Abstract
We introduce extensions of the Mangasarian-Fromovitz and Abadie constraint qualifications to nonsmooth optimization problems with feasibility given by means of lower-level sets. We do not assume directional differentiability, but only upper semicontinuity of the defining functions. By deriving and reviewing primal first-order optimality conditions for nonsmooth problems, we motivate the formulations of the constraint qualifications. Then, we study their interrelation, and we show how they are related to the Slater condition for nonsmooth convex problems, to nonsmooth reverse-convex problems, to the stability of parametric feasible set mappings, and to alternative theorems for the derivation of dual first-order optimality conditions.
In the literature on general semi-infinite programming problems, a number of formally different extensions of the Mangasarian-Fromovitz constraint qualification have been introduced recently under different structural assumptions. We show that all these extensions are unified by the constraint qualification presented here.
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References
Merkovsky, R. R., and Ward, D. E., General Constraint Qualifications in Nondifferentiable Programming, Mathematical Programming, Vol. 47, pp. 389–405, 1990.
Ward, D. E., A Constraint Qualification in Quasidifferentiable Programming, Optimization, Vol. 22, pp. 661–668, 1991.
Kuntz, L., and Scholtes, S., Constraint Qualifications is in Quasidifferentiable Optimization, Mathematical Programming, Vol. 60, pp. 339–347, 1993.
Demyanov, V. F., and Rubinov, A. M., Quasidifferential Calculus, Optimization Software, New York, NY, 1986.
Jourani, A., Constraint Qualifications and Lagrange Multipliers in Nondifferentiable Programming Problems, Journal of Optimization Theory and Applications, Vol. 81, pp. 533–548, 1994.
Kuntz, L., and Scholtes, S.,ANonsmooth Variant of the Mangasarian-Fromovitz Constraint Qualification, Journal of Optimization Theory and Applications, Vol. 82, pp. 59–75, 1994.
Laurent, P. J., Approximation et Optimization, Hermann, Paris, France, 1972.
Bonnans, J. F., and Shapiro, A., Perturbation Analysis of Optimization Problems, Springer, New York, NY, 2000.
Hettich, R., and Zencke, P., Numerische Methoden der Approximation und Semi-Infiniten Optimierung, Teubner, Stuttgart, Germany, 1982.
Stein, O., Bilevel Strategies in Semi-Infinite Programming, Kluwer Academic Publishers, Boston, MA, 2003.
Mangasarian, O. L., and Fromovitz, S., The Fritz John Necessary Optimality Conditions in the Presence of Equality and Inequality Constraints, Journal of Mathematical Analysis and Applications, Vol. 17, pp. 37–47, 1967.
Abadie, J. M., On the Kuhn-Tucker Theorem, Nonlinear Programming, Edited by J. Abadie, North Holland, Amsterdam, Holland, pp. 19–36, 1967 JOTA: VOL. 121, NO. 3, JUNE 2004 669
Peterson, D. W., A Review of Constraint Qualifications in Finite-Dimensional Spaces, SIAM Review, Vol. 15, pp. 639–654, 1973.
Rockafellar, R. T., Convex Analysis, Princeton University Press, Princeton, New Jersey, 1970.
Guddat, J., Jongen, H. T., Structural Stability in Nonlinear Optimization, Optimization, Vol. 18, pp. 617–631, 1987.
Guddat, J., Jongen, H. T., and Rückmann, J. J., On Stability and Stationary Points in Nonlinear Optimization, Journal of the Australian Mathematical Society, Vol. 28B, pp. 36–56, 1986.
Jongen, H. T., Twilt, F., and Weber, G. W., Semi-Infinite Optimization: Structure and Stability of the Feasible Set, Journal of Optimization Theory and Applications, Vol. 72, pp. 529–552, 1992.
Rockafellar, R. T., and Wets, R. J. B., Variational Analysis, Springer, Berlin, Germany, 1998
Berge, C., Topological Spaces, Oliver and Boyd, Edinburgh, Scotland, 1963.
Hogan, W. W., Point-to-Set Maps in Mathematical Programming, SIAM Review, Vol. 15, pp. 591–603, 1973.
Gould, F. J., and Tolle, J. W., Geometry of Optimality Conditions and Constraint Qualifications, Mathematical Programming, Vol. 2, pp. 1–18, 1972.
Guignard, M., Generalized Kuhn-Tucker Conditions for Mathematical Programming Problems in a Banach Space, SIAM Journal on Control, Vol. 7, pp. 232–241, 1969.
Wolkowicz, H.,Geometry of Optimality Conditions and Constraint Qualifications: The Convex Case, Mathematical Programming, Vol. 19, pp. 32–60, 1980.
Stein, O., First-Order Optimality Conditions for Degenerate Index Sets in Generalized Semi-Infinite Programming, Mathematics of Operations Research, Vol. 26, pp. 565–582, 2001.
