Quadratic third-order tensor optimization problem with quadratic constraints

Xiaoming Zhao,Qingzhi Yang,Lixing Yang

doi:10.19139/67

Abstract

Quadratically constrained quadratic programs (QQPs) problems play an important modeling role for many diverse problems. These problems are in general NP hard and numerically intractable. Semidenite programming (SDP) relaxations often provide good approximate solutions to these hard problems. For several special cases of QQP, e.g., convex programs and trust region subproblems, SDP relaxation provides the exact optimal value, i.e., there is a zero duality gap. However, this is not true for the general QQP, or even the QQP with two convex constraints, but a nonconvex objective.In this paper, we consider a certain QQP where the variable is neither vector nor matrix but a third-order tensor. This problem can be viewed as a generalization of the ordinary QQP with vector or matrix as it's variant. Under some mild conditions, we rst show that SDP relaxation provides exact optimal solutions for the original problem. Then we focus on two classes of homogeneous quadratic tensor programming problems which have no requirements on the constraints number. For one, we provide an easily implemental polynomial time algorithm to approximately solve the problem and discuss the approximation ratio. For the other, we show there is no gap between the SDP relaxation and itself.

Highlights

Tight semidefinite programming (SDP) relaxation is known to hold for only a few classes of nonconvex quadratically constrained quadratic programs (QQPs) such as trust region problem [5]
Many extensions of this problem were considered in [12, 15] while these results cannot be extended to QQPs involving two constraints [17]
The remaining of the paper is organized as follows: In section 2, we homogenize the quadratic third-order tensor programming (QTTP) problem and change it to a separable quadratic matrix programming (QMP) problem and show that the optimal values can be obtained exactly by its Semidefinite programming (SDP) relaxation under some mild conditions

Summary

Introduction

Tight semidefinite programming (SDP) relaxation is known to hold for only a few classes of nonconvex QQPs such as trust region problem [5] Many extensions of this problem were considered in [12, 15] while these results cannot be extended to QQPs involving two constraints [17]. The remaining of the paper is organized as follows: In section 2, we homogenize the QTTP problem and change it to a separable QMP problem and show that the optimal values can be obtained exactly by its SDP relaxation under some mild conditions.

Semidefinite relaxations of the QTTP problem and tightness results

Approximation algorithms for the homogeneous QTTP problem

Methods