Pointwise convergence of gradient‐like systems

Lageman, Christian

doi:10.1002/mana.200410564

Cited by 16 publications

(19 citation statements)

References 21 publications

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“…The fact that Algorithm 2 is an inexact version of the proximal algorithm is transparent: the first inequality (36) reflects the fact that a sufficient decrease of the value must be achieved, while the last lines (38), (39) both correspond to an inexact optimality condition.…”

Section: Convergence Of An Inexact Proximal Algorithm For Kl Functionsmentioning

confidence: 99%

“…First use Lemma 4.1 to conclude that condition (38) implies (39). Therefore, we assume that (36), (37) and (39) holds.…”

Section: Theorem 42 (Inexact Proximal Algorithm)mentioning

confidence: 99%

“…The reader is referred to [44,45,38] for the smooth cases, and to [15,17] for nonsmooth inequalities. Kurdyka-Lojasiewicz inequalities have been successfully used to analyze various types of asymptotic behavior: gradient-like systems [15,34,35,39], PDE [55,22], gradient methods [1,48], proximal methods [3], projection methods or alternating methods [5,14].…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Convergence Of An Inexact Proximal Algorithm For Kl Functionsmentioning

confidence: 99%

“…First use Lemma 4.1 to conclude that condition (38) implies (39). Therefore, we assume that (36), (37) and (39) holds.…”

Section: Theorem 42 (Inexact Proximal Algorithm)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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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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“…Subsequently, Łojasiewicz's idea and technique were used to study stability and convergence properties of gradient flows and gradient descent algorithms in [18,21,22]. Later extensions to nongradient systems and to a nonsmooth setting can be found in [23] and [24,25], respectively. The result that underlies all this work is the Łojasiewicz inequality between an analytic function and the norm of its gradient vector.…”

Section: Introductionmentioning

confidence: 98%

Arc-length-based Lyapunov tests for convergence and stability with applications to systems having a continuum of equilibria

Bhat

Bernstein

2010

Math. Control Signals Syst.

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In this paper, fundamental relationships are established between convergence of solutions, stability of equilibria, and arc length of orbits. More specifically, it is shown that a system is convergent if all of its orbits have finite arc length, while an equilibrium is Lyapunov stable if the arc length (considered as a function of the initial condition) is continuous at the equilibrium, and semistable if the arc length is continuous in a neighborhood of the equilibrium. Next, arc-length-based Lyapunov tests are derived for convergence and stability. These tests do not require the Lyapunov function to be positive definite. Instead, these results involve an inequality relating the right-hand side of the differential equation and the Lyapunov function derivative. This inequality makes it possible to deduce properties of the arc length function and thus leads to sufficient conditions for convergence and stability. Finally, it is shown that the converses of all the main results hold under additional assumptions. Examples are included to illustrate how our results are particularly suited for analyzing stability of systems having a continuum of equilibria.

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“…For example, a result of Lojasiewicz [20] establishes convergence of trajectories of gradient flows 1 to single equilibrium points for arbitrary real analytic functions f . This can be extended to gradient-like descent flows, see [1,3] and for more general results [17,18]. The pointwise convergence fails in the general smooth case.…”

Section: Gradient Algorithmsmentioning

confidence: 95%

Nonsmooth Riemannian Optimization with Applications to Sphere Packing and Grasping

Dirr

Helmke

Lageman

Lagrangian and Hamiltonian Methods for Nonlinear Control 2006

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Summary. This paper presents a survey on Riemannian geometry methods for smooth and nonsmooth constrained optimization. Gradient and subgradient descent algorithms on a Riemannian manifold are discussed. We illustrate the methods by applications from robotics and multi antenna communication. Gradient descent algorithms for dextrous hand grasping and for sphere packing problems on Grassmann manifolds are presented respectively.

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Pointwise convergence of gradient‐like systems

Cited by 16 publications

References 21 publications

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

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

Arc-length-based Lyapunov tests for convergence and stability with applications to systems having a continuum of equilibria

Nonsmooth Riemannian Optimization with Applications to Sphere Packing and Grasping

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