Proximal alternating linearized minimization for nonconvex and nonsmooth problems

By Jérôme Bolte, Shoham Sabach, Marc Teboulle. 2013

1,985 citations recorded by the source corpus.

Subfields: Computational Mechanics, Computational Theory and Mathematics, Numerical Analysis

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Citations in the corpus

Group Knowledge Transfer: Collaborative Training of Large CNNs on the Edge (2020, background): “The essence of GKT is that it reformulates FL as an alternating minimization (AM) approach [14, 15, 16, 17, 18, 19], which optimizes two random variables (the edge model and the server model) by alternatively fixing one and optimizing another.”
Coordinate descent algorithms (2015, methodology): “More recent work [1,6] focuses on CD with two blocks of variables, applied to functions that satisfy the so-called Kurdyka-(cid:32)Lojasiewicz (KL) property, such as semi-algebraic functions.”
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).”
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”
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”
Convergence analysis of alternating direction method of multipliers for a family of nonconvex problems (2014, background): “We note that there have been many recent works proposing new algorithms to solve nonconvex and nonsmooth problems, for example, [50, 51, 52, 53, 54].”
Semantic Pooling for Complex Event Analysis in Untrimmed Videos (2016, background): “Our isotonic regularizer is not convex, but as we show in Section 5, the popular proximal gradient algorithm can still be applied, hence enjoying the strong convergence guarantees recently established in [16].”
An Exact Penalty Method for Binary Optimization Based on MPEC Formulation (2017, background): “Jρ(x,v) = f(x) + ρ(n− xTv) s.t. − 1 ≤ x ≤ 1, ‖v‖22 ≤ n, x ∈ Ω (7) In each iteration, we minimize over x and v alternatingly (Tseng, 2001; Bolte, Sabach, and Teboulle, 2014), while fixing the parameter ρ.”
Joint Optimization of Radio and Computational Resources for Multicell Mobile-Edge Computing (2014, methodology): “M(inimization) algorithm [35], [36]; iii) alternating/successive minimization methods [37]–[39]; and iv) partial linearization methods [32], [40], [41].”
Gradient Primal-Dual Algorithm Converges to Second-Order Stationary Solutions for Nonconvex Distributed Optimization (2018, background): “We comment that a wide class of functions enjoys the KŁ property, for example a semi-algebraic function is a KL function; for detailed discussions of the KŁ property we refer the readers to (Bolte et al., 2014; Li & Pong, 2014a).”
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.”
Group Knowledge Transfer: Federated Learning of Large CNNs at the Edge (2020, background): “The essence of FedGKT is that it reformulates FL as an alternating minimization (AM) approach [13, 14, 15, 16, 17, 18], which optimizes two random variables (the edge model and the server model) by alternatively fixing one and optimizing another.”
Fast optimization via inertial dynamics with closed-loop damping (2020, background): “9) follows from the uniformized KL property Lemma 6 in [43] applied to the nonempty compact and connected set Ω = ω((xn)n≥0) × {0} (notice that according to [43, Remark 5] the connectedness of this set is generic for sequences satisfying limn→+∞(xn+1−xn) = 0).”
From error bounds to the complexity of first-order descent methods for convex functions (2015, methodology): “Convergence of subgradient descent sequences follows readily from [4] and [20, 32].”
An introduction to continuous optimization for imaging (2016, methodology): “Optimization for imaging 237 Algorithm 12 ‘PALM’ algorithm (Bolte et al. 2014).”
Hyperspectral Super-Resolution by Coupled Spectral Unmixing (2015, methodology): “Inspired by the PALM (proximal alternating linearized minimisation) algorithm [8], we propose to use a projected gradient method for both parts.”