Abstract

In this paper, some theoretical notions of well-posedness and of well-posedness in the generalized sense for scalar optimization problems are presented and some important results are analysed. Similar notions of well-posedness, respectively for a vector optimization problem and for a variational inequality of differential type, are discussed subsequently and, among the various vector well-posedness notions known in the literature, the attention is focused on the concept of pointwise well-posedness. Moreover, after a review of well-posedness properties, the study is further extended to a scalarizing procedure that preserves well-posedness of the notions listed, namely to a result, obtained with a special scalarizing function, which links the notion of pontwise well-posedness to the well-posedness of a suitable scalar variational inequality of differential type.

Highlights

  • The notion of well-posedness is significant for several mathematical problems and it is closely related to the stability of an optimization problem: it plays, a crucial role in the theoretical and in the numerical aspects of optimization theory [1] [2]

  • Similar notions of well-posedness, respectively for a vector optimization problem and for a variational inequality of differential type, are discussed subsequently and, among the various vector well-posedness notions known in the literature, the attention is focused on the concept of pointwise well-posedness

  • After a review of well-posedness properties, the study is further extended to a scalarizing procedure that preserves well-posedness of the notions listed, namely to a result, obtained with a special scalarizing function, which links the notion of pontwise well-posedness to the well-posedness of a suitable scalar variational inequality of differential type

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Summary

Introduction

The notion of well-posedness is significant for several mathematical problems and it is closely related to the stability of an optimization problem: it plays, a crucial role in the theoretical and in the numerical aspects of optimization theory [1] [2]. The notion of well-posedness for a vector optimization problem is, instead, less developed, less advanced; there is no commonly accepted definition of well-posed problem, in vector optimization Some attempts in this direction have been already done [8] [9] [10] [11] and have been made some comparisons with their scalar counterparts. [3] introduced the notion of well-posedness for variational inequality problems based on the fact that an optimization problem can be formulated as a variational inequality problem involving the derivative of the objective function. In all these cases, the idea is an extension of the concept of minimizing sequences seen as approximate solutions

Research Aims
Tykhonov and Hadamard Well-Posedness
Some Generalizations
Well-Posedness of Vector Optimization Problems
Main Results
Concluding Remarks and Future Perspectives for Research
Full Text
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