Barrier Operators and Associated Gradient-Like Dynamical Systems for Constrained Minimization Problems

Abstract

We study some continuous dynamical systems associated with constrained optimization problems. For that purpose, we introduce the concept of elliptic barrier operators and develop a unified framework to derive and analyze the associated class of gradient-like dynamical systems, called A-driven descent method (A-DM). Prominent methods belonging to this class include several continuous descent methods studied earlier in the literature such as steepest descent method, continuous gradient projection methods and Newton-type methods as well as continuous interior descent methods such as Lotka-Volterra-type differential equations and Riemannian gradient methods. Related discrete iterative methods such as proximal interior point algorithms based on Bregman functions and second order homogeneous kernels can also be recovered within our framework and allow for deriving some new and interesting dynamics. We prove global existence and strong viability results of the corresponding trajectories of (A-DM) for a smooth objective function. When the objective function is convex, we analyze the asymptotic behavior at infinity of the trajectory produced by the proposed class of dynamical systems (A-DM). In particular, we derive a general criterion ensuring the global convergence of the trajectory of (A-DM) to a minimizer of a convex function over a closed convex set. This result is then applied to several dynamics built upon specific elliptic barrier operators. Throughout the paper, our results are illustrated with many examples.