A Survey on Proximal Point Type Algorithms for Solving Vector Optimization Problems

Grad, Sorin‐Mihai

doi:10.1007/978-3-030-25939-6_11

Cited by 5 publications

(4 citation statements)

References 64 publications

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“…For anyx ∈ S, we have ψ F lev k−1 (x) ≤ 0, in view of the definition of S in (A1), (8), and (11). Hence, by combining (28) with y =x and the definition of (11), we obtain, for every w ∈ C and k ≥ 0,…”

Section: Lemma 43 Letmentioning

confidence: 92%

“…Other variants of the proximal gradient method were analyzed in the multiobjective and vector optimization setting in [8,17,44]. See, for instance, [28] for a review about vector proximal point method and some variants. Most recently, [34] proposed and analyzed a conjugate gradient method for solving vector optimization problems using different line-search strategies.…”

Section: Introductionmentioning

confidence: 99%

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A Strongly Convergent Proximal Point Method for Vector Optimization

Iusem

Melo

Serra

2021

J Optim Theory Appl

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In this paper, we propose and analyze a variant of the proximal point method for obtaining weakly efficient solutions of convex vector optimization problems in real Hilbert spaces, with respect to a partial order induced by a closed, convex and pointed cone with nonempty interior. The proposed method is a hybrid scheme that combines proximal point type iterations and projections onto some special halfspaces in order to achieve the strong convergence to a weakly efficient solution. To the best of our knowledge, this is the first time that proximal point type method with strong convergence has been considered in the literature for solving vector/multiobjective optimization problems in infinite dimensional Hilbert spaces.

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Section: Lemma 43 Letmentioning

confidence: 92%

Section: Introductionmentioning

confidence: 99%

A Strongly Convergent Proximal Point Method for Vector Optimization

Iusem

Melo

Serra

2021

J Optim Theory Appl

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“…Since no information is carried over between iterations of the algorithm, the trajectory (i.e., the sequence) generated by the system (17) is the same as the one generated by Algorithm 3. In particular, this means that the Pareto set of the MOP is contained in the set of fixed points of the system (17). Thus, the subdivision algorithm (which was originally designed to compute attractors of dynamical systems) can be used to compute (a superset of) the Pareto set.…”

Section: Globalization Using a Subdivision Algorithmmentioning

confidence: 99%

“…In [14,15], the subgradient method was generalized to the multiobjective case, but both articles report that their method is not suitable for real-life application due to inefficiency. In [16] (see also [17]), the proximal point method for single-objective optimization was generalized to convex vector optimization problems, where differentiability of the objectives is not required. In [18] (see also [19,20]), a multiobjective version of the proximal bundle method was proposed, which currently appears to be the most efficient solver.…”

Section: Introductionmentioning

confidence: 99%

An Efficient Descent Method for Locally Lipschitz Multiobjective Optimization Problems

Gebken

Peitz

2021

J Optim Theory Appl

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We present an efficient descent method for unconstrained, locally Lipschitz multiobjective optimization problems. The method is realized by combining a theoretical result regarding the computation of descent directions for nonsmooth multiobjective optimization problems with a practical method to approximate the subdifferentials of the objective functions. We show convergence to points which satisfy a necessary condition for Pareto optimality. Using a set of test problems, we compare our method with the multiobjective proximal bundle method by Mäkelä. The results indicate that our method is competitive while being easier to implement. Although the number of objective function evaluations is larger, the overall number of subgradient evaluations is smaller. Our method can be combined with a subdivision algorithm to compute entire Pareto sets of nonsmooth problems. Finally, we demonstrate how our method can be used for solving sparse optimization problems, which are present in many real-life applications.

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