A Trust-Region Method for Unconstrained Multiobjective Problems with Applications in Satisficing Processes

Abstract

ADInternational audienceMultiobjective optimization has a significant number of real-life applications. For this reason, in this paper we consider the problem of finding Pareto critical points for unconstrained multiobjective problems and present a trust-region method to solve it. Under certain assumptions, which are derived in a very natural way from assumptions used to establish convergence results of the scalar trust-region method, we prove that our trust-region method generates a sequence which converges in the Pareto critical way. This means that our generalized marginal function, which generalizes the norm of the gradient for the multiobjective case, converges to zero. In the last section of this paper, we give an application to satisficing processes in Behavioral Sciences. Multiobjective trust-region methods appear to be remarkable specimens of much more abstract satisficing processes, based on “variational rationality” concepts. One of their important merits is to allow for efficient computations. This is a striking result in Behavioral Sciences

A Trust-Region Algorithm for Heterogeneous Multiobjective Optimization

“…Another approach on which derivative-free methods are based on is the trust region method [6,7,8,9,10]. There are also multiobjective realizations of this approach [29,36]. Trust region methods are not initially designed for expensive functions but can easily be adapted to them.…”

“…It is an efficient and flexible approach for which many theoretical properties are documented in the literature. A basic generalization of such a method to multiobjective problems based on derivative information is given in [36]. They proof convergence to a Pareto critical point using a characterization of such points that is also used in multiobjective descent theory [14,20].…”

“…This paper will present a new trust region method that can regard heterogeneity. Like [36] we use the idea of generalizing the trust region approach to a multiobjective problem, but our algorithm differs in computing the descent direction and not needing the gradient of the objectives. The search direction is computed in the image space by using a local ideal point.…”

“…The differences in the determination of the search direction affect the convergence analysis such that it is not transferable from other trust region approaches without significant modifications. Still, we can use the same strategy to prove convergence to a Pareto critical point as [36] also using the characterization of such points from [20]. Since we also follow closely the basic scalar idea of trust region methods, the convergence analysis is also related to that in the scalar case [8].…”

A dynamic gradient approach to Pareto optimization with nonsmooth convex objective functions

“…To our knowledge, this work has never been done, and is a subject for further study. More recently, a trust-region method for unconstrained multiobjective problems involving smooth functions has been developed in [41], which uses the norm of the multiobjective steepest descent vector as a generalized marginal function. In [26,28], a Newton method for unconstrained strongly convex vector optimization has been developed, with a local superlinear convergence result.…”