Invited Session Tue.3.H 2051

Tuesday, 15:15 - 16:45 h, Room: H 2051

Cluster 24: Variational analysis [...]

Recent advances on linear complementarity problems


Chair: Héctor Ramírez



Tuesday, 15:45 - 16:10 h, Room: H 2051, Talk 2

Jean-Baptiste Hiriart-Urruty
A variational approach of the rank function

Coauthor: Hai Yen Le


We consider here the rank (of a matrix) from the variational viewpoint. Actually, besides being integer-valued, the rank function is lower-semicontinuous. We are interested in the rank function, because it appears as an objective (or constraint) function in various modern optimization problems, the so-called rank minimization problems (P).
A problem like (P) has some bizarre and/or interesting properties, from the optimization or variational viewpoint. The first one, well documented and used, concerns the "relaxed'' forms of it. We recall here some of these results and propose further developments:

  • (Global optimization) Every admissible point in (P) is a local minimizer.
  • (Moreau-Yosida approximation) The Moreau-Yosida approximate (or regularized version) of the objective function in (P), as well as the associated proximal mapping, can be explicitly calculated.
  • (Generalized subdifferentials) The generalized subdifferentials of the rank function can be determined. Actually, all the main ones coincide and their common value is a vector subspace!



    Tuesday, 16:15 - 16:40 h, Room: H 2051, Talk 3

    Héctor Ramírez
    Existence and stability results based on asymptotic analysis for semidefinite linear complementarity problems

    Coauthors: Julio López, Rúben López


    This talk is devoted to the study of existence and stability results of semidefinite linear complementarity problems (for short SDLCP). Our approach consists of approximating the variational inequality formulation of the SDLCP by a sequence of suitable chosen variational inequalities. This provides particular estimates for the asymptotic cone of the solution set of the SDLCP. We thus obtain new coercive and noncoercive existence results, as well as new properties related to the continuity of the solution sets of the SDLCP (such as outer/upper semicontinuity, Lipschitz-type continuity, among others). Moreover, this asymptotic approach leads to a natural extension of the class of García linear transformations, formerly defined in the context of linear complementarity problems, to this SDLCP setting.


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