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1. Some Methods for Solving Quasi-Equilibrium Problems
Van Hien Nguyen
Université de Namur, Belgique
vhnguyen@fundp.ac.be
Joint work with Jean-Jacques Strodiot (Université de Namur, Belgique),
Nguyen Thi Thu Van (Institute for Computational Science and Technology, HCMC, Vietnam)
Abstract
Let X be a given nonempty closed convex subset in I
Rn
and let f : X ×X −→ I
R be a given bifunction
with f(x, x) = 0 for all x ∈ X. We are interested in the following quasi-equilibrium problem
(
Find x⋆
∈ K(x⋆
) such that
f(x⋆
, y) ≥ 0 ∀ y ∈ K(x⋆
)
(QEP)
where K(·) is a multivalued mapping X −→ 2X
. When the mapping x −→ K(x) is such that K(x) = X
for all x ∈ X, (QEP) reduces naturally to the equilibrium problem:
(
Find x⋆
∈ X such that
f(x⋆
, y) ≥ 0 ∀ y ∈ X
(EP)
In the case when f(x, y) = F(x)T
(y − x) in (QEP) with F : X −→ I
Rn
, we obtain an important class of
problems, namely the class of quasi-variational inequality problems:
(
Find x⋆
∈ K(x⋆
) such that
F(x⋆
)T
(y − x⋆
) ≥ 0 ∀ y ∈ K(x⋆
)
(QV IP)
Note that the generalized Nash equilibrium problem (GNEP), which is an important model fruitfully
used in many different applications, can be reformulated as a (QEP) or (QV IP).
The (EP), (GNEP), (QV IP) and (V IP) have received considerable attention in recent years from
theoretical viewpoint, applications and solution methodologies. Compared with the (QEP), the literature
on the algorithms for the quasi-equilibrium problems is not as extensive. See, for example, the following
recent surveys and the references cited therein:
− M. Pappalardo, G. Mastroeni, M. Passacantando, Merit functions: a bridge between optimization
and equilibria, 4OR - A Quartely Journal of Operations Research 12 (2014) 1–33.
− G. Bigi, M. Castellani, M. Pappalardo, M. Passacantando, Existence and solution methods for
equilibria, European Journal of Operational Research 227 (2013) 1–11.
− F. Facchinei, C. Kanzow, Generalized Nash equilibrium problems, Annals of Operations Research
175 (2010) 177–211.
1

2. In this talk we focus on some numerical algorithms for solving quasi-equilibrium problems.1
Following are the assumptions we make.
Assumption (A):
(i) X is a nonempty closed convex subset in Rn
.
(ii) f : X × Λ −→ R, where Λ is an open convex set containing X, is a continuous function on X × Λ,
f(x, x) = 0, and y −→ f(x, y) is a convex function on Λ for all x ∈ X.
(iii) The set-valued mapping K : X −→ 2X
is continuous on X, where K(x) is a nonempty closed
convex set of X for all x ∈ X.
(iv) x ∈ K(x) for all x ∈ X.
(v) The set S⋆
= {x ∈ S | f(x, y) ≥ 0 ∀ y ∈ T} is nonempty, where S = ∩x∈XK(x) and T =
∪x∈XK(x).
(vi) The equilibrium bifunction f(·, ·) is pseudomonotone over X with respect to S⋆
, i.e.,
∀ x⋆
∈ S⋆
, ∀ y ∈ X f(y, x⋆
) ≤ 0.
Our general method can be expressed as follows.
General Method (QE):
Step 0 Let x0
∈ X, µ ∈ (0, 1), and γ ∈ (0, 2). Set k = 0.
Step 1 Compute yk
= arg miny∈K(xk)
f(xk
, y) + 1
2 ky − xk
k2
. If yk
= xk
then Stop.
Step 2 Otherwise, find a direction dk
such that hdk
, xk
− x∗
i ≥ µkxk
− yk
k2
0 for every x∗
∈ S∗
.
Step 3 Compute xk
(βk) = PK(xk)(xk
− βkdk
) where βk is such that
kxk
(βk) − x∗
k2
≤ kxk
− x∗
k2
− γ (2 − γ) µ2 kxk
− yk
k4
kdkk2
for every x∗
∈ S∗
. Set xk+1
= xk
(βk), k := k + 1 and go to Step 1.
Once we have the general method, we detail how to realize concretely the construction of a direction
dk
satisfying the inequality of Step 2 and how to perform a line search along the direction −dk
to reduce
the distance of xk
to the solution set of the quasi-equilibrium problem. A general convergence theorem
applicable to each algorithm is presented.
1J.J. Strodiot, Thi Thu Van Nguyen, Van Hien Nguyen, A new class of hybrid extragradient algorithms for solving
quasi-equilibrium problems, Journal of Global Optimization 56 (2013) 373–397.
2

