A New Scalarization and Numerical Method for Constructing Weak Pareto Front of Multi-objective Optimization Problems ∗ Joydeep Dutta † C. Yal¸ cın Kaya ‡ March 21, 2011 Abstract A numerical technique is presented for constructing an approximation of the weak Pareto front of nonconvex multi-objective optimization problems, based on a new Tchebychev-type scalarization and its equivalent representations. First, existing results on the standard Tcheby- chev scalarization, the weak Pareto and Pareto minima, as well as the uniqueness of the op- timal value in the Pareto front, are recalled and discussed for the case when the set of weak Pareto minima is the same as the set of Pareto minima, namely, when weak Pareto minima are also Pareto minima. Of the two algorithms we present, Algorithm 1 is based on this discussion. Algorithm 2, on the other hand, is based on the new scalarizations incorporating rays associated with the weights of the scalarization in the value (or objective) space, as con- straints. We prove two relevant results for the new scalarization. The new scalarizations and the resulting Algorithm 2 are particularly effective in constructing an approximation of the weak Pareto sections of the front. We illustrate the working and capability of both algorithms by means of smooth and nonsmooth test problems with connected and disconnected Pareto fronts. Key words : Multi-objective optimization, Pareto front, efficient set, Tchebychev scalarization, numerical methods, nonconvex optimization, nonsmooth optimiza- tion. 1 Introduction and Scalarization Techniques We consider the following multi-objective optimization problem. (P) min x∈X (f 1 (x),...,f p (x)) , where X ⊂ IR n , and the objective functions f i : IR n → IR, i =1,...,p, are continuous. Note that f i can in general be nonsmooth and nonconvex. There are two main solution concepts associated with Problem (P), namely the Pareto minimum and the weak Pareto minimum. A ∗ The authors are grateful to Regina Burachik for lengthy and insightful discussions. They are also indebted for her careful reading of the first draft of the manuscript. The authors thank two anonymous reviewers, whose comments improved the contents and exposition of the paper. † Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur, PIN 208016, India. E-mail: jdutta@iitk.ac.in . ‡ School of Mathematics and Statistics, University of South Australia, Mawson Lakes, S.A. 5095, Australia. E-mail: yalcin.kaya@unisa.edu.au .