Abstract
Combining results of Avakov about tangent directions to equality constraints given by smooth operators with results of Ben-Tal and Zowe, we formulate a second-order theory for optimality in the sense of Dubovitskii-Milyutin which gives nontrivial conditions also in the case of equality constraints given by nonregular operators. Secondorder feasible and tangent directions are defined to construct conical approximations to inequality and equality constraints which within a single construction lead to first- and second-order conditions of optimality for the problem also in the nonregular case. The definitions of secondorder feasible and tangent directions given in this paper allow for reparametrizations of the approximating curves and give approximating sets which form cones. The main results of the paper are a theorem which states second-order necessary condition of optimality and several corollaries which treat special cases. In particular, the paper generalizes the Avakov result in the smooth case.
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References
Ioffe, A. D., andTikhomirov, W. M.,Theory of Extremal Problems, North Holland, Amsterdam, Holland, 1979.
Girsanov, I. V.,Lectures on Mathematical Theory of Extremum Problems, Lecture Notes in Economics and Mathematical Systems, Springer, Berlin, Germany, 1972.
Ledzewicz-Kowalewska, U.,On Some Specification of the Dubovitskii-Milyutin Method, Nonlinear Analysis, Vol. 10, pp. 1367–1371, 1986.
Ben-Tal, A., andZowe, J.,A Unified Theory of First and Second-Order Conditions for Extremum Problems in Topological Vector Spaces, Mathematical Programming Study, Vol. 19, pp. 39–76, 1982.
Messerli, E. J., andPolak, E.,On Second-Order Necessary Conditions of Optimality, SIAM Journal on Control, Vol. 7, pp. 272–291, 1969.
Hoffmann, K. H., andKornstaedt, H. J.,Higher-Order Necessary Conditions in Abstract Mathematical Programming, Journal of Optimization Theory and Applications, Vol. 26, pp. 533–568, 1978.
Maurer, H., andZowe, J.,First- and Higher-Order Necessary and Sufficient Optimality Conditions for Infinite-Dimensional Programming Problems, Mathematical Programming, Vol. 16, pp. 98–110, 1979.
Tretyakov, A. A.,Necessary and Sufficient Conditions of Optimality of p-Order, USSR Computational Mathematics and Mathematical Physics, Vol. 24, pp. 123–127, 1984.
Levitin, E. E., Milyutin, A. A., andOsmolovskii, N. P.,Conditions of Higher Order for a Local Minimum in Problems with Constraints, Russian Mathematical Surveys, Vol. 33, pp. 97–168, 1978.
Ioffe, A. D.,Necessary and Sufficient Conditions for a Local Minimum, Part 3: Second-Order Conditions and Augmented Duality, SIAM Journal on Control and Optimization, Vol. 17, pp. 266–288, 1979.
Dmitruk, A. W., Milyutin, A. A., andOsmolovskii, N. P.,Lusternik Theorem and Theory of Extremum, Uspekhi Matematicheskich Nauk, Vol. 25, pp. 11–46, 1980.
Warga, J.,Second-Order Controllability and Optimization with Ordinary Controls, SIAM Journal on Control and Optimization, Vol. 23, pp. 49–60, 1985.
Avakov, E. R.,Extremum Conditions for Smooth Problems with Equality-Type Constraints, USSR Computational Mathematics and Mathematical Physics, Vol. 25, pp. 680–693, 1985.
Avakov, E. R.,Necessary Extremum Conditions for Smooth Abnormal Problems with Equality and Inequality-Type Constraints, Mathematicheskie Zametki, Vol. 45, pp. 3–11, 1989.
Avakov, E. R.,Necessary Conditions for the Minimum for Nonregular Problems in Banach Spaces: Maximum Principle for Abnormal Problems of Optimal Control, Trudy Matematicheskovo Instituta AN SSSR, Vol. 185, pp. 3–29, 1988 (in Russian).
Ledzewicz, U.,Euler-Lagrange Equation in the Case of Nonregular Equality Constraints, Journal of Optimization Theory and Applications, Vol. 71, pp. 549–568, 1991.
Ledzewicz, U.,On Abnormal Optimal Control Problems with Mixed Equality and Inequality Constraints, Journal of Mathematical Analysis and Applications, Vol. 173, pp. 18–42, 1993.
Ledzewicz, U.,Extension of the Local Maximum Principle to Abnormal Optimal Control Problems, Journal of Optimization Theory and Applications, Vol. 77, pp. 661–681, 1993.
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Communicated by T. S. Angell
This research was supported by NSF Grant DMS-91-009324, NSF Grant DMS-91-00043, SIUE Research Scholar Award and Fourth Quarter Fellowship, Summer 1992.
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Ledzewicz, U., Schaettler, H. Second-order conditions for extremum problems with nonregular equality constraints. J Optim Theory Appl 86, 113–144 (1995). https://doi.org/10.1007/BF02193463
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DOI: https://doi.org/10.1007/BF02193463