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

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Abstract

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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References

  1. Berge, G.,Topological Spaces, Macmillan, New York, New York, 1963.

    Google Scholar 

  2. Rockafellar, R. T.,Convex Analysis, Princeton University Press, Princeton, New Jersey, 1970.

    Google Scholar 

  3. Rockafellar, R. T.,Conjugate Duality and Optimization, SIAM, Philadelphia, Pennsylvania, 1974.

    Google Scholar 

  4. Brosowski, B.,Parametric Semi-Infinite Optimization, Verlag Peter Lang, Frankfurt am Main, Germany, 1982.

    Google Scholar 

  5. Bank, B., Guddat, J., Klatte, D., Kummer, B., andTammer, K.,Nonlinear Parametric Optimization, Akademie Verlag, Berlin, Germany, 1982.

    Google Scholar 

  6. Fiacco, A. V.,Introduction to Sensitivity and Stability Analysis in Nonlinear Programming, Academic Press, New York, New York, 1983.

    Google Scholar 

  7. Mangasarian, O. L.,Nonlinear Programming, McGraw-Hill, New York, New York, 1969.

    Google Scholar 

  8. Ortega, J. M., andRheinboldt, W. C.,Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, New York, 1970.

    Google Scholar 

  9. Avriel, M., Diewert, W. E., Schaible, S., andZiemba, W. T.,Introduction to Concave and Generalized Concave Functions, Generalized Concavity in Optimization and Economics, Edited by S. Schaible and W. T. Ziemba, Academic Press, New York, New York, 1981.

    Google Scholar 

  10. Kyparisis, J.,Optimal Value Bounds for Posynomial Geometric Programs, The George Washington University, Institute for Management Science and Engineering, Technical Paper Serial T-464, 1982.

  11. Everett, H.,Generalized Lagrange Multiplier Method for Solving Problems of Optimum Allocation of Resources, Operations Research, Vol. 18, pp. 107–118, 1963.

    Google Scholar 

  12. Rolewicz, S.,On Paraconvex Multifunctions, Operations Research Verfahren, Vol. 31, pp. 539–546, 1979.

    Google Scholar 

  13. Rolewicz, S.,On Graph γ-Paraconvex Multifunctions, Special Topics of Applied Mathematics, Edited by J. Frehse, D. Pallaschke, and U. Trottenberg, North-Holland, Amsterdam, Holland, 1980.

    Google Scholar 

  14. Rolewicz, S.,On Conditions Warranting Ψ 2 -Subdifferentiability, Mathematical Programming Study, Vol. 14, pp. 215–224, 1981.

    Google Scholar 

  15. Stern, M. H., andTopkis, D. M.,Rates of Stability in Nonlinear Programming, Operations Research, Vol. 24, pp. 462–476, 1976.

    Google Scholar 

  16. Dolecki, S.,Remarks on Semicontinuity, Bulletin de l'Académic Polonaise des Sciences, Vol. 25, pp. 863–867, 1977.

    Google Scholar 

  17. Dolecki, S.,Semicontinuity in Constrained Optimization, Part I.1, Metric Spaces; Part I.2, Normed Spaces; Part II, Optimization; Control and Cybernetics, Vol. 7, pp. 6–16, 1978; Vol. 7, pp. 18–26, 1978; and Vol. 7, pp. 51–68, 1978.

    Google Scholar 

  18. Robinson, S. M.,Some Continuity Properties of Polyhedral Multifunctions, Mathematical Programming Study, Vol. 14, pp. 206–214, 1981.

    Google Scholar 

  19. Tanino, T., andSawaragi, Y.,Duality Theory in Multiobjective Programming, Journal of Optimization Theory and Applications, Vol. 27, pp. 509–529, 1979.

    Google Scholar 

  20. Tanino, T., andSawaragi, Y.,Conjugate Maps and Duality in Multiobjective Optimization, Journal of Optimization Theory and Applications, Vol. 31, pp. 473–499, 1980.

    Google Scholar 

  21. Borwein, J. M.,Multivalued Convexity and Optimization: A Unified Approach to Inequality Constraints, Mathematical Programming, Vol. 13, pp. 183–199, 1977.

    Google Scholar 

  22. Martos, B.,Quasi-Convexity and Quasi-Monotonocity in Nonlinear Programming, Studia Scientiarum Mathematicarum Hungarica, Vol. 2, pp. 265–273, 1967.

    Google Scholar 

  23. Mangasarian, O. L., andRosen, J. B.,Inequalities for Stochastic Nonlinear Programming Problems, Operations Research, Vol. 12, pp. 143–154, 1964.

    Google Scholar 

  24. Luenberger, D. G.,Optimization by Vector Space Methods, John Wiley, New York, New York, 1969.

    Google Scholar 

  25. Geoffrion, A. M.,Duality in Nonlinear Programming: A Simplified Application Oriented Development, SIAM Review, Vol. 13, pp. 1–37, 1971.

    Google Scholar 

  26. Pol'yak, B. T.,Existence Theorems and Convergence of Minimizing Sequences in Extremum Problems with Restrictions, Soviet Mathematics, Vol. 7, pp. 72–75, 1966.

    Google Scholar 

  27. Rockafellar, R. T.,Monotone Processes of Convex and Concave Type, American Mathematical Society, Memoir No. 77, Providence, Rhode Island, 1967.

    Google Scholar 

  28. Borwein, J. M.,A Strong Duality Theorem for the Minimum of a Family of Convex Programs, Journal of Optimization Theory and Applications, Vol. 31, pp. 453–472, 1980.

    Google Scholar 

  29. Tagawa, S.,Optimierung mit Mengenwertige Abbildungen, University of Mannheim, PhD Dissertation, 1978.

  30. Wolfe, P.,A Duality Theorem for Nonlinear Programming, Quarterly of Applied Mathematics, Vol. 19, pp. 239–244, 1961.

    Google Scholar 

  31. Ioffe, A. D.,Differentielles Generalisées d'Applications Localement Lipschitziennes d'un Espace de Banach dans un Autre, Comptes Rendus des Séances de l'Académie des Sciences, Vol. 289, Serie A–B, pp. 637–639, 1979.

    Google Scholar 

  32. Penot, J. P.,Differentiability of Relations and Differential Stability of Perturbed Optimization Problems, Université de Pau, Preprint, 1982.

  33. Crouzeix, J. P.,Continuité des Applications Linéaires Multivoques, Revue Francaise d'Automatique, Informatique, et de Recherche Opérationelle, Vol. 7, Serie R1, pp. 62–67, 1973.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Operations Research, School of Engineering and Applied Science, The George Washington University, Washington, DC

    A. V. Fiacco (Professor)

  2. Department of Decision Sciences, College of Business Administration, Florida International University, Miami, Florida

    J. Kyparisis (Assistant Professor)

Authors
  1. A. V. Fiacco
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  2. J. Kyparisis
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Additional information

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). https://doi.org/10.1007/BF00938592

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