Reducing the complexity of equilibrium problems and applications to best approximation problems

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arXiv:2210.10831v1 [math.OC] 19 Oct 2022

Reducing the complexity of equilibrium

problems and applications to best

approximation problems

Valerian-Alin Fodor1Nicolae Popovici2,†

1,2Babes

,-Bolyai University of Cluj-Napoca, Romania

Faculty of Mathematics and Computer Science

Department of Mathematics

email1:valerian.fodor@ubbcluj.ro

Abstract

We consider scalar equilibrium problems governed by a bifunction

in a ﬁnite-dimensional framework. By using classical arguments in

Convex Analysis, we show that under suitable generalized convexity

assumptions imposed on the bifunction, the solutions of the equilib-

rium problem can be characterized by means of extreme or exposed

points of the feasible domain. Our results are relevant for diﬀerent

particular instances, such as variational inequalities and optimization

problems, especially for best approximation problems.

MSC 2010. 52A20, 41A50, 46N10, 90C33.

Key words. Extreme points, exposed points, equilibrium points.

1 Introduction and preliminaries

Throughout this paper Rnstands for the n-dimensional real Euclidean

space, whose norm k · k is induced by the usual inner product h·,·i.

For all x, y ∈S, we use the notations

[x, y] := {(1 −t)x+ty |t∈[0,1] },

]x, y[ := {(1 −t)x+ty |t∈]0,1[ }.

†Meanwhile, professor Nicolae Popovici passed away unexpectedly and prematurely.

Recall that a set S⊆Rnis called convex if

[x, y]⊆S, ∀x, y ∈S.

Of course, this is equivalent to say that

]x, y[⊆S, ∀x, y ∈S.

Given a convex set S⊆Rnwe denote by

ext S={x0∈S| ∀x, y ∈S:x0=1

2(x+y)⇒x=y=x0}

the set of extreme points of S. A point x0is said to be an exposed point of

Sif there is a supporting hyperplane Hwhich supports Sat x0such that

{x0}=H∩S. We denote the set of exposed points of Sby

exp S={x0∈S| ∃c∈Rnsuch that argmin

x∈S

hc, xi={x0}}

It is well-known that exp S⊆ext S.

The convex hull of a set M⊆Rn, i.e., the smallest convex set in Rn

containing Mis denoted by convM.

Next, we recall the following well-known theorems (see for example [1]

and [2]):

Theorem 1.1 (Minkowski (Krein-Milman)) Every compact convex set

in Rnis the convex hull of its extreme points.

Theorem 1.2 (Straszewicz) Every compact convex subset Mof Rnadmits

the representation:

M= cl(conv(exp M)).

Deﬁnition 1.1 Let Sbe a nonempty subset of Rnand let x∗∈Rn. A point

x0∈Sis said to be an element of best approximation to x∗from S(or a

nearest point to x∗from S) if

kx0−x∗k ≤ kx−x∗k,∀x∈S.

The problem of best approximation of x∗by elements of Sconsists in ﬁnding

all elements of best approximation to x∗from S. The solution set

PS(x∗) := {x0∈S| kx0−x∗k ≤ kx−x∗k,∀x∈S}

is called the metric projection of x∗on S.

Remark 1.1 The problem of best approximation is an optimization prob-

lem,

f(x)−→ min

x∈S,

whose objective function f:Rn→Ris deﬁned for all x∈Rnby

f(x) := kx−x∗k.

Actually, we have

PS(x∗) = argmin

x∈S

f(x).

Deﬁnition 1.2 Let Sbe a nonempty subset of Rnand let x∗∈Rn, we say

that x0∈Sis a farthest point from Sto x∗if

kx0−x∗k ≥ kx−x∗k,∀x∈S,

i.e.,

x0∈argmax

x∈S

kx−x∗k.

In this paper we will use the following well known results from Convex

Analysis (see for example [3]).

Proposition 1.1 Any farthest point from a nonempty set S⊆Rnto a point

x∗∈Rnis an exposed point of S, i.e.,

argmax

x∈S

kx−x∗k ⊆ exp S.

Theorem 1.3 (existence of elements of best approximation)

If Sis a nonempty closed subset of Rn, then for every x∗∈Rnthere is an

element of best approximation to x∗from S. In other words, we have

PS(x∗)6=∅, i.e., card(PS(x∗)) ≥1.

Theorem 1.4 (unicity of the element of best approximation)

If S⊆Rnis a nonempty convex set and x∗∈Rn, then there exists at most

one element of best approximation to x∗from S. In other words, we have

card(PS(x∗)) ≤1.

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摘要：

arXiv:2210.10831v1[math.OC]19Oct2022ReducingthecomplexityofequilibriumproblemsandapplicationstobestapproximationproblemsValerian-AlinFodor1NicolaePopovici2,†1,2Babes,-BolyaiUniversityofCluj-Napoca,RomaniaFacultyofMathematicsandComputerScienceDepartmentofMathematicsemail1:valerian.fodor@ubbcluj.roAbs...

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