Slice convergence: stability and optimization in nonreflexive spaces. (Slice convergence: stabilité et optimisation dans les espaces non réflexifs.) (French) Zbl 1072.49009
Summary: It is shown by D. Mentagui [ESAIM, Control Optim. Calc. Var. 9, 297–315 (2003; Zbl 1073.49006)] that, in the case of general Banach spaces, the Attouch-Wets convergence is stable by a class of classical operations of convex analysis, when the limits satisfy some natural qualification conditions. This fails with the slice convergence. We establish here uniform qualification conditions ensuring the stability of the slice convergence under the same operations which play a basic role in convex optimization. We obtain as consequences, some key stability results of epi-convergence established by L. McLinden and R. C. Bergstrom [Trans. Am. Math. Soc. 286, 127–142 (1981; Zbl 0468.90063)] in finite dimension. As an application, we give a model of convergence and stability for a wide class of problems in convex optimization and duality theory. The key ingredients in our methodology are, the horizon analysis, the notions of quasi-continuity and inf-local compactness of convex functions, and the bicontinuity of the Legendre-Fenchel transform relatively to the slice convergence.
MSC:
| 49J45 | Methods involving semicontinuity and convergence; relaxation |
| 49K40 | Sensitivity, stability, well-posedness |
| 49N15 | Duality theory (optimization) |
| 49J52 | Nonsmooth analysis |
| 90C25 | Convex programming |
| 90C46 | Optimality conditions and duality in mathematical programming |
Keywords:
convex optimization; slice convergence; Mosco convergence; epiconvergence; duality; cone horizonReferences:
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