Identifying Active Manifolds

Determining the “active manifold” for a minimization probl em is a large step towards solving the problem. Many researchers have studied under what conditions certain alg orithms identify active manifolds in a finite number of iterations. In this work we outline a unifying framework enc ompassing many earlier results on identification via the Subgradient (Gradient) Projection Method, Newton-like Me thods, and the Proximal Point Algorithm. This framework, prox-regular partial smoothness, has the advantage of not r equiring convexity for its conclusions, and therefore exte nds many of these earlier results.

[1]  R. Tyrrell Rockafellar,et al.  Generalized Hessian Properties of Regularized Nonsmooth Functions , 1996, SIAM J. Optim..

[2]  R. Mi,et al.  Proximal Points are on the Fast Track , 2002 .

[3]  J. Burke.  On identification of active constraints II: the nonconvex case , 1990 .

[4]  Sjur Didrik Flåm,et al.  On finite convergence and constraint identification of subgradient projection methods , 1992, Math. Program..

[5]  J. Dunn.  On the convergence of projected gradient processes to singular critical points , 1987 .

[6]  Paul H. Calamai,et al.  Projected gradient methods for linearly constrained problems , 1987, Math. Program..

[7]  Adrian S. Lewis,et al.  Active Sets, Nonsmoothness, and Sensitivity , 2002, SIAM J. Optim..

[8]  Michael C. Ferris,et al.  Finite termination of the proximal point algorithm , 1991, Math. Program..

[9]  J. Kyparisis,et al.  Finite convergence of algorithms for nonlinear programs and variational inequalities , 1991 .

[10]  R. Rockafellar,et al.  Local differentiability of distance functions , 2000 .

[11]  D. Bertsekas.  On the Goldstein-Levitin-Polyak gradient projection method , 1974, CDC 1974.

[12]  R. Rockafellar,et al.  Prox-regular functions in variational analysis , 1996 .

[13]  Warren Hare,et al.  A proximal method for identifying active manifolds , 2009, Comput. Optim. Appl..

[14]  J. J. Moré,et al.  On the identification of active constraints , 1988 .

[15]  R. Rockafellar.  Monotone Operators and the Proximal Point Algorithm , 1976 .

[16]  A. Lewis,et al.  Identifying active constraints via partial smoothness and prox-regularity , 2003 .

[17]  Warren Hare.  Functions and Sets of Smooth Substructure: Relationships and Examples , 2006, Comput. Optim. Appl..

[18]  Stephen J. Wright.  Identifiable Surfaces in Constrained Optimization , 1993 .