This paper proposes a finite-time converging proximal dynamic model (FPD) to deal with equilibrium problems. A distinctive feature of the FPD is its fast and finite-time convergence, in contrast to conventional proximal dynamic methods. It is shown that the solution of the proposed FPD converges to the solution of the corresponding equilibrium problems in finite-time under some mild conditions. Then the proposed FPD is applied to solve problems of nonsmooth composite optimization and absolute value equations. It is further shown in the case of solving composite optimization problems that the equilibrium point of the proposed proximal gradient dynamic model is globally finite-time stable under the so-called proximal Polyak-Lojasiewicz condition, which is weaker than strong convexity. Finally, numerical examples are presented to illustrate the effectiveness of the proposed methods.

In this article, several fixed-time (FT) neurodynamic algorithms with time-varying coefficients are introduced for composite optimization problems. The remarkable features of neurodynamic algorithms are FT convergence from arbitrary initial conditions with faster convergence rate by choosing different time-varying coefficients. The FT convergence of neurodynamic algorithms can be proved by the Polyak- ${\L}$ ojasiewicz condition, which is beyond strong convexity condition. The upper bounds of the settling time for time-varying neurodynamic algorithms are explicitly given. The robustness of neurodynamic algorithms under bounded noises are further studied. In addition, the proposed neurodynamic algorithms are also utilized for dealing with absolute value equations and sparse signal reconstruction problems. The circuit framework for FT neurodynamic algorithms is subsequently introduced, and an example simulated in Multisim 14.3 is provided to verify the practicability of the proposed analog circuits. Numerical experiments on image recovery and sparse logistic regression are conducted to validate the superiority of the proposed algorithms.

This article proposes two novel projection neural networks (PNNs) with fixed-time ( $\mathbf{FIX_t}$ ) convergence to deal with variational inequality problems (VIPs). The remarkable features of the proposed PNNs are $\mathbf{FIX_t}$ convergence and more accurate upper bounds for arbitrary initial conditions. The robustness of the proposed PNNs under bounded noises is further studied. In addition, the proposed PNNs are applied to deal with absolute value equations (AVEs), noncooperative games, and sparse signal reconstruction problems (SSRPs). The upper bounds of the settling time for the proposed PNNs are tighter than the bounds in the existing neural networks. The effectiveness and advantages of the proposed PNNs are confirmed by numerical examples.

In this article, several novel time-varying momentum-like neurodynamic optimization approaches are proposed for Nash equilibrium (NE) seeking of noncooperative games. It is shown that the dynamics trajectories converge to NE within fixed-time from arbitrary initial conditions, achieving a quicker convergence rate through the selection of distinct time-varying coefficients. Moreover, the upper bounds of the settling time for the proposed NE seeking neurodynamic approaches are explicitly provided. In addition, the study investigates the robustness of the designed neurodynamic approaches in the presence of bounded noises. The superior convergence properties and practicability of our approaches are demonstrated through a simulation example involving energy consumption games.