Convexity and concavity properties of the optimal value function in parametric nonlinear programming

Convexity and concavity properties of the optimal value functionf* are considered for the general parametric optimization problemP(ɛ) of the form minxf(x, ɛ), s.t.x εR(ɛ). Such properties off* and the solution set mapS* form an important part of the theoretical basis for sensitivity, stability, and parametric analysis in mathematical optimization. Sufficient conditions are given for several standard types of convexity and concavity off*, in terms of respective convexity and concavity assumptions onf and the feasible region point-to-set mapR. Specializations of these results to the general parametric inequality-equality constrained nonlinear programming problem and its right-hand-side version are provided. To the authors' knowledge, this is the most comprehensive compendium of such results to date. Many new results are given.

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