# Submodularity in conic quadratic mixed 0-1 optimization

## TL;DR

This paper introduces strong convex inequalities for conic quadratic mixed 0-1 problems, leveraging submodularity to improve relaxations and computational performance in various nonlinear discrete optimization applications.

## Contribution

It develops and proves convex inequalities based on submodularity that fully describe the convex hull of key conic quadratic constraints, enhancing solution methods.

## Key findings

- Inequalities significantly strengthen convex relaxations.
- Computational results show improved optimization performance.
- Applicable to diverse problems like risk minimization and queueing.

## 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.