The purpose of this paper is to study the dynamical behavior of the sequence produced by a forward-backward algorithm involving two random maximal monotone operators and a sequence of decreasing step sizes. Defining a mean monotone operator as an Aumann integral, and assuming that the sum of the two mean operators is maximal (sufficient maximality conditions are provided), it is shown that with probability one, the interpolated process obtained from the iterates is an asymptotic pseudo trajectory in the sense of Bena " m and Hirsch of the differential inclusion involving the sum of the mean operators. The convergence of the empirical means of the iterates towards a zero of the sum of the mean operators is shown, as well as the convergence of the sequence itself to such a zero under a demipositivity assumption. These results find applications in a wide range of optimization or variational inequality problems in random environments.
In the case where the constraining set has a complicated structure, an incremental minimization algorithm with random constraint updates has been proposed in @cite_1 , where a deterministic convex function @math is minimized on a finite intersection of closed and convex constraining sets. The algorithm developed in @cite_1 consists of a subgradient step over the objective @math followed by an update step towards a randomly chosen constraining set. Using the same principle, a distributed algorithm involving an additional consensus step has been proposed in @cite_22 . Random iterations involving proximal and subgradient operators were considered in @cite_10 and in @cite_20 . In @cite_20 , the functions @math are supposed to have a full domain, to satisfy @math for some constant @math which does not depend on @math and, finally, are such that @math . In the present paper, such conditions are not needed.
@cite_10 @cite_1 @cite_22 @cite_20
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This paper introduces a parallel and distributed algorithm for solving the following minimization problem with linear constraints: @math minimizef1(x1)+ź+fN(xN)subject toA1x1+ź+ANxN=c,x1źX1,ź,xNźXN,where @math Nź2, @math fi are convex functions, @math Ai are matrices, and @math Xi are feasible sets for variable @math xi. Our algorithm extends the alternating direction method of multipliers (ADMM) and decomposes the original problem into N smaller subproblems and solves them in parallel at each iteration. This paper shows that the classic ADMM can be extended to the N-block Jacobi fashion and preserve convergence in the following two cases: (i) matrices @math Ai are mutually near-orthogonal and have full column-rank, or (ii) proximal terms are added to the N subproblems (but without any assumption on matrices @math Ai). In the latter case, certain proximal terms can let the subproblem be solved in more flexible and efficient ways. We show that @math źxk+1-xkźM2 converges at a rate of o(1 k) where M is a symmetric positive semi-definte matrix. Since the parameters used in the convergence analysis are conservative, we introduce a strategy for automatically tuning the parameters to substantially accelerate our algorithm in practice. We implemented our algorithm (for the case ii above) on Amazon EC2 and tested it on basis pursuit problems with >300 GB of distributed data. This is the first time that successfully solving a compressive sensing problem of such a large scale is reported. In this paper, we consider conic programming problems whose constraints consist of linear equalities, linear inequalities, a nonpolyhedral cone, and a polyhedral cone. A convenient way for solving this class of problems is to apply the directly extended alternating direction method of multipliers (ADMM) to its dual problem, which has been observed to perform well in numerical computations but may diverge in theory. Ideally, one should find a convergent variant which is at least as efficient as the directly extended ADMM in practice. We achieve this goal by designing a convergent semiproximal ADMM (called sPADMM3c for convenience) for convex programming problems having three separable blocks in the objective function with the third part being linear. At each iteration, the proposed sPADMM3c takes one special block coordinate descent (BCD) cycle with the order @math , instead of the usual @math Gauss--Seidel BCD cycle used in the nonconvergent directly extended 3-block ADMM, for updating the variable blocks. Our numerical experiments demonstrate that the convergent method is at least 20 faster than the directly extended ADMM with unit step-length for the vast majority of about 550 large-scale doubly nonnegative semidefinite programming problems with linear equality and or inequality constraints. This confirms that at least for conic convex programming, one can design a convergent and efficient ADMM with a special BCD cycle of updating the variable blocks. This paper deals with iterative gradient and subgradient methods with random feasibility steps for solving constrained convex minimization problems, where the constraint set is specified as the intersection of possibly infinitely many constraint sets. Each constraint set is assumed to be given as a level set of a convex but not necessarily differentiable function. The proposed algorithms are applicable to the situation where the whole constraint set of the problem is not known in advance, but it is rather learned in time through observations. Also, the algorithms are of interest for constrained optimization problems where the constraints are known but the number of constraints is either large or not finite. We analyze the proposed algorithm for the case when the objective function is differentiable with Lipschitz gradients and the case when the objective function is not necessarily differentiable. The behavior of the algorithm is investigated both for diminishing and non-diminishing stepsize values. The almost sure convergence to an optimal solution is established for diminishing stepsize. For non-diminishing stepsize, the error bounds are established for the expected distances of the weighted averages of the iterates from the constraint set, as well as for the expected sub-optimality of the function values along the weighted averages. In this paper we consider the problem of minimizing a convex function using a randomized block coordinate descent method. One of the key steps at each iteration of the algorithm is determining the update to a block of variables. Existing algorithms assume that in order to compute the update, a particular subproblem is solved exactly. In his work we relax this requirement, and allow for the subproblem to be solved inexactly, leading to an inexact block coordinate descent method. Our approach incorporates the best known results for exact updates as a special case. Moreover, these theoretical guarantees are complemented by practical considerations: the use of iterative techniques to determine the update as well as the use of preconditioning for further acceleration.