On Some Quadratic Optimization Problems Arising in Computer Vision

The goal of this paper is to find methods for solving various quadratic optimization problems, mostly arising from computer vision (image segmentation and contour grouping). We consider mainly two problems: Problem 1. Let A be an n× n Hermitian matrix and let b ∈ C be any vector, maximize z∗Az + z∗b+ b∗z subject to z∗z = 1, z ∈ C. Problem 2. If A is a real n× n symmetric matrix and b ∈ R is any vector, maximize x>Ax+ 2x>b subject to x>x = 1, x ∈ R. First, we show that Problem 1 reduces to Problem 2. We reduce Problem 2 to the problem of finding the intersection of an algebraic curve generalizing the hyperbola to R with the unit sphere. This allows us to analyze the number of solutions of Problem 2 in terms of the nature of the eigenvalues of the (symmetric) matrix A. As a consequence, we prove that the maximum of the function f(x) = x>Ax + 2x>b on the unit sphere is achieved for all the critical points (x, λ) of the Lagrangian L(x, λ) = x>Ax+ 2x>b− λ(x>x− 1), such that λ ≥ σi, for all eigenvalues σi of A. Problem 2 has been considered before, but our approach involving a simple algebraic curve sheds some new light on the problem and simplifies some proofs. We provide an extensive discussion of related works.