Centralized and Collective Neurodynamic Optimization Approaches for Sparse Signal Reconstruction via L₁-Minimization

Inertial primal-dual projection neurodynamic approaches for constrained convex optimization problems and application to sparse recovery

Second-order (inertial) neurodynamic approaches are excellent tools for solving convex optimization problems in an accelerated manner, while the majority of existing approaches to neurodynamic approaches focus on unconstrained and simple constrained convex optimization problems. This paper presents a centralized primal - dual projection neurodynamic approach with time scaling (CPDPNA-TS). Built upon the heavy - ball method, this approach is tailored for convex optimization problems characterized by set and affine constraints, which contains a second-order projection ODE (ordinary differential equation) with derivative feedback for the primal variables and a first-order ODE for the dual variables. We prove a strong global solution to CPDPNA-TS in terms of existence, uniqueness and feasibility. Subsequently, we demonstrate that CPDPNA-TS has a nonergodic exponential and an ergodic O1t convergence properties when choosing suitable time scaling parameters, without strong convexity assumption on the objective functions. In addition, we extend the CPDPNA-TS to a case that CPDPNA-TS with a small perturbation and a case that has a distributed framework, and prove that two versions of the extension enjoy the similar convergence properties of CPDPNA-TS. Finally, we perform numerical experiments on sparse recovery in order to illustrate the effectiveness and superiority of the presented projection neurodynamic approaches.

A novel two-layer fuzzy neural network for solving inequality-constrained ℓ1-minimization problem with applications

In this paper, we propose a novel two-layer fuzzy neural network model (TLFNN) for solving the inequality-constrained ℓ1-minimization problem. The stability and global convergence of the proposed TLFNN model are detailedly analyzed using the Lyapunov theory. Compared with the existing three-layer neural network model (TLNN) recently designed by Yang et al., the proposed TLFNN model possesses less storage, stronger robustness, faster convergence rate and higher convergence accuracy. These advantages are illustrated by some numerical experiments, where it is shown that the TLFNN model can achieve a convergence accuracy of 10-13 within 5s while the TLNN model can only acquire 10-6 in 105s when some random coefficient matrices are applied. Since the linear equality-constrained conditions can be equivalently transformed into double inequality-constrained ones, some simulation experiments for sparse signal reconstruction show that the proposed TLFNN model also has less convergence time and stronger robustness than the existing state-of-the-art neural network models for the equality-constrained ℓ1-minimization problem.

Distributed continuous-time accelerated neurodynamic approaches for sparse recovery via smooth approximation to L1-minimization

This paper develops two continuous-time distributed accelerated neurodynamic approaches for solving sparse recovery via smooth approximation to L1-norm minimization problem. First, the L1-norm minimization problem is converted into a distributed smooth optimization problem by utilizing multiagent consensus theory and smooth approximation. Then, a distributed primal-dual accelerated neurodynamic approach is designed by using Karush-Kuhn-Tucker (KKT) condition and Nesterov's accelerated method. Furthermore, in order to reduce the structure complexity of the presented neurodynamic approach, based on the projection matrix, we eliminate a dual variable in the KKT condition and propose a distributed accelerated neurodynamic approach with a simpler structure. It is proved that the two proposed distributed neurodynamic approaches both achieve O(1t2) convergence rate. Finally, the simulation results of sparse recovery are given to demonstrate the effectiveness of the proposed approaches.