Barrier operators and associated gradient-like dynamical systems for constrained minimization problems. J´ erˆ ome Bolte 1 and Marc Teboulle 2 Abstract. We study some continuous dynamical systems associated with constrained optimization prob- lems. For that purpose, we introduce the concept of elliptic barrier operators and develop a unified frame- work 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, Newton type methods as well as continuous interior descent methods such as Lotka-Volterra type differ- ential 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. Key words: Dynamical systems, continuous gradient-like systems, elliptic barrier opera- tors, Lotka-Volterra differential equations, asymptotic analysis, viability, Lyapunov func- tionals, explicit and implicit discrete schemes, interior proximal algorithms, global conver- gence, constrained convex minimization, Riemannian gradient methods. 1 Introduction This paper proposes to study some continuous dynamical systems in relation with the constrained optimization problem (P ) inf {f (x): x ∈ C }, where C is a nonempty open convex subset of IR n , n ≥ 1, f : IR n → IR is a convex function and C denotes the closure of C . Our first aim is to give a unified framework to smooth continuous interior descent methods studied earlier in the literature: steepest descent method, Lotka-Volterra type equations, 1 D´ epartement de Math´ ematiques, Case 51, ACSIOM-CNRS FRE 2311, Universit´ e Montpellier II, 34095 Montpellier, Cedex 5, France email: bolte@math.univ-montp2.fr. 2 School of Mathematical Sciences,Tel-Aviv University, Ramat-Aviv 69978, Israel email: teboulle@post.tau.ac.il 1