Smooth representation of a parametric polyhedral convex set with application to sensitivity in optimization

Luc, Dinh

doi:10.1090/S0002-9939-97-03507-7

Smooth representation of a parametric polyhedral convex set with application to sensitivity in optimization
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Proc. Amer. Math. Soc. 125 (1997), 555-567 Request permission

Abstract:

We show in this paper that if a polyhedral convex set is defined by a parametric linear system with smooth entries, then it possesses local smooth representation almost everywhere. This result is then applied to study the differentiability of the solutions and the marginal functions of several classes of parametric optimization problems.

References

Jean-Pierre Aubin and Hélène Frankowska, Set-valued analysis, Systems & Control: Foundations & Applications, vol. 2, Birkhäuser Boston, Inc., Boston, MA, 1990. MR 1048347
B. Bank, J. Guddat, D. Klatte, B. Kummer, and K. Tammer, Nonlinear parametric optimization, Birkhäuser Verlag, Basel-Boston, Mass., 1983. MR 701243
Claude Berge, Espaces topologiques: Fonctions multivoques, Collection Universitaire de Mathématiques, Vol. III, Dunod, Paris, 1959 (French). MR 0105663
George B. Dantzig, Jon Folkman, and Norman Shapiro, On the continuity of the minimum sets of a continuous function, J. Math. Anal. Appl. 17 (1967), 519–548. MR 207426, DOI 10.1016/0022-247X(67)90139-4
Anthony V. Fiacco, Introduction to sensitivity and stability analysis in nonlinear programming, Mathematics in Science and Engineering, vol. 165, Academic Press, Inc., Orlando, FL, 1983. MR 721641
Reiner Horst and Hoang Tuy, Global optimization, Springer-Verlag, Berlin, 1990. Deterministic approaches. MR 1102239, DOI 10.1007/978-3-662-02598-7
A. B. Levy and R. T. Rockafellar, Sensitivity analysis of solutions to generalized equations, Trans. Amer. Math. Soc. 345 (1994), no. 2, 661–671. MR 1260203, DOI 10.1090/S0002-9947-1994-1260203-5
Đinh Thế Lục, Random version of the theorems of the alternative, Math. Nachr. 129 (1986), 149–155. MR 864629, DOI 10.1002/mana.19861290113
D. T. Luc and P. H. Dien, Differentiable selection of optimal solutions in parametric linear programming, Proc. Amer. Math. Soc. (in press).
J.-P. Penot, Preservation of persistence and stability under intersections and operations, Parts I, II, J. Optim. Theory Appl. 79 (1993), 525–550, 551–561.
Jean-Paul Penot, Compact nets, filters, and relations, J. Math. Anal. Appl. 93 (1983), no. 2, 400–417. MR 700155, DOI 10.1016/0022-247X(83)90184-1
Stephen M. Robinson, Stability theory for systems of inequalities. I. Linear systems, SIAM J. Numer. Anal. 12 (1975), no. 5, 754–769. MR 410521, DOI 10.1137/0712056
Stephen M. Robinson, An implicit-function theorem for a class of nonsmooth functions, Math. Oper. Res. 16 (1991), no. 2, 292–309. MR 1106803, DOI 10.1287/moor.16.2.292
R. Tyrrell Rockafellar, Convex analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970. MR 0274683