Abstract
The past decade has seen the introduction of a number of classes of nonsmooth functions possessing smooth substructure, e.g., “amenable functions”, “partly smooth functions”, and “g ∘ F decomposable functions”. Along with these classes a number of structural properties have been proposed, e.g., “identifiable surfaces”, “fast tracks”, and “primal-dual gradient structures”. In this paper we examine the relationships between these various classes of functions and their smooth substructures.
In the convex case we show that the definitions of identifiable surfaces, fast tracks, and partly smooth functions are equivalent. In the nonconvex case we discuss when a primal-dual gradient structure or g ∘ F decomposition implies the function is partly smooth, and vice versa. We further provide examples to show these classes are not equal.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J.F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems. Springer Series in Operations Research, Springer-Verlag, New York, 2000.
J.V. Burke and J.J. Moré, “On the identification of active constraints,” SIAM J. Numer. Anal., vol. 25, no. 5, pp. 1197–1211, 1988.
F.H. Clarke, “Nonsmooth analysis and optimization,” in Proceedings of the International Congress of Mathematicians (Helsinki, 1978), pp. 847–853, Helsinki, 1980. Acad. Sci. Fennica.
R. Fletcher, Practical Methods of Optimization, Volume 2. John Wiley & Sons Ltd., Chichester, 1981. Constrained optimization, A Wiley-Interscience Publication.
W.L. Hare and A.S. Lewis, “Identifying active constraints via partial smoothness and prox-regularity,” Journal of Convex Analysis, vol. 11, pp. 251–266, 2004.
W.L. Hare and R.A. Poliquin, “The quadratic sub-Lagrangian of a prox-regular function,” in Proceedings of the Third World Congress of Nonlinear Analysts, Part 2 (Catania, 2000), vol. 47, pp. 1117–1128, 2001.
A.D. Ioffe, “Necessary and sufficient conditions for a local minimum. I. A reduction theorem and first order conditions,” SIAM J. Control Optim., vol. 17, no. 2, pp. 245–250, 1979.
A.D. Ioffe, “Necessary and sufficient conditions for a local minimum. II. Conditions of Levitin-Miljutin-Osmolovskii type,” SIAM J. Control Optim., vol. 17, no. 2, pp. 251–265, 1979.
A.D. Ioffe, “Necessary and sufficient conditions for a local minimum. III. Second order conditions and augmented duality,” SIAM J. Control Optim., vol. 17, no. 2, pp. 266–288, 1979.
A.D. Ioffe and V.L. Levin, “Subdifferentials of convex functions,” Trudy Moskov. Mat. Obšč., vol. 26, pp. 3–73, 1972.
C. Lemaréchal, F. Oustry, and C. Sagastizábal, “The U-Lagrangian of a convex function,” Trans. Amer. Math. Soc., vol. 352, no. 2, pp. 711–729, 2000.
A.S. Lewis, “Active sets, nonsmoothness, and sensitivity,” SIAM J. Optim., vol. 13, no. 3, pp. 702–725 (electronic) (2003), 2002.
R. Mifflin and C. Sagastizábal, “On VU-theory for functions with primal-dual gradient structure,” SIAM J. Optim., vol. 11, no. 2, pp. 547–571 (electronic), 2000.
R. Mifflin and C. Sagastizábal, “Proximal points are on the fast track,” J. Convex Anal., vol. 9, no. 2, pp. 563–579, 2002. Special issue on optimization (Montpellier, 2000).
R. Mifflin and C. Sagastizábal, “Primal-dual gradient structured functions: Second-order results; links to epi-derivatives and partly smooth functions,” SIAM Journal on Optimization, vol. 13, no. 4, pp. 1174–1194, 2003.
R. Mifflin and C. Sagastizábal, “A vu-algorithm for convex minimzation,” Math. Prog., (online), July 24th 2005.
E.A. Nurminski, “On \(\varepsilon\)-subgradient methods of nondifferentiable optimization,” in International Symposium on Systems Optimization and Analysis (Rocquencourt, 1978), volume 14 of Lecture Notes in Control and Information Sci., Springer, Berlin, 1979, pp. 187–195.
R.A. Poliquin and R.T. Rockafellar, “Amenable functions in optimization,” in Nonsmooth Optimization: Methods and Applications (Erice, 1991), Gordon and Breach, Montreux, 1992, pp. 338–353.
R.A. Poliquin and R.T. Rockafellar, “A calculus of epi-derivatives applicable to optimization,” Canad. J. Math., vol. 45, no. 4, pp. 879–896, 1993.
R.T. Rockafellar, “Favorable classes of Lipschitz-continuous functions in subgradient optimization,” in Progress in Nondifferentiable Optimization, volume 8 of IIASA Collaborative Proc. Ser. CP-82, Internat. Inst. Appl. Systems Anal., Laxenburg, 1982, pp. 125–143.
R.T. Rockafellar, “First- and second-order epi-differentiability in nonlinear programming,” Trans. Amer. Math. Soc., vol. 307, no. 1, pp. 75–108, 1988.
R.T. Rockafellar and R.J.-B. Wets, Variational Analysis, volume 317 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1998.
A. Shapiro, “On a class of nonsmooth composite functions,” Mathematics of Operations Research, 2003, to appear.
S.J. Wright, “Identifiable surfaces in constrained optimization,” SIAM J. Control Optim., vol. 31, no. 4, pp. 1063–1079, 1993.
Author information
Authors and Affiliations
Corresponding author
Additional information
The author would like to thank Adrian Lewis, Bob Mifflin and the two anonymous referees for their many helpful contributions.
Rights and permissions
About this article
Cite this article
Hare, W.L. Functions and Sets of Smooth Substructure: Relationships and Examples. Comput Optim Applic 33, 249–270 (2006). https://doi.org/10.1007/s10589-005-3059-4
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10589-005-3059-4