On the convergence of the proximal algorithm for nonsmooth functions involving analytic features

By Hédy Attouch, Jérôme Bolte. 2008

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Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

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Citations in the corpus

Linear Convergence of Gradient and Proximal-Gradient Methods Under the Polyak-Łojasiewicz Condition (2016, background): “4 Proximal-Gradient Generalization Attouch and Bolte [2009] consider a generalization of the the PL inequality due to Kurdyka to give conditions under which the classic proximal-point algorithm achieves a linear convergence rate for non-smooth problems (called the KL inequality).”
On Nonconvex Decentralized Gradient Descent (2016, background): “Then its derivative g′(x) = (1 + x) −1 − 2 < 0, ∀x ∈ [0, 1].”
Proximal alternating linearized minimization for nonconvex and nonsmooth problems (2013, background): “where c is positive real number and θ belongs to [0, 1) (see [1] for more details).”
A regularization interpretation of the proximal point method for weakly convex functions (2020, background): “We do not discuss about the rate of convergence and we refer to [4, 19].”
Error Bounds and Hölder Metric Subregularity (2014, background): “ed to be important in sensitivity analysis and quantifying linear/sublinear convergence rates for the proximal point and alternating projection algorithms in optimization and variational inequalities [2,3,9,16,45,47,48,58]. One can easily see that Ho¨lder metric subregularity property (2) is equival”
Convergence Analysis of a Momentum Algorithm with Adaptive Step Size for Non Convex Optimization (2019, methodology): “The K L inequality has been used to show the convergence of several first-order optimization methods towards critical points (Attouch and Bolte, 2009; Attouch et al., 2010; Bolte et al., 2014; Li et al., 2017).”
A Globally Convergent Algorithm for Nonconvex Optimization Based on Block Coordinate Update (2014, background): “The proof follows that of Theorem 2 of [3].”
Analysis of nonsmooth stochastic approximation: the differential inclusion approach (2018, background): “s therein.1 In this paper, we will also consider proximal algorithms, which have become an important tool in nonsmooth optimization problems; the literature in this ﬁeld is also huge, see for example [4, 7, 31, 33, 13]). Proximal algorithmswith stochastic updates have been proposed and studied in re”
Modern regularization methods for inverse problems (2018, background): “It is important to emphasize that even for nonsmooth, nonconvex optimization there is a vast amount of recent publications, ranging from forward-backward, respectively proximaltype, schemes [8, 9, 10, 49, 50], over linearized proximal schemes [365, 47, 366, 298], to inertial methods [299, 309], prim”
Characterizations of Lojasiewicz inequalities and applications (2008, background): “hown in a section devoted to applications (Section 3.4), this reformulation is particularly adapted for the analysis of proximal methods involving nonconvex criteria: these results are in the line of [14, 5]. In the second part of this work (Section 3), His a proper real Hilbert space and fis assume”
iPiano: Inertial Proximal Algorithm for Nonconvex Optimization (2014, background): “However, with slightly more assumptions to the objective, namely, that it satisfies the Kurdyka– Lojasiewicz inequality [30, 31, 26], several algorithms have been shown to converge [14, 5, 3, 4].”
Characterizations of Lojasiewicz inequalities: Subgradient flows, talweg, convexity (2009, background): “4), this reformulation is particularly adapted for the analysis of proximal methods involving nonconvex criteria: these results are in the line of [15, 5].”
Proximal Alternating Minimization and Projection Methods for Nonconvex Problems: An Approach Based on the Kurdyka-Lojasiewicz Inequality (2008, methodology): “In [4] Lojasiewicz inequality is used to derive the convergence of the usual proximal method but alternating algorithms are not considered.”
A NUMERICAL ANALYSIS OF THE CAHN-HILLIARD EQUATION WITH DYNAMIC BOUNDARY CONDITIONS (2010, background): “It is based on the Lojasiewicz gradient inequality (see also [1]).”
On Convergence of Model Parallel Proximal Gradient Algorithm for Stale Synchronous Parallel System (2016, methodology): “Then, inspired by the recent KurdykaLojasiewicz (K L) inequality [3, 5, 9, 11, 20], we further prove in Theorem 2 that the whole sequence of msPG in fact converges to a critical point, under mild technical assumptions that we verify for many familiar examples.”
Super-efficiency of automatic differentiation for functions defined as a minimum (2020, methodology): “A function f with minimizer z ∗ is p -Łojasiewicz [Attouch and Bolte, 2009] when µ ( f ( z ) − f ( z ∗ )) p − 1 ≤ (cid:107)∇ f ( z ) p for some µ > 0 .”
A Block Coordinate Descent Method for Regularized Multiconvex Optimization with Applications to Nonnegative Tensor Factorization and Completion (2013, methodology): “for ν = 1−2θ 1−θ < 0, some constant μ > 0 and any N > K with sufficiently large K by the same argument as in the proof of Theorem 2 of [3].”