On the Convergence of the Iterates of the “Fast Iterative Shrinkage/Thresholding Algorithm”
We discuss here the convergence of the iterates of the “Fast Iterative Shrinkage/Thresholding Algorithm,” which is an algorithm proposed by Beck and Teboulle for minimizing the sum of two convex, lower-semicontinuous, and proper functions (defined in a Euclidean or Hilbert space), such that one is d...
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| Vydané v: | Journal of optimization theory and applications Ročník 166; číslo 3; s. 968 - 982 |
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| Hlavní autori: | , |
| Médium: | Journal Article |
| Jazyk: | English |
| Vydavateľské údaje: |
New York
Springer US
01.09.2015
Springer Nature B.V |
| Predmet: | |
| ISSN: | 0022-3239, 1573-2878 |
| On-line prístup: | Získať plný text |
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| Abstract | We discuss here the convergence of the iterates of the “Fast Iterative Shrinkage/Thresholding Algorithm,” which is an algorithm proposed by Beck and Teboulle for minimizing the sum of two convex, lower-semicontinuous, and proper functions (defined in a Euclidean or Hilbert space), such that one is differentiable with Lipschitz gradient, and the proximity operator of the second is easy to compute. It builds a sequence of iterates for which the objective is controlled, up to a (nearly optimal) constant, by the inverse of the square of the iteration number. However, the convergence of the iterates themselves is not known. We show here that with a small modification, we can ensure the same upper bound for the decay of the energy, as well as the convergence of the iterates to a minimizer. |
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| AbstractList | We discuss here the convergence of the iterates of the "Fast Iterative Shrinkage/Thresholding Algorithm," which is an algorithm proposed by Beck and Teboulle for minimizing the sum of two convex, lower-semicontinuous, and proper functions (defined in a Euclidean or Hilbert space), such that one is differentiable with Lipschitz gradient, and the proximity operator of the second is easy to compute. It builds a sequence of iterates for which the objective is controlled, up to a (nearly optimal) constant, by the inverse of the square of the iteration number. However, the convergence of the iterates themselves is not known. We show here that with a small modification, we can ensure the same upper bound for the decay of the energy, as well as the convergence of the iterates to a minimizer. |
| Author | Dossal, Ch Chambolle, A. |
| Author_xml | – sequence: 1 givenname: A. orcidid: 0000-0002-9465-4659 surname: Chambolle fullname: Chambolle, A. email: antonin.chambolle@cmap.polytechnique.fr organization: CMAP, Ecole Polytechnique, CNRS – sequence: 2 givenname: Ch surname: Dossal fullname: Dossal, Ch organization: IMB, Univ. Bordeaux 1, CNRS |
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| ContentType | Journal Article |
| Copyright | Springer Science+Business Media New York 2015 |
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| References | Combettes, Wajs (CR1) 2005; 4 Gabay, Mercier (CR4) 1976; 2 Beck, Teboulle (CR6) 2009; 2 Nesterov (CR14) 2004 Nesterov (CR7) 1983; 269 Nesterov (CR9) 2005; 103 CR5 CR16 CR15 Güler (CR8) 1992; 2 Lions, Mercier (CR2) 1979; 16 CR11 CR10 Eckstein, Bertsekas (CR3) 1992; 55 Alvarez, Attouch (CR12) 2001; 9 Moudafi, Oliny (CR13) 2003; 155 J Eckstein (746_CR3) 1992; 55 Y Nesterov (746_CR9) 2005; 103 A Beck (746_CR6) 2009; 2 PL Combettes (746_CR1) 2005; 4 A Moudafi (746_CR13) 2003; 155 746_CR10 746_CR11 Y Nesterov (746_CR14) 2004 P-L Lions (746_CR2) 1979; 16 D Gabay (746_CR4) 1976; 2 746_CR5 Y Nesterov (746_CR7) 1983; 269 746_CR15 746_CR16 O Güler (746_CR8) 1992; 2 F Alvarez (746_CR12) 2001; 9 |
