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Inexact Variants of the Proximal Point Algorithm without Monotonicity
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Convergence of the Iterates of Descent Methods for Analytic Cost Functions
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Clarke Subgradients of Stratifiable Functions
SIAM Journal on Optimization
A New Class of Alternating Proximal Minimization Algorithms with Costs-to-Move
SIAM Journal on Optimization
Computing proximal points of nonconvex functions
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On the convergence of the proximal algorithm for nonsmooth functions involving analytic features
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Finding Best Approximation Pairs Relative to a Convex and Prox-Regular Set in a Hilbert Space
SIAM Journal on Optimization
Alternating Projections on Manifolds
Mathematics of Operations Research
Optimization Algorithms on Matrix Manifolds
Optimization Algorithms on Matrix Manifolds
Local Linear Convergence for Alternating and Averaged Nonconvex Projections
Foundations of Computational Mathematics
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Applying Metric Regularity to Compute a Condition Measure of a Smoothing Algorithm for Matrix Games
SIAM Journal on Optimization
A New Sequential Optimality Condition for Constrained Optimization and Algorithmic Consequences
SIAM Journal on Optimization
Fast regularization of matrix-valued images
ECCV'12 Proceedings of the 12th European conference on Computer Vision - Volume Part III
Prox-Regularity of Rank Constraint Sets and Implications for Algorithms
Journal of Mathematical Imaging and Vision
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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 and g are proper lower semicontinuous functions, defined on Euclidean spaces, and Q is a smooth function that 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-Łojasiewicz 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-Łojasiewicz 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 $Q(x,y)=\Vert x-y \Vert ^2$ 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. To illustrate our results with concrete problems, we provide a convergent proximal reweighted ℓ1 algorithm for compressive sensing and an application to rank reduction problems.