3. Complex Variational Inequalities and Applications
Francisco Facchinei
We introduce and study variational inequalities in the complex domain, along with some technical
tools useful in their study. In particular we discuss and clarify several issues related to the so called
Wirtinger derivatives of functions of several complex variables.
We then extend to the complex domain some recent developments in the field of the distributed
solution of (generalized) Nash equilibrium problems and briefly discuss the problem of selecting a
particular solution in monotone games of complex variables when multiple solutions exist.
In order to illustrate our techniques we consider some new MIMO games over vector Gaussian
Interference Channels, modeling some distributed resource allocation problems in MIMO
Cognitive Radio systems and femtocells. These games are examples of Nash equilibrium problems
that can not be handled by current methodologies.

4. A SENSITIVITY ANALYSIS OF A PARAMETRIC GENERALIZED EQUATION
M. Bianchi
We will provide a global regularity result for the solution map of the generalized equation
0 ∈ H(x, p), by assuming openness and Aubin properties with respect to some sets U and V
of the map H. As a consequence, we infer the regularity of the solution map of an inclusion
problem that is controlled by the composition of a parametric map with another map which
fulfills suitable regularity assumptions. The result can be applied to investigate a global
condition number for convex vector-valued problems in the Euclidean framework.
(Joint work with G. Kassay and R. Pini.)

5. LINEAR OPENNESS OF THE COMPOSITION OF SET-VALUED MAPS
AND APPLICATIONS TO VARIATIONAL SYSTEMS
R. Pini
Given the generalized equation 0 ∈ H(x, p), where H : X ×P → 2W and X, W, P are metric
spaces, our aim is to investigate the regularity properties of the solution map S = S(p).
In particular, we are interested in the case H(x, p) = G(F1(x, p), F2(x, p)), and we infer
Lipschitz continuity of the set-valued map S via suitable properties of the maps F1, F2 and
G. The main tools are the Nadler fixed point theorem and the Lim lemma. We obtain, as
a special case, the well-known result concerning the sum of two set-valued maps.
(Joint work with M. Bianchi and G. Kassay.)

6. Does convexity arise in
optimization naturally?
Fabián Flores-Bazán
Departamento de Ingenierı́a Matemática and CI2
MA, Universidad de Concepción,
Chile
fflores(at)ing-mat.udec.cl
The convex world offers the most desirable properties in optimization, and the lack
of that provides an interesting challenge in mathematics. In this lecture we show
various instances from mathematical programming to calculus of variations where
convexity is present in one way or in another. Among the issues to be described lie:
KKT optimality conditions; Dines-type convex theorem and the S-lemma in quadratic
programming; optimal value functions; local optimality implies global.
This research is supported in part by CONICYT-Chile through FONDECYT 112-
0980 and by Center for Mathematical Modeling (CMM), Universidad de Chile.
References
[1] Flores-Bazán F., Mastroeni G., Strong duality in cone constrained nonconvex
optimization, SIAM Journal on Optimization, 23 No 1 (2013), 153–169.
[2] Flores-Bazán F., Cárcamo G., A geometric characterization of strong duality in
nonconvex quadratic programming with linear and nonconvex quadratic constraints”,
Math. Program., Ser. A, 2014.
[3] Flores-Bazán F., Mastroeni G., Characterizing FJ and KKT condition in noncon-
vex mathematical programming with applications, SIAM Journal on Optimization,
2014. Submitted.
[4] Flores-Bazán F., Jourani A., Mastroeni G., On the convexity of the value function
for a class of nonconvex variational problems: existence and optimality conditions,
2014. Submitted.
[5] Flores-Bazán F., Opazo F., Dines theorem for inhomogeneous quadratic functions
and nonconvex optimization, 2014. Submitted.

