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Partial Smoothness,Tilt Stability, and Generalized Hessians

Adrian Lewis(aslewis***at***orie.cornell.edu)
Shanshan Zhang(sz254***at***cornell.edu)

Abstract: We compare two recent variational-analytic approaches to second-order conditions and sensitivity analysis for nonsmooth optimization. We describe a broad setting where computing the generalized Hessian of Mordukhovich is easy. In this setting, the idea of tilt stability introduced by Poliquin and Rockafellar is equivalent to a classical smooth second-order condition.

Keywords: variational analysis, nonsmooth optimization, second-order, sensitivity analysis, prox-regular, subdifferential continuity, partial smoothness, generalized Hessian, tilt stability

Category 1: Convex and Nonsmooth Optimization (Nonsmooth Optimization )

Citation: [1] F. J. A. Artacho and M. H. Geoffroy, Characterization of metric regularity of subdi erentials, Convex Anal., 15 (2008), pp. 365–380. [2] J. Bolte, A. Daniilidis, and A. S. Lewis, Tame mappings are semismooth, Math. Program., Vol. 117 (2009). [3] J. F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems, Springer, New York, 2000. [4] A. L. Dontchev and R. T. Rockafellar, Implicit Functions and Solutions Mappings, Springer, New York, 2009. [5] W. L. Hare and A. S. Lewis, Identifying active constraints via partial smoothness and prox-regularity, Convex Anal., 11 (2004), pp. 251–266. [6] R. Henrion, J. Outrata, and T. Surowiec, On the co-derivative of normal cone mappings to inequality systems, Nonlinear Anal., 71 (2009), pp. 1213–1226. [7] J. M. Lee, Introduction to Smooth Manifolds, Springer, New York, 2006. [8] A. S. Lewis, Active sets, nonsmoothness and sensitivity, SIAM J. Optim., 13 (2003), pp. 702–725. [9] R. Mifflin and C. Sagastiz´abal, A VU-algorithm for convex minimization, Math. Program., 104 (2005), pp. 583–608. [10] S. A. Miller and J. Malick, Newton methods for nonsmooth convex minimization: connections among U- lagrangian, Riemannian Newton and SQP methods, Math. Progam., 104 (2005), pp. 609–633. [11] B. S.Mordukhovich, Sensitivity analysis in nonsmooth optimization, SIAM J. Appl. Math., 58 (1992), pp. 32–46. [12] , Variational Analysis and Generalized Di erentiation, Springer, Berlin, 2006. [13] B. S. Mordukhovich and J. Outrata, On second-order subdi erentials and their applications, SIAM J. Optim., 1 (2001), pp. 139–169. [14] B. S. Mordukhovich and R. T. Rockafellar, Second-order subdi erential calculus with applications to optimization, (2011). Manuscript. [15] F. Oustry, The U-lagrangian of the maximum eigenvalue function, SIAM J. Optim., 9 (1999), pp. 526–549. [16] J. V. Outrata and H. Ram´rez C., On the Aubin property of critical points to perturbed second-order cone programs, SIAM J. Optim., 21 (2011), pp. 798–823. [17] R. A. Poliquin and R. T. Rockafellar, Prox-regular functions in variational analysis, Trans. Amer. Math. Soc., 348 (1996), pp. 1805–1838. [18] , Tilt stability of a local minimum, SIAM J. Optim., 8 (1998), pp. 287–299. [19] R. A. Poliquin, R. T. Rockafellar, and L. Thibault, Local di erentiability of distance functions, Trans. Amer. Math. Soc., 352 (2000), pp. 5231–5249. [20] R. T. Rockafellar and R. J.-B. Wets, Variational Analysis, Springer-Verlag, Berlin, 1998. [21] S. J. Wright, Identifiable surfaces in constrained optimization, SIAM J. Control Optim., 31 (1993), pp. 1063– 1079.

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Entry Submitted: 10/17/2011
Entry Accepted: 10/17/2011
Entry Last Modified: 10/17/2011

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