Calmness of the Feasible Set Mapping for Linear Inequality Systems
- 190 Downloads
In this paper we deal with parameterized linear inequality systems in the n-dimensional Euclidean space, whose coefficients depend continuosly on an index ranging in a compact Hausdorff space. The paper is developed in two different parametric settings: the one of only right-hand-side perturbations of the linear system, and that in which both sides of the system can be perturbed. Appealing to the backgrounds on the calmness property, and exploiting the specifics of the current linear structure, we derive different characterizations of the calmness of the feasible set mapping, and provide an operative expresion for the calmness modulus when confined to finite systems. In the paper, the role played by the Abadie constraint qualification in relation to calmness is clarified, and illustrated by different examples. We point out that this approach has the virtue of tackling the calmness property exclusively in terms of the system’s data.
KeywordsCalmness Local error bounds Variational analysis Semi-infinite programming Linear programming Feasible set mapping
Mathematics Subject Classifications (2010)90C34 90C31 49J53 90C05
Unable to display preview. Download preview PDF.
- 5.Fabian, M., Henrion, R., Kruger, A.Y., Outrata, J.: About error bounds in metric spaces. In: Operations Research Proceeding 2011. Selected Paper of the International Conference Operations Research (OR 2011), August 30, September 2, 2011, Zurich, Switzerland, pp. 33–38. Springer, Berlin (2012)Google Scholar
- 9.Henrion, R., Klatte, D.: Regularity and stability in nonlinear semi-infinite optimization. In: Reemtsen, R., Rückmann, J.-J. (eds.) Semi-Infinite Programming. Kluwer (1998)Google Scholar
- 12.Ioffe, A.D.: Metric regularity and subdifferential calculus. Uspehi Mat. Nauk 55(3), 103–162 (2000). (in Russian), English translation: Russian Math. Surveys 55(3), 501–558 (2000)Google Scholar
- 14.Klatte, D., Kummer, B.: Nonsmooth equations in optimization: regularity, calculus, methods and applications. Nonconvex Optimization and Applications 60. Kluwer, Dordrecht (2002)Google Scholar
- 20.Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, I: Basic Theory. Springer, Berlin (2006)Google Scholar
- 21.Robinson, S.M.: Some continuity properties of polyhedral multifunctions. In: Mathematical programming at Oberwolfach (Proc. Conf. Math. Forschungsinstitut, Oberwolfach, 1979). Math. Program. Stud. No. 14, pp. 206–214 (1981)Google Scholar