Computing proximal points of nonconvex functions

Hare, Warren; Sagastizábal, Claudia

doi:10.1007/s10107-007-0124-6

Cited by 97 publications

(79 citation statements)

References 28 publications

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“…In Attouch and Bolte [4], the approach is based on the Kurdyka-Lojasiewicz inequality, which is valid for a large class of possibly nonsmooth functions including real analytic or semialgebraic functions. In Hare and Sagastizabal [29], and Iusem et al [31], the approach is based on a local control of the defect of convexity. Kaplan and Tichatschke [33] consider proximal point approach to general variational inequality problems.…”

Section: Comparison With Proximal Algorithmsmentioning

confidence: 99%

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Local Search Proximal Algorithms as Decision Dynamics with Costs to Move

Attouch

Soubeyran

2010

Set-Valued Anal

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Acceptable moves for the "worthwhile-to-move" incremental principle are such that "advantages-to-move" are higher than some fraction of "costs-to-move". When combined with optimization, this principle gives raise to adaptive local search proximal algorithms. Convergence results are given in two distinctive cases, namely low local costs-to-move and high local costs-to-move. In this last case, one obtains a dynamic cognitive approach to Ekeland's -variational principle. Introduction of costs-to-move in the algorithms yields robustness and stability properties. Mathematics Subject Classifications (2010)Partially supported by ANR-08-BLAN-0294-03.H. Attouch (B)

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Section: Comparison With Proximal Algorithmsmentioning

confidence: 99%

“…The same holds concerning (42) and y n+1 . Following (29), for n large enough, let us say n ≥ n 0 the exploration constraint is not active, and the following optimality properties hold…”

Section: Theorem 41 Let H Be a Real Hilbert Space Let G()mentioning

confidence: 99%

Local Search Proximal Algorithms as Decision Dynamics with Costs to Move

Attouch

Soubeyran

2010

Set-Valued Anal

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“…-A sequence of null steps of a bundle subroutine can approximate proximal points within any desired accuracy [3] (see also [10] for the nonconvex case).…”

Section: (Persistence and Stability Of The Nondegeneracy Condition) mentioning

confidence: 99%

Identifying Structure of Nonsmooth Convex Functions by the Bundle Technique

Daniilidis¹,

Sagastizábal²,

Solodov³

2009

SIAM J. Optim.

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Abstract. We consider the problem of minimizing nonsmooth convex functions, defined piecewise by a finite number of functions each of which is either convex quadratic or twice continuously differentiable with positive definite Hessian on the set of interest. This is a particular case of functions with primal-dual gradient structure, a notion closely related to the so-called VU space decomposition: At a given point, nonsmoothness is locally restricted to the directions of the subspace V, while along the subspace U the behavior of the function is twice differentiable. Constructive identification of the two subspaces is important, because it opens the way to devising fast algorithms for nonsmooth optimization (by following iteratively the manifold of smoothness on which superlinear U -Newton steps can be computed). In this work we show that, for the class of functions in consideration, the information needed for this identification can be obtained from the output of a standard bundle method for computing proximal points, provided a minimizer satisfies the nondegeneracy and strong transversality conditions.

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“…While the present article is concerned with the numerical computation of the Moreau envelope, contrary to [25] we do not consider computing its value at one point but instead we tackle the problem of computing the Moreau envelope on a grid.…”

Section: Introductionmentioning

confidence: 98%

Fast Moreau envelope computation I: numerical algorithms

Lucet

2007

Numer Algor

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The present article summarizes the state of the art algorithms to compute the discrete Moreau envelope, and presents a new linear-time algorithm, named NEP for NonExpansive Proximal mapping. Numerical comparisons between the NEP and two existing algorithms: The Linear-time Legendre Transform (LLT) and the Parabolic Envelope (PE) algorithms are performed. Worst-case time complexity, convergence results, and examples are included. The fast Moreau envelope algorithms first factor the Moreau envelope as several one-dimensional transforms and then reduce the brute force quadratic worst-case time complexity to linear time by using either the equivalence with Fast Legendre Transform algorithms, the computation of a lower envelope of parabolas, or, in the convex case, the non expansiveness of the proximal mapping.

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Computing proximal points of nonconvex functions

Cited by 97 publications

References 28 publications

Local Search Proximal Algorithms as Decision Dynamics with Costs to Move

Local Search Proximal Algorithms as Decision Dynamics with Costs to Move

Identifying Structure of Nonsmooth Convex Functions by the Bundle Technique

Fast Moreau envelope computation I: numerical algorithms

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