Convergence analysis of a nonmonotone projected gradient method for multiobjective optimization problems

Fazzio, Nadia Soledad; Schuverdt, María Laura

doi:10.1007/s11590-018-1353-8

Cited by 25 publications

(14 citation statements)

References 22 publications

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“…At the same time, Qu et al [17] studied the max-type nonmonotone scheme for multiobjective gradient methods, with a different scheme for computing search directions (the subproblem associated to the search directions considers a bound constraint, while here we follow the original works [4,5], which makes the subproblem unconstrained). Parallel to these works, Fazzio and Schuverdt [3] also obtained convergence results, this time for the average-type nonmonotone projected gradient method. However, no numerical experiments were presented.…”

Section: Introductionmentioning

confidence: 90%

“…However, no numerical experiments were presented. Although the papers [3,17] have intersections with ours, the search directions and the scheme used in the convergence proofs are different, as well as the numerical experiments. The hybrid-type nonmonotone line search, which performs better among all the considered line searches, is also brand new as far as we know.…”

Section: Introductionmentioning

confidence: 99%

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Nonmonotone line searches for unconstrained multiobjective optimization problems

2019

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In the last two decades, many descent methods for multiobjective optimization problems were proposed. In particular, the steepest descent and the Newton methods were studied for the unconstrained case. In both methods, the search directions are computed by solving convex subproblems, and the stepsizes are obtained by an Armijo-type line search. As a consequence, the objective function values decrease at each iteration of the algorithms. In this work, we consider nonmonotone line searches, i.e., we allow the increase of objective function values in some iterations. Two well-known types of nonmonotone line searches are considered here: the one that takes the maximum of recent function values, and the one that takes their average. We also propose a new nonmonotone technique specifically for multiobjective problems. Under reasonable assumptions, we prove that every accumulation point of the sequence produced by the nonmonotone version of the steepest descent and Newton methods is Pareto critical. Moreover, we present some numerical experiments, showing that the nonmonotone technique is also efficient in the multiobjective case.

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Section: Introductionmentioning

confidence: 90%

Section: Introductionmentioning

confidence: 99%

Nonmonotone line searches for unconstrained multiobjective optimization problems

2019

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“…Qu et al [26] proposed two nonmonotone gradient methods for convex vector optimization based on the nonmonotone line search given by Grippo et al [18] in the scalar context. In [27], Fazzio and Schuverdt [27] considered a projected gradient method based on the nonmonotone line search technique of [22].…”

Section: Introductionmentioning

confidence: 99%

Convergence of a nonmonotone projected gradient method for nonconvex multiobjective optimization

2021

J. Nonlinear Var. Anal.

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We consider a nonmonotone projected gradient method for nonconvex multiobjective optimization problem. Under some mild assumptions, we prove that each accumulation point of the sequence generated by the algorithm, if exists, is Pareto stationary. Furthermore, when the nonmonotone line search in the algorithm is replaced by the monotone one, i.e., the classical Armijo search rule, the full convergence of the generated sequence to a Pareto stationary point (respectively, a weak Pareto optimal point) is established when the multiobjective function is quasiconvex (respectively, pseudoconvex). Numerical experiments are also presented to illustrate the effectiveness of this method.

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“…In this paper, we center on a numerical algorithm, i.e., the projected gradient method, for solving multiobjective optimization problems. As a classical and efficient method, the projected gradient method and its variant were extensively investigated by researchers for solving scalar optimization problems [8,9,10,11,12,13,14], and they were further extended to the multiobjective optimization setting successfully; see, e.g., [15,16,17,18] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

“…Recently, Fazzio and Schuverdt [16] extended this projected gradient method to include a nonmonotone line search for constrained convex multiobjective optimization problems. They associated a variable steplength instead of the original fixed parameter to compute the search direction.…”

Section: Introductionmentioning

confidence: 99%

Convergence analysis of a projected gradient method for multiobjective optimization problems

2021

J. Nonlinear Var. Anal.

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Convergence analysis of a nonmonotone projected gradient method for multiobjective optimization problems

Cited by 25 publications

References 22 publications

Nonmonotone line searches for unconstrained multiobjective optimization problems

Nonmonotone line searches for unconstrained multiobjective optimization problems

Convergence of a nonmonotone projected gradient method for nonconvex multiobjective optimization

Convergence analysis of a projected gradient method for multiobjective optimization problems

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