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

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Bolte, Jrme Teboulle, Marc
Copyright Year	2003
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.
Starting Page	1266
Ending Page	1292
Page Count	27
File Format	PDF
ISSN	03630129
DOI	10.1137/S0363012902410861
e-ISSN	10957138
Issue Number	4
Volume Number	42
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2006-07-26
Access Restriction	Subscribed
Subject Keyword	elliptic barrier operators Nonlinear equations explicit and implicit discrete schemes Asymptotic properties global convergence asymptotic analysis Riemannian gradient methods Convex programming Dynamical systems in optimization and economics viability dynamical systems Lyapunov functionals Lotka-Volterra differential equations Methods of Riemannian geometry continuous gradient-like systems Convex functions and convex programs interior proximal algorithms Vector fields, flows, ordinary differential equations constrained convex minimization Problems involving ordinary differential equations
Content Type	Text
Resource Type	Article
Subject	Control and Optimization Applied Mathematics

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