Facial reduction heuristics and the motivational example of mixed-integer conic optimization

Facial reduction heuristics are developed in the interest of added performance and reliability in methods for mixed-integer conic optimization. Specifically, the process of branch-and-bound is shown to spawn subproblems for which the conic relaxations are difficult to solve, and the objective bounds of linear relaxations are arbitrarily weak. While facial reduction algorithms already exist to deal with these issues, heuristic variants represent a very potent supplement due to their inherent speed and accuracy. The paper covers a family of heuristics based on linear optimization, subgradient matching, single-cone analysis, and cone factorization.