[Paper Review] Gradient Dynamic Approach to the Tensor Complementarity Problem

Xuezhong Wang, Maolin Che+2 more|arXiv (Cornell University)|Aug 9, 2018

Tensor decomposition and applications30 references3 citations

TL;DR

This paper proposes a nonlinear gradient dynamic system (NGDS) for solving the tensor complementarity problem (TCP), leveraging activation functions to ensure convergence. The method demonstrates superior performance over linear activation functions, with numerical simulations confirming robust convergence across various tensor types and parameter settings.

ABSTRACT

Nonlinear gradient dynamic approach for solving the tensor complementarity problem (TCP) is presented. Theoretical analysis shows that each of the defined dynamical system models ensures the convergence performance. The computer simulation results further substantiate that the considered dynamical system can solve the tensor complementarity problem (TCP).

Motivation & Objective

• To address the lack of dynamic system approaches for solving tensor complementarity problems (TCP) with high-order tensors.
• To design a nonlinear gradient dynamical system (NGDS) that ensures convergence for TCP under various tensor classes.
• To investigate the impact of different activation functions on the convergence and performance of the NGDS for TCP.
• To provide a numerically efficient and hardware-implementable method for online TCP solution via dynamical systems.

Proposed method

• A nonlinear gradient dynamic system (NGDS) is formulated using an energy function that decreases monotonically toward equilibrium, corresponding to the TCP solution.
• The system employs activation functions—linear, bipolar-sigmoid, power-sigmoid, and smooth power-sigmoid—to enhance convergence and stability.
• The dynamical system is modeled as a first-order ordinary differential equation, with the gradient of the energy function guiding state evolution.
• The convergence of the trajectory to the equilibrium point is theoretically proven using Lyapunov stability and asymptotic stability analysis.
• The method is applied to TCP with P-tensors, strong P-tensors, and strictly semi-positive tensors, ensuring solution existence and uniqueness under given conditions.
• Numerical simulations use varying parameters (γ, q) and initial conditions to validate performance across diverse test cases.

Experimental results

Research questions

• RQ1 Can a nonlinear gradient dynamic system effectively solve the tensor complementarity problem (TCP) with general m-order tensors?
• RQ2 How do different activation functions influence the convergence speed and accuracy of the NGDS for TCP?
• RQ3 Does the NGDS ensure global convergence to the solution of TCP under various tensor classes such as P-tensors and strictly semi-positive tensors?
• RQ4 How does the choice of the parameter γ affect the convergence behavior of the NGDS in numerical experiments?
• RQ5 Can the NGDS be implemented efficiently in hardware due to its first-order ODE structure and online computation capability?

Key findings

• The NGDS with nonlinear activation functions (e.g., bipolar-sigmoid, power-sigmoid) consistently achieves faster and more robust convergence than the linear activation function.
• For all tested examples, the residual error in the NGDS with γ = 1×10⁶ converged to near-zero values, confirming solution accuracy.
• The system demonstrated convergence to the correct solution across multiple initial conditions and q vectors in Examples 5.1–5.3, including cases with non-unique solutions.
• Theoretical analysis confirms asymptotic and exponential stability of the equilibrium point, ensuring convergence under the proposed energy function framework.
• The use of smooth power-sigmoid functions (e.g., f_sps) yielded the most consistent convergence performance across different test scenarios.
• The method successfully solved TCP instances with P-tensors, strong P-tensors, and strictly semi-positive tensors, validating its broad applicability.