Bregman Forward-Backward Operator Splitting

Bùi, Minh N.; Combettes, Patrick L.

doi:10.1007/s11228-020-00563-z

Cited by 17 publications

(4 citation statements)

References 18 publications

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“…This algorithm was first studied in [45] (see also [14]). Now let A = ∂ϕ for some proper lower semicontinuous convex function ϕ : H → ]−∞, +∞], let B = ∇ψ for some differentiable convex function ψ : H → R, and set θ = ϕ + ψ.…”

Section: Direct Iterationsmentioning

confidence: 99%

Warped Proximal Iterations for Monotone Inclusions

Bùi¹,

Combettes²

2019

Preprint

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Resolvents play a central role in the design and the analysis of splitting algorithms for solving monotone inclusions. We investigate a generalization of this notion, called warped resolvent, which is constructed with the help of an auxiliary operator. Iterations involving warped resolvents are shown to capture a wide range of algorithmic schemes and to lead to new monotone operator splitting methods. Particular attention is given to cutting plane algorithms, which operate with successive outer approximations to the solution set. Weakly and strongly convergent algorithms are devised in this context.

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Section: Direct Iterationsmentioning

confidence: 99%

Warped Proximal Iterations for Monotone Inclusions

Bùi¹,

Combettes²

2019

Preprint

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“…The conditions on the kernel have been further relaxed in [14], which allows for non-self-adjoint linear kernels. In the multiple works on Bregman-distance based resolvents, for instance [15][16][17][18][19][20][21], the linearity condition is dropped altogether by allowing the kernel to be the gradient of some differentiable convex function. These relaxations allow the resolvent to be adapted to a particular problem, either to improve the speed of convergence or to make an otherwise intractable resolvent evaluation tractable.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear Forward-Backward Splitting with Momentum Correction

Morin¹,

Banert²,

Giselsson³

2021

Preprint

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The nonlinear, or warped, resolvent recently explored by Giselsson and Bùi-Combettes has been used to model a large set of existing and new monotone inclusion algorithms. To establish convergent algorithms based on these resolvents, corrective projection steps are utilized in both works. We present a different way of ensuring convergence by means of a nonlinear momentum term, which in many cases leads to cheaper per-iteration cost. The expressiveness of our method is demonstrated by deriving a wide range of special cases. These cases cover and expand on the forward-reflectedbackward method of Malitsky-Tam, the primal-dual methods of Vũ-Condat and Chambolle-Pock, and the forward-reflected-Douglas-Rachford method of Ryu-Vũ. A new primal-dual method that uses an extra resolvent step is also presented as well as a general approach for adding Polyak momentum to any special case of our nonlinear forward-backward method, in particular all the algorithms listed above.

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“…This concept plays an important role in various field of applied mathematics and information theory. Numerical optimization methods with Bregman proximity operator are investigated widely in the literature; see [11,2,9,17,20,26,28,30,37,22,32] for instances. In particular, the stochastic numerical methods with Bregman distance for saddle point problems are of great interest; see [3,30,36,12] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

A Stochastic Variance Reduction Algorithm with Bregman Distances for Structured Composite Problems

Dũng¹,

Vũ²

2021

Preprint

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We develop a novel stochastic primal dual splitting method with Bregman distances for solving a structured composite problems involving infimal convolutions in non-Euclidean spaces. The sublinear convergence in expectation of the primal-dual gap is proved under mild conditions on stepsize for the general case. The linear convergence rate is obtained under additional condition like the strong convexity relative to Bregman functions.

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Bregman Forward-Backward Operator Splitting

Cited by 17 publications

References 18 publications

Warped Proximal Iterations for Monotone Inclusions

Warped Proximal Iterations for Monotone Inclusions

Nonlinear Forward-Backward Splitting with Momentum Correction

A Stochastic Variance Reduction Algorithm with Bregman Distances for Structured Composite Problems

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