7. Some applications of variational inequalities to equilibrium problems
Patrizia Daniele – Università degli Studi di Catania
In this talk we show how many problems of topical interest such as: transportation networks,
spatially distributed economic markets, supply chain network models, financial equilibrium
problems, static electric power supply chain networks, supply chain network with recycled
materials, … can be studied in the framework of variational inequalities, which express in a
compact and handy form the equilibrium conditions. Moreover, we shall present, in a Hilbert space
setting, a general random traffic equilibrium problem, giving a random generalized Wardrop
equilibrium condition.

8. ❱❛r✐❛t✐♦♥❛❧ ✐♥❡q✉❛❧✐t✐❡s✱ ❜✐❧❡✈❡❧ ♠♦❞❡❧s
❛♥❞ t❤❡ ♦♣t✐♠❛❧ ♣♦❧❧✉t✐♦♥ ❡♠✐ss✐♦♥ ♣r✐❝❡
♣r♦❜❧❡♠
▲❛✉r❛ ❙❝r✐♠❛❧✐
❉✐♣❛rt✐♠❡♥t♦ ❞✐ ▼❛t❡♠❛t✐❝❛ ❡ ■♥❢♦r♠❛t✐❝❛
❯♥✐✈❡rs✐tà ❞✐ ❈❛t❛♥✐❛
❱✐❛❧❡ ❆✳ ❉♦r✐❛ ✻✱ ✾✺✶✷✺ ❈❛t❛♥✐❛✱ ■t❛❧②
s❝r✐♠❛❧✐❅❞♠✐✳✉♥✐❝t✳✐t
❆❜str❛❝t
❚❤❡ ❝♦♥❝❡r♥ ❢♦r ❡♥✈✐r♦♥♠❡♥t❛❧ q✉❛❧✐t② ❤❛s ❜❡❝♦♠❡ ♦♥❡ ♦❢ t❤❡ ♠♦st ✐♠♣♦rt❛♥t
✐ss✉❡ ✐♥ t❤❡ ❧❛st ❞❡❝❛❞❡s✳ ❘❛♣✐❞ ✉r❜❛♥✐③❛t✐♦♥✱ ♣♦♣✉❧❛t✐♦♥ ❣r♦✇t❤✱ ❛♥❞ ✐♥❞✉s✲
tr✐❛❧✐③❛t✐♦♥ ❤❛✈❡ ❧❡❞ t♦ s❡✈❡r❡ ❡♥✈✐r♦♥♠❡♥t❛❧ ♣r♦❜❧❡♠s✱ ❡s♣❡❝✐❛❧❧② ✐♥ ❞❡✈❡❧♦♣✐♥❣
❝♦✉♥tr✐❡s✳
❘❡❝❡♥t❧②✱ ❜✐❧❡✈❡❧ ♣r♦❣r❛♠♠✐♥❣ ❤❛s ❜❡❡♥ r❡❝♦❣♥✐③❡❞ t♦ ❜❡ ❛ s✉❝❝❡ss❢✉❧ ❛♣♣r♦❛❝❤
✐♥ ♠♦❞❡❧✐♥❣ ♣♦❧❧✉t✐♦♥ ❛❜❛t❡♠❡♥t ♣r♦❜❧❡♠s✳ ■♥s♣✐r❡❞ ❜② ❬✷❪✱ ✇❡ ✐♥✈❡st✐❣❛t❡ t❤❡
