Error analysis of approximation algorithm for standard bi-quadratic programming

TL;DR

This paper develops a polynomial time approximation scheme for standard bi-quadratic programming by reformulating it as copositive tensor programming and approximating the copositive cone, enabling near-optimal solutions.

Contribution

It introduces a PTAS for StBQP by approximating copositive tensor cones, extending to multi-quadratic programming, with numerical validation.

Findings

Existence of a PTAS for StBQP

Reformulation as copositive tensor programming

Numerical examples demonstrating effectiveness

Abstract

We consider the problem of approximately solving a standard bi-quadratic programming (StBQP), which is NP-hard. After reformulating the original problem as an equivalent copositive tensor programming, we show how to approximate the optimal solution by approximating the cone of copositive tensors via a serial polyhedral cones. The established quality of approximation shows that, a polynomial time approximation scheme (PTAS) for solving StBQP exists and can be extended to solving standard multi-quadratic programming. Some numerical examples are provided to illustrate our approach.