Sensitivity analysis of solutions to generalized equations

Levy, A. B.; Rockafellar, R. T.

doi:10.1090/S0002-9947-1994-1260203-5

Sensitivity analysis of solutions to generalized equations
HTML articles powered by AMS MathViewer

by A. B. Levy and R. T. Rockafellar

Trans. Amer. Math. Soc. 345 (1994), 661-671

DOI: https://doi.org/10.1090/S0002-9947-1994-1260203-5

PDF | Request permission

Abstract:

Generalized equations are common in the study of optimization through nonsmooth analysis. For instance, variational inequalities can be written as generalized equations involving normal cone mappings, and have been used to represent first-order optimality conditions associated with optimization problems. Therefore, the stability of the solutions to first-order optimality conditions can be determined from the differential properties of the solutions of parameterized generalized equations. In finite-dimensions, solutions to parameterized variational inequalities are known to exhibit a type of generalized differentiability appropriate for multifunctions. Here it is shown, in a Banach space setting, that solutions to a much broader class of parameterized generalized equations are "differentiable" in a similar sense.

References

Implicit B-differentiability in generalized equations

Alan J. King and R. Tyrrell Rockafellar, Sensitivity analysis for nonsmooth generalized equations, Math. Programming 55 (1992), no. 2, Ser. A, 193–212. MR 1167597, DOI 10.1007/BF01581199
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

On Robinson’s implicit function theorem

R. T. Rockafellar, Proto-differentiability of set-valued mappings and its applications in optimization, Ann. Inst. H. Poincaré C Anal. Non Linéaire 6 (1989), no. suppl., 449–482. Analyse non linéaire (Perpignan, 1987). MR 1019126, DOI 10.1016/S0294-1449(17)30034-3

Partical extension of Attouch’s theorem with application to proto-derivatives of subgradient mappings

Second-order variational analysis with applications to sensitivity in optimization

R. A. Poliquin and R. T. Rockafellar, Tilt stability of a local minimum, SIAM J. Optim. 8 (1998), no. 2, 287–299. MR 1618790, DOI 10.1137/S1052623496309296

Amenable functions in optimization

A. B. Levy, Second-order epi-derivatives of integral functionals, Set-Valued Anal. 1 (1993), no. 4, 379–392. MR 1267204, DOI 10.1007/BF01027827
A. Shapiro, On concepts of directional differentiability, J. Optim. Theory Appl. 66 (1990), no. 3, 477–487. MR 1080259, DOI 10.1007/BF00940933
Stephen M. Robinson, Local structure of feasible sets in nonlinear programming. III. Stability and sensitivity, Math. Programming Stud. 30 (1987), 45–66. Nonlinear analysis and optimization (Louvain-la-Neuve, 1983). MR 874131, DOI 10.1007/bfb0121154
R. T. Rockafellar, Nonsmooth analysis and parametric optimization, Methods of nonconvex analysis (Varenna, 1989) Lecture Notes in Math., vol. 1446, Springer, Berlin, 1990, pp. 137–151. MR 1079762, DOI 10.1007/BFb0084934
A. Haraux, How to differentiate the projection on a convex set in Hilbert space. Some applications to variational inequalities, J. Math. Soc. Japan 29 (1977), no. 4, 615–631. MR 481060, DOI 10.2969/jmsj/02940615
F. Mignot, Contrôle dans les inéquations variationelles elliptiques, J. Functional Analysis 22 (1976), no. 2, 130–185 (French). MR 0423155, DOI 10.1016/0022-1236(76)90017-3
R. T. Rockafellar, First- and second-order epi-differentiability in nonlinear programming, Trans. Amer. Math. Soc. 307 (1988), no. 1, 75–108. MR 936806, DOI 10.1090/S0002-9947-1988-0936806-9
Anthony V. Fiacco and Jerzy Kyparisis, Sensitivity analysis in nonlinear programming under second order assumptions, Systems and optimization (Enschede, 1984) Lect. Notes Control Inf. Sci., vol. 66, Springer, Berlin, 1985, pp. 74–97. MR 878584, DOI 10.1007/BFb0043393