Didier Aussel

Subsmooth sets: Functional characterizations and related concepts

Prox-regularity of a set (Poliquin-Rockafellar-Thibault, 2000), or its global version, proximal smoothness (Clarke-Stern-Wolenski, 1995) plays an important role in variational analysis, not only because it is associated with some fundamental properties as the local continuous differentiability of the function dist (C;.), or the local uniqueness of the projection mapping, but also because in the case where C is the epigraph of a locally Lipschitz function, it is equivalent to the weak convexity (lower-C 2 property) of the function. In this paper we provide an adapted geometrical concept, called sub smoothness, which permits an epigraphic characterization of the approximate convex functions (or lower-C 1 property). Subsmooth sets turn out to be naturally situated between the classes of prox-regular and of nearly radial sets. This latter class has been recently introduced by Lewis in 2002. We hereby relate it to the Mifflin semismooth functions.

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.

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.

Quasiconvex programming with locally starshaped constraint region and applications to quasiconvex MPEC

In this article we prove an existence result, necessary and sufficient conditions for quasiconvex programming problem with a locally starshaped constraint region. Our optimality conditions are different from the usual optimality conditions, in that the subdifferential of the objective function is replaced by a normal cone operator. Such an optimality condition has advantage over the usual one, i.e. it becomes sufficient even when the objective function is only quasiconvex. As a special case we derive the corresponding results for the class of ‘Quasiconvex-quasiaffine’ MPEC which is a class of mathematical programs with complementarity constraints where the objective function is quasiconvex, the inequality constraint is quasiconvex and the rest of constraints are quasiaffine.

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.New notions of gap functions are defined for set-valued variational inequalities. We prove finiteness and error bounds properties, i.e. upper estimates for the distance to the solution set of the variational inequality.

Water integration in eco-industrial parks using a multi-leader-follower approach

The design and optimization of industrial water networks in eco-industrial parks are studied by formulating and solving multi-leader-follower game problems. The methodology is explained by demonstrating its advantages against multi-objective optimization approaches. Several formulations and solution methods for MLFG are discussed in detail. The approach is validated on a case study of water integration in EIP without and with regeneration units. In the latter, multi-leader-single-follower and single-leader-multi-follower games are studied. Each enterprises objective is to minimize the total annualized cost, while the EIP authority objective is to minimize the consumption of freshwater within the ecopark. The MLFG is transformed into a MOPEC and solved using GAMS® as an NLP. Obtained results are compared against the MOO approach and between different MLFG formulations. The methodology proposed is proved to be very reliable in multi-criteria scenarios compared to MOO approaches, providing numerical Nash equilibrium solutions and specifically in EIP design and optimization.

Stability of Quasimonotone Variational Inequality Under Sign-Continuity

Whenever the data of a Stampacchia variational inequality, that is, the set-valued operator and/or the constraint map, are subject to perturbations, then the solution set becomes a solution map, and the study of the stability of this solution map concerns its regularity. An important literature exists on this topic, and classical assumptions, for monotone or quasimonotone set-valued operators, are some upper or lower semicontinuity. In this paper, we limit ourselves to perturbations on the constraint map, and it is proved that regularity results for the solution maps can be obtained under some very weak regularity hypothesis on the set-valued operator, namely the lower or upper sign-continuity.

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.

Normal Cones to Sublevel Sets: An Axiomatic Approach

An axiomatic approach of normal operators to sublevel sets is given. Considering the Clarke-Rockafellar subdifferential (resp. quasiconvex functions), the definition given in [4] (resp. [5]) is recovered. Moreover, the results obtained in [4] are extended in this more general setting. Under mild assumptions, quasiconvex continuous functions are classified, establishing an equivalence relation between functions with the same normal operator. Applications in pseudoconvexity are also discussed.

Deregulated electricity markets with thermal losses and production bounds: models and optimality conditions

A multi-leader-common-follower game formulation has been recently used by many authors to model deregulated electricity markets. In our work, we first propose a model for the case of electricity market with thermal losses on transmission and with production bounds, a situation for which we emphasize several formulations based on different types of revenue functions of producers. Focusing on a problem of one particular producer, we provide and justify an MPCC reformulation of the producer’s problem. Applying the generalized differential calculus, the so-called M-stationarity conditions are derived for the reformulated electricity market model. Finally, verification of suitable constraint qualification that can be used to obtain first order necessary optimality conditions for the respective MPCCs are discussed.