Abstract
In this study, we present a unifying framework for the cones of tangents to an arbitrary set and some of its applications. We highlight the significance of these cones and their polars both from the point of view of differentiability and subdifferentiability theory and the point of view of mathematical programming. This leads to a generalized definition of a subgradient which extends the well-known definition of a subgradient of a convex function to the nonconvex case. As an application, we develop necessary optimality conditions for a min-max problem and show that these conditions are also sufficient under moderate convexity assumptions.
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References
Bouligand, G.,Introduction à la Géométrie Infinitésimale Directe, Gauthier-Villars, Paris, France, 1932.
Flett, T. M.,Mathematical Analysis, McGraw-Hill, Book Company, New York, New York, 1966.
Flett, T. M.,On Differentiation in Normed Vector Spaces, Journal of the London Mathematical Society, Vol. 42, 1967.
Roetman, E. L.,Tangent Planes and Differentiation, Mathematics Magazine, Vol. 43, No. 1, 1970.
Hestenes, M. R.,Calculus of Variations and Optimal Control Theory, John Wiley and Sons, New York, New York, 1966.
Nashed, M. Z.,Differentiability and Related Properties of Nonlinear Operators: Some Aspects of the Role of Differentials in Nonlinear Functional Analysis, Nonlinear Functional Analysis and Applications, Edited by L. B. Rall, Academic Press, New York, New York, 1971.
Abadie, J.,On the Kuhn-Tucker Theorem, Nonlinear Programming, Edited by J. Abadie, North Holland Publishing Company, Amsterdam, Holland, 1967.
Varaiya, P. P.,Nonlinear Programming in Banach Space, SIAM Journal on Applied Mathematics, Vol. 15, No. 2, 1967.
Guignard, M.,Generalized Kuhn-Tucker Conditions for Mathematical Programming Problems in a Banach Space, SIAM Journal on Control, Vol. 7, No. 12, 1969.
Gould, F. J., andTolle, J. W.,A Necessary and Sufficient Constraint Qualification for Constrained Optimization, SIAM Journal on Applied Mathematics, Vol. 20, No. 2, 1971.
Bazaraa, M. S., Goode, J. J., andShetty, C. M.,Constraint Qualifications Revisited, Management Science, Vol. 18, No. 9, 1972.
Bazaraa, M. S., Goode, J. J., Nashed, M. Z., andShetty, C. M.,Nonlinear Programming Without Differentiability in Banach Spaces: Necessary and Sufficient Constraint Qualification, Applicable Analysis (to appear).
Neustadt, L. W.,A General Theory of Extremals, Journal of Computer and System Sciences, Vol. 3, No. 1, 1969.
Halkin, H.,A Satisfactory Treatment of Equality and Operator Constraints in the Dubovitskii-Milyutin Optimization Formalism, Journal of Optimization Theory and Applications, Vol. 6, No. 2, 1970.
Dubovitskii, A. Ya., andMilyutin, A. A.,Extremum Problems in the Presence of Restrictions, USSR Computational Mathematics and Mathematical Physics, Vol. 5, No. 1, 1965.
Rockafellar, R. T.,Duality and Stability in Extremum Problems Involving Convex Functions, Pacific Journal of Mathematics, Vol. 21, No. 1, 1967.
Rockafellar, R. T.,Duality in Nonlinear Programming, Mathematics of the Decision Sciences, Edited by G. B. Dantzig and A. F. Veinott, American Mathematical Society, Providence, Rhode Island, 1968.
Demyanov, V. F.,Algorithms for Some Minimax Problems. Journal of Computer and System Sciences, Vol. 2, No. 4, 1968.
Demyanov, V. F., andRubinov, A. M.,Approximate Methods for Solving Extremal Problems, American Elsevier Publishing Company, New York, New York, 1970.
Valentine, F. A.,Convex Sets, McGraw-Hill, Book Company, New York, New York, 1964.
Rockafellar, R. T.,Convex Analysis, Princeton University Press, Princeton, New Jersey, 1970.
Van Slyke, R. M., andWets, R. J. B.,A Duality Theorem for Abstract Mathematical Programming with Applications to Optimal Control Theory, Journal of Mathematical Analysis and Applications, Vol. 22, No. 3, 1968.
Brøndsted, A., andRockafellar, R. T.,On Subdifferentiability of Convex Functions, Proceedings of American Mathematical Society, Vol. 16, No. 4, 1965.
Bazaraa, M. S., Goode, J. J., andShetty, C. M.,Optimality Criteria in Nonlinear Programming without Differentiability, Journal of Operations Research, Vol. 19, No. 1, 1971.
Bazaraa, M. S., Goode, J. J., andNashed, M. Z.,Cone Differentiability in Normed Linear Spaces with Applications (to appear).
Mangasarian, O. L.,Nonlinear Programming, McGraw-Hill Book Company, New York, New York, 1969.
Fan, K.,On Systems of Linear Inequalities, Linear Inequalities and Related Systems, Edited by H. W. Kuhn and A. W. Tucker, Annals of Mathematics Study, 38, Princeton University Press, Princeton, New Jersey, 1956.
Hartman, P.,Ordinary Differential Equations, John Wiley and Sons, New York, New York, 1964.
Moreau, J. J.,Fonctionelles Convexes, Lecture Notes, Seminar on Partial Differential Equations, College de France, Montpellier, France, 1966–1967.
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Communicated by M. R. Hestenes
The work of the third author was sponsored by the United States Army, Contract No. DA-31-124-ARO-D-462, while he was at the Mathematics Research Center, The University of Wisconsin, Madison, Wisconsin.
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Bazaraa, M.S., Goode, J.J. & Nashed, M.Z. On the cones of tangents with applications to mathematical programming. J Optim Theory Appl 13, 389–426 (1974). https://doi.org/10.1007/BF00934938
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DOI: https://doi.org/10.1007/BF00934938