Reducing the complexity of equilibrium problems and applications to best approximation problems
2025-04-29
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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.
1
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.
2
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.
3
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arXiv:2210.10831v1[math.OC]19Oct2022ReducingthecomplexityofequilibriumproblemsandapplicationstobestapproximationproblemsValerian-AlinFodor1NicolaePopovici2,†1,2Babes,-BolyaiUniversityofCluj-Napoca,RomaniaFacultyofMathematicsandComputerScienceDepartmentofMathematicsemail1:valerian.fodor@ubbcluj.roAbs...
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时间：2025-04-29
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