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

 

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

Coauthor: Hai Yen Le

 

Abstract:
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:
\begin{compactitem}[-]

  • (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!
    \end{compactitem}

     

    Talk 2 of the invited session Tue.3.H 2051
    "Recent advances on linear complementarity problems" [...]
    Cluster 24
    "Variational analysis" [...]

     

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