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.

在本文中，我们考虑一类 NP-hard 的四次多项式问题。首先，我们关注紧集上的双二次优化问题 (Bi-QOP)，当 Bi-QOP 的目标函数是凹的时，它被证明等效于多线性优化问题 (MOP)。然后，我们引入了一个增强的 Bi-QOP（也可以看作是正则化的 Bi-QOP），以保证底层目标函数的凹度。理论上，当紧集被指定为单位球时，增强的 Bi-QOP 和原始问题共享相同的最优解。通过利用所得 MOP 的多块结构，我们相应地提出了一种近端交替最小化算法，以获得所考虑问题的近似最优值。所提出算法的收敛性是在温和条件下建立的。最后，报告了合成数据集上的一些初步计算结果，以显示所提出算法的效率。