Solving a new type of quadratic optimization problem having a joint numerical range constraint

TL;DR

This paper introduces a versatile quadratic optimization framework with joint numerical range constraints, enabling solutions to previously unsolved problems like QSIC and AQP through SDP and analysis of convexity.

Contribution

It formulates a general quadratic optimization model incorporating joint numerical range constraints, unifying and solving several complex quadratic problems efficiently.

Findings

Convex case solvable via SDP and $ ext{S}$-procedure.

Non-convex case reduces to linear dependence analysis.

Effective approximation method using bisection on $[0,2\pi]$.

Abstract

We propose a new formulation of quadratic optimization problems. The objective function $F(f(x),g(x))$ is given as composition of a quadratic function $F(z)$ with two $n$-variate quadratic functions $z_1=f(x)$ and $z_2=g(x).$ In addition, it incorporates with a set of linear inequality constraints in $z=(z_1,z_2)^T,$ while having an implicit constraint that $z$ belongs to the joint numerical range of $(f,g).$ The formulation is very general in the sense that it covers quadratic programming with a single quadratic constraint of all types, including the inequality-type, the equality-type, and the interval-type. Even more, the composition of "quadratic with quadratics" as well as the joint numerical range constraint all together allow us to formulate existing unsolved (or not solved efficiently) problems into the new model. In this paper, we solve the quadratic hypersurfaces intersection problem (QSIC) proposed by P$\acute{{\rm o}}$lik and Terlaky; and the problem (AQP) to minimize the absolute value of a quadratic function over a quadratic constraint proposed by Ye and Zhang. We show that, when $F(z)$ and the joint numerical range constraint are both convex, the optimal value of the convex optimization problem can be obtained by solving an SDP followed from a new development of the $\mathcal{S}$-procedure. The optimal solution can be approximated by conducting a bisection method on $[0,2\pi].$ On the other hand, if the joint numerical range of $f(x)$ and $g(x)$ is non-convex, the respective quadratic matrices of $f(x)$ and $g(x)$ must be linearly dependent. The linear dependence property enables us to solve (QSIC) and (AQP) accordingly by elementary analysis.