Regularized Lotka-Volterra Dynamical System as Continuous Proximal-Like Method in Optimization

We introduce and study a new type of dynamical system which combines the continuous gradient method with a nonlinear Lotka-Volterra (LV) type of differential system within a logarithmic-quadratic proximal scheme. We prove a global existence and viability result for the resulting trajectory which holds for a general smooth function. The asymptotic behavior of the produced trajectory is analyzed and global convergence of the trajectory to a minimizer of the convex minimization problem over the nonnegative orthant is established. The implicit discretization which is at the origin of the proposed continuous dynamical system is an interior proximal scheme for minimizing a closed proper convex function, and convergence results and properties of the resulting discrete scheme are also established. We show finally that the trajectories of the family of regularized Lotka-Volterra systems, parametrized by the positive parameter associated with the quadratic proximal term, are uniformly convergent to the solution of the classical LV-dynamical system, as the parameter associated with the proximal term approaches zero.