Differentiable selection of optimal solutions in parametric linear programming

Luc, Dinh; Pham Huy Dien

doi:10.1090/S0002-9939-97-03090-6

Differentiable selection of optimal solutions in parametric linear programming
HTML articles powered by AMS MathViewer

by Dinh The Luc and Pham Huy Dien PDF

Proc. Amer. Math. Soc. 125 (1997), 883-892 Request permission

Abstract:

In the present paper we prove that if the data of a parametric linear optimization problem are smooth, the solution map admits a local smooth selection “almost” everywhere. This in particular shows that the set of points where the marginal function of the problem is nondifferentiable is nowhere dense.

References

Jean-Pierre Aubin, Further properties of Lagrange multipliers in nonsmooth optimization, Appl. Math. Optim. 6 (1980), no. 1, 79–90. MR 557056, DOI 10.1007/BF01442884
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
C. Berge, Topological spaces, Macmillan, New York 1963.
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
Pham Huy Dien and Dinh The Luc, On the calculation of generalized gradients for a marginal function, Acta Math. Vietnam. 18 (1993), no. 2, 309–326. MR 1292088
J. P. Evans and F. J. Gould, Stability in nonlinear programming, Operations Res. 18 (1970), 107–118. MR 264984, DOI 10.1287/opre.18.1.107
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
Jacques Gauvin and François Dubeau, Differential properties of the marginal function in mathematical programming, Math. Programming Stud. 19 (1982), 101–119. Optimality and stability in mathematical programming. MR 669727, DOI 10.1007/bfb0120984
J.-B. Hiriart-Urruty, Gradients généralisés de fonctions marginales, SIAM J. Control Optim. 16 (1978), no. 2, 301–316 (French, with English summary). MR 493610, DOI 10.1137/0316019
K. Kuratovskiĭ, Topologiya. Tom I, Izdat. “Mir”, Moscow, 1966 (Russian). Translated from the English by M. Ja. Antonovskiĭ; With a preface by P. S. Aleksandrov. MR 0217750
J. V. Outrata, On generalized gradients in optimization problems with set-valued constraints, Math. Oper. Res. 15 (1990), no. 4, 626–639. MR 1080469, DOI 10.1287/moor.15.4.626
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
J.-P. Penot, Preservation of persistence and stability under intersections and operations. I. Persistence, J. Optim. Theory Appl. 79 (1993), no. 3, 525–550. MR 1255285, DOI 10.1007/BF00940557
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, A characterization of stability in linear programming, Operations Res. 25 (1977), no. 3, 435–447. MR 446490, DOI 10.1287/opre.25.3.435
R. T. Rockafellar, Lagrange multipliers and subderivatives of optimal value functions in nonlinear programming, Math. Programming Stud. 17 (1982), 28–66. MR 654690, DOI 10.1007/bfb0120958
J. E. Spingarn, Fixed and variable constraints in sensitivity analysis, SIAM J. Control Optim. 18 (1980), no. 3, 297–310. MR 569019, DOI 10.1137/0318021
Lionel Thibault, On subdifferentials of optimal value functions, SIAM J. Control Optim. 29 (1991), no. 5, 1019–1036. MR 1110085, DOI 10.1137/0329056
D. E. Ward, Differential stability in non-Lipschitzian optimization, J. Optim. Theory Appl. 73 (1992), no. 1, 101–120. MR 1152238, DOI 10.1007/BF00940081
N. D. Yen and P. H. Dien, On differential estimations for marginal functions in mathematical programming problems with inclusion constraints, Lectures Notes in Control and Inform. Sci. vol. 143, Springer-Verlag, Berlin, 1990, pp. 244–251.