Fast convex optimization via inertial dynamics with Hessian driven damping
We first study the fast minimization properties of the trajectories of the second-order evolution equationx¨(t)+αtx˙(t)+β∇2Φ(x(t))x˙(t)+∇Φ(x(t))=0, where Φ:H→R is a smooth convex function acting on a real Hilbert space H, and α, β are positive parameters. This inertial system combines an isotropic v...
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| Published in: | Journal of Differential Equations Vol. 261; no. 10; pp. 5734 - 5783 |
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| Format: | Journal Article |
| Language: | English |
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15.11.2016
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| ISSN: | 0022-0396, 1090-2732 |
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| Abstract | We first study the fast minimization properties of the trajectories of the second-order evolution equationx¨(t)+αtx˙(t)+β∇2Φ(x(t))x˙(t)+∇Φ(x(t))=0, where Φ:H→R is a smooth convex function acting on a real Hilbert space H, and α, β are positive parameters. This inertial system combines an isotropic viscous damping which vanishes asymptotically, and a geometrical Hessian driven damping, which makes it naturally related to Newton's and Levenberg–Marquardt methods. For α≥3, and β>0, along any trajectory, fast convergence of the valuesΦ(x(t))−minHΦ=O(t−2) is obtained, together with rapid convergence of the gradients ∇Φ(x(t)) to zero. For α>3, just assuming that argminΦ≠∅, we show that any trajectory converges weakly to a minimizer of Φ, and that Φ(x(t))−minHΦ=o(t−2). Strong convergence is established in various practical situations. In particular, for the strongly convex case, we obtain an even faster speed of convergence which can be arbitrarily fast depending on the choice of α. More precisely, we have Φ(x(t))−minHΦ=O(t−23α). Then, we extend the results to the case of a general proper lower-semicontinuous convex function Φ:H→R∪{+∞}. This is based on the crucial property that the inertial dynamics with Hessian driven damping can be equivalently written as a first-order system in time and space, allowing to extend it by simply replacing the gradient with the subdifferential. By explicit–implicit time discretization, this opens a gate to new − possibly more rapid − inertial algorithms, expanding the field of FISTA methods for convex structured optimization problems. |
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| AbstractList | We first study the fast minimization properties of the trajectories of the second-order evolution equationx¨(t)+αtx˙(t)+β∇2Φ(x(t))x˙(t)+∇Φ(x(t))=0, where Φ:H→R is a smooth convex function acting on a real Hilbert space H, and α, β are positive parameters. This inertial system combines an isotropic viscous damping which vanishes asymptotically, and a geometrical Hessian driven damping, which makes it naturally related to Newton's and Levenberg–Marquardt methods. For α≥3, and β>0, along any trajectory, fast convergence of the valuesΦ(x(t))−minHΦ=O(t−2) is obtained, together with rapid convergence of the gradients ∇Φ(x(t)) to zero. For α>3, just assuming that argminΦ≠∅, we show that any trajectory converges weakly to a minimizer of Φ, and that Φ(x(t))−minHΦ=o(t−2). Strong convergence is established in various practical situations. In particular, for the strongly convex case, we obtain an even faster speed of convergence which can be arbitrarily fast depending on the choice of α. More precisely, we have Φ(x(t))−minHΦ=O(t−23α). Then, we extend the results to the case of a general proper lower-semicontinuous convex function Φ:H→R∪{+∞}. This is based on the crucial property that the inertial dynamics with Hessian driven damping can be equivalently written as a first-order system in time and space, allowing to extend it by simply replacing the gradient with the subdifferential. By explicit–implicit time discretization, this opens a gate to new − possibly more rapid − inertial algorithms, expanding the field of FISTA methods for convex structured optimization problems. |
| Author | Attouch, Hedy Peypouquet, Juan Redont, Patrick |
| Author_xml | – sequence: 1 givenname: Hedy surname: Attouch fullname: Attouch, Hedy email: hedy.attouch@univ-montp2.fr organization: Institut Montpelliérain Alexander Grothendieck, UMR 5149 CNRS, Université Montpellier 2, place Eugène Bataillon, 34095 Montpellier cedex 5, France – sequence: 2 givenname: Juan orcidid: 0000-0002-8551-0522 surname: Peypouquet fullname: Peypouquet, Juan email: juan.peypouquet@usm.cl organization: Univesidad Técnica Federico Santa María, Av España 1680, Valparaiso, Chile – sequence: 3 givenname: Patrick surname: Redont fullname: Redont, Patrick email: patrick.redont@univ-montp2.fr organization: Institut Montpelliérain Alexander Grothendieck, UMR 5149 CNRS, Université Montpellier 2, place Eugène Bataillon, 34095 Montpellier cedex 5, France |
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| Keywords | Gradient flows Inertial dynamics Convex optimization Hessian-driven damping Fast convergent methods Forward–backward algorithm Gradient flowsInertial dynamics |
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| Snippet | We first study the fast minimization properties of the trajectories of the second-order evolution equationx¨(t)+αtx˙(t)+β∇2Φ(x(t))x˙(t)+∇Φ(x(t))=0, where Φ:H→R... |
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| SubjectTerms | Convex optimization Fast convergent methods Forward–backward algorithm Gradient flows Hessian-driven damping Inertial dynamics Mathematics |
| Title | Fast convex optimization via inertial dynamics with Hessian driven damping |
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