New computable algorithms for smooth multiobjective optimization problems

Abstract

We propose new practical algorithms for solving smooth multiobjective optimization problems based on determining joint decreasing directions via suitable linear programming problems. The presented iterative method is specialized for unconstrained, sign constrained and linearly constrained multiobjective optimization problems. In all cases discussed in this paper, we show that the objective function values sequence is decreasing with respect to the considered nonnegative orthant while the iterates are feasible. Furthermore, we prove that every accumulation point of the sequence generated by the algorithm, if any, is a substationary point to the considered multiobjective optimization problem, and, under convexity assumptions, it is actually a weakly Pareto efficient (also known as weakly Pareto-optimal) point. Different to similar algorithms from the literature, the ones proposed in this work involve joint decreasing directions that are easily computable in polynomial time by solving linear programming problems. Numerical experiments on unconstrained and linearly constrained convex multiobjective optimization test problems and on portfolio optimization problems show the applicability of the proposed algorithms.