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Bounded convergence for perturbed minimization problems. (English) Zbl 1153.49316

Summary: Bounded convergence results are established for sequences of convex functions of the form \(F(\cdot ,0)\). As particular cases, one obtains convergence results for functions of the following types: \(F(\cdot ,T(\cdot))\),\( f+g\circ T\), \(\max\{f,g\circ T\}\), \(g\circ T\), \(f + g\), \(\max\{f,g\}\). In the case of the sum we obtain the well-known theorem of G. Beer and R. Lucchetti [Math. Oper. Res. 17, No. 3, 715–726 (1992; Zbl 0767.49011)] as well as the recent result obtained by A. Eberhard and R. Wenczel [J. Convex Anal. 7, No. 1, 47–71 (2000; Zbl 0961.49005)] (and reduce to that last result when the space is complete).

MSC:

49K40 Sensitivity, stability, well-posedness
90C31 Sensitivity, stability, parametric optimization
46N10 Applications of functional analysis in optimization, convex analysis, mathematical programming, economics
52A40 Inequalities and extremum problems involving convexity in convex geometry
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