On the strong metric subregularity in mathematical programming

• Autorzy: Osmolovskii Nikolai P. , Veliov Vladimir M.

Identyfikatory

DOI

10.2478/candc-2021-0027

• Języki publikacji: EN

Abstrakty

EN

This note presents sufficient conditions for the property of strong metric subregularity (SMSr) of the system of first order optimality conditions for a mathematical programming problem in a Banach space (the Karush-Kuhn-Tucker conditions). The constraints of the problem consist of equations in a Banach space setting and a finite number of inequalities. The conditions, under which SMSr is proven, assume that the data are twice continuously Fréchet differentiable, the strict Mangasarian-Fromovitz constraint qualification is satisfied, and the second-order sufficient optimality condition holds. The obtained result extends the one known for finite-dimensional problems. Although the applicability of the result is limited to the Banach space setting (due to the twice Fréchet differentiability assumptions and the finite number of inequality constraints), the paper can be valuable due to the self-contained exposition, and provides a ground for extensions. One possible extension was recently implemented in Osmolovskii and Veliov (2021).

Słowa kluczowe

EN

optimization; mathematical programming; Karush-Kuhn-Tucker conditions; metric regularity

• Wydawca: Systems Research Institute, Polish Academy of Sciences
• Czasopismo: Control and Cybernetics
• Rocznik: 2021
• Tom: Vol. 50, No. 4
• Strony: 457--471
• Opis fizyczny: Bibliogr. 15 poz.

Twórcy

autor

Osmolovskii Nikolai P.

• Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

autor

Veliov Vladimir M.

• Institute of Statistics and Mathematical Methods in Economics, Vienna University of Technology, Austria

Bibliografia

• Bonnans, J.F. and Shapiro, A. (2000) Perturbation Analysis of Optimization Problems. Springer.
• Cibulka, R. Dontchev, A.L. and Kruger, A.Y. (2018) Strong metric subregularity of mappings in variational analysis and optimization. Journal of Mathematical Analysis and Application, 457: 1247–1282.
• Dmitruk, A.V. and Osmolovskii, N.P. (2018) A General Lagrange Multipliers Theorem and Related Questions. In: Control Systems and Mathematical Methods in Economics (Feichtinger et al., eds.), Lecture Notes in Economics and Mathematical Systems, 687: 165–194, Springer, Berlin.
• Dontchev, A. L. and Rockafellar, R. T. (1998) Characterizations of Lipschitz stability in nonlinear programming. Mathematical Programming with Data Perturbations, 65–82, Lecture Notes in Pure and Appl. Math., 195, Dekker, New York.
• Dontchev, A. L. and Rockafellar, R. T. (2004) Regularity and conditioning of solution mappings in variational analysis. Set-Valued Analysis, 12: 79–109.
• Dontchev, A. L. and Rockafellar, R. T. (2014) Implicit Functions and Solution Mappings: A View from Variational Analysis. Second edition. Springer, New York.
• Dubovitskii, A. Ya. and Milyutin, A.A. (1965) Extremum problems in the presence of restrictions. USSR Comput. Math. and Math. Phys., 5(3): 1–80.
• Hoffman, A.J. (1952) On approximate solutions of systems of linear inequalities. J. Res. Nat’l Bur. Standards 49: 263–265.
• Ioffe, A.D. (1979) Regular points of Lipschitz functions. Trans. Am. Math. Soc. 251, 61–69.
• Ioffe, A.D. (2017) Variational Analysis and Regular Mappings. Springer.
• Ioffe, A.D and Tikhomorov, V.M. (1974) Theory of Extremal Problems. Nauka, Moscow (in Russian); English translation: North Holland, 1979.
• Klatte, D. and Kummer, B. (2002) Nonsmooth Equations in Optimization. Kluwer Academic Publisher.
• Kyparisis, J. (1985) On uniqueness of Kuhn-Tucker multiplies in non-linear programming. Math. Programming, 32: 242–246.
• Levitin, E.S., Milyutin, A.A. and Osmolovskii, N.P. (1978) Higherorder local minimum conditions in problems with constraints. Uspekhi Mat. Nauk, 33: 8–148; English translation in Russian Math. Surveys, 33: 9–168.
• Osmolovskii, N.P. and Veliov, V.M. (2021) On the strong subregularity of the optimality mapping in mathematical programming and calculus of variations. J. Math. Anal. Appl. 500(1), Article 125077.