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Extended radial epiderivatives of non-convex vector-valued maps and parametric quasiconvex programming. (English) Zbl 1312.49013

Summary: In this paper, we consider extended vector-valued mappings defined on a normed linear space. Based on the recent semicontinuous regularizations related to hypographical and/or epigraphical profile mappings of the considered function, we define semicontinuous radial epiderivatives. We then demonstrate that the properties of these epiderivatives amount to properties of hypographical and/or epigraphical profile mappings of the corresponding difference quotient of the underlying function, which simplify fairly well the proofs in the radial epiderivative formulaes. In particular, we stress the impact of semicontinuity, hence we characterize with new arguments the radial epiderivatives in terms of the suprema and/or infima of the interiorly radial cone of the hypograph and/or epigraph of the considered function. Finally, we obtain optimality conditions for general non-convex constrained vector optimization problems. Thereafter we apply the obtained pattern to a parametric quasiconvex programming problem for which we derive necessary and sufficient optimality conditions that are not sensitive to perturbations at the nominal level, yielding henceforth more – and strong (at least under asymptotically regular constraints) – information than the recent stability results obtained under additional conditions on the regularity of the normal cone to the adjusted sublevel sets of the underlying function.

MSC:

49J52 Nonsmooth analysis
49J53 Set-valued and variational analysis
90C46 Optimality conditions and duality in mathematical programming
90C29 Multi-objective and goal programming
90C26 Nonconvex programming, global optimization
58C07 Continuity properties of mappings on manifolds
06A06 Partial orders, general
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