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Tensor Based Proximal Alternating Minimization Method for A Kind of Inhomogeneous Quartic Optimization Problem

In this paper, we propose an efficient numerical approach for solving a specific type of quartic inhomogeneous polynomial optimization problem inspired by practical applications. The primary contribution of this work lies in establishing an inherent equivalence between the quartic inhomogeneous polynomial optimization problem and a multilinear optimization problem (MOP). This result extends the equivalence between fourth-order homogeneous polynomial optimization and multilinear optimization in the existing literature to the equivalence between fourth-order inhomogeneous polynomial optimization and multilinear optimization. By leveraging the multi-block structure embedded within the MOP, a tensor-based proximal alternating minimization algorithm is proposed to approximate the optimal value of the quartic problem. Under mild assumptions, the convergence of the algorithm is rigorously proven. Finally, the effectiveness of the proposed algorithm is demonstrated through preliminary computational results obtained using synthetic datasets.