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SIAM Journal on Matrix Analysis and Applications
On the Best Rank-1 Approximation of Higher-Order Supersymmetric Tensors
SIAM Journal on Matrix Analysis and Applications
Global Optimization with Polynomials and the Problem of Moments
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New Results on Quadratic Minimization
SIAM Journal on Optimization
Semidefinite Approximations for Global Unconstrained Polynomial Optimization
SIAM Journal on Optimization
Minimizing Polynomials via Sum of Squares over the Gradient Ideal
Mathematical Programming: Series A and B
On the complexity of Putinar's Positivstellensatz
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Approximation Bounds for Quadratic Optimization with Homogeneous Quadratic Constraints
SIAM Journal on Optimization
Semidefinite Relaxation Bounds for Indefinite Homogeneous Quadratic Optimization
SIAM Journal on Optimization
Z-eigenvalue methods for a global polynomial optimization problem
Mathematical Programming: Series A and B
Eigenvalues of a real supersymmetric tensor
Journal of Symbolic Computation
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This paper studies the so-called biquadratic optimization over unit spheres $\min_{x\in \mathbb{R}^n,y\in \mathbb{R}^m}\sum_{1\leq i,k\leq n,\,1\leq j,l \leq m}b_{ijkl}x_{i}y_{j}x_{k}{y}_l$, subject to $\|x\| = 1$, $\|y\| =1$. We show that this problem is NP-hard, and there is no polynomial time algorithm returning a positive relative approximation bound. Then, we present various approximation methods based on semidefinite programming (SDP) relaxations. Our theoretical results are as follows: For general biquadratic forms, we develop a $\frac{1}{2\max\{m,n\}^2}$-approximation algorithm under a slightly weaker approximation notion; for biquadratic forms that are square-free, we give a relative approximation bound $\frac{1}{nm}$; when $\min\{n,m\}$ 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.