The Home-Away Assignment Problems and Break Minimization/Maximization Problems in Sports Scheduling

Suppose that we have a timetable of a round-robin tournament with a number of teams, and distances among their homes. The home-away assignment problem is to find a home-away assignment that minimizes the total traveling distance of the teams. This paper also deals with the break minimization (maximization) problem, which finds a home-away assignment that minimizes (maximizes) the number of breaks, i.e., the number of occurrences of consecutive matches held either both at away or both at home for a team. Part of this aim of this paper is to give a unified view to the three problems, the break minimization/maximization problems and the home-away assignment problem. We see that optimal solutions of the break minimization/maximization problems are obtained by solving the home-away assignment problem. For the home-away assignment problem, we propose a formulation as an integer program, and some rounding algorithms. We also provide a technique to transform the home-away assignment problem to MIN RES CUT and apply Goemans and Williamson\u27s algorithm for MAX RES CUT, which is based on a positive semidefinite programming relaxation, to the obtained MIN RES CUT instances. Computational experiments show that our approaches quickly generate solutions of good approximation rations

Scalable Semidefinite Relaxation for Maximum A Posterior Estimation

Maximum a posteriori (MAP) inference over discrete Markov random fields is a fundamental task spanning a wide spectrum of real-world applications, which is known to be NP-hard for general graphs. In this paper, we propose a novel semidefinite relaxation formulation (referred to as SDR) to estimate the MAP assignment. Algorithmically, we develop an accelerated variant of the alternating direction method of multipliers (referred to as SDPAD-LR) that can effectively exploit the special structure of the new relaxation. Encouragingly, the proposed procedure allows solving SDR for large-scale problems, e.g., problems on a grid graph comprising hundreds of thousands of variables with multiple states per node. Compared with prior SDP solvers, SDPAD-LR is capable of attaining comparable accuracy while exhibiting remarkably improved scalability, in contrast to the commonly held belief that semidefinite relaxation can only been applied on small-scale MRF problems. We have evaluated the performance of SDR on various benchmark datasets including OPENGM2 and PIC in terms of both the quality of the solutions and computation time. Experimental results demonstrate that for a broad class of problems, SDPAD-LR outperforms state-of-the-art algorithms in producing better MAP assignment in an efficient manner.Comment: accepted to International Conference on Machine Learning (ICML 2014

Spatial and spin symmetry breaking in semidefinite-programming-based Hartree-Fock theory

The Hartree-Fock problem was recently recast as a semidefinite optimization over the space of rank-constrained two-body reduced-density matrices (RDMs) [Phys. Rev. A 89, 010502(R) (2014)]. This formulation of the problem transfers the non-convexity of the Hartree-Fock energy functional to the rank constraint on the two-body RDM. We consider an equivalent optimization over the space of positive semidefinite one-electron RDMs (1-RDMs) that retains the non-convexity of the Hartree-Fock energy expression. The optimized 1-RDM satisfies ensemble $N$-representability conditions, and ensemble spin-state conditions may be imposed as well. The spin-state conditions place additional linear and nonlinear constraints on the 1-RDM. We apply this RDM-based approach to several molecular systems and explore its spatial (point group) and spin ($S^2$ and $S_3$) symmetry breaking properties. When imposing $S^2$ and $S_3$ symmetry but relaxing point group symmetry, the procedure often locates spatial-symmetry-broken solutions that are difficult to identify using standard algorithms. For example, the RDM-based approach yields a smooth, spatial-symmetry-broken potential energy curve for the well-known Be--H$_2$ insertion pathway. We also demonstrate numerically that, upon relaxation of $S^2$ and $S_3$ symmetry constraints, the RDM-based approach is equivalent to real-valued generalized Hartree-Fock theory.Comment: 9 pages, 6 figure

Convex relaxation of mixture regression with efficient algorithms

We develop a convex relaxation of maximum a posteriori estimation of a mixture of regression models. Although our relaxation involves a semidefinite matrix variable, we reformulate the problem to eliminate the need for general semidefinite programming. In particular, we provide two reformulations that admit fast algorithms. The first is a max-min spectral reformulation exploiting quasi-Newton descent. The second is a min-min reformulation consisting of fast alternating steps of closed-form updates. We evaluate the methods against Expectation-Maximization in a real problem of motion segmentation from video data