Alternating direction method for bi-quadratic programming (Q652699)

The authors investigate bi-quadratic programming problems (Bi-QP) as those studied by \textit{C. Ling, J. Nie, L. Qi} and \textit{Y. Ye} [SIAM J. Optim. 20, No. 3, 1286--1310 (2009; Zbl 1221.90074)], namely \[ \text{minimize}\quad \sum_{i=1}^m \sum_{j=1}^n \sum_{k=1}^m \sum_{l=1}^n b_{ijkl} x_i y_j x_k y_l\quad \text{s.t.} \quad \|x\|=1, \|y\|=1, \] where \(b_{ijkl}\) are real numbers for all \(i, k \in \{1,\dots,m\}\) and \(j, l \in \{1,\dots,n\}\), while \(x=(x_1,\dots,x_m)\) and \(y=(y_1,\dots,y_n)\) are vectors in the Euclidean spaces \((\mathbb{R}^m,\|\cdot\|)\) and \((\mathbb{R}^n,\|\cdot\|)\), respectively. By means of a reformulation of (Bi-QP), the authors introduce a quadratic semidefinite programming relaxation for (Bi-QP) and develop an alternating direction method for solving it. The convergence of the proposed method is established and numerical experiments are provided.