Approximate proper efficiency in vector optimization

Ke-quan, Zhao; Chen, Guang-Ya; Yang, Xinmin

doi:10.1080/02331934.2014.979818

Cited by 12 publications

(8 citation statements)

References 37 publications

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“…where ε > 0 and E is an improvement set with respect to C; (vi) intE, where E is a solid (i.e., intE = ∅) improvement set with respect to C. For more details and examples of improvement sets, one can refer to [23,14,39,40,41,33,28]. 3.…”

Section: Definition 22 [23 14]mentioning

confidence: 99%

“…Gutiérrez et al [23] generalized the concepts of improvement set and E-optimality to a general topological vector space. Subsequent works about this aspect one can also refer to Zhao and Yang [39,40], Zhao et al [41], Oppezzi and Rossi [33], Lalitha and Chatterjee [28]. As mentioned in [23], E-optimality via improvement sets has been shown to be very suitable to deal in a unified way with well-known exact and approximate nondominated concepts.…”

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confidence: 99%

“…Conclusions. In this paper, motivated by the recent development of improvement sets in vector optimization (see [23,14,39,40,41,33,28]), we have introduced a new vector quasiequilibrium problem (UVQEP) via improvement set D and its weak version (UWVQEP), also their Minty-type dual problems and the corresponding set-valued cases. By taking different improvement sets, these models reduce to well-known exact and approximate vector quasiequilibrium problems with vectorvalued or set-valued mappings in the literature.…”

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confidence: 99%

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Unified vector quasiequilibrium problems via improvement sets and nonlinear scalarization with stability analysis

Wei¹,

Zuo²,

Chen³

2020

Numerical Algebra, Control &Amp; Optimization

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This paper has two objectives. The first one is to propose a new vector quasiequilibrium problem where the ordering relation is defined via an improvement set D, and its weak version, also their Minty-type dual problems and the corresponding set-valued cases. These models provide unified frameworks to deal with well-known exact and approximate vector quasiequilibrium problems with vector-valued or set-valued mappings. The second one is to study solution stability in the sense of Hölder continuity of the unique solution to parametric unified (resp. weak) vector quasiequilibrium problems, by employing the Gerstewitz scalarization techniques. In particular, we deduce a new stability result for the typical vector optimization problem related with (resp. weak) D-optimality, by considering perturbations of both the objective function and the feasible set.

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Section: Definition 22 [23 14]mentioning

confidence: 99%

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confidence: 99%

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confidence: 99%

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Unified vector quasiequilibrium problems via improvement sets and nonlinear scalarization with stability analysis

Wei¹,

Zuo²,

Chen³

2020

Numerical Algebra, Control &Amp; Optimization

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“…We denoted this by Ge( f, S, R p + , ). Definition 3.6 [20]. Let Z be a normed linear space and ∈ K. A feasible pointx ∈ S is called an −Super proper efficient solution of (MOP) if there exist λ ∈ (0, 1) and M > 0 such that…”

Section: Cone Characterizations Of Approximate Solutionsmentioning

confidence: 99%

Approximate proper efficiency for multiobjective optimization problems

Gao

2019

Filomat

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This paper is devoted to the study of a new kind of approximate proper efficiency in terms of proximal normal cone and co-radiant set for multiobjective optimization problem. We derive some properties of the new approximate proper efficiency and discuss the relations with the existing approximate concepts, such as approximate efficiency and approximate Benson proper efficiency. At last, we study the linear scalarizations for the new approximate proper efficiency under the generalized convexity assumption and give some examples to illustrate the main results.

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“…Scalarization for cone-ordered optimization, including the Pareto case, has been studied extensively. For example, see [23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39], which present scalarizations of varying degrees of abstraction. Few involve polyhedral cones.…”

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Scalarizations for Maximization with Respect to Polyhedral Cones

Corley¹

2017

JAAM

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Abstract. Efficient points are obtained for cone-ordered maximizations in n R using the method of scalarization. Various scalarizations are presented for ordering cones in general and then for the important special case of polyhedral cones. For polyhedral cones, it is shown how to find vectors in the positive dual cone that are needed for a scalarized objective function. Instructive examples are presented. Keywords:Multiobjective optimization, polyhedral cones, cone maximization, Pareto maximization, scalarization. IntroductionMultiobjective optimization is applied in various fields of science, engineering, and economics when decisions involve two or more conflicting objectives. For example, a pharmaceutical company may wish to determine a dose for a new drug that would maximize its therapeutic benefits while minimizing its deleterious side effects. The result would be a multiobjective optimization problem. In many other situations, however, a decision may not be adequately characterized by such standard criteria as Pareto optimality, goal programming, or lexicographic maximization. For this reason, cone-ordered maximization was developed, of which the previously mentioned criteria are special cases. . Such results are difficult to implement, however. For this reason, scalarization is the method of choice for actually obtaining a numerical solution to a cone-ordered maximization. A scalarization is an optimization problem with a single scalar objective function for which an optimal solution solves the cone-ordered problem. Examples are the scalarizations of Borwein [6] and Jahn [10] in terms of the dual cone and that of Soland [11] for Pareto maximization involving both the dual cone and aspiration levels for the individual objective functions. References [12][13][14][15][16][17][18] consolidate the theory and solution techniques for cone-ordered maximizations. However, specific results for polyhedral cones are scarce. For such cones, Yu [5] proposes a specialized scalarization based on finding cone extreme points. Sawaragi et al. [1985] reduce the problem to finding Pareto efficient points of a set in the objective function's image space. More recently, for example, Gutierrez, et al. [19] define a scalarization by perturbing the same matrix.We consider a decision problem with an n-dimensional objective function, where the optimization criterion is maximization with respect to a partial order induced by a pointed convex cone in .n R The purpose here is to present scalarizations for such problems in general, with the emphasis on polyhedral cones. Methods are also developed for finding dual vectors of a polyhedral cone as required for scalarized objective functions. One such approach involves linear programming. The paper is organized as follows. In Section 2 the requisite preliminaries concerning cones are established, while the notion of coneordered maximization is defined in Section 3. In Section 4 various scalarizations are presented. Section 5 then addresses the problem of obtaining the required dual vector...

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Approximate proper efficiency in vector optimization

Cited by 12 publications

References 37 publications

Unified vector quasiequilibrium problems via improvement sets and nonlinear scalarization with stability analysis

Unified vector quasiequilibrium problems via improvement sets and nonlinear scalarization with stability analysis

Approximate proper efficiency for multiobjective optimization problems

Scalarizations for Maximization with Respect to Polyhedral Cones

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