An Efficient and Globally Optimal Algorithm for Nonconvex QCQP with One Equality Constraint

TL;DR

This paper introduces a fast, non-iterative algorithm for solving a specific nonconvex QCQP problem with one equality constraint, achieving global optimality efficiently compared to traditional methods.

Contribution

The paper proposes a novel two-stage algorithm combining Simultaneous Diagonalization and Bisection Search for globally optimal solutions to nonconvex QCQP with one equality constraint, improving computational efficiency.

Findings

Achieves global optimality with reduced computational time

Scales to larger problem sizes than traditional methods

Maintains high solution quality in numerical experiments

Abstract

In this paper, we concentrate on a particular category of quadratically constrained quadratic programming (QCQP): nonconvex QCQP with one equality constraint. This type of QCQP problem optimizes a quadratic objective under a fixed second-order cost and has various engineering applications. It often serves as a subproblem in an iterative algorithm framework. However, the development of a high-quality and efficient solution remains an open problem in the existing literature. Traditionally, the Semidefinite Relaxation (SDR) technique is applied for an optimal solution with a prohibitively high order of time complexity. To improve computational efficiency, we propose a fast and non-iterative algorithm to reach a globally optimal solution. This algorithm consists of two consecutive stages: Simultaneous Diagonalization (SD) and Bisection Search (BS). The SD stage decouples the original problem through an affine mapping and the BS stage finds the optimal Lagrange multiplier by solving an equation induced from first- and second-order Karush-Kuhn-Tucker (KKT) conditions. In addition, we enrich the proposed algorithm with further extensions on the problem structure, namely, rank-deficient parameter, indefiniteness, constraint augmentation, and matrix-format variable. Numerical simulations show that the proposed algorithm achieves good numerical performance in terms of constraint satisfaction, optimality gap, and computational time, and scales to problem sizes at least ten times those supported by the traditional benchmarks.