An ε-Minimax Theorem for Bi-Lower-Semicontinuous Set-Valued Mappings

2019 ◽  
Vol 40 (7) ◽  
pp. 825-843
Author(s):  
Caiyun Jin
Keyword(s):  
Lower Semicontinuous ◽  
Minimax Theorem ◽  
Set Valued Mappings

A Characterization of Proximal Subgradient Set-Valued Mappings

1993 ◽  
Vol 36 (1) ◽  
pp. 116-122 ◽  
Author(s):  
R. A. Poliquin
Keyword(s):  
Lower Semicontinuous ◽  
Lower Semicontinuous Function ◽  
Semicontinuous Function ◽  
Set Valued Mapping ◽  
Selection Property ◽  
Set Valued Mappings ◽  
Bounded Below

AbstractIn this paper we tackle the problem of identifying set-valued mappings that are subgradient set-valued mappings. We show that a set-valued mapping is the proximal subgradient mapping of a lower semicontinuous function bounded below by a quadratic if and only if it satisfies a monotone selection property.

scholarly journals Ekeland-type variational principle with applications to nonconvex minimization and equilibrium problems

2019 ◽  
Vol 24 (3) ◽  
pp. 407-432
Author(s):  
Iram Iqbal ◽  
Nawab Hussain
Keyword(s):  
Fixed Point ◽  
Variational Principle ◽  
Metric Spaces ◽  
Equilibrium Problems ◽  
Lower Semicontinuous ◽  
Minimax Theorem ◽  
Nonconvex Minimization ◽  
Equilibrium Theorem

The aim of the present paper is to establish a variational principle in metric spaces without assumption of completeness when the involved function is not lower semicontinuous. As consequences, we derive many fixed point results, nonconvex minimization theorem, a nonconvex minimax theorem, a nonconvex equilibrium theorem in noncomplete metric spaces. Examples are also given to illustrate and to show that obtained results are proper generalizations.

scholarly journals A Minimax Theorem forL-0-Valued Functions on Random Normed Modules

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Shien Zhao ◽  
Yuan Zhao
Keyword(s):  
Lower Semicontinuous ◽  
Saddle Points ◽  
Minimax Theorem ◽  
Lower Semicontinuous Function ◽  
Semicontinuous Function ◽  
Minimax Theorems ◽  
Random Normed Module ◽  
Normed Module ◽  
Random Normed Modules ◽  
Definition Of

We generalize the well-known minimax theorems toL¯0-valued functions on random normed modules. We first give some basic properties of anL0-valued lower semicontinuous function on a random normed module under the two kinds of topologies, namely, the (ε,λ)-topology and the locallyL0-convex topology. Then, we introduce the definition of random saddle points. Conditions for anL0-valued function to have a random saddle point are given. The most greatest difference between our results and the classical minimax theorems is that we have to overcome the difficulty resulted from the lack of the condition of compactness. Finally, we, using relations between the two kinds of topologies, establish the minimax theorem ofL¯0-valued functions in the framework of random normed modules and random conjugate spaces.

scholarly journals Systems of variational relations with lower semicontinuous set-valued mappings

2015 ◽  
Vol 31 (3) ◽  
pp. 269-275
Author(s):  
MIRCEA BALAJ ◽  
Keyword(s):  
Fixed Point ◽  
Lower Semicontinuous ◽  
Variational Relations ◽  
Set Valued Mappings ◽  
Existence Criteria ◽  
Fixed Point Techniques

In this paper, we use fixed point techniques to establish existence criteria of the solution for a system of two variational relations with lower semicontinuous set-valued mappings.

scholarly journals Equivalence of viscosity and weak solutions for a p-parabolic equation

2021 ◽  
Author(s):  
Jarkko Siltakoski
Keyword(s):  
Parabolic Equation ◽  
Weak Solutions ◽  
Lower Semicontinuity ◽  
Lower Semicontinuous ◽  
Relationship Of ◽  
The Relationship

AbstractWe study the relationship of viscosity and weak solutions to the equation $$\begin{aligned} \smash {\partial _{t}u-\varDelta _{p}u=f(Du)}, \end{aligned}$$ ∂ t u - Δ p u = f ( D u ) , where $$p>1$$ p > 1 and $$f\in C({\mathbb {R}}^{N})$$ f ∈ C ( R N ) satisfies suitable assumptions. Our main result is that bounded viscosity supersolutions coincide with bounded lower semicontinuous weak supersolutions. Moreover, we prove the lower semicontinuity of weak supersolutions when $$p\ge 2$$ p ≥ 2 .

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.

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