Geometry of quadratic maps via convex relaxation

TL;DR

This paper explores the geometry of quadratic map images, revealing hidden convexity and developing algorithms to analyze their convexity properties, with applications in optimization and physical systems.

Contribution

It introduces algorithms for analyzing quadratic map images' convexity, including verification, boundary detection, and convex subset identification, implemented in an open-source MATLAB library.

Findings

Algorithms effectively verify non-membership of points in the image.

Methods identify boundary points in specific directions.

Stochastic checks determine convexity and find maximal convex subsets.

Abstract

We consider several basic questions pertaining to the geometry of image of a general quadratic map. In general the image of a quadratic map is non-convex, although there are several known classes of quadratic maps when the image is convex. Remarkably, even when the image is not convex it often exhibits hidden convexity: a surprising efficiency of convex relaxation to address various geometric questions by reformulating them in terms of convex optimization problems. In this paper we employ this strategy and put forward several algorithms that solve the following problems pertaining to the image: verify if a given point does not belong to the image; find the boundary point of the image lying in a particular direction; stochastically check if the image is convex, and if it is not, find a maximal convex subset of the image. Proposed algorithms are implemented in the form of an open-source MATLAB library CAQM, which accompanies the paper. Our results can be used for various problems of discrete optimization, uncertainty analysis, physical applications, and study of power flow equations.