Parametric Vector Optimization Problems Research Articles

Mathematical Analysis of P-Stability Maps for Parametric Conic Vector Optimization

Stability analysis for nonlinear programming systems deals with the possible changes of the system parameters and/or equations that maintain the stability of the solutions. It is a crucial requirement to study the nonlinear system and its practical values, specifically the economic impact in most real-world applications. This paper presents some outcomes in connection with stability analysis corresponding to parametric conic vector optimization problems. For these last optimization problems, two novel types of P-Stability maps, which are the P-Stability notion map and the P-Stability perturbation map, are considered based on six kinds of sets: P-feasible set, P-solvability set, the first, second, third, and fourth kinds of P-Stability notion sets with respect to a specific domination cone P. Furthermore, qualitative characteristics of the P-Stability maps under some continuity and convexity assumptions on the objective function are provided and proved. Specifically, the connections between the P-Stability maps and the P-Stability notion set are investigated. Accordingly, these characteristics were extended to the P-perturbation maps. In addition, the idea of Pstability has heavily used in different applications like network privacy, engineering fields, and some business financial models.

On Second-Order Semi-Differentiability of Index γ of Perturbation Maps in Parametric Vector Optimization Problems

In this paper, we prove that the Henig global proper efficient solution map and the Henig global proper efficient perturbation map of a parametric vector optimization problem are second-order semi-differentiable with index [Formula: see text] under some suitable qualification conditions. Several examples are given to illustrate the obtained results.

Conditions for the stability of ideal efficient solutions in parametric vector optimization via set-valued inclusions

In present paper, an analysis of the stability behaviour of ideal efficient solutions to parametric vector optimization problems is conducted. A sufficient condition for the existence of ideal efficient solutions to locally perturbed problems and their nearness to a given reference value is provided by refining recent results on the stability theory of parameterized set-valued inclusions. More precisely, the Lipschitz lower semicontinuity property of the solution mapping is established, with an estimate of the related modulus. A notable consequence of this fact is the calmness behaviour of the ideal value mapping associated to the parametric class of vector optimization problems. Within such an analysis, a refinement of a recent existence result, specific for ideal efficient solutions to unperturbed problem and enhanced by related error bounds, is discussed. Some connections with the concept of robustness in multi-objective optimization are also sketched.

On second-order radial-asymptotic proto-differentiability of the borwein perturbation maps

This paper deals with second-order sensitivity analysis of parameterized vector optimization problems. We prove that the Borwein efficient solution map and the Borwein efficient perturbation map of a parametric vector optimization problem are second-order radial-asymptotic proto-differentiable under some suitable qualification conditions. Some examples are also given for illustrating the obtained results.

Semicontinuity of the minimal solution mappings to parametric set optimization problems on Banach lattices

This paper aims at investigating the stability of minimal solutions to set optimization problems on Banach lattices. We introduce the concepts of strictly quasi cone-convexlikeness and supremum mapping of the set-valued mappings and obtain an existence theorem of minimal solutions to set optimization problems by using Zorn's lemma. We first derive the continuity of supremum mapping of the set-valued mapping. Then, we establish the upper semicontinuity and lower semicontinuity of the minimal solution mapping and weak minimal solution mapping to parametric set optimization problems by utilizing strictly quasi cone-convexlikeness and the continuity of supremum mapping. Finally, our main results are applied to semicontinuity of the solution mappings to parametric vector optimization problems.

