18 research outputs found
A Ky Fan minimax inequality for quasiequilibria on finite dimensional spaces
Several results concerning existence of solutions of a quasiequilibrium
problem defined on a finite dimensional space are established. The proof of the
first result is based on a Michael selection theorem for lower semicontinuous
set-valued maps which holds in finite dimensional spaces. Furthermore this
result allows one to locate the position of a solution. Sufficient conditions,
which are easier to verify, may be obtained by imposing restrictions either on
the domain or on the bifunction. These facts make it possible to yield various
existence results which reduce to the well known Ky Fan minimax inequality when
the constraint map is constant and the quasiequilibrium problem coincides with
an equilibrium problem. Lastly, a comparison with other results from the
literature is discussed
Including Social Nash Equilibria in Abstract Economies
We consider quasi-variational problems (variational problems having constraint sets depending on their own solutions) which appear in concrete economic models such as social and economic networks, financial derivative models, transportation network congestion and traffic equilibrium. First, using an extension of the classical Minty lemma, we show that new upper stability results can be obtained for parametric quasi-variational and linearized quasi-variational problems, while lower stability, which plays a fundamental role in the investigation of hierarchical problems, cannot be achieved in general, even on very restrictive conditions. Then, regularized problems are considered allowing to introduce approximate solutions for the above problems and to investigate their lower and upper stability properties. We stress that the class of quasi-variational problems include social Nash equilibrium problems in abstract economies, so results about approximate Nash equilibria can be easily deduced.quasi-variational, social Nash equilibria, approximate solution, closed map, lower semicontinuous map, upper stability, lower stability
On Hölder calmness of solution mappings in parametric equilibrium problems
We consider parametric equilibrium problems in metric spaces. Sufficient conditions for the Hölder calmness of solutions are established. We also study the Hölder well-posedness for equilibrium problems in metric spaces
Projected solutions of generalized quasivariational problems in Banach spaces
This paper focuses on the analysis of generalized quasivariational inequalities with non-self map. In Aussel et al., (2016), introduced the concept of the projected solution to study such problems. Subsequently, in the literature, this concept has attracted great attention and has been developed from different perspectives. The main contribution of this paper is to prove new existence results of the projected solution for generalized quasivariational inequality problems with non-self map in suitable infinite dimensional spaces. As an application, a quasiconvex quasioptimization problem is studied through a normal cone approach
Existence Theorems of ε
A new existence result of ε-vector equilibrium problem is first obtained. Then, by using the existence theorem of ε-vector equilibrium problem, a weakly ε-cone saddle point theorem is also obtained for vector-valued mappings
The Michaelis-Menten-Stueckelberg Theorem
We study chemical reactions with complex mechanisms under two assumptions:
(i) intermediates are present in small amounts (this is the quasi-steady-state
hypothesis or QSS) and (ii) they are in equilibrium relations with substrates
(this is the quasiequilibrium hypothesis or QE). Under these assumptions, we
prove the generalized mass action law together with the basic relations between
kinetic factors, which are sufficient for the positivity of the entropy
production but hold even without microreversibility, when the detailed balance
is not applicable. Even though QE and QSS produce useful approximations by
themselves, only the combination of these assumptions can render the
possibility beyond the "rarefied gas" limit or the "molecular chaos"
hypotheses. We do not use any a priori form of the kinetic law for the chemical
reactions and describe their equilibria by thermodynamic relations. The
transformations of the intermediate compounds can be described by the Markov
kinetics because of their low density ({\em low density of elementary events}).
This combination of assumptions was introduced by Michaelis and Menten in 1913.
In 1952, Stueckelberg used the same assumptions for the gas kinetics and
produced the remarkable semi-detailed balance relations between collision rates
in the Boltzmann equation that are weaker than the detailed balance conditions
but are still sufficient for the Boltzmann -theorem to be valid. Our results
are obtained within the Michaelis-Menten-Stueckelbeg conceptual framework.Comment: 54 pages, the final version; correction of a misprint in Attachment
About regularity properties in variational analysis and applications in optimization
Regularity properties lie at the core of variational analysis because of their importance for stability analysis of optimization and variational problems, constraint qualications, qualication conditions in coderivative and subdierential calculus and convergence analysis of numerical algorithms. The thesis is devoted to investigation of several research questions related to regularity properties in variational analysis and their applications in convergence analysis and optimization. Following the works by Kruger, we examine several useful regularity properties of collections of sets in both linear and Holder-type settings and establish their characterizations and relationships to regularity properties of set-valued mappings. Following the recent publications by Lewis, Luke, Malick (2009), Drusvyatskiy, Ioe, Lewis (2014) and some others, we study application of the uniform regularity and related properties of collections of sets to alternating projections for solving nonconvex feasibility problems and compare existing results on this topic. Motivated by Ioe (2000) and his subsequent publications, we use the classical iteration scheme going back to Banach, Schauder, Lyusternik and Graves to establish criteria for regularity properties of set-valued mappings and compare this approach with the one based on the Ekeland variational principle. Finally, following the recent works by Khanh et al. on stability analysis for optimization related problems, we investigate calmness of set-valued solution mappings of variational problems.Doctor of Philosoph