Efficient Semidefinite Branch-and-Cut for MAP-MRF Inference

We propose a Branch-and-Cut (B&C) method for solving general MAP-MRF inference problems. The core of our method is a very efficient bounding procedure, which combines scalable semidefinite programming (SDP) and a cutting-plane method for seeking violated constraints. In order to further speed up the computation, several strategies have been exploited, including model reduction, warm start and removal of inactive constraints. We analyze the performance of the proposed method under different settings, and demonstrate that our method either outperforms or performs on par with state-of-the-art approaches. Especially when the connectivities are dense or when the relative magnitudes of the unary costs are low, we achieve the best reported results. Experiments show that the proposed algorithm achieves better approximation than the state-of-the-art methods within a variety of time budgets on challenging non-submodular MAP-MRF inference problems.Comment: 21 page

A new Lenstra-type Algorithm for Quasiconvex Polynomial Integer Minimization with Complexity 2^O(n log n)

We study the integer minimization of a quasiconvex polynomial with quasiconvex polynomial constraints. We propose a new algorithm that is an improvement upon the best known algorithm due to Heinz (Journal of Complexity, 2005). This improvement is achieved by applying a new modern Lenstra-type algorithm, finding optimal ellipsoid roundings, and considering sparse encodings of polynomials. For the bounded case, our algorithm attains a time-complexity of s (r l M d)^{O(1)} 2^{2n log_2(n) + O(n)} when M is a bound on the number of monomials in each polynomial and r is the binary encoding length of a bound on the feasible region. In the general case, s l^{O(1)} d^{O(n)} 2^{2n log_2(n) +O(n)}. In each we assume d>= 2 is a bound on the total degree of the polynomials and l bounds the maximum binary encoding size of the input.Comment: 28 pages, 10 figure

Approximate Convex Optimization by Online Game Playing

Lagrangian relaxation and approximate optimization algorithms have received much attention in the last two decades. Typically, the running time of these methods to obtain a $\epsilon$ approximate solution is proportional to $\frac{1}{\epsilon^2}$. Recently, Bienstock and Iyengar, following Nesterov, gave an algorithm for fractional packing linear programs which runs in $\frac{1}{\epsilon}$ iterations. The latter algorithm requires to solve a convex quadratic program every iteration - an optimization subroutine which dominates the theoretical running time. We give an algorithm for convex programs with strictly convex constraints which runs in time proportional to $\frac{1}{\epsilon}$. The algorithm does NOT require to solve any quadratic program, but uses gradient steps and elementary operations only. Problems which have strictly convex constraints include maximum entropy frequency estimation, portfolio optimization with loss risk constraints, and various computational problems in signal processing. As a side product, we also obtain a simpler version of Bienstock and Iyengar's result for general linear programming, with similar running time. We derive these algorithms using a new framework for deriving convex optimization algorithms from online game playing algorithms, which may be of independent interest

Large-scale Binary Quadratic Optimization Using Semidefinite Relaxation and Applications

In computer vision, many problems such as image segmentation, pixel labelling, and scene parsing can be formulated as binary quadratic programs (BQPs). For submodular problems, cuts based methods can be employed to efficiently solve large-scale problems. However, general nonsubmodular problems are significantly more challenging to solve. Finding a solution when the problem is of large size to be of practical interest, however, typically requires relaxation. Two standard relaxation methods are widely used for solving general BQPs--spectral methods and semidefinite programming (SDP), each with their own advantages and disadvantages. Spectral relaxation is simple and easy to implement, but its bound is loose. Semidefinite relaxation has a tighter bound, but its computational complexity is high, especially for large scale problems. In this work, we present a new SDP formulation for BQPs, with two desirable properties. First, it has a similar relaxation bound to conventional SDP formulations. Second, compared with conventional SDP methods, the new SDP formulation leads to a significantly more efficient and scalable dual optimization approach, which has the same degree of complexity as spectral methods. We then propose two solvers, namely, quasi-Newton and smoothing Newton methods, for the dual problem. Both of them are significantly more efficiently than standard interior-point methods. In practice, the smoothing Newton solver is faster than the quasi-Newton solver for dense or medium-sized problems, while the quasi-Newton solver is preferable for large sparse/structured problems. Our experiments on a few computer vision applications including clustering, image segmentation, co-segmentation and registration show the potential of our SDP formulation for solving large-scale BQPs.Comment: Fixed some typos. 18 pages. Accepted to IEEE Transactions on Pattern Analysis and Machine Intelligenc