On generalized second-order proto-differentiability of the Benson proper perturbation maps in parametric vector optimization problems

On generalized second-order proto-differentiability of the Benson proper perturbation maps in parametric vector optimization problems

Parametric approach for approximate efficiency of robust multiobjective fractional programming problems

In this paper, a methodology is developed to solve an uncertain multiobjective fractional programming problem in the face of data uncertainty in the objective and constraint functions. We use the robust optimization approach (worst‐case approach) for finding ɛ‐efficient solutions of the associated robust multiobjective fractional programming problem defined as a robust (worst‐case) counterpart. For the robust multiobjective fractional programming problem constructed in the robust approach, In fact, in the parametric Dinkelbach approach. We define for the given multiobjective fractional programming problem its corresponding parametric robust vector optimization problem. Then, we establish both necessary and sufficient optimality conditions for a feasible solution to be an ‐efficient solution (an approximate efficient solution) of the parametric robust vector optimization problem. Also we prove the relationship between ɛ‐efficiency of the robust multiobjective fractional programming problem (and thus ɛ‐efficiency of the considered uncertain multiobjective fractional programming problem) and ‐efficiency of its corresponding parametric robust vector optimization problem constructed in the Dinkelbach approach. In order to prove this result, we also use a scalarization method. Thus, the equivalence between an approximate efficient solution of the parametric robust vector optimization problem and an approximate minimizer of its associated scalar optimization problem constructed in the used scalarization method is also established.

Coderivatives and Aubin property of efficient point and efficient solution set-valued maps in parametric vector optimization

The aim of this paper is computing the coderivatives of efficient point and efficient solution set-valued maps in a parametric vector optimization problem. By using a method different from the existing literature we establish an upper estimate and explicit expression for the coderivatives of an efficient point set-valued map where the independent variable can take values in the whole space. As an application, we give some characterizations on the Aubin property of an efficient point map and an explicit expression of the coderivative for an efficient solution map. We provide several examples illustrating the main results.

On Second-Order Composed Proto-Differentiability of Proper Perturbation Maps in Parametric Vector Optimization Problems

The main aim of this paper is to study second-order sensitivity analysis in parametric vector optimization problems. We prove that the proper perturbation maps and the proper efficient solution maps of parametric vector optimization problems are second-order composed proto-differentiable under some appropriate qualification conditions. Some examples are provided to illustrate our results.

On higher-order proto-differentiability and higher-order asymptotic proto-differentiability of weak perturbation maps in parametric vector optimization

The main purpose of this paper is to study higher-order sensitivity analysis in parametric vector optimization problems. Firstly, the higher-order proto-differentiability/the higher-order asymptotic proto-differentiability of the feasible map of a parametric vector optimization problem are considered. Then, we verify that the weak efficient solution map and the weak perturbation map of a parameterized vector optimization problem are higher-order proto-differentiable/higher-order asymptotic proto-differentiable under some suitable qualification conditions.

Sensitivity analysis in parametric vector optimization in Banach spaces viaτw-contingent derivatives

This paper is concerned with sensitivity analysis in parametric vector optimization problems via ${\tau}^w$-contingent derivatives. Firstly, relationships between the ${\tau}^w$-contingent derivative of the Borwein proper perturbation map and the ${\tau}^w$-contingent derivative of feasible map in objective space are considered. Then, the formulas for estimating the ${\tau}^w$-contingent derivative of the Borwein proper perturbation map via the ${\tau}^w$-contingent of the constraint map and the Hadamard derivative of the objective map are obtained.

On generalized $\tau^w$-contingent epiderivatives in parametric vector optimization problems

On generalized $\tau^w$-contingent epiderivatives in parametric vector optimization problems

Semicontinuity of the Minimal Solution Set Mappings for Parametric Set-Valued Vector Optimization Problems

With the help of a level mapping, this paper mainly investigates the semicontinuity of minimal solution set mappings for set-valued vector optimization problems. First, we introduce a kind of level mapping which generalizes one given in Han and Gong (Optimization 65:1337–1347, 2016). Then, we give a sufficient condition for the upper semicontinuity and the lower semicontinuity of the level mapping. Finally, in terms of the semicontinuity of the level mapping, we establish the upper semicontinuity and the lower semicontinuity of the minimal solution set mapping to parametric set-valued vector optimization problems under the C-Hausdorff continuity instead of the continuity in the sense of Berge.