| References_xml | – year: 2004 ident: CR14 publication-title: Introductory Lectures on Convex Optimization, Volume 87 of Applied Optimization doi: 10.1007/978-1-4419-8853-9 – ident: CR15 – volume: 16 start-page: 964 issue: 6 year: 1979 end-page: 979 ident: CR2 article-title: Splitting algorithms for the sum of two nonlinear operators publication-title: SIAM J. Numer. Anal. doi: 10.1137/0716071 – volume: 55 start-page: 293 issue: 3, Ser. A year: 1992 end-page: 318 ident: CR3 article-title: On the Douglas–Rachford splitting method and the proximal point algorithm for maximal monotone operators publication-title: Math. Program. doi: 10.1007/BF01581204 – ident: CR16 – volume: 9 start-page: 3 issue: 1–2 year: 2001 end-page: 11 ident: CR12 article-title: An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping publication-title: Set-Valued Anal. doi: 10.1023/A:1011253113155 – volume: 103 start-page: 127 issue: 1, Ser. A year: 2005 end-page: 152 ident: CR9 article-title: Smooth minimization of non-smooth functions publication-title: Math. Program. doi: 10.1007/s10107-004-0552-5 – volume: 4 start-page: 1168 issue: 4 year: 2005 end-page: 1200 ident: CR1 article-title: Signal recovery by proximal forward–backward splitting publication-title: Multiscale Model. Simul. doi: 10.1137/050626090 – ident: CR10 – ident: CR11 – volume: 269 start-page: 543 issue: 3 year: 1983 end-page: 547 ident: CR7 article-title: A method for solving the convex programming problem with convergence rate publication-title: Dokl. Akad. Nauk SSSR – ident: CR5 – volume: 2 start-page: 183 issue: 1 year: 2009 end-page: 202 ident: CR6 article-title: A fast iterative shrinkage-thresholding algorithm for linear inverse problems publication-title: SIAM J. Imaging Sci. doi: 10.1137/080716542 – volume: 2 start-page: 649 issue: 4 year: 1992 end-page: 664 ident: CR8 article-title: New proximal point algorithms for convex minimization publication-title: SIAM J. Optim. doi: 10.1137/0802032 – volume: 2 start-page: 17 issue: 1 year: 1976 end-page: 40 ident: CR4 article-title: A dual algorithm for the solution of nonlinear variational problems via finite element approximation publication-title: Comput. Math. Appl. doi: 10.1016/0898-1221(76)90003-1 – volume: 155 start-page: 447 issue: 2 year: 2003 end-page: 454 ident: CR13 article-title: Convergence of a splitting inertial proximal method for monotone operators publication-title: J. Comput. Appl. Math. doi: 10.1016/S0377-0427(02)00906-8 – volume: 103 start-page: 127 issue: 1, Ser. A year: 2005 ident: 746_CR9 publication-title: Math. Program. doi: 10.1007/s10107-004-0552-5 – ident: 746_CR16 doi: 10.1007/978-1-4419-9467-7 – ident: 746_CR15 – ident: 746_CR5 doi: 10.1007/978-1-4419-9569-8_10 – ident: 746_CR11 doi: 10.1007/s10851-014-0523-2 – volume: 9 start-page: 3 issue: 1–2 year: 2001 ident: 746_CR12 publication-title: Set-Valued Anal. doi: 10.1023/A:1011253113155 – volume: 155 start-page: 447 issue: 2 year: 2003 ident: 746_CR13 publication-title: J. Comput. Appl. Math. doi: 10.1016/S0377-0427(02)00906-8 – volume: 16 start-page: 964 issue: 6 year: 1979 ident: 746_CR2 publication-title: SIAM J. Numer. Anal. doi: 10.1137/0716071 – volume: 269 start-page: 543 issue: 3 year: 1983 ident: 746_CR7 publication-title: Dokl. Akad. Nauk SSSR – volume: 2 start-page: 649 issue: 4 year: 1992 ident: 746_CR8 publication-title: SIAM J. Optim. doi: 10.1137/0802032 – volume-title: Introductory Lectures on Convex Optimization, Volume 87 of Applied Optimization year: 2004 ident: 746_CR14 doi: 10.1007/978-1-4419-8853-9 – volume: 4 start-page: 1168 issue: 4 year: 2005 ident: 746_CR1 publication-title: Multiscale Model. Simul. doi: 10.1137/050626090 – volume: 2 start-page: 17 issue: 1 year: 1976 ident: 746_CR4 publication-title: Comput. Math. Appl. doi: 10.1016/0898-1221(76)90003-1 – ident: 746_CR10 – volume: 2 start-page: 183 issue: 1 year: 2009 ident: 746_CR6 publication-title: SIAM J. Imaging Sci. doi: 10.1137/080716542 – volume: 55 start-page: 293 issue: 3, Ser. A year: 1992 ident: 746_CR3 publication-title: Math. Program. doi: 10.1007/BF01581204 |
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| Snippet | We discuss here the convergence of the iterates of the “Fast Iterative Shrinkage/Thresholding Algorithm,” which is an algorithm proposed by Beck and Teboulle... We discuss here the convergence of the iterates of the "Fast Iterative Shrinkage/Thresholding Algorithm," which is an algorithm proposed by Beck and Teboulle... |
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| SubjectTerms | Algorithms Applications of Mathematics Calculus of Variations and Optimal Control; Optimization Construction Convergence Engineering Euclidean geometry Euclidean space Hilbert space Inverse Iterative methods Mathematical analysis Mathematics Mathematics and Statistics Operations Research/Decision Theory Optimization Optimization algorithms Shrinkage Studies Theory of Computation |
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| Title | On the Convergence of the Iterates of the “Fast Iterative Shrinkage/Thresholding Algorithm” |
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