The problem of obtaining the maximum a posteriori (MAP) estimate of a discrete random field is of fundamental importance in many areas of Computer Science. In this work, we build on the tree reweighted message passing (TRW) framework of Kolmogorov and Wainwright et al. TRW iteratively optimizes the Lagrangian dual of a linear programming relaxation for MAP estimation. We show how the dual formulation of TRW can be extended to include linear cycle inequalities. We then consider the inclusion of some recently proposed second order cone (SOC) constraints in the dual. We propose efficient iterative algorithms for solving the resulting duals. Similar to the method described by Kolmogorov, these methods are guaranteed to converge. We test our algorithms on a large set of synthetic data, as well as real data. Our experiments show that the additional constraints (i.e.\ cycle inequalities and SOC constraints) provide better results in cases where the TRW framework fails (namely MAP estimation for non-submodular energy functions).

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