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Binary optimization is a central problem in mathematical optimization and itsapplications are abundant. To solve this problem, we propose a new class ofcontinuous optimization techniques which is based on Mathematical Programmingwith Equilibrium Constraints (MPECs). We first reformulate the binary programas an equivalent augmented biconvex optimization problem with a bilinearequality constraint, then we propose two penalization/regularization methods(exact penalty and alternating direction) to solve it. The resulting algorithmsseek desirable solutions to the original problem via solving a sequence oflinear programming convex relaxation subproblems. In addition, we prove thatboth the penalty function and augmented Lagrangian function, induced by addingthe complementarity constraint to the objectives, are exact, i.e., they havethe same local and global minima with those of the original binary program whenthe penalty parameter is over some threshold. The convergence of bothalgorithms can be guaranteed, since they essentially reduce to block coordinatedescent in the literature. Finally, we demonstrate the effectiveness andversatility of our methods on several important problems, including graphbisection, constrained image segmentation, dense subgraph discovery, modularityclustering and Markov random fields. Extensive experiments show that ourmethods outperform existing popular techniques, such as iterative hardthresholding, linear programming relaxation and semidefinite programmingrelaxation.

Binary Optimization via Mathematical Programming with Equilibrium Constraints