scholarly journals Minimax Theorem for Upper and Lower Semicontinuous Payoffs

10.7249/rm478 ◽  
1950 ◽  
Author(s):  
Irving Glicksberg
Keyword(s):  
Lower Semicontinuous ◽  
Minimax Theorem

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 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 A Projected Forward-Backward Algorithm for Constrained Minimization with Applications to Image Inpainting

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 890
Author(s):  
Suthep Suantai ◽  
Kunrada Kankam ◽  
Prasit Cholamjiak
Keyword(s):  
Image Processing ◽  
Minimization Problem ◽  
Convex Functions ◽  
Lower Semicontinuous ◽  
Image Inpainting ◽  
Constrained Minimization ◽  
Convex Minimization Problem ◽  
Mild Conditions ◽  
Constrained Minimization Problem ◽  
Weak Convergence Theorem

In this research, we study the convex minimization problem in the form of the sum of two proper, lower-semicontinuous, and convex functions. We introduce a new projected forward-backward algorithm using linesearch and inertial techniques. We then establish a weak convergence theorem under mild conditions. It is known that image processing such as inpainting problems can be modeled as the constrained minimization problem of the sum of convex functions. In this connection, we aim to apply the suggested method for solving image inpainting. We also give some comparisons to other methods in the literature. It is shown that the proposed algorithm outperforms others in terms of iterations. Finally, we give an analysis on parameters that are assumed in our hypothesis.

scholarly journals Connected perimeter of planar sets

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
François Dayrens ◽  
Simon Masnou ◽  
Matteo Novaga ◽  
Marco Pozzetta
Keyword(s):  
Minimization Problem ◽  
Existence Of Solutions ◽  
Lower Semicontinuous ◽  
Representation Formula ◽  
Steiner Trees ◽  
Lower Semicontinuous Envelope ◽  
Simply Connected

AbstractWe introduce a notion of connected perimeter for planar sets defined as the lower semicontinuous envelope of perimeters of approximating sets which are measure-theoretically connected. A companion notion of simply connected perimeter is also studied. We prove a representation formula which links the connected perimeter, the classical perimeter, and the length of suitable Steiner trees. We also discuss the application of this notion to the existence of solutions to a nonlocal minimization problem with connectedness constraint.

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