This paper establishes convergence of generalized Bregman-function-based proximal point algorithms when the iterates are computed only approximately. The problem being solved is modeled as a general maximal monotone operator, and need not reduce to minimization of a function. The accuracy conditions on the iterates resemble those required for the classical "linear" proximal point algorithm, but are slightly stronger; they should be easier to verify or enforce in practice than conditions conditions given in earlier analyses of approximate generalized proximal methods. Subject to these practically enforceable accuracy restrictions, convergence is obtained under the same conditions currently established for exact Bregman-function-based proximal methods.

Abstract. We study a class of generalized bundle methods for which the stabilizing term can be any closed convex function satisfying certain properties. This setting covers several algorithms from the literature that have been so far regarded as distinct. Under a different hypothesis on the stabilizing term and/or the function to be minimized, we prove finite termination, asymptotic convergence, and finite convergence to an optimal point, with or without limits on the number of serious steps and/or requiring the proximal parameter to go to infinity. The convergence proofs leave a high degree of freedom in the crucial implementative features of the algorithm, i.e., the management of the bundle of subgradients (β-strategy) and of the proximal parameter (t-strategy). We extensively exploit a dual view of bundle methods, which are shown to be a dual ascent approach to one nonlinear problem in an appropriate dual space, where nonlinear subproblems are approximately solved at each step with an inner linearization approach. This allows us to precisely characterize the changes in the subproblems during the serious steps, since the dual problem is not tied to the local concept of ε-subdifferential. For some of the proofs, a generalization of inf-compactness, called ∗-compactness, is required; this concept is related to that of asymptotically well-behaved functions. Key words. nondifferentiable optimization, bundle methods AMS subject classifications. 90C25 PII. S1052623498342186 Introduction. We are concerned with the numerical solution of the primal problem

The problem of computing a maximum a posteriori (MAP) configuration is a central computational challenge associated with Markov random fields. A line of work has focused on “tree-based ” linear programming (LP) relaxations for the MAP problem. This paper develops a family of super-linearly convergent algorithms for solving these LPs, based on proximal minimization schemes using Bregman divergences. As with standard messagepassing on graphs, the algorithms are distributed and exploit the underlying graphical structure, and so scale well to large problems. Our algorithms have a double-loop character, with the outer loop corresponding to the proximal sequence, and an inner loop of cyclic Bregman divergences used to compute each proximal update. Different choices of the Bregman divergence lead to conceptually related but distinct LP-solving algorithms. We establish convergence guarantees for our algorithms, and illustrate their performance via some simulations. We also develop two classes of graph-structured rounding schemes, randomized and deterministic, for obtaining integral configurations from the LP solutions. Our deterministic rounding schemes use a “re-parameterization ” property of our algorithms so that when the LP solution is integral, the MAP solution can be obtained even before the LP-solver converges to the optimum. We also propose a graph-structured randomized rounding scheme that applies to iterative LP solving algorithms in general. We analyze the performance of our rounding schemes, giving bounds on the number of iterations required, when the LP is integral, for the rounding schemes to obtain the MAP solution. These bounds are expressed in terms of the strength of the potential functions, and the energy gap, which measures how well the integral MAP solution is separated from other integral configurations. We also report simulations comparing these rounding schemes. 1

We analyze proximal methods based on entropy-like distances for the minimization of convex functions subject to nonnegativity constraints. We prove global convergence results for the methods with approximate minimization steps and an ergodic convergence result for the case of finding a zero of a maximal monotone operator. We also consider linearly constrained convex problems and establish a quadratic convergence rate result for linear programs. Our analysis allows us to simplify and extend the available convergence results for these methods.

In this paper, we propose an infeasible interior proximal method for solving a convex programming problem with linear constraints. The interior proximal method proposed by Auslender and Haddou is a proximal method using a distance-like barrier function, and it has a global convergence property under mild assumptions. However this method is applicable only to problems whose feasible region has an interior point, because an initial point for the method must be chosen from the interior of the feasible region. The algorithm proposed in this paper is based on the idea underlying the infeasible interior point method for linear programming. This algorithm is applicable to problems whose feasible region may not have a nonempty interior, and it can be started from an arbitrary initial point. We establish global convergence of the proposed algorithm under appropriate assumptions.

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