Solving the many-body problem with linear programing

Guangtai Qu, Shaoze Li, Zhibin Deng +1 more

The multi-linearly constrained separable binary quadratic programming problem (MLSBQP) is a direct extension of the classical singly-linearly constrained separable binary quadratic programming problem (SLSBQP). In the literature, several algorithms have been proposed for SLSBQP which can solve problem instances with millions of variables globally. However, for the case of MLSBQP, the current state-of-the-art solver can only solve problems with up to 40000 binary variables within a reasonable computational time. In this paper, we propose an extremely efficient branch-and-bound algorithm for solving MLSBQP. The main feature of the proposed algorithm is that an efficient dual coordinate ascent algorithm is applied to obtain lower bounds, and the complexity of each inner loop of the dual coordinate ascent algorithm is $ \mathcal{O}(n\log n) $. Our numerical experiments show that the proposed branch-and-bound algorithm can solve MLSBQP with up to millions of variables within only several minutes, and performs much faster than Gurobi when the number of linear constraints is small.

Giampaolo Co’

Jerome K. Percus

Pascal O. Vontobel

Interior-point algorithms constitute a very interesting class of algorithms for solving linear-programming problems. In this paper we study efficient implementations of such algorithms for solving the linear program that appears in the linear-programming decoder formulation.

Christian Frønsdal

Chin-Yao Chang, Eric B. Jones, Peter A Graf

Quantum computing (QC) is emerging as a new computing resource that could be superior to conventional computing (CC) for certain classes of optimization problems. However, in principle QC can only solve unconstrained binary programming problems, while mixed-integer linear programming (MIP) is of most interest in practice. We attempt to bridge the gap between the capability of QC and real-world applications by developing a new approach for MIP. The idea is decomposing the MIP into binary programming and linear programming (LP) problems, which are respectively solved by QC and conventional computing. We formalize a decomposition approach that ensures that with a sufficient number of back and forth iterations, the algorithm can reach the optimal solution of the original MIP problem. The algorithm is tested on a 2000Q D-Wave quantum processing units (QPU) and is shown to be effective for small-scaled test cases.