Cheney, E. W., Introduction to Approximation Theory, McGraw-Hill, New York, NY, 1966.
Goberna, M. A., and López, M. A., Linear Semi-Infinite Optimization,Wiley, Chichester, England, 1998.
John, F., Extremum Problems with Inequalities as Subsidiary Conditions, Studies and Essays, R. Courant Anniversary Volume, Interscience, New York, NY, pp. 187–204, 1948.
Hettich, R., and Kortanek, K. O., Semi-Infinite Programming: Theory, Methods and Applications, SIAM Review, Vol. 35, pp. 380–429, 1993.
Danskin, J. M., The Theory of Max-Min and Its Applications to Weapons Allocation Problems, Springer, New York, NY, 1967.
Guerra vasquez, F., and Orozco, J. A., On Constraint Qualifications in Semi-Infinite Optimization, Preprint, Departamento de Física y Matemáticas, Escuela de Ciencias, Universidad de las Américas, Puebla, Mexico, 2001.
Kuhn, H. W., Nonlinear Programming: A Historical View, Nonlinear Programming, Edited by R. W. Cottle and C. E. Lemke, American Mathematical Society, Providence, Rhode Island, pp. 1–26, 1976. 670 JOTA: VOL. 121, NO. 3, JUNE 2004
Lempio, F., and Maurer, H., Differential Stability in Infinite-Dimensional Nonlinear Programming, Applied Mathematics and Optimization, Vol. 6, pp. 139–152, 1980.
Borwein, J. M., and Lewis, A. S., Partially-Finite Convex Programming, Part 1: Duality Theory, Mathematical Programming, Vol. 57, pp. 15–48, 1992.
Borwein, J. M., and Lewis, A. S., Partially-Finite Convex Programming, Part 2: Explicit Lattice Models, Mathematical Programming, Vol. 57, pp. 49–84, 1992.
Borwein, J. M., and Wolkowicz, H., Characterization of Optimality for the Abstract Convex Program with Finite-Dimensional Range, Journal of the Australian Mathematical Society, Vol. 30A, pp. 390–411, 1980/81.
Borwein, J. M., and Wolkowicz, H., A Simple Constraint Qualification in Infinite-Dimensional Programming, Mathematical Programming, Vol. 35, pp. 83–96, 1986.
Karney, D. F., A Duality Theorem for Semi-Infinite Convex Programs and Their Finite Subprograms, Mathematical Programming, Vol. 27, pp. 75–82, 1983.
Li, W., Nahak, C., and Singer, I., Constraint Qualifications for Semi-Infinite Systems of Convex Inequalities, SIAM Journal on Optimization, Vol. 11, pp. 31–52, 2000.
Hogan, W. W., Directional Derivatives for Extremal Value Functions with Applications to the Completely Convex Case, Operations Research, Vol. 21, pp. 188–209, 1973.
Still, G., Generalized Semi-Infinite Programming: Numerical Aspects, Optimization, Vol. 49, pp. 223–242, 2001.
Gauvin, J., and Dubeau, F., Differential Properties of the Marginal Function in Mathematical Programming, Mathematical Programming Study, Vol. 19, pp. 101–119, 1982.
Kyparisis, J., On Uniqueness of Kuhn-Tucker Multipliers in Nonlinear Programming, Mathematical Programming, Vol. 32, pp. 242–246, 1985.
Gauvin, J., and Dubeau, F., Some Examples and Counterexamples for the Stability Analysis of Nonlinear Programming Problems, Mathematical Programming Study, Vol. 21, pp. 69–78, 1983.
Gauvin, J., and Tolle, J. W., Differential Stability in Nonlinear Programming, SIAM Journal on Control and Optimization, Vol. 15, pp. 294–311, 1977
Stein, O., and Still, G., On Optimality Conditions for Generalized Semi-Infinite Programming Problems, Journal of Optimization Theory and Applications, Vol. 104, pp. 443–458, 2000.
Weber, G. W., Structural Stability in Generalized Semi-Infinite Optimization, Vychislitel'nye Tekhnologii, Vol. 6, pp. 25–46, 2001.
Rückmann, J. J., and Shapiro, A., First-Order Optimality Conditions in Generalized Semi-Infinite Programming, Journal of Optimization Theory and Applications, Vol. 101, pp. 677–691, 1999.
Jongen, H. T., Rückmann, J. J., and Stein, O., Generalized Semi-Infinite Optimization: A First Order Optimality Condition and Examples, Mathematical Programming, Vol. 83, pp. 145–158, 1998. JOTA: VOL. 121, NO. 3, JUNE 2004 671
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Stein, O. On Constraint Qualifications in Nonsmooth Optimization. Journal of Optimization Theory and Applications 121, 647–671 (2004). https://doi.org/10.1023/B:JOTA.0000037607.48762.45
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DOI: https://doi.org/10.1023/B:JOTA.0000037607.48762.45