Further properties of the forward–backward envelope with applications to difference-of-convex programming

Liu, Tianxiang; Pong, Ting Kei

doi:10.1007/s10589-017-9900-2

Cited by 44 publications

(40 citation statements)

References 36 publications

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“…Nevertheless, under a local strong convexity and Lipschitz differentiability assumption onĥ, yet with no requirement on its domain, it is possible to locally identify the Bregman proximal mapping and its Moreau envelope with Euclidean objects, namely the forward-backward mapping and the corresponding FBE function [68,83]. This result complements what was first observed in [51], namely that the Euclidean FBE is, in fact, a Bregman-Moreau envelope. The advantage of this identification will be revealed in the next subsections, where local differentiability results will be deduced with virtually no effort based on already established results in the Euclidean setting.…”

Section: Fixed Pointsmentioning

confidence: 53%

A Bregman Forward-Backward Linesearch Algorithm for Nonconvex Composite Optimization: Superlinear Convergence to Nonisolated Local Minima

Ahookhosh¹,

Themelis²,

Patrinos³

2021

SIAM J. Optim.

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We introduce Bella, a locally superlinearly convergent Bregman forward-backward splitting method for minimizing the sum of two nonconvex functions, one of which satisfies a relative smoothness condition and the other one is possibly nonsmooth. A key tool of our methodology is the Bregman forward-backward envelope (BFBE), an exact and continuous penalty function with favorable first-and second-order properties, which enjoys a nonlinear error bound when the objective function satisfies a Lojasiewicz-type property. The proposed algorithm is of linesearch type over the BFBE along user-defined update directions and converges subsequentially to stationary points and globally under the Kurdyka-Lojasiewicz condition. Moreover, when the update directions are superlinear in the sense of Facchinei and Pang [Finite-Dimensional Variational Inequalities and Complementarity Problems, Volume I, Springer, New York, 2003], owing to the given nonlinear error bound unit stepsize is eventually always accepted and the algorithm attains superlinear convergence rates even when the limit point is a nonisolated minimum.

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Section: Fixed Pointsmentioning

confidence: 53%

A Bregman Forward-Backward Linesearch Algorithm for Nonconvex Composite Optimization: Superlinear Convergence to Nonisolated Local Minima

Ahookhosh¹,

Themelis²,

Patrinos³

2021

SIAM J. Optim.

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“…After analyzing the objective function (9) of DSCA, we can see that it is a difference combination of trace terms regarding A. Such a objective function may be nonconvex in practice [36], [37], since its Hessian matrix with regard to A is a difference combination of positive definite matrices and consequently may not be positive definite. Although its optimal solutions may be obtained via the generalized eigendecomposition algorithm, its optimality may be limited since it is solved in a conditionally-convex solution space spanned by a set of linear orthogonal eigenvectors (see (11)), whose optimality is dominated by the convexity of (9).…”

Section: B Convex Dsca (C-dsca)mentioning

confidence: 99%

A Convex Discriminant Semantic Correlation Analysis for Cross-View Recognition

Qiu

Cao

et al. 2022

IEEE Trans. Cybern.

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Canonical correlation analysis (CCA) is a typical statistical model used to analyze the correlation components between different view representations of the same objects. When the label information is available with the data representations, CCA can be extended to its discriminative counterparts by incorporating supervision in the analysis. Although most discriminative variants of CCA have achieved improved results, nearly all of their objective functions are nonconvex, implying that optimal solutions are difficult to obtain. More importantly, that crossview representations from the same sample should be consistent, i.e., the cross-view semantic consistency, has however not been modelled. To overcome these drawbacks, in this paper we propose a Discriminant Semantic Correlation Analysis (DSCA) model by modelling the cross-view semantic consistency for each object in the sample space rather than in the commonly used feature space. To boost the nonlinear discriminating capability of DSCA, we extend it from the Euclidean to the geodesic space by transforming the metric and incorporating both the cross-view semantic and representation correlation information and consequently obtain our final model with convex objective, namely Convex DSCA (C-DSCA). Finally, with extensive experiments and comparisons we validate the effectiveness and superiority of the proposed method.

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“…It is rooted to the sparse signal under tight frame and the constrained and unconstrained x 1 − α x 2 minimizations, which has recently attracted a lot of attention. The constrained x 1 − α x 2 minimization [18,28,33,36,37,52] is recovery of x. The unconstrained x 1 − α x 2 minimization [19,33,36,37,52] is…”

Section: Contributionsmentioning

confidence: 99%

Signal and Image Reconstruction with Tight Frames via Unconstrained $\ell_1-α\ell_2$-Analysis Minimizations

Ge¹,

Geng²

2021

Preprint

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In the paper, we introduce an unconstrained analysis model based on the ℓ 1 − αℓ 2 (0 < α ≤ 1) minimization for the signal and image reconstruction. We develop some new technology lemmas for tight frame, and the recovery guarantees based on the restricted isometry property adapted to frames. The effective algorithm is established for the proposed nonconvex analysis model. We illustrate the performance of the proposed model and algorithm for the signal and compressed sensing MRI reconstruction via extensive numerical experiments. And their performance is better than that of the existing methods.

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Further properties of the forward–backward envelope with applications to difference-of-convex programming

Cited by 44 publications

References 36 publications

A Bregman Forward-Backward Linesearch Algorithm for Nonconvex Composite Optimization: Superlinear Convergence to Nonisolated Local Minima

A Bregman Forward-Backward Linesearch Algorithm for Nonconvex Composite Optimization: Superlinear Convergence to Nonisolated Local Minima

A Convex Discriminant Semantic Correlation Analysis for Cross-View Recognition

Signal and Image Reconstruction with Tight Frames via Unconstrained $\ell_1-α\ell_2$-Analysis Minimizations

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