Submodularity in Conic Quadratic Mixed 0–1 Optimization

Abstract

We describe strong convex valid inequalities for conic quadratic mixed 0-1 optimization. These inequalities can be utilized for solving numerous practical nonlinear discrete optimization problems from value-at-risk minimization to queueing system design, from robust interdiction to assortment optimization through appropriate conic quadratic mixed 0-1 relaxations. The inequalities exploit the submodularity of the binary restrictions and are based on the polymatroid inequalities over binaries for the diagonal case. We prove that the convex inequalities completely describe the convex hull of a single conic quadratic constraint as well as the rotated cone constraint over binary variables and unbounded continuous variables. We then generalize and strengthen the inequalities by incorporating additional constraints of the optimization problem. Computational experiments on mean-risk optimization with correlations, assortment optimization, and robust conic quadratic optimization indicate that the new inequalities strengthen the convex relaxations substantially and lead to significant performance improvements.

Strong formulations for quadratic optimization with M-matrices and indicator variables

“…The process outlined here to find subgradient cuts (9) for f can be utilized for any convex piecewise nonlinear function, and will be used for other functions in the rest of the paper. Convex piecewise nonlinear functions also arise in strong formulations for mixed-integer conic quadratic optimization [5], and subgradient linear cuts for such functions were recently used in the context of the pooling problem [36]. As Theorem 1 below states, inequality (7) and bound constraints for the binary variables describe the convex hull of X U .…”

Lifted polymatroid inequalities for mean-risk optimization with indicator variables

“…Separation. The separation problem for inequalities (6) and conv(K σ ) is solved exactly and fast due to Edmond's greedy algorithm for optimization over polymatroids. We do not have such an exact separation algorithm for the lifted polymatroid inequalities and, therefore, use an inexact approach.…”

“…For the pure binary case, Atamtürk and Narayanan [9] exploit the submodularity of the underlying set function to describe its convex lower envelope via polymatroid inequalities. Atamtürk and Gómez [6] describe a variety of applications for this model and give strong valid inequalities for the mixed 0 − 1 case without the on-off constraints. The ideal (convex hull) representation for the conic quadratic mixed 0 − 1 set with indicator variables F remains an open question.…”

Successive Quadratic Upper-Bounding for Discrete Mean-Risk Minimization and Network Interdiction

“…In order to strengthen the convex relaxation of 0-1 problems with a mean-risk objective, one can utilize the polymatroid inequalities [6]. Polymatroid inequalities exploit the submodularity of the mean-risk objective for the diagonal case.…”