Abstract
We present an overview to Lipschitz-type properties of mappings associated with solutions of optimization problems including variational inequalities and mathematical programs. We show that these properties are inherited in various ways by the mapping acting from parameters of the problem and the starting point to the set of sequences generated by Newton’s method. Some new insights into convergence of Newton’s/SQP method are also presented.
This research was supported by the National Science Foundation.
The original version of this chapter was revised: The copyright line was incorrect. This has been corrected. The Erratum to this chapter is available at DOI: 10.1007/978-0-387-35514-6_15
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References
J.-P. Aubin (1984), Lipschitz behavior of solutions to convex minimization problems, Math. Oper. Res., 9, pp. 87–111.
R. Cominetti (1990), Metric regularity, tangent sets, and second-order optimality conditions, Appl. Math. Optim., 21, pp. 265–287.
A.V. Dmitruk, A.A. Milyutin and N.P. Osmolovskii (1980), The Lyusternik theorem and the theory of extremum, Uspekhi Math. Nauk, 35, pp. 11–46.
A.L. Dontchev (1995), Characterizations of Lipschitz stability in optimization, in Recent developments in well-posed variational problems„ Math. Appl., 331, Kluwer, Dordrecht, pp. 95–115.
A.L. Dontchev (1996), The Graves theorem revisited, J. Convex Anal., 3, pp. 45–53.
A.L. Dontchev, W.W. Hager and V. Veliov, Uniform convergence and mesh independence of Newton’s method for discretized variational problems, SIAM J. Control and Optim.,to appear.
A.L. Dontchev and R.T. Rockafellar (1996), Characterizations of strong regularity for variational inequalities over polyhedral convex sets, SIAM J. Optim., 6, pp. 1087–1105.
A.L. Dontchev and R.T. Rockafellar (1998), Characterizations of Lipschitzian stability in nonlinear programming, in Mathematical programming with data perturbations, Lecture Notes in Pure and Appl. Math., 195, Dekker, New York, pp. 65–82.
W.W. Hager and M.S. Gowda (1999), Stability in the presence of degeneracy and error estimation, Math. Programming, 85, pp. 181192.
A.D. Ioffe, Metric regularity and subdifferential calculus, to appear.
A.J. King and R.T. Rockafellar (1992), Sensitivity analysis for non-smooth generalized equations, Math. Programming, 5, Ser. A, pp. 193–212.
D. Klatte, Upper Lipschitz behavior of solutions to perturbed C1,1 programs preprint.
D. Klatte and B. Kummer (1999), Strong stability in nonlinear programming revisited, J. Austral. Math. Soc., 40, pp. 336–352.
B. Kummer (1998), Lipschitzian and pseudo-Lipschitzian inverse functions and applications to nonlinear optimization, in Mathematical programming with data perturbations, Lecture Notes in Pure and Appl. Math., 195, Dekker, New York, pp. 201–222.
A.B. Levy (1996), Implicit multifunction theorems for the sensitivity analysis of variational conditions, Math. Programming, 74, pp. 333–350.
A. Levy, Solution sensitivity from general principles, preprint.
R.A. Poliquin and R.T. Rockafellar (1998), Tilt stability of local minima, SIAM J. Optim., 2, pp. 287–299.
A. Levy, R.A. Poliquin and R.T. Rockafellar, Stability of locally optimal solutions, SIAM J. Optim., to appear.
J. Renegar (1995), Linear programming, complexity theory and elementary functional analysis, Math. Programming, 70, pp. 279–351.
S.M. Robinson (1979), Generalized equations and their solution, Part I: Basic Theory, Math. Programming Study, 10, pp. 128–1. 41.
S.M. Robinson (1980), Strongly regular generalized equations, Math. Oper. Res., 5, pp. 43–62.
S.M. Robinson (1981), Some continuity properties of polyhedral multifunctions, Math. Programming Study, 14, pp. 206–214.
R.T. Rockafellar (1989), Proto-differentiability of set-valued mappings and its applications in optimization, Analyse non linéaire (Perpignan, 1987), Ann. Inst. H. Poincaré Anal. Non Linéaire 6, suppl., pp. 449–482.
R.T. Rockafellar, Extended nonlinear programming, in Nonlinear Optimization and Applications 2, Kluwer, Dordrecht, to appear.
R.T. Rockafellar and R.J.-B. Wets (1997), Variational analysis, Springer-Verlag, Berlin.
H. Xu (1999), Set-valued approximations and Newton’s methods, Math. Programming, 84, pp. 401–420.
T. Zolezzi (1998), Well-posedness and conditioning of optimization problems, Proc. 4th International Conf. Math. Methods Oper. Research and 6th Workshop on Well-posedness and Stability of Optimization Problems, Pliska Stud. Math. Bulgar., 12, pp. 267–280.
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Dontchev, A.L. (2000). Lipschitzian Stability of Newton’s Method for Variational Inclusions. In: Powell, M.J.D., Scholtes, S. (eds) System Modelling and Optimization. CSMO 1999. IFIP — The International Federation for Information Processing, vol 46. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-35514-6_6
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