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

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Convexity and concavity properties of the optimal value functionf* are considered for the general parametric optimization problemP(ɛ) of the form min x f(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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This paper is based on results presented in the PhD Thesis of the second author completed at The George Washington University under the direction of the first author.

This work was partly supported by the Office of Naval Research, Program in Logistics, Contract No. N00014-75-C-0729 and by the National Science Foundation, Grant No. ECS-82-01370 to the Institute for Management Science and Engineering, The George Washington University, Washington, DC.

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Fiacco, A.V., Kyparisis, J. Convexity and concavity properties of the optimal value function in parametric nonlinear programming. J Optim Theory Appl 48, 95–126 (1986).

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Key Words

  • Nonlinear programming
  • parametric analysis
  • convexity
  • optimal value function
  • point-to-set mappings