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Fast convergence of inertial dynamics and algorithms with asymptotic vanishing viscosity.

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Title: Fast convergence of inertial dynamics and algorithms with asymptotic vanishing viscosity.
Authors: Attouch, Hedy1, Redont, Patrick1, Chbani, Zaki2, Peypouquet, Juan3
Source: Mathematical Programming. Mar2018, Vol. 168 Issue 1/2, p123-175. 53p.
Subjects: Robust convex optimization, Dynamical systems, Dynamics, Viscosity, Stochastic convergence
Abstract: In a Hilbert space setting H, we study the fast convergence properties as t→+∞ of the trajectories of the second-order differential equation x¨(t)+αtx˙(t)+∇Φ(x(t))=g(t),where ∇Φ is the gradient of a convex continuously differentiable function Φ:H→R,α is a positive parameter, and g:[t0,+∞[→H is a small perturbation term. In this inertial system, the viscous damping coefficient αt vanishes asymptotically, but not too rapidly. For α≥3, and ∫t0+∞t‖g(t)‖dt<+∞, just assuming that argminΦ≠∅, we show that any trajectory of the above system satisfies the fast convergence property Φ(x(t))-minHΦ≤Ct2.Moreover, for α>3, any trajectory converges weakly to a minimizer of Φ. The strong convergence is established in various practical situations. These results complement the O(t-2) rate of convergence for the values obtained by Su, Boyd and Candès in the unperturbed case 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. [ABSTRACT FROM AUTHOR]
Copyright of Mathematical Programming is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: &lt;searchLink fieldCode=&quot;JN&quot; term=&quot;%22Mathematical+Programming%22&quot;&gt;Mathematical Programming&lt;/searchLink&gt;. Mar2018, Vol. 168 Issue 1/2, p123-175. 53p.
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  Data: In a Hilbert space setting H&lt;inline-graphic&gt;&lt;/inline-graphic&gt;, we study the fast convergence properties as t→+∞&lt;inline-graphic&gt;&lt;/inline-graphic&gt; of the trajectories of the second-order differential equation x&#168;(t)+αtx˙(t)+∇Φ(x(t))=g(t),&lt;graphic&gt;&lt;/graphic&gt;where ∇Φ&lt;inline-graphic&gt;&lt;/inline-graphic&gt; is the gradient of a convex continuously differentiable function Φ:H→R,α&lt;inline-graphic&gt;&lt;/inline-graphic&gt; is a positive parameter, and g:[t0,+∞[→H&lt;inline-graphic&gt;&lt;/inline-graphic&gt; is a &lt;italic&gt;small&lt;/italic&gt; perturbation term. In this inertial system, the viscous damping coefficient αt&lt;inline-graphic&gt;&lt;/inline-graphic&gt; vanishes asymptotically, but not too rapidly. For α≥3&lt;inline-graphic&gt;&lt;/inline-graphic&gt;, and ∫t0+∞t‖g(t)‖dt&lt;+∞&lt;inline-graphic&gt;&lt;/inline-graphic&gt;, just assuming that argminΦ≠∅&lt;inline-graphic&gt;&lt;/inline-graphic&gt;, we show that any trajectory of the above system satisfies the fast convergence property Φ(x(t))-minHΦ≤Ct2.&lt;graphic&gt;&lt;/graphic&gt;Moreover, for α&gt;3&lt;inline-graphic&gt;&lt;/inline-graphic&gt;, any trajectory converges weakly to a minimizer of Φ&lt;inline-graphic&gt;&lt;/inline-graphic&gt;. The strong convergence is established in various practical situations. These results complement the O(t-2)&lt;inline-graphic&gt;&lt;/inline-graphic&gt; rate of convergence for the values obtained by Su, Boyd and Cand&#232;s in the unperturbed case g=0&lt;inline-graphic&gt;&lt;/inline-graphic&gt;. 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. [ABSTRACT FROM AUTHOR]
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  Data: &lt;i&gt;Copyright of Mathematical Programming is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder&#39;s express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.&lt;/i&gt; (Copyright applies to all Abstracts.)
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      – Type: doi
        Value: 10.1007/s10107-016-0992-8
    Languages:
      – Code: eng
        Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 53
        StartPage: 123
    Subjects:
      – SubjectFull: Robust convex optimization
        Type: general
      – SubjectFull: Dynamical systems
        Type: general
      – SubjectFull: Dynamics
        Type: general
      – SubjectFull: Viscosity
        Type: general
      – SubjectFull: Stochastic convergence
        Type: general
    Titles:
      – TitleFull: Fast convergence of inertial dynamics and algorithms with asymptotic vanishing viscosity.
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            NameFull: Attouch, Hedy
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            NameFull: Redont, Patrick
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            NameFull: Chbani, Zaki
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              M: 03
              Text: Mar2018
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              Y: 2018
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