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 ( 1 ) exponential and an ergodic O 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.