An efficient alternating minimization method for fourth degree polynomial optimization

In this paper, we consider a class of fourth degree polynomial problems, which are NP-hard. First, we are concerned with the bi-quadratic optimization problem (Bi-QOP) over compact sets, which is proven to be equivalent to a multi-linear optimization problem (MOP) when the objective function of Bi-QOP is concave. Then, we introduce an augmented Bi-QOP (which can also be regarded as a regularized Bi-QOP) for the purpose to guarantee the concavity of the underlying objective function. Theoretically, both the augmented Bi-QOP and the original problem share the same optimal solutions when the compact sets are specified as unit spheres. By exploiting the multi-block structure of the resulting MOP, we accordingly propose a proximal alternating minimization algorithm to get an approximate optimal value of the problem under consideration. Convergence of the proposed algorithm is established under mild conditions. Finally, some preliminary computational results on synthetic datasets are reported to show the efficiency of the proposed algorithm.