Generalized Nash equilibrium problem, variational inequality and quasiconvexity

It is well known that the generalized Nash equilibrium problem, a model for multi-leader–follower games, can be reformulated as a quasivariational inequality. We show that, in fact, a reformulation in terms of a variational inequality can be obtained in the general setting of quasiconvex nondifferentiable decision functions. An existence result is deduced.     

Semicontinuity of the solution map of quasivariational inequalities

We investigate continuity properties (closedness and lower semicontinuity) of the solution map of a quasivariational inequality which is subjet to perturbations. Perturbations are here considered both on the set-valued operator and on the constraint map defining the quasivariational inequality. Two concepts of solution map will be considered.     

Maximal quasimonotonicity and dense single-directional properties of quasimonotone operators

The concept of maximal quasimonotonicity of set-valued map is introduced and studied. Regularity properties of this class of operators is investigated, in particular the ccvc property, an adaptation to the quasimonotone case of the classical notion of cusco map. In an Asplund space, we provide sufficient conditions for a ccvc quasimonotone operator to be single-directional on a $G_\delta $ G δ -dense subset.     

Preface to “Optimization,Convex and Variational Analysis”

Set-Valued and Variational Analysis -     

On Gap Functions for Multivalued Stampacchia Variational Inequalities

Gap functions have proved to be efficient tools to study single-valued variational inequalities. This approach allows us to reformulate the problem into an optimization problem.     

252Cf-plasma desorption mass spectrometry of unmodified lipid A: fragmentation patterns and localization of fatty acids.

The fragmentation patterns of synthetic Escherichia coli-type lipid A in plasma desorption mass spectrometry (PDMS) in both negative- and positive-ion modes were determined. Negative-ion spectra gave signals for the main diphosphorylated (intact) molecular species in their native proportions. Intact and alkaline-treated lipid A in this mode gave, for the glucosamine I moiety, easily identified signals that have not been previously reported in PDMS. These spectra gave enough information to localize the fatty acids. The procedure was verified with relatively homogeneous lipids A prepared from Salmonella minnesota R595 and Neisseria meningitidis lipopolysaccharides, and then applied to the previously unstudied Yersinia entercolitica O:11,24 lipid A to obtain the localization of its fatty acids. The possibility of obtaining this much information from two negative-ion spectra was attributed to the method, described earlier, of preparing the samples. In the positive-ion mode, about half of the E. coli ions containing diglucosamine appeared as monodephosphorylated species and/or as Na adducts. The intact glucosamine II moiety and its fragment ions gave signals none of which were Na adducts. With lipids A prepared from S. minnesota, N. meningitidis, and Y. enterocolitica, similar fragmentation patterns were observed. For lipid A structure determination, the positive-ion mode could play a confirmatory role. The above results and some of those reported by others were compared.     

Subdifferential Properties of Quasiconvex and Pseudoconvex Functions: Unified Approach

In this paper, we are mainly concerned with the characterization of quasiconvex or pseudoconvex nondifferentiable functions and the relationship between those two concepts. In particular, we characterize the quasiconvexity and pseudoconvexity of a function by mixed properties combining properties of the function and properties of its subdifferential. We also prove that a lower semicontinuous and radially continuous function is pseudoconvex if it is quasiconvex and satisfies the following optimality condition: 0f(x)f has a global minimum at x. The results are proved using the abstract subdifferential introduced in Ref. 1, a concept which allows one to recover almost all the subdifferentials used in nonsmooth analysis.     

Existence Results for Quasi-variational Inequalities with Applications to Radner Equilibrium Problems

Set-Valued and Variational Analysis - A quasi-variational inequality corresponds to a variational inequality in which the constraint set depends on the current value of the variable....     

Gap Functions for Quasivariational Inequalities and Generalized Nash Equilibrium Problems

The gap function (or merit function) is a classic tool for reformulating a Stampacchia variational inequality as an optimization problem. In this paper, we adapt this technique for quasivariational inequalities, that is, variational inequalities in which the constraint set depends on the current point. Following Fukushima (J. Ind. Manag. Optim. 3:165–171, 2007), an axiomatic approach is proposed. Error bounds for quasivariational inequalities are provided and an application to generalized Nash equilibrium problems is also considered.