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 same vein as our paper, @cite_28 considered a collection @math of @math maximal monotone operators, and studied the iterations [ y_ n+1 A( n+1 (1), x_n) , , x_ n+1 = i=2 ^N (I + n+1 A( n+1 (i), ))^ -1 (x_n - n+1 y_ n+1 ) , , ] where @math , and where @math is a sequence of permutations of the set @math . The convergence of @math to a zero of @math is established in @cite_28 . In the recent paper @cite_12 , a relaxed version of Algorithm is considered, where @math is cocoercive and where its output, as well as the output of the resolvent of @math , are subjected to random errors. The convergence of the iterates to a zero of @math is established under summability assumptions on these errors.
@cite_28 @cite_12
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The purpose of this paper is to establish the almost sure weak ergodic convergence of a sequence of iterates @math given by @math where @math is a collection of maximal monotone operators on a separable Hilbert space, @math is an independent identically distributed sequence of random variables on @math and @math is a positive sequence in @math . The weighted averaged sequence of iterates is shown to converge weakly to a zero (assumed to exist) of the Aumann expectation @math under the assumption that the latter is maximal. We consider applications to stochastic optimization problems of the form @math , where @math is a normal convex integrand and @math is a collection of closed convex sets. In this case, the iterations are closely related to a stochastic proximal algorithm recently proposed by Wang and Bertsekas [Increment... For the problem of minimizing a lower semicontinuous proper convex function f on a Hilbert space, the proximal point algorithm in exact form generates a sequence @math by taking @math to be the minimizes of @math , where @math . This algorithm is of interest for several reasons, but especially because of its role in certain computational methods based on duality, such as the Hestenes-Powell method of multipliers in nonlinear programming. It is investigated here in a more general form where the requirement for exact minimization at each iteration is weakened, and the subdifferential @math is replaced by an arbitrary maximal monotone operator T. Convergence is established under several criteria amenable to implementation. The rate of convergence is shown to be “typically” linear with an arbitrarily good modulus if @math stays large enough, in fact superlinear if @math . The case of @math is treated in extra detail. Applicati... 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. Let f be a random Boolean formula that is an instance of 3-SAT. We consider the problem of computing the least real number k such that if the ratio of the number of clauses over the number of variables of f strictly exceeds k , then f is almost certainly unsatisfiable. By a well-known and more or less straightforward argument, it can be shown that kF5.191. This upper bound was improved by to 4.758 by first providing new improved bounds for the occupancy problem. There is strong experimental evidence that the value of k is around 4.2. In this work, we define, in terms of the random formula f, a decreasing sequence of random variables such that, if the expected value of any one of them converges to zero, then f is almost certainly unsatisfiable. By letting the expected value of the first term of the sequence converge to zero, we obtain, by simple and elementary computations, an upper bound for k equal to 4.667. From the expected value of the second term of the sequence, we get the value 4.601q . In general, by letting the U This work was performed while the first author was visiting the School of Computer Science, Carleton Ž University, and was partially supported by NSERC Natural Sciences and Engineering Research Council . of Canada , and by a grant from the University of Patras for sabbatical leaves. The second and third Ž authors were supported in part by grants from NSERC Natural Sciences and Engineering Research . Council of Canada . During the last stages of this research, the first and last authors were also partially Ž . supported by EU ESPRIT Long-Term Research Project ALCOM-IT Project No. 20244 . †An extended abstract of this paper was published in the Proceedings of the Fourth Annual European Ž Symposium on Algorithms, ESA’96, September 25]27, 1996, Barcelona, Spain Springer-Verlag, LNCS, . pp. 27]38 . That extended abstract was coauthored by the first three authors of the present paper. Correspondence to: L. M. Kirousis Q 1998 John Wiley & Sons, Inc. CCC 1042-9832r98r030253-17 253