❝♦♠♣❧❡① ❜❡❤❛✈✐♦r ♦❢ ♠❛♥✉❢❛❝t✉r❡rs ♦❢ ❛ s✉♣♣❧② ❝❤❛✐♥ ✐♥ t❤❡ ♣r❡s❡♥❝❡ ♦❢ ❝❡♥tr❛❧
♣❧❛♥♥✐♥❣ ❞❡❝✐s✐♦♥s✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ✇❡ ♣r❡s❡♥t ❛ ❝♦♥t✐♥✉♦✉s✲t✐♠❡ ❝❡♥tr❛❧ ♣❧❛♥♥✐♥❣
♠♦❞❡❧ ✐♥ ✇❤✐❝❤ t❤❡ ❣♦✈❡r♥♠❡♥t ❝❤♦♦s❡s t❤❡ ♦♣t✐♠❛❧ ♣r✐❝❡ ♦❢ t❤❡ ♣♦❧❧✉t✐♦♥ ❡♠✐s✲
s✐♦♥ ✇✐t❤ ❝♦♥s✐❞❡r❛t✐♦♥ t♦ ♠❛♥✉❢❛❝t✉r❡rs✬ r❡s♣♦♥s❡ t♦ t❤❡ ♣r✐❝❡✳ ❖♥ t❤❡ ♦t❤❡r
❤❛♥❞✱ t❤❡ ♠❛♥✉❢❛❝t✉r❡rs ❝❤♦♦s❡ t❤❡ ♦♣t✐♠❛❧ q✉❛♥t✐t✐❡s ♦❢ ♣r♦❞✉❝t✐♦♥ t♦ ♠❛①✐♠✐③❡
t❤❡✐r ♣r♦✜ts✱ ❣✐✈❡♥ t❤❡ ♣r✐❝❡ ♦❢ ♣♦❧❧✉t✐♦♥ ❡♠✐ss✐♦♥✳ ❙✉❝❤ ❛ s✐t✉❛t✐♦♥ ❝❛♥ ❜❡st ❜❡
❢♦r♠✉❧❛t❡❞ ❛s ❛ ❜✐❧❡✈❡❧ ♠♦❞❡❧✱ ✇❤❡r❡ t❤❡ ❣♦✈❡r♥♠❡♥t✬s ♣r♦❜❧❡♠ ✐s t❤❡ ❧❡❛❞❡r✬s
♣r♦❜❧❡♠ ✇✐t❤ t❤❡ ❣♦❛❧ ♦❢ ♠❛①✐♠✐③✐♥❣ t❤❡ s♦❝✐❛❧ ✇❡❧❢❛r❡ ✈✐❛ t❛①❛t✐♦♥✱ ❛♥❞ t❤❡
♠❛♥✉❢❛❝t✉r❡r✬s ♣r♦❜❧❡♠ ❞❡✜♥❡s t❤❡ ❢♦❧❧♦✇❡r✬s ♣r♦❜❧❡♠✳
❯s✐♥❣ s♦♠❡ ♥❡✇ r❡❝❡♥t r❡s✉❧ts ♦♥ ✈❛r✐❛t✐♦♥❛❧ ✐♥❡q✉❛❧✐t② ❛♥❞ ✐♥✜♥✐t❡ ❞✐♠❡♥✲
s✐♦♥❛❧ ❞✉❛❧✐t② t❤❡♦r②✱ ✇❡ r❡❢♦r♠✉❧❛t❡ t❤❡ ❜✐❧❡✈❡❧ ♣r♦❣r❛♠♠✐♥❣ ♣r♦❜❧❡♠ ✐♥t♦ ❛
♦♥❡ ❧❡✈❡❧ ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ✇❡ ❣✐✈❡ t❤❡ ❡q✉✐❧✐❜r✐✉♠ ❝♦♥❞✐t✐♦♥s
✉♥❞❡r❧②✐♥❣ t❤❡ ♠❛♥✉❢❛❝t✉r❡r✬ ♣r♦❜❧❡♠✱ ✐♥ ✇❤✐❝❤ ▲❛❣r❛♥❣❡ ♠✉❧t✐♣❧✐❡rs ❛ss♦❝✐❛t❡❞
✇✐t❤ ♠♦❞❡❧ ❝♦♥str❛✐♥ts ❛r❡ ♣r❡s❡♥t✳ ❚❤❡♥✱ ✇❡ ♣r♦✈❡ t❤❡ ❡q✉✐✈❛❧❡♥❝❡ ❜❡t✇❡❡♥ t❤❡
❧♦✇❡r✲❧❡✈❡❧ ♣r♦❜❧❡♠ ❛♥❞ ❛♥ ♦♣♣♦rt✉♥❡ ✈❛r✐❛t✐♦♥❛❧ ✐♥❡q✉❛❧✐t② ✇❤❡r❡ ▲❛❣r❛♥❣❡ ♠✉❧✲
t✐♣❧✐❡rs ❞♦ ♥♦t ❛♣♣❡❛r✳ ❋✐♥❛❧❧②✱ ✇❡ s❤♦✇ t❤❛t t❤❡ r❡s✉❧t✐♥❣ ✈❛r✐❛t✐♦♥❛❧ ✐♥❡q✉❛❧✐t②
✐s ❡q✉✐✈❛❧❡♥t t♦ t❤❡ ❡q✉✐❧✐❜r✐✉♠ ❝♦♥❞✐t✐♦♥s❀ ❤❡♥❝❡ t❤❡ ❧♦✇❡r✲❧❡✈❡❧ ♣r♦❜❧❡♠ ❝❛♥ ❜❡
✶

