On the Convexity of Image of a Multidimensional Quadratic Map

TL;DR

This paper investigates the convexity properties of the image of multidimensional quadratic maps, providing conditions for convexity, methods for identifying convex regions, and applications to power flow equations in electrical networks.

Contribution

It introduces a new sufficient condition for convexity of the quadratic map's image and extends it to the full image, also relating to convexity of joint numerical ranges.

Findings

Established a sufficient condition for convexity of the compact part of the image.

Extended the convexity condition to the entire image of the quadratic map.

Proved convexity of the solvability set for Power Flow equations in DC networks.

Abstract

We study convexity of image of a general multidimensional quadratic map. We split the full image into two parts by an appropriate hyperplane such that one part is compact, and formulate a sufficient condition for convexity of the compact part. We propose a way to identify such convex parts of the full image which can be used in practical applications. By shifting the hyperplane to infinity we extend the sufficient condition for convexity to apply to the full image of the quadratic map. As a related result, we formulate a novel condition for the joint numerical range of m-tuple of hermitian matrices to be convex. Finally, we illustrate our findings by considering several examples. In particular we prove convexity of solvability set for the Power Flow equations in case of DC networks.