Identifying Structure of Nonsmooth Convex Functions by the Bundle Technique

We consider a class of smoothing methods for minimization problems where the feasible set is convex but the objective function is not convex, not differentiable and perhaps not even locally Lipschitz at the solutions. Such optimization problems arise from wide applications including image restoration, signal reconstruction, variable selection, optimal control, stochastic equilibrium and spherical approximations. In this paper, we focus on smoothing methods for solving such optimization problems, which use the structure of the minimization problems and composition of smoothing functions for the plus function (x) + . Many existing optimization algorithms and codes can be used in the inner iteration of the smoothing methods. We present properties of the smoothing functions and the gradient consistency of subdifferential associated with a smoothing function. Moreover, we describe how to update the smoothing parameter in the outer iteration of the smoothing methods to guarantee convergence of the smoothing methods to a stationary point of the original minimization problem.

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“…Numerical methods for solving nonsmooth, nonconvex optimization problems have been studied extensively [7,10,12,35,37,51,59,68,77,102].…”

Section: Smoothing Algorithmsmentioning

confidence: 99%

Smoothing methods for nonsmooth, nonconvex minimization

2012

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“…This hope motivated the UV-decomposition idea, and was pursued further by Mifflin and Sagastizábal [25,26] and recently by Daniilidis, Sagastizábal and Solodov [9]. A general approach, based on a proximal algorithm for composite optimization, is sketched by Lewis and Wright in [22], and broad techniques for estimating identifiable surfaces computationally are discussed in [23].…”

Section: Introductionmentioning

confidence: 99%

Generic Optimality Conditions for Semialgebraic Convex Programs

Bolte¹,

Daniilidis

Lewis

2011

Mathematics of OR

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We consider linear optimization over a nonempty convex semialgebraic feasible region F . Semidefinite programming is an example. If F is compact, then for almost every linear objective there is a unique optimal solution, lying on a unique "active" manifold, around which F is "partly smooth", and the second-order sufficient conditions hold. Perturbing the objective results in smooth variation of the optimal solution. The active manifold consists, locally, of these perturbed optimal solutions; it is independent of the representation of F , and is eventually identified by a variety of iterative algorithms such as proximal and projected gradient schemes. These results extend to unbounded sets F .

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“…In the expression of G k in (12), this means that QP solver gives a solution with no more than n +1 positive simplicial multipliers (such a solution always exists by the Carathéodory Theorem). A similar assumption/property for a QP solver had been used for a different QP-based method in [13,Sec.5], and specifically for a bundle procedure in [7].…”

Section: Remark 2 (Uniformly Bounded Number Of Active Indices In Subpmentioning

confidence: 98%

A proximal bundle method for nonsmooth nonconvex functions with inexact information

2015

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For a class of nonconvex nonsmooth functions, we consider the problem of computing an approximate critical point, in the case when only inexact information about the function and subgradient values is available. We assume that the errors in function and subgradient evaluations are merely bounded, and in principle need not vanish in the limit. We examine the redistributed proximal bundle approach in this setting, and show that reasonable convergence properties are obtained. We further consider a battery of difficult nonsmooth nonconvex problems, made even more difficult by introducing inexactness in the available information. We verify that very satisfactory outcomes are obtained in our computational implementation of the inexact algorithm.

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Identifying Structure of Nonsmooth Convex Functions by the Bundle Technique

Cited by 20 publications

References 24 publications

Smoothing methods for nonsmooth, nonconvex minimization

Smoothing methods for nonsmooth, nonconvex minimization

Generic Optimality Conditions for Semialgebraic Convex Programs

A proximal bundle method for nonsmooth nonconvex functions with inexact information

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