Well-posedness in vector optimization and scalarization results

In this paper, we give a survey on well-posedness notions of Tykhonov's type for vector optimization problems and the links between them with respect to the classification proposed by Miglierina, Molho and Rocca in [33]. We consider also the notions of extended well-posedness introduced by X.X. Huang ([19],[20]) in the nonparametric case to complete the hierchical structure characterizing these concepts. Finally we propose a review of some theoretical results in vector optimization mainly related to different notions of scalarizing functions, linear and nonlinear, introduced in the last decades, to simplify the study of various well-posedness properties.

Generalized Levitin--Polyak well-posedness in constrained optimization

Author name used in this publication: X. Q. Yang2006-2007 > Academic research: refereed > Publication in refereed journalVersion of RecordPublishe

Levitin-Polyak well-posedness of split multivalued variational inequalities

We introduce and study the split multivalued variational inequality problem (SMVIP) and the parametric SMVIP. We examine, in particular, Levitin-Polyak well-posedness of SMVIPs and parametric SMVIPs in Hilbert spaces. We provide several examples to illustrate our theoretical results. We also discuss several important special cases.Comment: arXiv admin note: text overlap with arXiv:2208.0712

Pointwise well-posedness in vector optimization and variational inequalities

In this note we consider some notions of well-posedness for scalar and vector variational inequalities and we recall their connections with optimization problems. Subsequently, we investigate similar connections between well-posedness of a vector optimization problem and a related variational inequality problem and we present an result obtained with scalar characterizations of vector optimality concepts

α

The concepts of α-well-posedness, α-well-posedness in the generalized sense, L-α-well-posedness and L-α-well-posedness in the generalized sense for mixed quasi variational-like inequality problems are investigated. We present some metric characterizations for these well-posednesses

Directional Tykhonov well-posedness for optimization problems and variational inequalities

By using the so-called minimal time function, we propose and study a novel notion of directional Tykhonov well-posedness for optimization problems, which is an extension of the widely acknowledged notion of Tykhonov. In this way, we first provide some characterizations of this notion in terms of the diameter of level sets and admissible functions. Then, we investigate relationships between the level sets and admissible functions mentioned above. Finally, we apply the technology developed before to study directional Tykhonov well-posedness for variational inequalities. Several examples are presented as well to illustrate the applicability of our results.Comment: 2

Well-posedness for generalized mixed vector variational-like inequality problems in Banach space

In this article, we focus to study about well-posedness of a generalized mixed vector variational-like inequality and optimization problems with aforesaid inequality as constraint. We establish the metric characterization of well-posedness in terms of approximate solution set.Thereafter, we prove the sufficient conditions of generalized well-posedness by assuming the boundedness of approximate solution set. We also prove that the well-posedness of considered optimization problems is closely related to that of generalized mixed vector variational-like inequality problems. Moreover, we present some examples to investigate the results established in this paper