Fast convergence of inertial dynamics and algorithms with asymptotic vanishing viscosity

Fast convergence of inertial dynamics and algorithms with asymptotic vanishing viscosity In a Hilbert space setting $${{\mathcal {H}}}$$ H , we study the fast convergence properties as $$t \rightarrow + \infty $$ t → + ∞ of the trajectories of the second-order differential equation $$\begin{aligned} \ddot{x}(t) + \frac{\alpha }{t} \dot{x}(t) + \nabla \Phi (x(t)) = g(t), \end{aligned}$$ x ¨ ( t ) + α t x ˙ ( t ) + ∇ Φ ( x ( t ) ) = g ( t ) , where $$\nabla \Phi $$ ∇ Φ is the gradient of a convex continuously differentiable function $$\Phi : {{\mathcal {H}}} \rightarrow {{\mathbb {R}}}, \alpha $$ Φ : H → R , α is a positive parameter, and $$g: [t_0, + \infty [ \rightarrow {{\mathcal {H}}}$$ g : [ t 0 , + ∞ [ → H is a small perturbation term. In this inertial system, the viscous damping coefficient $$\frac{\alpha }{t}$$ α t vanishes asymptotically, but not too rapidly. For $$\alpha \ge 3$$ α ≥ 3 , and $$\int _{t_0}^{+\infty } t \Vert g(t)\Vert dt < + \infty $$ ∫ t 0 + ∞ t ‖ g ( t ) ‖ d t < + ∞ , just assuming that $${{\mathrm{argmin\,}}}\Phi \ne \emptyset $$ argmin Φ ≠ ∅ , we show that any trajectory of the above system satisfies the fast convergence property $$\begin{aligned} \Phi (x(t))- \min _{{{\mathcal {H}}}}\Phi \le \frac{C}{t^2}. \end{aligned}$$ Φ ( x ( t ) ) - min H Φ ≤ C t 2 . Moreover, for $$\alpha > 3$$ α > 3 , any trajectory converges weakly to a minimizer of $$\Phi $$ Φ . The strong convergence is established in various practical situations. These results complement the $${{\mathcal {O}}}(t^{-2})$$ O ( t - 2 ) rate of convergence for the values obtained by Su, Boyd and Candès in the unperturbed case $$g=0$$ g = 0 . Time discretization of this system, and some of its variants, provides new fast converging algorithms, expanding the field of rapid methods for structured convex minimization introduced by Nesterov, and further developed by Beck and Teboulle with FISTA. This study also complements recent advances due to Chambolle and Dossal. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mathematical Programming Springer Journals

Fast convergence of inertial dynamics and algorithms with asymptotic vanishing viscosity

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Publisher
Springer Berlin Heidelberg
Copyright
Copyright © 2016 by Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Mathematics of Computing; Numerical Analysis; Combinatorics; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics
ISSN
0025-5610
eISSN
1436-4646
D.O.I.
10.1007/s10107-016-0992-8
Publisher site
See Article on Publisher Site

Abstract

In a Hilbert space setting $${{\mathcal {H}}}$$ H , we study the fast convergence properties as $$t \rightarrow + \infty $$ t → + ∞ of the trajectories of the second-order differential equation $$\begin{aligned} \ddot{x}(t) + \frac{\alpha }{t} \dot{x}(t) + \nabla \Phi (x(t)) = g(t), \end{aligned}$$ x ¨ ( t ) + α t x ˙ ( t ) + ∇ Φ ( x ( t ) ) = g ( t ) , where $$\nabla \Phi $$ ∇ Φ is the gradient of a convex continuously differentiable function $$\Phi : {{\mathcal {H}}} \rightarrow {{\mathbb {R}}}, \alpha $$ Φ : H → R , α is a positive parameter, and $$g: [t_0, + \infty [ \rightarrow {{\mathcal {H}}}$$ g : [ t 0 , + ∞ [ → H is a small perturbation term. In this inertial system, the viscous damping coefficient $$\frac{\alpha }{t}$$ α t vanishes asymptotically, but not too rapidly. For $$\alpha \ge 3$$ α ≥ 3 , and $$\int _{t_0}^{+\infty } t \Vert g(t)\Vert dt < + \infty $$ ∫ t 0 + ∞ t ‖ g ( t ) ‖ d t < + ∞ , just assuming that $${{\mathrm{argmin\,}}}\Phi \ne \emptyset $$ argmin Φ ≠ ∅ , we show that any trajectory of the above system satisfies the fast convergence property $$\begin{aligned} \Phi (x(t))- \min _{{{\mathcal {H}}}}\Phi \le \frac{C}{t^2}. \end{aligned}$$ Φ ( x ( t ) ) - min H Φ ≤ C t 2 . Moreover, for $$\alpha > 3$$ α > 3 , any trajectory converges weakly to a minimizer of $$\Phi $$ Φ . The strong convergence is established in various practical situations. These results complement the $${{\mathcal {O}}}(t^{-2})$$ O ( t - 2 ) rate of convergence for the values obtained by Su, Boyd and Candès in the unperturbed case $$g=0$$ g = 0 . Time discretization of this system, and some of its variants, provides new fast converging algorithms, expanding the field of rapid methods for structured convex minimization introduced by Nesterov, and further developed by Beck and Teboulle with FISTA. This study also complements recent advances due to Chambolle and Dossal.

Journal

Mathematical ProgrammingSpringer Journals

Published: Mar 24, 2016

References

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