Short description of the main scientific results of the project:

S. Adly, A. Hantoute, B.K. Le, Well-posedness, stability and robustness for a class of set-valued Lure dynamical systems, submitted to Set-Valued Analysis.

The well-posedness, stability and robust stability analysis for a class of Lur’e dynamical systems are studied. This class includes linear time-invariant dynamical systems (A, B, C, D) under a static set-valued negative feedback F where the matrix D is supposed to be nonzero (which complicates the analysis). The results are established for maximal monotone set-valued part F and extended to the hypo-monotone setting. To illustrate the theoretical developments, some examples in non-regular electrical circuits are discussed. The methodology used in this paper is based on tools from set-valued and non-smooth analysis.

S. Adly, B.K. Le, Unbounded Second Order State Dependent Moreaus Sweeping Processes in Hilbert Spaces, submitted to JOTA.

An existence and uniqueness result of a second order sweeping process with velocity in the moving set under perturbation in infinite-dimensional Hilbert spaces is studied by using an implicit discretization scheme. It is assumed that the moving set depends on the time, the state and is allowed to be unbounded. The existence result is new even in finite-dimensional space. Our methodology is based on convex and variational analysis.

D. Aussel, M. Cervinka, M. Marechal, Deregulated electricity markets with termal losses and production bounds: models and optimality conditions, accepted for publication in RAIRO.

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.

D. Aussel, Y. Garcia, On Extensions of Kenderov’s Single-Valuedness Result for Monotone Maps and Quasimonotone Maps, SIAM J. Optim., 24 (2014), 702-713.

One of the most famous single-valuedness results for set-valued maps is due to P. Kenderov, Fund. Math. LXXXVIII(1975), 61-69. and states that a monotone set-valued operator is single-valued at any point where it is lower semi-continuous. This has been extended in [Christensen-Kenderov , Math. Scand. 54 (1984)] to monotone maps satisfying a so-called ∗−property. Our aim in this work is twofold: first to prove that the ∗−property assumption can be weakened; second to emphasize that these classical single- valuedness results for monotone operators can be obtained, in very simple way, as direct consequences of counterpart results proved for quasimonotone operators in terms of single-directionality.

 H. Attouch, L. M. Briceño-Arias, P. L. Combettes, A strongly convergent primal-dual method for nonover- lapping domain decomposition, submitted. Numerische Mathematik

We propose a primal–dual parallel proximal splitting method for solving domain decomposition prob- lems for partial differential equations. The problem is formulated via minimization of energy functions on the subdomains with coupling constraints which model various properties of the solution at the in- terfaces. The proposed method can handle a wide range of linear and nonlinear problems, with flexible, possibly nonlinear, transmission conditions across the interfaces. Strong convergence in the energy spaces is established in this general setting, and without any additional assumption on the energy functions or the geometry of the problem. Several examples are presented.

 A. Jourani, L. Thibault, D. Zagrodny, The NSLUC property and Klee envelope, to appear for publication in Mathematische Annalen (2015).

A notion called norm subdifferential local uniform convexity (NSLUC) is introduced and studied. It is shown that the property holds for all subsets of a Banach space whenever the norm is either locally uniformly convex or k-fully convex. The property is also valid for all subsets of the Banach space if the norm is Kadec-Klee and its dual norm is Gaˆteaux differentiable off zero. The NSLUC concept allows us to obtain new properties of the Klee envelope, for example a connection between attainment sets of the Klee envelope of a function and its convex hull. It is also proved that the Klee envelope with unit power plus an appropriate distance function is equal to some constant on an open convex subset as large as we need. As a consequence of obtained results, the subdifferential properties of the Klee envelope can be inherited from subdifferential properties of the opposite of the distance function to the complement of the bounded convex open set. Moreover the problem of singleton property of sets with unique farthest point is reduced to the problem of convexity of Chebyshev sets.

A. Jofré, A. Jourani, A characterization of free disposal hypothesis in nonconvex economies with infinite- dimensional commodity spaces, SIAM Journal on Optimization, 25 (2015), 699–712.

Our aim in this paper is to prove geometric characterizations of the free disposal condition for nonconvex economies on infinite-dimensional commodity spaces even if the cone and the production set involved in the condition have empty interior such as in L1 with the positive cone L1+. We then use this characterization to prove existence of Pareto and weak Pareto optimal points. We also explore a notion of extremal systems a` la Kruger-Mordukhovich. We show that the free disposal hypothesis alone assures extremality of the production set with respect to some set.

R. Correa, A. Hantoute, A. Jourani, Characterizations of convex approximate subdifferential calculus in Banach spaces, to appear for publication in Transactions of the American Mathematical Society, (2015). 


We establish subdifferential calculus rules for the sum of convex functions defined on normed spaces. This is achieved by means of a condition relying on the continuity behaviour of the inf-convolution of their corresponding conjugates, with respect to any given topology intermediate between the norm and the weak* topologies on the dual space. Such a condition turns out to be also necessary in Banach spaces. These results extend both the classical formulas by Hiriart Urruty-Phelps in 1993 and by Thibault in 1997.

F. Flores-Bazán, A. Jourani, G. Mastroeni, On the convexity of the value function for a class of nonconvex variational problems: existence and optimality conditions, SIAM Journal on Control and Optimization, 52 (2014), 3673–3693.

In this paper we study a class of perturbed constrained nonconvex variational problems depending on either time/state or time/state’s derivative variables. Its (optimal) value function is proved to be convex and then several related properties are obtained. Existence, strong duality results and necessary/sufficient optimality conditions are established. Moreover, via a necessary optimality condition in terms of Mor- dukhovich’s normal cone, it is shown that local minima are global. Such results are given in terms of the Hamiltonian function. Finally various examples are exhibited showing the wide applicability of our main results.

S. Adly, A. Hantoute, M. Théra, Nonsmooth Lyapunov pairs for differential inclusions governed by operators with nonempty interior domain, accepted in Math. Programming, 2015

The general theory of Lyapunov stability of first-order differential inclusions in Hilbert spaces has been studied by the authors in a previous paper. This new contribution focuses on the case when the interior of the domain of the maximally monotone operator governing the given differential inclusion is nonempty; this includes in a natural way the finite-dimensional case. The current setting leads to simplified, more explicit, criteria and permits some flexibility in the choice of the generalized subdifferentials. Some consequences of the viability of closed sets are given. Our Analysis makes uses of standard tolos from convex and nonsmooth analysis.