Piecewise Linear Multicriteria Programs: The Continuous Case and Its Discontinuous Generalization
Ya Ping Fang, Kaiwen Meng, Xiao Qi Yang
Operations Research · 2012 · Volume 60 · Issue 2
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.
Keywords
multicriteria program, piecewise linear function, the structure of (weak) Pareto solution set, bi-criteria program, algorithm
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