2018
DOI: 10.1137/16m107534x
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The Proximal Point Method for Locally Lipschitz Functions in Multiobjective Optimization with Application to the Compromise Problem

Abstract: This paper studies the constrained multiobjective optimization problem of finding Pareto critical points of vector-valued functions. The proximal point method considered by Bonnel, Iusem, and Svaiter [SIAM J. Optim., 15 (2005), pp. 953-970] is extended to locally Lipschitz functions in the finite dimensional multiobjective setting. To this end, a new (scalarization-free) approach for convergence analysis of the method is proposed where the first-order optimality condition of the scalarized problem is replaced … Show more

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Cited by 26 publications
(10 citation statements)
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“…Our non-convex descent property also implies the one obtained by Ji et al [34]. It is worth to mention that even if DC function is a locally Lipschitz function our method is different to the one considered by Bento et al [4] because our algorithm minimizes at each iteration a linear approximation of the objective function instead of minimizing the objetive function as [4] does. As we will see in Section 6 our approach have important applications in Behavioral Sciences.…”
Section: Introductionmentioning
confidence: 54%
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“…Our non-convex descent property also implies the one obtained by Ji et al [34]. It is worth to mention that even if DC function is a locally Lipschitz function our method is different to the one considered by Bento et al [4] because our algorithm minimizes at each iteration a linear approximation of the objective function instead of minimizing the objetive function as [4] does. As we will see in Section 6 our approach have important applications in Behavioral Sciences.…”
Section: Introductionmentioning
confidence: 54%
“…Remark 8 It is known that every DC function is a locally Lipschitz function; see [30]. The proximal point method for locally Lipschitz functions in multiobjective optimization was analyzed by Bento et al [4] who firstly considered a possibly non-convex vector improving constraints Ω k . However, we mention that Algorithm 1 (applied for DC functions) offers an additional flexibility for considering, at each step, a linear approximation G(x) − JH(x k )(x − x k ) instead of minimize the non-convex function…”
Section: Remarkmentioning
confidence: 99%
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“…. , f m , that have to be optimized at the same time on M. In recent years, there has been a significant increase in the number of papers addressing this class of problems; for example, see [1][2][3][4][5][6][7]. Here, among the methods designed for solving multiobjective optimization problems, we are interested in the steepest descent method.…”
Section: Introductionmentioning
confidence: 99%