Efficiency in quasiconvex multiobjective nondifferentiable optimization on the real line

Abstract

ABSTRACT A realistic solution concept associated with a multiobjective optimization problem is that named Pareto (or efficient) solution, which is more difficult to be treated from a mathematical point of view than the notion of weak Pareto (or weakly efficient) solution. This work provides a complete description of the efficient solution set, when the objective functions are defined on the real line. This is motivated, besides theoretical aspects, also by a numerical point of view, since most algorithms in scalar minimization involve the solvability of a one-dimensional optimization problem to find the next iterate. It is expected that the same situation occurs in the multiobjective optimization problem. We first consider the case when all the objective functions are semistrictly quasiconvex, and afterwards we consider the same problem under quasiconvexity along with some additional assumptions. The latter allows us to deal with the general bicriteria optimization problem under quasiconvexity. Several examples showing the applicability of our results are presented, and an algorithm is proposed to compute the whole efficient solution set.