Haibin Chen

This paper addresses biquadratic polynomial programming (BPP), an NP-hard optimization problem closely related to biquadratic tensors. We first establish several necessary and sufficient conditions for the positive semi-definiteness and positive definiteness of biquadratic Cauchy tensors. Leveraging the structured properties of these tensors, we then prove that the BPP and its equivalent multilinear formulation share the same set of optimal solutions. This result allows us to establish the global sequence convergence of the proximal alternating minimization (PAM) algorithm via the Kurdyka- Lojasiewicz (KL) property, extending the analysis in [8]. Furthermore, by reformulating the equivalent multilinear problem as an unconstrained optimization model, we enable the analysis of its KL exponent and derive an explicit expression for the convergence rate of PAM. Finally, numerical experiments are conducted on both biquadratic Cauchy tensors and general biquadratic tensor instances to evaluate the efficiency, stability, and practical performance of the proposed algorithm.

Polynomial optimization encompasses a broad class of problems in which both the objective function and constraints are polynomial functions of the decision variables. In recent years, a substantial body of research has focused on reformulating polynomial optimization problems (POPs) as conic programs over the cone of completely positive tensors (CPTs). In this article, we propose several new completely positive reformulations for a class of POPs with linear inequality constraints. Our approach begins by lifting these problems into a novel convex optimization framework, wherein the variables are represented as combinations of symmetric rank-one tensors. Based on this lifted formulation, we present a general characterization of POPs with linear inequality constraints that can be reformulated as conic programs over the CPT cone. Additionally, we construct the dual formulations of the resulting completely positive programs. Under mild assumptions, we prove that these dual problems are strictly feasible and strong duality holds.

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.

In this paper, we systemically introduce completely positive biquadratic (CPB) tensors and copositive biquadratic tensors. We show that all weakly CPB tensors are sum of squares tensors, the CPB tensor cone and the copositive biquadratic tensor cone are dual cone to each other. We also show that the outer product of two completely positive matrices is a CPB tensor, and the outer product of two copositive matrices is a copositive biquadratic tensor. We then study two easily checkable subclasses of CPB tensors, namely positive biquadratic Cauchy tensors and biquadratic Pascal tensors. We show that a biquadratic Pascal tensor is both strongly CPB and positive definite.

In this paper, we propose a novel tensor-based Dinkelbach--Type method for computing extremal tensor generalized eigenvalues. We show that the extremal tensor generalized eigenvalue can be reformulated as a critical subproblem of the classical Dinkelbach--Type method, which can subsequently be expressed as a multilinear optimization problem (MOP). The MOP is solved under a spherical constraint using an efficient proximal alternative minimization method, in which we rigorously establish the global convergence. Additionally, the equivalent MOP is reformulated as an unconstrained optimization problem, allowing for the analysis of the Kurdyka-Lojasiewicz (KL) exponent and providing an explicit expression for the convergence rate of the proposed algorithm. Preliminary numerical experiments on solving extremal tensor generalized eigenvalues and minimizing high-order trust-region subproblems are provided, validating the efficacy and practical utility of the proposed method.

In this paper, a modified dynamical system is proposed to address the split feasibility problem. For any initial point, it is proved that the proposed approach converges in a fixed time. Furthermore, by introducing time-varying coefficients, the proposed method can be accelerated. It is shown that the settling time for the proposed fixed-time converging approach has a uniform upper bound for all initial conditions. The proposed approach has the advantage that it is robust against bounded vanishing perturbation. Numerical experiments on signal processing and image restoration tasks validate its effectiveness and demonstrate superior performance. These properties make the method particularly suitable for real-time applications—such as signal recovery and image enhancement—where both computational efficiency and predictable convergence are critical.