Abstract not found

We develop an approach for computing provably exact maximum a posteriori (MAP) configurations for a subclass of problems on graphs with cycles. By decomposing the original problem into a convex combination of tree-structured problems, we obtain an upper bound on the optimal value of the original problem (i.e., the log probability of the MAP assignment) in terms of the combined optimal values of the tree problems. We prove that this upper bound is met with equality if and only if the tree problems share an optimal configuration in common. An important implication is that any such shared configuration must also be a MAP configuration for the original problem. Next we present and analyze two methods for attempting to obtain tight upper bounds: (a) a tree-reweighted messagepassing algorithm that is related to but distinct from the max-product (min-sum) algorithm; and (b) a tree-relaxed linear program (LP), which is derived from the Lagrangian dual of the upper bounds. Finally, we discuss the conditions that govern when the relaxation is tight, in which case the MAP configuration can be obtained. The analysis described here generalizes naturally to convex combinations of hypertree-structured distributions.

We develop and analyze methods for computing provably optimal maximum a posteriori (MAP) configurations for a subclass of Markov random fields defined on graphs with cycles. By decomposing the original distribution into a convex combination of tree-structured distributions, we obtain an upper bound on the optimal value of the original problem (i.e., the log probability of the MAP assignment) in terms of the combined optimal values of the tree problems. We prove that this upper bound is tight if and only if all the tree distributions share an optimal configuration in common. An important implication is that any such shared configuration must also be a MAP configuration for the original distribution. Next we develop two approaches to attempting to obtain tight upper bounds: (a) a tree-relaxed linear program (LP), which is derived from the Lagrangian dual of the upper bounds; and (b) a tree-reweighted max-product messagepassing algorithm that is related to but distinct from the max-product algorithm. In this way, we establish a connection between a certain LP relaxation of the modefinding problem, and a reweighted form of the max-product (min-sum) message-passing algorithm.

The subgradient method is frequently used to optimize dual functions in Lagrangian relaxation for separable integer programming problems. In the method, all subproblems must be optimally solved to obtain a subgradient direction. In this paper, the "surrogate subgradient method" is developed, where a proper direction can be obtained without optimally solving all the subproblems. In fact, only approximate optimization of one subproblem is needed to get a proper "surrogate subgradient direction," and the directions are smooth for problems of large size. The convergence of the algorithm is proved. Compared with methods that take effort to find better directions, this method can obtain good directions with much less effort, and provides a new approach that is especially powerful for problems of very large size.

Semidefinite relaxations of quadratic 0-1 programming or graph partitioning problems are well known to be of high quality. However, solving them by primaldual interior point methods can take much time even for problems of moderate size. The recent spectral bundle method of Helmberg and Rendl can solve quite efficiently large structured equality-constrained semidefinite programs if the trace of the primal matrix variable is fixed, as happens in many applications. We extend the method so that it can handle inequality constraints without seriously increasing computation time. Encouraging preliminary computational results are reported.

Abstract. Image inpainting is a fundamental problem in image processing and has many applications. Motivated by the recent tight frame based methods on image restoration in either the image or the transform domain, we propose an iterative tight frame algorithm for image inpainting. We consider the convergence of this framelet-based algorithm by interpreting it as an iteration for minimizing a special functional. The proof of the convergence is under the framework of convex analysis and optimization theory. We also discuss the relationship of our method with other wavelet-based methods. Numerical experiments are given to illustrate the performance of the proposed algorithm. Key words. Tight frame, inpainting, convex analysis 1. Introduction. The problem of inpainting [2] occurs when part of the pixel data in a picture is missing or over-written by other means. This arises for example in restoring ancient drawings, where a portion of the picture is missing or damaged due to aging or scratch; or when an image is transmitted through a noisy channel. The task of inpainting is to recover the missing region from the incomplete data observed. Ideally, the restored image should possess shapes and patterns consistent

by convex risk minimization

Computable general equilibrium models and other types of variational inequalities play a key role in computational economics. This paper describes the design and implementation of a pathsearch-damped Newton method for solving such problems. Our algorithm improves on the typical Newton method (which generates and solves a sequence of LCP's) in both speed and robustness. The underlying complementarity problem is reformulated as a normal map so that standard algorithmic enchancements of Newton's method for solving nonlinear equations can be easily applied. The solver is implemented as a GAMS subsystem, using an interface library developed for this purpose. Computational results obtained from a number of test problems arising in economics are given.

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Abstract. Linear programming (LP) relaxation is a common technique used to find good solutions to complex optimization problems. We present the method of “LP decoding”: applying LP relaxation to the problem of maximum-likelihood (ML) decoding. An arbitrary binary-input memoryless channel is considered. This treatment of the LP decoding method places our previous work on turbo codes [6] and low-density parity-check (LDPC) codes [8] into a generic framework. We define the notion of a proper relaxation, and show that any LP decoder that uses a proper relaxation exhibits many useful properties. We describe the notion of pseudocodewords under LP decoding, unifying many known characterizations for specific codes and channels. The fractional distance of an LP decoder is defined, and it is shown that LP decoders correct a number of errors equal to half the fractional distance. We also discuss the application of LP decoding to binary linear codes. We define the notion of a relaxation being symmetric for a binary linear code. We show that if a relaxation is symmetric, one may assume that the all-zeros codeword is transmitted. 1