An adaptive primal-dual framework for nonsmooth convex minimization

Tran-Dinh, Quoc; Alacaoglu, Ahmet; Fercoq, Olivier; Cevher, Volkan

doi:10.1007/s12532-019-00173-3

Cited by 21 publications

(28 citation statements)

References 108 publications

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“…There is a rich literature on primal-dual algorithms searching for a saddle point of L (see [45] and references therein). In the special case where f = 0, the alternating direction method of multipliers (ADMM) proposed by Glowinsky and Marroco [25], Gabay and Mercier [23] and the algorithm of Chambolle and Pock [12] are amongst the most celebrated ones.…”

Section: Motivationmentioning

confidence: 99%

A Coordinate-Descent Primal-Dual Algorithm with Large Step Size and Possibly Nonseparable Functions

Fercoq¹,

Bianchi²

2019

SIAM J. Optim.

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This paper introduces a randomized coordinate descent version of the Vũ-Condat algorithm. By coordinate descent, we mean that only a subset of the coordinates of the primal and dual iterates is updated at each iteration, the other coordinates being maintained to their past value. Our method allows us to solve optimization problems with a combination of differentiable functions, constraints as well as non-separable and non-differentiable regularizers.We show that the sequences generated by our algorithm almost surely converge to a saddle point of the problem at stake, for a wider range of parameter values than previous methods. In particular, the condition on the step-sizes depends on the coordinate-wise Lipschitz constant of the differentiable function's gradient, which is a major feature allowing classical coordinate descent to perform so well when it is applicable. We then prove a sublinear rate of convergence in general and a linear rate of convergence if the objective enjoys strong convexity properties.We illustrate the performances of the algorithm on a total-variation regularized least squares regression problem and on large scale support vector machine problems.

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Section: Motivationmentioning

confidence: 99%

A Coordinate-Descent Primal-Dual Algorithm with Large Step Size and Possibly Nonseparable Functions

Fercoq¹,

Bianchi²

2019

SIAM J. Optim.

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“…As we can see from (8) of Lemma 1 that the bound on dist K (Ax + By − c) depends on λ instead of λ − λ 0 from an initial dual variable λ 0 . We use the idea of "restarting the prox-center point" from [47,48] to adaptively update λ 0 . This idea has been recently used in [38,49] as a restarting strategy and it has significantly improved the performance of the algorithms.…”

Section: Shifting the Initial Dual Variable And Restartingmentioning

confidence: 99%

“…where ∇ϕρ is given by (47). Since proving the convergence of this variant is out of scope of this paper, we refer to our forthcoming work [46] for the full theory of restarting.…”

Section: Shifting the Initial Dual Variable And Restartingmentioning

confidence: 99%

“…If we set it too big, then it does not improve the performance and we need to run with a large number of iterations. In [46], we provide a full theory on how to adaptively choose the frequency to guarantee the convergence of the restarting ASGARD methods, which can also be applied to PAPA.…”

Section: Dense Convex Quadratic Programsmentioning

confidence: 99%

“…We also discuss restarting strategies for our methods to significantly improve their practical performance. The convergence guarantee of this strategy will be presented in our forthcoming work [46]. Let us clarify the following points of our contribution.…”

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confidence: 98%

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Proximal alternating penalty algorithms for nonsmooth constrained convex optimization

Tran-Dinh

2018

Comput Optim Appl

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We develop two new proximal alternating penalty algorithms to solve a wide range class of constrained convex optimization problems. Our approach mainly relies on a novel combination of the classical quadratic penalty, alternating minimization, Nesterov's acceleration, and adaptive strategy for parameters. The first algorithm is designed to solve generic and possibly nonsmooth constrained convex problems without requiring any Lipschitz gradient continuity or strong convexity, while achieving the best-known O 1 k -convergence rate in a non-ergodic sense, where k is the iteration counter. The second algorithm is also designed to solve non-strongly convex, but semi-strongly convex problems. This algorithm can achieve the best-known O 1 k 2convergence rate on the primal constrained problem. Such a rate is obtained in two cases: (i) averaging only on the iterate sequence of the strongly convex term, or (ii) using two proximal operators of this term without averaging. In both algorithms, we allow one to linearize the second subproblem to use the proximal operator of the corresponding objective term. Then, we customize our methods to solve different convex problems, and lead to new variants. As a byproduct, these algorithms preserve the same convergence guarantees as in our main algorithms. We verify our theoretical development via different numerical examples and compare our methods with some existing state-of-the-art algorithms.Keywords Proximal alternating algorithm · quadratic penalty method · accelerated scheme · constrained convex optimization · first-order methods · convergence rate.

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Optimal ecological transition path of a credit portfolio distribution, based on multidate Monge–Kantorovich formulation

Gobet

Lage

2023

Ann Oper Res

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Accounting for climate transition risks is one of the most important challenges in the transition to a low-carbon economy. Banks are encouraged to align their investment portfolios to CO2 trajectories fixed by international agreements, showing the necessity of a quantitative methodology to implement it. We propose a mathematical formulation for this problem and a multistage optimization criterion for a transition between the current bank portfolio and a target one. The optimization problem combines the Monge-Kantorovich formulation of optimal transport, for which the cost is defined according to the financial context, and a credit risk measure. We show that the problem is well-posed, and can be embedded into a saddle-point problem for which Primal-Dual algorithms can be used. We design a numerical scheme that is able to solve the problem in available time, with nice scalability properties according to the number of decision times; its numerical convergence is analysed. Last we test the model using real financial data, illustrating that the optimal portfolio alignment may di er from the naive interpolation between the initial portfolio and the target.

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An adaptive primal-dual framework for nonsmooth convex minimization

Cited by 21 publications

References 108 publications

A Coordinate-Descent Primal-Dual Algorithm with Large Step Size and Possibly Nonseparable Functions

A Coordinate-Descent Primal-Dual Algorithm with Large Step Size and Possibly Nonseparable Functions

Proximal alternating penalty algorithms for nonsmooth constrained convex optimization

Optimal ecological transition path of a credit portfolio distribution, based on multidate Monge–Kantorovich formulation

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