Inexact proximal $$\epsilon $$-subgradient methods for composite convex optimization problems

Millán, R. Díaz; Machado, Majela Pentón

doi:10.1007/s10898-019-00808-8

Cited by 10 publications

(5 citation statements)

References 33 publications

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“…This algorithm is proposed by Lions and Mercier [10] and Passty in [17]. For further reading see [14] and references therein.…”

Section: -Endmentioning

confidence: 99%

The proximal methods for solving absolute value equation

Shahsavari¹,

Ketabchi²

2021

NACO

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In this paper, by considering that the objective function of the least squares NP-hard absolute value equations (AVE) Ax − |x| = b, is nonconvex and non-smooth, two types of proximal algorithms are proposed to solve it. One of them is the proximal difference-of-convex algorithm with extrapolation and another is the proximal subgradient method. The convergence results of the proposed methods are proved under certain assumptions. Moreover, a numerical comparison is presented to demonstrate the effectiveness of the suggested methods.

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“…This algorithm is proposed by Lions and Mercier [10] and Passty in [17]. For further reading see [14] and references therein.…”

Section: -Endmentioning

confidence: 99%

The proximal methods for solving absolute value equation

Shahsavari¹,

Ketabchi²

2021

NACO

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“…One can reformulate this requirement as ∃ε ≥ 0: v ∈ ∂ ε h(x) and 1 2 x − z + λv 2 + λε ≤ σ. This criterion is used among others in the hybrid approximate extragradient (HPE) framework [99,101,102,104], in its inertial/accelerated versions [73,12,4], or for forwardbackward splittings [71,11]. This criterion is generalized in the (monotone) operator world, through the notion of ε-enlargements [19,16], generalizing the notion of ε-subdifferentials.…”

Section: 1mentioning

confidence: 99%

“…One way of dealing with this problem is to approximate the proximal operator of the discrete total variation. As in [112,71], we apply FISTA [10] on the dual of the proximal subproblem (which is provided e.g., in [21, Example 3.1]), which we use in the accelerated inexact forward-backward method.…”

Section: Total Variation Regularizationmentioning

confidence: 99%

“…In particular, those notions were applied to the proximal point algorithm [16,39,99,73], inexact splitting scheme such as forward-backward splitting (and its accelerated variants) [97,112,71,11], Douglas-Rachford [43,44,106,2], three-operator splitting [114], online optimization [31,1,8], and for designing metaalgorithms such as the hybrid approximate extragradient method [99,72,73,4], and "Catalyst" [63,64]. Inexact proximal operations are also closely related to the theory of ε-subdifferentials, introduced in [14], and to their inexact gradient and subgradient methods (see e.g., [98,71,28,29]). Finally, let us mention higher-order proximal methods, that are introduced in [80,81], and also used together with notions for approximating them.…”

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confidence: 99%

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Principled Analyses and Design of First-Order Methods with Inexact Proximal Operators

Barré¹,

Taylor²,

Bach³

2020

Preprint

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Proximal operations are among the most common primitives appearing in both practical and theoretical (or high-level) optimization methods. This basic operation typically consists in solving an intermediary (hopefully simpler) optimization problem. In this work, we survey notions of inaccuracies that can be used when solving those intermediary optimization problems. Then, we show that worst-case guarantees for algorithms relying on such inexact proximal operations can be systematically obtained through a generic procedure based on semidefinite programming. This methodology is primarily based on the approach introduced by Drori and Teboulle [36] and on convex interpolation results, and allows producing non-improvable worst-case analyzes. In other words, for a given algorithm, the methodology generates both worst-case certificates (i.e., proofs) and problem instances on which those bounds are achieved.Relying on this methodology, we provide three new methods with conceptually simple proofs: (i) an optimized relatively inexact proximal point method, (ii) an extension of the hybrid proximal extragradient method of Monteiro and Svaiter [73], and (iii) an inexact accelerated forward-backward splitting supporting backtracking line-search, and both (ii) and (iii) supporting possibly strongly convex objectives. Finally, we use the methodology for studying a recent inexact variant of the Douglas-Rachford splitting due to Eckstein and Yao [44].We showcase and compare the different variants of the accelerated inexact forward-backward method on a factorization and a total variation problem.

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“…Indeed, we can easily handle constraints via projections or a regularization term h(x) with proximal operators. Recent advances in proximal subgradient descent methods and their analysis can be found in [21] and [22]. Convergence may be slow, but is remedied by using a suitable warm start.…”

Section: Algorithmsmentioning

confidence: 99%