Some problems concerning semi-continuous set-valued mappings

1970 ◽  
pp. 45-48 ◽  
Author(s):  
K. Kuratowski
Keyword(s):  
Set Valued Mappings

scholarly journals R-Regularity of Set-Valued Mappings Under the Relaxed Constant Positive Linear Dependence Constraint Qualification with Applications to Parametric and Bilevel Optimization

2021 ◽  
Author(s):  
Patrick Mehlitz ◽  
Leonid I. Minchenko
Keyword(s):  
Linear Dependence ◽  
Constraint Qualification ◽  
Parametric Optimization ◽  
Optimization Problems ◽  
Sufficient Conditions ◽  
Bilevel Optimization ◽  
Regularity Property ◽  
Constant Rank Constraint Qualification ◽  
Existence Criterion ◽  
Set Valued Mappings

AbstractThe presence of Lipschitzian properties for solution mappings associated with nonlinear parametric optimization problems is desirable in the context of, e.g., stability analysis or bilevel optimization. An example of such a Lipschitzian property for set-valued mappings, whose graph is the solution set of a system of nonlinear inequalities and equations, is R-regularity. Based on the so-called relaxed constant positive linear dependence constraint qualification, we provide a criterion ensuring the presence of the R-regularity property. In this regard, our analysis generalizes earlier results of that type which exploited the stronger Mangasarian–Fromovitz or constant rank constraint qualification. Afterwards, we apply our findings in order to derive new sufficient conditions which guarantee the presence of R-regularity for solution mappings in parametric optimization. Finally, our results are used to derive an existence criterion for solutions in pessimistic bilevel optimization and a sufficient condition for the presence of the so-called partial calmness property in optimistic bilevel optimization.

scholarly journals On a Class of Differential Variational Inequalities in Infinite-Dimensional Spaces

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 266 ◽  
Author(s):  
Savin Treanţă
Keyword(s):  
Optimal Control ◽  
Variational Inequality ◽  
Variational Inequalities ◽  
Optimal Control Theory ◽  
Infinite Dimensional ◽  
Nash Games ◽  
New Class ◽  
Set Valued Mappings ◽  
Differential Variational Inequalities ◽  
Theoretical Developments

A new class of differential variational inequalities (DVIs), governed by a variational inequality and an evolution equation formulated in infinite-dimensional spaces, is investigated in this paper. More precisely, based on Browder’s result, optimal control theory, measurability of set-valued mappings and the theory of semigroups, we establish that the solution set of DVI is nonempty and compact. In addition, the theoretical developments are accompanied by an application to differential Nash games.

scholarly journals Applications of Fixed Point Theorems to Generalized Saddle Points of Bifunctions on Chain-Complete Posets

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Jinlu Li ◽  
Ying Liu ◽  
Hongya Gao
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Topological Structure ◽  
Fixed Point Theorems ◽  
Equilibrium Problems ◽  
Saddle Points ◽  
Order Relation ◽  
Partial Order Relation ◽  
Chain Complete ◽  
Set Valued Mappings

We apply the extensions of the Abian-Brown fixed point theorem for set-valued mappings on chain-complete posets to examine the existence of generalized and extended saddle points of bifunctions defined on posets. We also study the generalized and extended equilibrium problems and the solvability of ordered variational inequalities on posets, which are equipped with a partial order relation and have neither an algebraic structure nor a topological structure.

A concept of inner prederivative for set-valued mappings and its applications

2018 ◽  
Vol 24 (3) ◽  
pp. 1059-1074
Author(s):  
Michel H. Geoffroy ◽  
Yvesner Marcelin
Keyword(s):  
Optimality Conditions ◽  
Necessary Optimality Conditions ◽  
Welfare Economics ◽  
Inverse Mapping ◽  
First Order ◽  
Inverse Mapping Theorem ◽  
Homogeneous Set ◽  
Optimal Allocations ◽  
Variational Perturbations ◽  
Set Valued Mappings

We introduce a class of positively homogeneous set-valued mappings, called inner prederivatives, serving as first order approximants to set-valued mappings. We prove an inverse mapping theorem involving such prederivatives and study their stability with respect to variational perturbations. Then, taking advantage of their properties we establish necessary optimality conditions for the existence of several kind of minimizers in set-valued optimization. As an application of these last results, we consider the problem of finding optimal allocations in welfare economics. Finally, to emphasize the interest of our approach, we compare the notion of inner prederivative to the related concepts of set-valued differentiation commonly used in the literature.

Hadamard types of well-posedness of non-self set-valued mappings for coincide points

2005 ◽  
Vol 63 (5-7) ◽  
pp. e2427-e2436 ◽  
Author(s):  
Yong-hui Zhou ◽  
Jian Yu ◽  
Hui Yang ◽  
Shu-wen Xiang
Keyword(s):  
Well Posedness ◽  
Set Valued Mappings
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