A Revisit to Quadratic Programming with One Inequality Quadratic Constraint via Matrix Pencil

TL;DR

This paper revisits quadratic programming with one quadratic inequality constraint by analyzing the associated matrix pencil, focusing on challenging cases where standard assumptions do not hold, and proposes alternative solution methods.

Contribution

It introduces a matrix pencil analysis approach to handle problematic QP1QC cases, providing new insights and alternative solution techniques beyond traditional SDP methods.

Findings

Identifies conditions for no Slater point, unboundedness, and unattainability in QP1QC.

Provides an alternative computational approach using matrix pencil analysis.

Enhances understanding of the structure of QP1QC problems.

Abstract

The quadratic programming over one inequality quadratic constraint (QP1QC) is a very special case of quadratically constrained quadratic programming (QCQP) and attracted much attention since early 1990's. It is now understood that, under the primal Slater condition, (QP1QC) has a tight SDP relaxation (PSDP). The optimal solution to (QP1QC), if exists, can be obtained by a matrix rank one decomposition of the optimal matrix X? to (PSDP). In this paper, we pay a revisit to (QP1QC) by analyzing the associated matrix pencil of two symmetric real matrices A and B, the former matrix of which defines the quadratic term of the objective function whereas the latter for the constraint. We focus on the \undesired" (QP1QC) problems which are often ignored in typical literature: either there exists no Slater point, or (QP1QC) is unbounded below, or (QP1QC) is bounded below but unattainable. Our analysis is conducted with the help of the matrix pencil, not only for checking whether the undesired cases do happen, but also for an alternative way to compute the optimal solution in comparison with the usual SDP/rank-one-decomposition procedure.