We investigate the asymptotic behavior of a stochastic version of the forward-backward splitting algorithm for finding a zero of the sum of a maximally monotone set-valued operator and a cocoercive operator in Hilbert spaces. Our general setting features stochastic approximations of the cocoercive operator and stochastic perturbations in the evaluation of the resolvents of the … Read more
We propose new primal-dual decomposition algorithms for solving systems of inclusions involving sums of linearly composed maximally monotone operators. The principal innovation in these algorithms is that they are block-iterative in the sense that, at each iteration, only a subset of the monotone operators needs to be processed, as opposed to all operators as in … Read more
Computational methods are proposed for solving a convex quadratic program (QP). Active-set methods are defined for a particular primal and dual formulation of a QP with general equality constraints and simple lower bounds on the variables. In the first part of the paper, two methods are proposed, one primal and one dual. These methods generate … Read more
Operator splitting schemes are a class of powerful algorithms that solve complicated monotone inclusion and convex optimization problems that are built from many simpler pieces. They give rise to algorithms in which all simple pieces of the decomposition are processed individually. This leads to easily implementable and highly parallelizable or distributed algorithms, which often obtain … Read more
We consider the total variation minimization model with consistent finite element discretization. It has been shown in the literature that this model can be reformulated as a saddle-point problem and be efficiently solved by the primal-dual method. The convergence for this application of the primal-dual method has also been analyzed. In this paper, we focus … Read more
The primal-dual algorithm recently proposed by Chambolle \& Pock (abbreviated as CPA) for structured convex optimization is very efficient and popular. It was shown by Chambolle \& Pock in \cite{CP11} and also by Shefi \& Teboulle in \cite{ST14} that CPA and variants are closely related to preconditioned versions of the popular alternating direction method of … Read more
We revisit the proofs of convergence for a first order primal-dual algorithm for convex optimization which we have studied a few years ago. In particular, we prove rates of convergence for a more general version, with simpler proofs and more complete results. ArticleDownload View PDF
Primal-dual splitting schemes are a class of powerful algorithms that solve complicated monotone inclusions and convex optimization problems that are built from many simpler pieces. They decompose problems that are built from sums, linear compositions, and infimal convolutions of simple functions so that each simple term is processed individually via proximal mappings, gradient mappings, and … Read more
The alternating direction method of multipliers (ADM or ADMM) breaks a complex optimization problem into much simpler subproblems. The ADM algorithms are typically short and easy to implement yet exhibit (nearly) state-of-the-art performance for large-scale optimization problems. To apply ADM, we first formulate a given problem into the “ADM-ready” form, so the final algorithm depends … Read more
Regularized and stabilized sequential quadratic programming (SQP) methods are two classes of methods designed to resolve the numerical and theoretical difficulties associated with ill-posed or degenerate nonlinear optimization problems. Recently, a regularized SQP method has been proposed that allows convergence to points satisfying certain second-order KKT conditions (SIAM J. Optim., 23(4):1983–2010, 2013). The method is … Read more