Stability of Implicit Multifunctions via Point-Based Criteria and Applications

We first establish new point-based sufficient conditions for an implicit multifunction to achieve the metric subregularity. These conditions are expressed in terms of limit set critical for metric subregularity of the corresponding parametric multifunction formulated the implicit multifunction. We then show that the sufficient conditions obtained turn out to be also necessary for the metric subregularity of the implicit multifunction in the case, where the corresponding parametric multifunction is (locally) convex and closed. In this way, we give criteria ensuring the calmness for the implicit multifunction. As applications, we derive point-based sufficient and necessary conditions for a multifunction (resp., its inverse multifunction) to have the metric subregularity (resp., the calmness) and for the efficient solution map of a parametric vector optimization problem to admit the metric subregularity as well as the calmness.

On higher-order proto-differentiability of perturbation maps

This paper is concerned with higher-order sensitivity analysis in parametric vector optimization problems. Firstly, higher-order proto-differentiability of a set-valued mapping from one Euclidean space to another is defined. Then, we prove that the perturbation map/the proper perturbation map/the weak perturbation map of a parameterized vector optimization problem are higher-order proto-differentiable under some suitable qualification conditions.

Continuity of solutions mappings of parametric set optimization problems

<p style='text-indent:20px;'>Set optimization is an indispensable part of theory and method of optimization, and has been received wide attentions due to its extensive applications in group decision and group game problems. This paper focus on the continuity of the strict (weak) minimal solution set mapping of parametric set-valued vector optimization problems with the lower set less order relation. We firstly introduce a concept of strict lower level mapping of parametric set-valued vector optimization problems. Moreover, the upper and lower semicontinuity of the strict lower level mapping are obtained under some suitable conditions. Lastly, the sufficient condition for the continuity of the strict minimal solution set mappings of parametric set optimization problems are established by a new proof method, which is different from that in [<xref ref-type="bibr" rid="Xu2014">27</xref>,<xref ref-type="bibr" rid="Xu2016">28</xref>].

Lipschitz Continuity of Approximate Solution Mappings to Parametric Generalized Vector Equilibrium Problems

In this paper, we establish Lipschitz continuity of strongly efficient approximate solution mapping to parametric generalized vector equilibrium problems without using monotonicity and any information of the solution mappings. Moreover, we make a new attempt to establish Lipschitz continuity of weakly efficient approximate solution mapping and efficient approximate solution mapping to parametric generalized vector equilibrium problems by using a scalarization method and a density result, respectively. As an application of the main results, we obtain Lipschitz continuity of strongly efficient approximate solution mapping, weakly efficient approximate solution mapping and efficient approximate solution mapping to parametric vector optimization problems.

Stability of solution mappings for parametric bilevel vector equilibrium problems

In this paper, we first revisit the parametric bilevel vector equilibrium problems in Hausdorff topological vector spaces. Then we study the stability conditions such as (Hausdorff) upper semicontinuity, (Hausdorff) lower semicontinuity, outer-continuity and outer-openness of solutions for such problems. Many examples are provided to illustrate the essentialness of the imposed assumptions. For the applications, we obtain the stability results for the parametric vector variational inequality problems with equilibrium constraints and parametric vector optimization problems with equilibrium constraints.

Upper Hölder continuity of parametric vector optimization problems

This paper is concerned with upper Hölder continuity and Hölder calmness of a perturbed vector optimization problem. We establish some new sufficient conditions for upper Hölder continuity and Hölder calmness of the perturbed solution mappings and the perturbed optimal value mappings of a vector optimization problem under the case that the objective function and the feasible set are, respectively, perturbed by parameters. Our results generalize and extend the corresponding ones of Li and Li (Appl. Math. Comput. 232:908-918, 2014).

On the solution continuity of parametric set optimization problems

The aim of this paper is to investigate the continuity of the solution set maps of set-valued vector optimization problems with set optimization criterion. First, we introduce a new concept, which is called a u-lower level map. Then, we give some sufficient conditions for the upper and lower semicontinuities of the generalized lower level map. Finally, by virtue of the semicontinuity of the u-lower level map, we obtain the continuity of the minimal solution set map to parametric set-valued vector optimization problems with set optimization criterion.