## Convergence of convex-concave saddle functions: Applications to convex programming and mechanics.(English)Zbl 0667.49009

In the framework of the epi-convergence theory the authors generalize to reflexive Banach spaces some results on the bicontinuity with respect to the epitopology of the Legendre-Fenchel transform on the space of proper, lower semicontinuous, convex functions defined on $$R^ n.$$
It is shown that the appropriate notion of convergence for saddle function is that of epi/hypo-convergence.
The results are applied to study the stability of the optimal solutions and associated multipliers of convex programs. Applications in mechanics are also considered.
The first one concerns homogenization of elasticity, where primal and dual variable are respectively equal to displacement vector fields and stress tensor fields. The homogenization process is studied by introduction of the associated Lagrangians, with their epi/hypo- convergence that provides the convergence of their saddle points. The second one deals with the convergence of the primal/dual solutions in a reinforcement problem when the thickness of the reinforced zone goes to zero.
Reviewer: M.Codegone

### MSC:

 49J45 Methods involving semicontinuity and convergence; relaxation 74E05 Inhomogeneity in solid mechanics 90C25 Convex programming 90C31 Sensitivity, stability, parametric optimization 90C55 Methods of successive quadratic programming type 54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)
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