of Mathematics, Shanghai University, Shanghai 200444, China (email)

Abstract: Multicriterion optimization and Pareto optimality are fundamental tools in economics. In this paper we propose a new relaxation method for solving multiple objective quadratic programming problems. Exploiting the technique of the linear weighted sum method, we reformulate the original multiple objective quadratic programming problems into a single objective one. Since such single objective quadratic programming problem is still nonconvex and NP-hard in general. By using the techniques of lifting and doubly nonnegative relaxation, respectively, this single objective quadratic programming problem is transformed to a computable convex doubly nonnegative programming problem. The optimal solutions of this computable convex problem are (weakly) Pareto optimal solutions of the original problem under some mild conditions. Moreover, the proposed method is tested with two examples and a practical portfolio selection example. The test examples are solved by CVX package which is a solver for convex optimization. The numerical results show that the proposed method is effective and promising.

Keywords:  Multiple objective programming, quadratic programming, linear weighted sum method, copositive programming, completely positive programming.
Mathematics Subject Classification:  Primary: 90C29, 90C26; Secondary: 49M20.

Received: October 2012;      Revised: August 2013;      Available Online: October 2013.

 References