The ABC of DC Programming

Oliveira, Welington de

doi:10.1007/s11228-020-00566-w

Cited by 26 publications

(11 citation statements)

References 31 publications

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“…) we conclude that ν k+1 satisfies (15). Therefore, the strategy defined in (19) is a particular instance of (A1) which turns Algorithm 3 into a non-monotone boosted version of the DCA employing the non-monotone line search proposed in [21].…”

Section: Asymptotic Convergence Analysismentioning

confidence: 78%

“…Thus, we can take ν k+1 ≥ 0 satisfying (15). Furthermore, if (ν k ) k∈N satisfies (A1) with δ min > 0, then (ν k ) k∈N also satisfies (A2).…”

Section: Asymptotic Convergence Analysismentioning

confidence: 99%

“…To the best of our knowledge, DCA was the first algorithm directly designed to solve (1); see [34,35]. Since then, several variants of DCA have arisen and its theoretical and practical properties have been investigated over the years, resulting in a wide literature on the subject; see [15,26]. Algorithmic aspects of DC programming have been received significant attention lately such as subgradient-type (see [8,25]), proximal-subgradient [12,28,32,33], proximal bundle [14], double bundle [24], codifferential [5] and inertial method [16].…”

Section: Introductionmentioning

confidence: 99%

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A boosted DC algorithm for non-differentiable DC components with non-monotone line search

Ferreira¹,

Santos²,

Souza³

2021

Preprint

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We introduce a new approach to apply the boosted difference of convex functions algorithm (BDCA) for solving non-convex and non-differentiable problems involving difference of two convex functions (DC functions). Supposing the first DC component differentiable and the second one possibly non-differentiable, the main idea of BDCA is to use the point computed by the DC algorithm (DCA) to define a descent direction and perform a monotone line search to improve the decreasing the objetive function accelerating the convergence of the DCA. However, if the first DC component is non-differentiable, then the direction computed by BDCA can be an ascent direction and a monotone line search cannot be performed. Our approach uses a non-monotone line search in the BDCA (nmBDCA) to enable a possible growth in the objective function values controlled by a parameter. Under suitable assumptions, we show that any cluster point of the sequence generated by the nmBDCA is a critical point of the problem under consideration and provide some iteration-complexity bounds. Furthermore, if the first DC component is differentiable, we present different iteration-complexity bounds and prove the full convergence of the sequence under the Kurdyka-Lojasiewicz property of the objective function. Some numerical experiments show that the nmBDCA outperforms the DCA such as its monotone version.

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Section: Asymptotic Convergence Analysismentioning

confidence: 78%

“…Thus, we can take ν k+1 ≥ 0 satisfying (15). Furthermore, if (ν k ) k∈N satisfies (A1) with δ min > 0, then (ν k ) k∈N also satisfies (A2).…”

Section: Asymptotic Convergence Analysismentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A boosted DC algorithm for non-differentiable DC components with non-monotone line search

Ferreira¹,

Santos²,

Souza³

2021

Preprint

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“…Proof. First we fix an arbitrary subgradient v ∈ ∂f (x) and deduce from (11) applied to x = x and v = v that…”

Section: Remark 23mentioning

confidence: 99%

Coderivative-Based Semi-Newton Method in Nonsmooth Difference Programming

Artacho¹,

Mordukhovich²,

Pérez-Aros³

2023

Preprint

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This paper addresses the study of a new class of nonsmooth optimization problems, where the objective is represented as a difference of two generally nonconvex functions. We propose and develop a novel Newton-type algorithm to solving such problems, which is based on the coderivative generated second-order subdifferential (generalized Hessian) and employs advanced tools of variational analysis. Well-posedness properties of the proposed algorithm are derived under fairly general requirements, while constructive convergence rates are established by using additional assumptions including the Kurdyka-Lojasiewicz condition. We provide applications of the main algorithm to solving a general class of nonsmooth nonconvex problems of structured optimization that encompasses, in particular, optimization problems with explicit constraints. Finally, applications and numerical experiments are given for solving practical problems that arise in biochemical models, con-

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“…DC optimization algorithms have been proved to be particularly successful for analyzing and solving a variety of highly structured and practical problems; see for instance [4,17,32]. To the best of our knowledge Souza and Oliveira [50] was the first work dealing with DC functions in Riemannian manifolds, more precisely, the authors proposed the proximal point algorithm for DC functions studying the convergence of the method in Hadamard manifolds.…”

Section: Introductionmentioning

confidence: 99%

The difference of convex algorithm on Riemannian manifolds

Ferreira¹,

Santos²,

Souza³

2021

Preprint

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In this paper we propose a Riemannian version of the difference of convex algorithm (DCA) to solve a minimization problem involving the difference of convex (DC) function. Equivalence between the classical and a simplified Riemannian version of the DCA is stablished. We also prove that under mild assumptions the Riemannian version of the DCA is well defined and every cluster point of the sequence generated by the proposed method, if any, is a critical point of the objective DC function. Some duality relations between the DC problem and its dual are also established. Applications to the problem of maximizing a convex function in a compact set and manifold-valued image denoising are presented in the DC context.

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The ABC of DC Programming

Cited by 26 publications

References 31 publications

A boosted DC algorithm for non-differentiable DC components with non-monotone line search

A boosted DC algorithm for non-differentiable DC components with non-monotone line search

Coderivative-Based Semi-Newton Method in Nonsmooth Difference Programming

The difference of convex algorithm on Riemannian manifolds

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