Alternating Projections on Manifolds

Lewis, Adrian S.; Malick, Jérôme

doi:10.1287/moor.1070.0291

Cited by 184 publications

(228 citation statements)

References 36 publications

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“…We notice that this result generalizes nicely for "spectral sets"; see [LM08]. Note also that the numerical cost of computing this projection is essentially that of computing the spectral decomposition of C, the matrix to project.…”

Section: Projection Onto Semidefinite Positive Matricessupporting

confidence: 52%

Projection Methods in Conic Optimization

Henrion

Malick

2011

International Series in Operations Research &Amp; Management Science

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There exist efficient algorithms to project a point onto the intersection of a convex cone and an affine subspace. Those conic projections are in turn the work-horse of a range of algorithms in conic optimization, having a variety of applications in science, finance and engineering. This chapter reviews some of these algorithms, emphasizing the so-called regularization algorithms for linear conic optimization, and applications in polynomial optimization. This is a presentation of the material of several recent research articles; we aim here at clarifying the ideas, presenting them in a general framework, and pointing out important techniques.

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Section: Projection Onto Semidefinite Positive Matricessupporting

confidence: 52%

Projection Methods in Conic Optimization

Henrion

Malick

2011

International Series in Operations Research &Amp; Management Science

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“…quadratic functions, polynomial functions, real analytic functions, but it can also be captured by adequate analytic assumptions, e.g. metric regularity [2,41,42], cohypomonotonicity [51,36], selfconcordance [49], partial smoothness [40,59]. In this paper, our central assumption for the study of such algorithms is that the function f satisfies the (nonsmooth) Kurdyka-Lojasiewicz inequality, which means, roughly speaking, that the functions under consideration are sharp up to a reparametrization (see Section 2.2).…”

Section: Introductionmentioning

confidence: 99%

“…We do not give a precise definition of definability in this work, but the flexibility of this concept is briefly illustrated in Example 5.4(b). Functions that are not necessarily tame but that satisfy Lojasiewicz inequality are given in [5], basic assumptions involve metric-regularity and transversality (see also [41,42] and Example 5.5).…”

Section: Introductionmentioning

confidence: 99%

Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward–backward splitting, and regularized Gauss–Seidel methods

2011

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To cite this version:Hedy Attouch, Jérôme Bolte, Benar Svaiter. Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward-backward splitting, and regularized Gauss-Seidel methods. Math. Program., Ser. A, 2011, 137 (1), pp.91-124. <10.1007/s10107-011-0484-9>. Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward-backward splitting, and regularized Gauss-Seidel methods , 15, 2010; revised: July, 21, 2011 Abstract In view of the minimization of a nonsmooth nonconvex function f , we prove an abstract convergence result for descent methods satisfying a sufficient-decrease assumption, and allowing a relative error tolerance. Our result guarantees the convergence of bounded sequences, under the assumption that the function f satisfies the Kurdyka-Lojasiewicz inequality. This assumption allows to cover a wide range of problems, including nonsmooth semi-algebraic (or more generally tame) minimization. The specialization of our result to different kinds of structured problems provides several new convergence results for inexact versions of the gradient method, the proximal method, the forward-backward splitting algorithm, the gradient projection and some proximal regularization of the Gauss-Seidel method in a nonconvex setting. Our results are illustrated through feasibility problems, or iterative thresholding procedures for compressive sensing.2010 Mathematics Subject Classification: 34G25, 47J25, 47J30, 47J35, 49M15, 49M37, 65K15, 90C25, 90C53.

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“…Specific examples related to feasibility problems are provided, they involve (possibly tangent) realanalytic manifolds, transverse manifolds (see [36]), semialgebraic sets or more generally tame sets.…”

Section: Introductionmentioning

confidence: 99%

“…Some nonconvex cases are easily computable and amount to explicit or standard computations: projections onto spheres or onto ellipsoids, constant rank matrices, or even one real variable second order equations. A good reference for some of these aspects is [36]. In order to provide a more realistic and flexible tool for solving real-world problems, it would be natural to consider inexact versions of algorithm (3) (see [34,23] for some work in that direction); this subject is out of the scope of the present paper but it is a matter for future research.…”

Section: Introductionmentioning

confidence: 99%

Proximal Alternating Minimization and Projection Methods for Nonconvex Problems: An Approach Based on the Kurdyka-Łojasiewicz Inequality

et al. 2010

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Abstract. We study the convergence properties of an alternating proximal minimization algorithm for nonconvex structured functions of the type: L(x, y) = f (x)+Q(x, y)+g(y), where f : R n → R∪{+∞} and g : R m → R∪{+∞} are proper lower semicontinuous functions, and Q : R n × R m → R is a smooth C 1 function which couples the variables x and y. The algorithm can be viewed as a proximal regularization of the usual Gauss-Seidel method to minimize L.We work in a nonconvex setting, just assuming that the function L satisfies the Kurdyka-Lojasiewicz inequality. An entire section illustrates the relevancy of such an assumption by giving examples ranging from semialgebraic geometry to "metrically regular" problems.Our main result can be stated as follows: If L has the Kurdyka-Lojasiewicz property, then each bounded sequence generated by the algorithm converges to a critical point of L. This result is completed by the study of the convergence rate of the algorithm, which depends on the geometrical properties of the function L around its critical points. When specialized to Q(x, y) = x − y 2 and to f , g indicator functions, the algorithm is an alternating projection mehod (a variant of Von Neumann's) that converges for a wide class of sets including semialgebraic and tame sets, transverse smooth manifolds or sets with "regular "intersection. In order to illustrate our results with concrete problems, we provide a convergent proximal reweighted ℓ 1 algorithm for compressive sensing and an application to rank reduction problems.

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Alternating Projections on Manifolds

Cited by 184 publications

References 36 publications

Projection Methods in Conic Optimization

Projection Methods in Conic Optimization

Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward–backward splitting, and regularized Gauss–Seidel methods

Proximal Alternating Minimization and Projection Methods for Nonconvex Problems: An Approach Based on the Kurdyka-Łojasiewicz Inequality

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