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
Summary
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
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