×
Fiacco, A. V.; Kyparisis, J.

Convexity and concavity properties of the optimal value function in parametric nonlinear programming. (English) Zbl 0556.90085

J. Optimization Theory Appl. 48, 95-126 (1986).
Convexity and concavity properties of the optimal value function \(f^*\) are considered for the general parametric optimization problem P(\(\epsilon)\) of the form \(\min_{x} f(x,\epsilon)\), s.t. \(x\in {\mathbb{R}}(\epsilon)\). Such properties of \(f^*\) and the solution set map \(S^*\) 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 of \(f^*\), in terms of respective convexity and concavity assumptions on f and the feasible region point-to-set map R. Specializations of these results to the general parametric inequality- equality constrained nonlinear programming problem and its right-hand- side version are provided.

Cited in 1 Review
Cited in 38 Documents

MathOverflow Questions:

Literature request: proving or disproving convexity of the optimal value function of semidefinite program (SDP) or convex optimization in general

MSC:

90C31 Sensitivity, stability, parametric optimization
90-02 Research exposition (monographs, survey articles) pertaining to operations research and mathematical programming
49M37 Numerical methods based on nonlinear programming

Keywords:

convexity conditions; set-valued mappings; survey; optimal value function; parametric optimization
Cite Review PDF
Full Text: DOI

References:

[1] Berge, G.,Topological Spaces, Macmillan, New York, New York, 1963.
[2] Rockafellar, R. T.,Convex Analysis, Princeton University Press, Princeton, New Jersey, 1970. · Zbl 0193.18401
[3] Rockafellar, R. T.,Conjugate Duality and Optimization, SIAM, Philadelphia, Pennsylvania, 1974. · Zbl 0296.90036
[4] Brosowski, B.,Parametric Semi-Infinite Optimization, Verlag Peter Lang, Frankfurt am Main, Germany, 1982. · Zbl 0502.49001
[5] Bank, B., Guddat, J., Klatte, D., Kummer, B., andTammer, K.,Nonlinear Parametric Optimization, Akademie Verlag, Berlin, Germany, 1982. · Zbl 0502.49002
[6] Fiacco, A. V.,Introduction to Sensitivity and Stability Analysis in Nonlinear Programming, Academic Press, New York, New York, 1983. · Zbl 0543.90075
[7] Mangasarian, O. L.,Nonlinear Programming, McGraw-Hill, New York, New York, 1969.
[8] Ortega, J. M., andRheinboldt, W. C.,Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, New York, 1970. · Zbl 0241.65046
[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. · Zbl 0539.90087
[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. · Zbl 0113.14202
[12] Rolewicz, S.,On Paraconvex Multifunctions, Operations Research Verfahren, Vol. 31, pp. 539–546, 1979. · Zbl 0403.49021
[13] Rolewicz, S.,On Graph {\(\gamma\)}-Paraconvex Multifunctions, Special Topics of Applied Mathematics, Edited by J. Frehse, D. Pallaschke, and U. Trottenberg, North-Holland, Amsterdam, Holland, 1980.
[14] Rolewicz, S.,On Conditions Warranting {\(\Psi\)} 2 -Subdifferentiability, Mathematical Programming Study, Vol. 14, pp. 215–224, 1981. · Zbl 0444.90106 · doi:10.1007/BFb0120930
[15] Stern, M. H., andTopkis, D. M.,Rates of Stability in Nonlinear Programming, Operations Research, Vol. 24, pp. 462–476, 1976. · Zbl 0352.90062 · doi:10.1287/opre.24.3.462
[16] Dolecki, S.,Remarks on Semicontinuity, Bulletin de l’Académic Polonaise des Sciences, Vol. 25, pp. 863–867, 1977. · Zbl 0372.54008
[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. · Zbl 0401.90085
[18] Robinson, S. M.,Some Continuity Properties of Polyhedral Multifunctions, Mathematical Programming Study, Vol. 14, pp. 206–214, 1981. · Zbl 0449.90090 · doi:10.1007/BFb0120929
[19] Tanino, T., andSawaragi, Y.,Duality Theory in Multiobjective Programming, Journal of Optimization Theory and Applications, Vol. 27, pp. 509–529, 1979. · Zbl 0378.90100 · doi:10.1007/BF00933437
[20] Tanino, T., andSawaragi, Y.,Conjugate Maps and Duality in Multiobjective Optimization, Journal of Optimization Theory and Applications, Vol. 31, pp. 473–499, 1980. · Zbl 0418.90080 · doi:10.1007/BF00934473
[21] Borwein, J. M.,Multivalued Convexity and Optimization: A Unified Approach to Inequality Constraints, Mathematical Programming, Vol. 13, pp. 183–199, 1977. · Zbl 0375.90062 · doi:10.1007/BF01584336
[22] Martos, B.,Quasi-Convexity and Quasi-Monotonocity in Nonlinear Programming, Studia Scientiarum Mathematicarum Hungarica, Vol. 2, pp. 265–273, 1967. · Zbl 0178.22901
[23] Mangasarian, O. L., andRosen, J. B.,Inequalities for Stochastic Nonlinear Programming Problems, Operations Research, Vol. 12, pp. 143–154, 1964. · Zbl 0132.13806 · doi:10.1287/opre.12.1.143
[24] Luenberger, D. G.,Optimization by Vector Space Methods, John Wiley, New York, New York, 1969. · Zbl 0176.12701
[25] Geoffrion, A. M.,Duality in Nonlinear Programming: A Simplified Application Oriented Development, SIAM Review, Vol. 13, pp. 1–37, 1971. · Zbl 0232.90049 · doi:10.1137/1013001
[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.
[27] Rockafellar, R. T.,Monotone Processes of Convex and Concave Type, American Mathematical Society, Memoir No. 77, Providence, Rhode Island, 1967. · Zbl 0189.19602
[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. · Zbl 0418.90070 · doi:10.1007/BF00934472
[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. · Zbl 0109.38406 · doi:10.1090/qam/135625
[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.
[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. · Zbl 0258.46015 · doi:10.1051/m2an/197307R100621
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.