Biquadratic Optimization Over Unit Spheres and Semidefinite Programming Relaxations - Ling, Chen; Nie, Jiawang; Qi, Liqun; Ye, Yinyu (SIAM journal on optimization, 2009)
This paper studies the so-called biquadratic optimization over unit spheres ... , subject to ... . The authors show that this problem is NP-hard, and there is no polynomial time algorithm returning a positive relative approximation bound. Then, they present various approximation methods based on semidefinite programming (SDP) relaxations.Their theoretical results are as follows: For general biquadratic forms, they develop a ... -approximation algorithm under a slightly weaker approximation notion; for biquadratic forms that are square-free, we give a relative approximation bound ... when min ... is a constant, we present two polynomial time approximation schemes (PTASs) which are based on sum of squares (SOS) relaxation hierarchy and grid sampling of the standard simplex. For practical computational purposes, we propose the first order SOS relaxation, a convex quadratic SDP relaxation, and a simple minimum eigenvalue method and show their error bounds. Some illustrative numerical examples are also included.(ProQuest: ... denotes formulae/symbols omitted.)