100,877 research outputs found
Belief Propagation for Linear Programming
Belief Propagation (BP) is a popular, distributed heuristic for performing
MAP computations in Graphical Models. BP can be interpreted, from a variational
perspective, as minimizing the Bethe Free Energy (BFE). BP can also be used to
solve a special class of Linear Programming (LP) problems. For this class of
problems, MAP inference can be stated as an integer LP with an LP relaxation
that coincides with minimization of the BFE at ``zero temperature". We
generalize these prior results and establish a tight characterization of the LP
problems that can be formulated as an equivalent LP relaxation of MAP
inference. Moreover, we suggest an efficient, iterative annealing BP algorithm
for solving this broader class of LP problems. We demonstrate the algorithm's
performance on a set of weighted matching problems by using it as a cutting
plane method to solve a sequence of LPs tightened by adding ``blossom''
inequalities.Comment: To appear in ISIT 201
Minimum cost b-matching problems with neighborhoods
In this paper, we deal with minimum cost b-matching problems on graphs where the nodes are assumed to belong to non-necessarily convex regions called neighborhoods, and the costs are given by the distances between points of the neighborhoods. The goal in the proposed problems is twofold: (i) finding a b-matching in the graph and (ii) determining a point in each neighborhood to be the connection point among the edges defining the b-matching. Different variants of the minimum cost b-matching problem are considered depending on the criteria to match neighborhoods: perfect, maximum cardinality, maximal and the a-b-matching problems. The theoretical complexity of solving each one of these problems is analyzed. Different mixed integer non-linear programming formulations are proposed for each one of the considered problems and then reformulated as Second Order Cone formulations. An extensive computational experience shows the efficiency of the proposed formulations to solve the problems under study
A Method for Finding Structured Sparse Solutions to Non-negative Least Squares Problems with Applications
Demixing problems in many areas such as hyperspectral imaging and
differential optical absorption spectroscopy (DOAS) often require finding
sparse nonnegative linear combinations of dictionary elements that match
observed data. We show how aspects of these problems, such as misalignment of
DOAS references and uncertainty in hyperspectral endmembers, can be modeled by
expanding the dictionary with grouped elements and imposing a structured
sparsity assumption that the combinations within each group should be sparse or
even 1-sparse. If the dictionary is highly coherent, it is difficult to obtain
good solutions using convex or greedy methods, such as non-negative least
squares (NNLS) or orthogonal matching pursuit. We use penalties related to the
Hoyer measure, which is the ratio of the and norms, as sparsity
penalties to be added to the objective in NNLS-type models. For solving the
resulting nonconvex models, we propose a scaled gradient projection algorithm
that requires solving a sequence of strongly convex quadratic programs. We
discuss its close connections to convex splitting methods and difference of
convex programming. We also present promising numerical results for example
DOAS analysis and hyperspectral demixing problems.Comment: 38 pages, 14 figure
Partial policy iteration for L1-robust Markov decision processes
Robust Markov decision processes (MDPs) compute reliable solutions for dynamic decision problems with partially-known transition probabilities. Unfortunately, accounting for uncertainty in the transition probabilities significantly increases the computational complexity of solving robust MDPs, which limits their scalability. This paper describes new, efficient algorithms for solving the common class of robust MDPs with s- and sa-rectangular ambiguity sets defined by weighted L1 norms. We propose partial policy iteration, a new, efficient, flexible, and general policy iteration scheme for robust MDPs. We also propose fast methods for computing the robust Bellman operator in quasi-linear time, nearly matching the ordinary Bellman operator's linear complexity. Our experimental results indicate that the proposed methods are many orders of magnitude faster than the state-of-the-art approach, which uses linear programming solvers combined with a robust value iteration
Minimum Weight Perfect Matching via Blossom Belief Propagation
Max-product Belief Propagation (BP) is a popular message-passing algorithm
for computing a Maximum-A-Posteriori (MAP) assignment over a distribution
represented by a Graphical Model (GM). It has been shown that BP can solve a
number of combinatorial optimization problems including minimum weight
matching, shortest path, network flow and vertex cover under the following
common assumption: the respective Linear Programming (LP) relaxation is tight,
i.e., no integrality gap is present. However, when LP shows an integrality gap,
no model has been known which can be solved systematically via sequential
applications of BP. In this paper, we develop the first such algorithm, coined
Blossom-BP, for solving the minimum weight matching problem over arbitrary
graphs. Each step of the sequential algorithm requires applying BP over a
modified graph constructed by contractions and expansions of blossoms, i.e.,
odd sets of vertices. Our scheme guarantees termination in O(n^2) of BP runs,
where n is the number of vertices in the original graph. In essence, the
Blossom-BP offers a distributed version of the celebrated Edmonds' Blossom
algorithm by jumping at once over many sub-steps with a single BP. Moreover,
our result provides an interpretation of the Edmonds' algorithm as a sequence
of LPs
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Zero-one IP problems: Polyhedral descriptions & cutting plane procedures
A systematic way for tightening an IP formulation is by employing classes of linear inequalities that define facets of the convex hull of the feasible integer points of the respective problems. Describing as well as identifying these inequalities will help in the efficiency of the LP-based cutting plane methods. In this report, we review classes of inequalities that partially described zero-one poly topes such as the 0-1 knapsack polytope, the set packing polytope and the travelling salesman polytope. Facets or valid inequalities derived from the 0-1 knapsack and the set packing polytopes are algorithmically identifie
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