A general double-proximal gradient algorithm for d.c. programming

Banert, Sebastian; Boţ, Radu Ioan

doi:10.1007/s10107-018-1292-2

Cited by 25 publications

(30 citation statements)

References 30 publications

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“…This together with Theorem 4.2 and Remark 4.2 shows that the sequence {x t } generated by pDCA e is globally convergent to a stationary point of (5.3). 6 These λ satisfy λ < 1 2 A T b ∞ for all our random instances. 7 In the tables, "max" means the number of iterations hits 5000.…”

Section: Least Squares Problems With Logarithmic Regularizermentioning

confidence: 92%

“…We will show that the sequence {x t } generated by pDCA e is convergent to a stationary point of F under suitable assumptions. Our analysis follows a similar line of arguments to other convergence analysis based on KL property (see, for example, [3,4,5,6]), but has to make extensive use of the following auxiliary function: (i) lim t→∞ dist((0, 0), ∂E(x t , x t−1 )) = 0.…”

Section: Convergence Analysis I: Global Subsequential Convergence Of mentioning

confidence: 99%

“…The computational results are presented in Tables 1 and 2, which correspond to problem (5.2) with λ = 5 × 10 −4 and λ = 1 × 10 −3 respectively. 6 We report the time for computing λ max (A T A) (t λmax ), the number of iterations (iter), 7 CPU times in seconds (CPU time), 8 and the function values at termination (fval), averaged over the 30 random instances. We can see that pDCA e always outperforms pDCA and GIST.…”

Section: Least Squares Problems With ℓ 1−2 Regularizermentioning

confidence: 99%

“…A classical algorithm for solving DC optimization problems is the so-called DC algorithm (DCA), which was proposed by Tao and An [26]; see also [6,18,30,31,32] for more recent developments. 1 In each iteration, this algorithm replaces the concave part of the objective by a linear majorant and solves the resulting convex optimization problem.…”

Section: Introductionmentioning

confidence: 99%

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A proximal difference-of-convex algorithm with extrapolation

2017

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We consider a class of difference-of-convex (DC) optimization problems whose objective is levelbounded and is the sum of a smooth convex function with Lipschitz gradient, a proper closed convex function and a continuous concave function. While this kind of problems can be solved by the classical difference-of-convex algorithm (DCA) [26], the difficulty of the subproblems of this algorithm depends heavily on the choice of DC decomposition. Simpler subproblems can be obtained by using a specific DC decomposition described in [27]. This decomposition has been proposed in numerous work such as [18], and we refer to the resulting DCA as the proximal DCA. Although the subproblems are simpler, the proximal DCA is the same as the proximal gradient algorithm when the concave part of the objective is void, and hence is potentially slow in practice. In this paper, motivated by the extrapolation techniques for accelerating the proximal gradient algorithm in the convex settings, we consider a proximal difference-of-convex algorithm with extrapolation to possibly accelerate the proximal DCA. We show that any cluster point of the sequence generated by our algorithm is a stationary point of the DC optimization problem for a fairly general choice of extrapolation parameters: in particular, the parameters can be chosen as in FISTA with fixed restart [15]. In addition, by assuming the Kurdyka-Lojasiewicz property of the objective and the differentiability of the concave part, we establish global convergence of the sequence generated by our algorithm and analyze its convergence rate. Our numerical experiments on two difference-of-convex regularized least squares models show that our algorithm usually outperforms the proximal DCA and the general iterative shrinkage and thresholding algorithm proposed in [17].

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Section: Least Squares Problems With Logarithmic Regularizermentioning

confidence: 92%

Section: Convergence Analysis I: Global Subsequential Convergence Of mentioning

confidence: 99%

Section: Least Squares Problems With ℓ 1−2 Regularizermentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A proximal difference-of-convex algorithm with extrapolation

2017

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“…see, e.g., [2,Proposition 1]. We call (â,η) a critical point of Φ if K * η ∈ ∂G(â) and Kâ ∈ ∂F * (η).…”

Section: The Case γmentioning

confidence: 99%

Variational Models for Color Image Correction Inspired by Visual Perception and Neuroscience

Batard¹,

Hertrich²,

Steidl³

2020

J Math Imaging Vis

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Reproducing the perception of a real-world scene on a display device is a very challenging task which requires the understanding of the camera processing pipeline, the display process, and the way the human visual system processes the light it captures. Mathematical models based on psychophysical and physiological laws on color vision, named Retinex, provide efficient tools to handle degradations produced during the camera processing pipeline like the reduction of the contrast. In particular, Batard and Bertalmío [J Math. Imag. Vis. 60(6), 849-881 (2018)] described some psychophysical laws on brightness perception as covariant derivatives, included them into a variational model, and observed that the quality of the color image correction is correlated with the accuracy of the vision model it includes. Based on this observation, we postulate that this model can be improved by including more accurate data on vision with a special attention on visual neuroscience here. Then, inspired by the presence of neurons responding to different visual attributes in the area V1 of the visual cortex as orientation, color or movement, to name a few, and horizontal connections modeling the interactions between those neurons, we construct two variational models to process both local (edges,

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Second-Order Numerical Variational Analysis

Mordukhovich

2024

Springer Series in Operations Research and Financial Engineering

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A general double-proximal gradient algorithm for d.c. programming

Cited by 25 publications

References 30 publications

A proximal difference-of-convex algorithm with extrapolation

A proximal difference-of-convex algorithm with extrapolation

Variational Models for Color Image Correction Inspired by Visual Perception and Neuroscience

Second-Order Numerical Variational Analysis

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