# Simplex QP-based methods for minimizing a conic quadratic objective over polyhedra

## TL;DR

This paper introduces simplex QP-based methods for efficiently solving conic quadratic optimization problems over polyhedra, outperforming interior point methods especially in branch-and-bound contexts for discrete problems.

## Contribution

It proposes a reformulation using the perspective of the quadratic function, enabling simple coordinate descent and bisection algorithms that are suitable for warm starts and branch-and-bound algorithms.

## Key findings

- Up to 22x faster for large convex instances
- Higher precision solutions compared to interior point methods
- Significant speed-ups over barrier-based and LP-based branch-and-bound algorithms

## Abstract

We consider minimizing a conic quadratic objective over a polyhedron. Such problems arise in parametric value-at-risk minimization, portfolio optimization, and robust optimization with ellipsoidal objective uncertainty; and they can be solved by polynomial interior point algorithms for conic quadratic optimization. However, interior point algorithms are not well-suited for branch-and-bound algorithms for the discrete counterparts of these problems due to the lack of effective warm starts necessary for the efficient solution of convex relaxations repeatedly at the nodes of the search tree. In order to overcome this shortcoming, we reformulate the problem using the perspective of the quadratic function. The perspective reformulation lends itself to simple coordinate descent and bisection algorithms utilizing the simplex method for quadratic programming, which makes the solution methods amenable to warm starts and suitable for branch-and-bound algorithms. We test the simplex-based quadratic programming algorithms to solve convex as well as discrete instances and compare them with the state-of-the-art approaches. The computational experiments indicate that the proposed algorithms scale much better than interior point algorithms and return higher precision solutions. In our experiments, for large convex instances, they provide up to 22x speed-up. For smaller discrete instances, the speed-up is about 13x over a barrier-based branch-and-bound algorithm and 6x over the LP-based branch-and-bound algorithm with extended formulations.