Optimization Online


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.

Download: [PDF]

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

Modify/Update this entry

  Visitors Authors More about us Links
  Subscribe, Unsubscribe
Digest Archive
Search, Browse the Repository


Coordinator's Board
Classification Scheme
Give us feedback
Optimization Journals, Sites, Societies
Mathematical Optimization Society