PSEUDO-LIPSCHITZIAN CONTINUITY OF SOLUTION MAPPINGS IN PARAMETRICAL OPTIMIZATION PROBLEMS
Abstract
The study of the properties of solution mappings in parametrical optimization problems represents an urgent problem. Particularly, considerable efforts are directed to finding the conditions of different types of generalized Lipschitzian continuity of solution mappings, namely their calmness and pseudo-Lipschitzian continuity (also referred to as the Aubin property) [1]. A new interesting approach to investigating the calmness of solution mappings has recently been proposed by Canovas et al. [2] for parametrical linear programming problems and applied to a much wider range of problems by Klatte and Kummer [3]. In this approach, the calmness of solution mappings is related to the calmness of an associated system representing a constraint on the level set of the objective function on the domain of the problem. In our note, we propose to expand the use of the approach [3] for investigating the pseudo-Lipschitzian continuity of solution mappings. Several sufficient conditions for the pseudo-Lipschitzian continuity of solution mappings, as well as the generalization of the Hoffman lemma are presented.
About the Authors
L. I. Minchenko
Belarusian State University of Informatics and Radioelectronics
Belarus
D. Sc. (Physics and Mathematics), Professor, Professor of the Department of Informatics
Belarus
D. Sc. (Physics and Mathematics), Professor, Professor of the Department of Informatics
D. E. Berezhnov
Belarusian State University of Informatics and Radioelectronics
Belarus
Assistant of the Department of Informatics
Belarus
Assistant of the Department of Informatics
References
1. Rockafellar R. T., Wets R. J.¬B. Variational analysis. Springer, Berlin, 1998. 732 p. Doi: 10.1007/978¬3¬642¬02431¬3
2. Canovas M. J., Hantoute A., Parra J., Toledo F. J. Calmness of the argmin mapping in linear semi¬infinite optimization. Journal on Optimization Theory and Applications, 2014, vol. 160, pp. 111–126. Doi: 10.1007/s10957¬013-0371¬z
3. Klatte D., Kummer B. On calmness of the argmin mapping in parametric optimization problems. Journal on Op ti¬mization Theory and Applications, 2014, vol. 165, no. 3, pp. 708–719. Doi: 10.1007/s10957¬014¬0643¬2
4. Henrion R., Outrata J. Calmness of constraint systems with applications. Mathematical Programming, 2005, vol. 104, no. 2¬3, pp. 437–464. Doi: 10.1007/s10107¬005¬0623¬2
5. Ioffe A. D., Outrata J. On metric and calmness qualification conditions in subdifferential calculus. Set¬Valued Analysis, 2008, vol. 16, pp. 199–227. Doi: 10.1007/s11228¬008¬0076¬x
6. Dontchev A. L., Rockafellar R. T. Implicit functions and solution mappings. New York, Springer, 2009. 574 p. Doi: 10.1007/978¬0¬387¬87821¬8
7. Shu Lu. Implications of the constant rank constraint qualification. Mathematical Programming, 2009, vol. 126, pp. 365–392. Doi: 10.1007/s10107¬009¬0288¬3
8. Klatte D., Kummer B. Aubin property and uniqueness of solutions in cone constrained optimization. Mathematical Methods of Operational Research, 2013, vol. 77, no. 3, pp. 291–304. Doi: 10.1007/s00186¬013¬0429¬6
9. Fedorov V. V. Numerical Maximilian Methods. Мoscow, Nauka Publ., 1979. 280 p. (in Russian).
10. Luderer B., Minchenko L., Satsura T. Multivalued analysis and nonlinear programming problems with perturbations. Dordrecht, Kluwer Acad. Publ., 2002. 207 p. Doi: 10.1007/978¬1¬4757¬3468¬3
11. Hoffman A. J. On approximate solutions of systems of linear inequalities. Journal of Research of the National Bureau of Standards, 1952, vol. 49, no. 4, pp. 263–265.
12. Pshenichnyi B. N. Convex Analysis and Extremum Problems. Moscow, Nauka Publ., 1980. 320 p. (in Russian).
13. Minchenko L. I., Tarakanov A. N. On second order derivatives of value functions. Optimization, 2015, vol. 64, no. 2, pp. 389–407.
Review
Views: 697
Email this article 