9. r❡♣❧❛❝❡❞ ❜② t❤❡ ❡q✉✐❧✐❜r✐✉♠ ❝♦♥❞✐t✐♦♥s✳ ❲❡ ❛❧s♦ ❡♥s✉r❡ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ▲❛❣r❛♥❣❡
♠✉❧t✐♣❧✐❡rs✱ s♦ t❤❛t t❤❡ ❡q✉✐❧✐❜r✐✉♠ ❝♦♥❞✐t✐♦♥s ❛r❡ ✇❡❧❧✲❞❡✜♥❡❞✳ ▼♦r❡♦✈❡r✱ ✇❡
✐♥✈❡st✐❣❛t❡ t❤❡ ❡①✐st❡♥❝❡ ♦❢ s♦❧✉t✐♦♥s ❛♥❞ ♣r♦✈✐❞❡ ❛ ♥✉♠❡r✐❝❛❧ ❡①❛♠♣❧❡✳
❘❡❢❡r❡♥❝❡s
❬✶❪ ❉❡♠♣❡✱ ❙✳ ✷✵✵✷✳ ❋♦✉♥❞❛t✐♦♥s ♦❢ ❇✐❧❡✈❡❧ Pr♦❣r❛♠♠✐♥❣✳ ❑❧✉✇❡r✱ ❉♦r❞r❡❝❤t✳
❬✷❪ ❲❛♥❣ ●✲▼✳✱ ▲✲▼✳ ▼❛✱ ▲✲▲✳ ▲✐✳ ✷✵✶✶✳ ❆♥ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ ❜✐❧❡✈❡❧ ♣r♦❣r❛♠♠✐♥❣
♣r♦❜❧❡♠ ✐♥ ♦♣t✐♠❛❧ ♣♦❧❧✉t✐♦♥ ❡♠✐ss✐♦♥ ♣r✐❝❡✳ ❏✳ ❙❡r✈✳ ❙❝✐✳ ▼❛♥❣✳ ✹ ✸✸✹✕✸✸✽✳
✷

10. Existence Results for Generalized Vector
Equilibrium Problems on Unbounded Sets
E. Allevi1
, I.V. Konnov2
, and M. Rocco3
Abstract
Many problems of practical interest in optimization, economics and engi-
neering involve equilibrium in their description; this fact has motivated re-
searchers to establish general results on the existence of solutions for equi-
librium problems; see e.g. [2, 3]. In fact there is a vast literature on equi-
librium problems and their treatment in optimization, variational and quasi-
variational inequalities, and complementarity problems. Many authors inves-
tigated different equilibrium models, extending scalar equilibrium problems
to the vector-valued and set-valued cases; see e.g. [4, 11, 12, 13].
In most of the papers on the existence of solutions of generalized vector
equilibrium problems either boundedness of the feasible set or a certain coer-
civity condition is assumed. The purpose of the present paper is to provide
some existence theorems concerning solutions of generalized vector equilib-
rium problems on an unbounded set with set-valued maps defined on reflexive
Banach spaces, exploiting a new coercivity condition, which was introduced
in [7] for scalar bifunctions and in [8] for vector-valued functions.
Key words: Generalized vector equilibrium problems, set-valued bifunc-
tions, coercivity condition, existence results.
References
[1] Q. H. Ansari and F. Flores-Bazn. Recession methods for generalized vector
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1
Department of Economics and Management, Brescia University, Contrada
S.Chiara, 50, Brescia 25122, Italy.
2
Department of System Analysis and Information Technologies, Kazan Federal
University, ul. Kremlevskaya, 18, Kazan 420008, Russia.
3
European Central Bank, Kaiserstrasse 29, 60311 Frankfurt am Main, Germany.
1

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2

12. An existence result for quasiequilibrium
problems in separable Banach spaces
Marco Castellani, Massimiliano Giuli
Department of Information Engineering, Computer Science and Mathematics,
University of L’Aquila, Italy
Abstract
A quasiequilibrium problem is an equilibrium problem in which the constraint set
is subject to modifications depending on the considered point. This model encom-
passes many relevant problems as special cases, among which variational and qua-
sivariational inequalities, social (or generalized) Nash equilibrium problems, mixed
quasivariational-like inequalities and so on. We furnish an existence result which
proof relies on an important selection theorem due to Michael. In our result the
usual assumption of closedness of the constraint map is not required and also the
finite-dimensionality of the space is dropped.