Piecewise Linear Multicriteria Programs

Xiaoqi Yang

Department of Applied Mathematics, The Hong Kong Polytechnic University

Time: 16:00-17:00, June 21

Venue：A1-512, Main Building

Abstract

In this talk we study multicriteria programs with piecewise linear objective functions and a polyhedron set constraint. Here a piecewise linear function may be continuous or discontinuous. In the discontinuous case, the domain is the union of some closed polyhedra and some semi-closed polyhedra, where the later are defined as the intersection of some closed and/or open half spaces. The application of discontinuous piecewise linear functions can be found in transaction cost and sprse constraint. We obtain an algebraic representation of a 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. This is a generalization of the well-known Arrow, Barankin and Blackwell theorem, which says that the (weak) Pareto solution/point set of a linear multicriteria program are the unions of finitely many polyhedra. We will investigate their weak sharp minimum property. 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.

Brief Biography:

Professor Yang received his BSc degree in Mathematics from the Chongqing Jianzhu University, Chongqing, a MSc degree in Operations Research and Control Theory from the Institute of System Science at Chinese Academy of Science, China and a PhD degree in Applied Mathematics at University of New South Wales, Australia. He then spent five years at the School of Mathematics and Statistics in the University of Western Australia as a Research Associate (1994-1996), an Australian Research Council postdoctoral fellow (1997) and a lecturer (1998). Since 1999 he has been with the Department of Applied Mathematics, Hong Kong Polytechnic University, as an Assistant Professor (1999-2001), Associate Professor (2002-2004), Professor (2005- ). In 2000 he received the ISI Citation Classic for the paper "The vector complementary problem and its equivalences with the weak minimal element in ordered spaces", Journal of Mathematical Analysis and Applications, Vol. 153 (1990) pp. 136-158. In the same year he received the President's Award for Outstanding Performance / Achievement in the category of Research and Scholarly Activities, Hong Kong Polytechnic University. In 2006 he received The First Prize of Natural Science from the Chongqing Municipal Government in 2006.