Piecewise Linear Multicriteria Programs: The Continuous Case and Its Discontinuous Generalization

Authors:
Ya Ping Fang;Kaiwen Meng;Xiao Qi Yang
Affiliations:
Department of Mathematics, Sichuan University, Chengdu, Sichuan, China;School of Economics and Management, Southwest Jiaotong University, Chengdu 610031, China;Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hum, Kowloon, Hong Kong
Venue:
Operations Research
Year:
2012

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Abstract

In this paper we study piecewise linear multicriteria programs, that is, multicriteria programs with either a continuous or discontinuous piecewise linear objective function and a polyhedron set constraint. We obtain an algebraic representation of a semi-closed polyhedron and apply it to show that the image of a semi-closed polyhedron under a continuous linear function is always one semi-closed polyhedron. We establish that the (weak) Pareto solution/point set of a piecewise linear multicriteria program is the union of finitely many semi-closed polyhedra. We propose an algorithm for finding the Pareto point set of a continuous piecewise linear bi-criteria program and generalize it to the discontinuous case. We apply our algorithm to solve the discontinuous bi-criteria portfolio selection problem with an l∞ risk measure and transaction costs and show that this algorithm can be improved by using an ideal point strategy.