en:intro:researches:optimization

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en:intro:researches:optimization [2019/12/17 10:18]
Hideaki IIDUKA [Decentralized Optimization Algorithms]
en:intro:researches:optimization [2020/02/21 11:33] (現在)
Hideaki IIDUKA
ライン 9: ライン 9:
 In particular, it is assumed that $C^{(i)}$ can be expressed as the **fixed point set** of a **nonexpansive mapping**. This implies the metric projection onto $C^{(i)}$ cannot be efficiently computed (e.g., $C^{(i)}$ is the intersection of many closed convex sets or the set of minimizers of a convex function).((The metric projection onto $C^{(i)}$, denoted by $P_{C^{(i)}}$,​ is defined for all $x\in H$ by $P_{C^{(i)}}(x) \in C^{(i)}$ and $\|x - P_{C^{(i)}}(x)\| = \inf_{y\in C^{(i)}} \|x-y\|$.)) Please see the [[en:​intro:​researches:​fixedpoint|Fixed Point Algorithms]] page for the details of fixed point sets.\\ In particular, it is assumed that $C^{(i)}$ can be expressed as the **fixed point set** of a **nonexpansive mapping**. This implies the metric projection onto $C^{(i)}$ cannot be efficiently computed (e.g., $C^{(i)}$ is the intersection of many closed convex sets or the set of minimizers of a convex function).((The metric projection onto $C^{(i)}$, denoted by $P_{C^{(i)}}$,​ is defined for all $x\in H$ by $P_{C^{(i)}}(x) \in C^{(i)}$ and $\|x - P_{C^{(i)}}(x)\| = \inf_{y\in C^{(i)}} \|x-y\|$.)) Please see the [[en:​intro:​researches:​fixedpoint|Fixed Point Algorithms]] page for the details of fixed point sets.\\
  
-Here, we divide the problem into the three categories.+Here, we divide the problem into the four categories.
   - **Smooth Convex Optimization Problem**:​\\ It assumes that $f^{(i)}$ $(i\in \mathcal{I})$ is smooth and convex. The problem includes [[http://​ieeexplore.ieee.org/​xpl/​login.jsp?​tp=&​arnumber=1206687&​url=http%3A%2F%2Fieeexplore.ieee.org%2Fiel5%2F78%2F27152%2F01206687.pdf%3Farnumber%3D1206687|signal recovery problem]], [[http://​ieeexplore.ieee.org/​xpl/​login.jsp?​tp=&​arnumber=4291862&​url=http%3A%2F%2Fieeexplore.ieee.org%2Fiel5%2F78%2F4291841%2F04291862.pdf%3Farnumber%3D4291862|beamforming problem]], [[http://​ieeexplore.ieee.org/​xpl/​login.jsp?​tp=&​arnumber=4604754&​url=http%3A%2F%2Fieeexplore.ieee.org%2Fiel5%2F49%2F4604726%2F04604754.pdf%3Farnumber%3D4604754|storage allocation problem]], [[http://​epubs.siam.org/​doi/​abs/​10.1137/​110850542|optimal control problem]], and [[http://​epubs.siam.org/​doi/​abs/​10.1137/​120866877|bandwidth allocation problem]].   - **Smooth Convex Optimization Problem**:​\\ It assumes that $f^{(i)}$ $(i\in \mathcal{I})$ is smooth and convex. The problem includes [[http://​ieeexplore.ieee.org/​xpl/​login.jsp?​tp=&​arnumber=1206687&​url=http%3A%2F%2Fieeexplore.ieee.org%2Fiel5%2F78%2F27152%2F01206687.pdf%3Farnumber%3D1206687|signal recovery problem]], [[http://​ieeexplore.ieee.org/​xpl/​login.jsp?​tp=&​arnumber=4291862&​url=http%3A%2F%2Fieeexplore.ieee.org%2Fiel5%2F78%2F4291841%2F04291862.pdf%3Farnumber%3D4291862|beamforming problem]], [[http://​ieeexplore.ieee.org/​xpl/​login.jsp?​tp=&​arnumber=4604754&​url=http%3A%2F%2Fieeexplore.ieee.org%2Fiel5%2F49%2F4604726%2F04604754.pdf%3Farnumber%3D4604754|storage allocation problem]], [[http://​epubs.siam.org/​doi/​abs/​10.1137/​110850542|optimal control problem]], and [[http://​epubs.siam.org/​doi/​abs/​10.1137/​120866877|bandwidth allocation problem]].
-  - **Nonsmooth Convex Optimization Problem**:​\\ It assumes that $f^{(i)}$ $(i\in \mathcal{I})$ is nonsmooth and convex. The problem includes [[http://​ieeexplore.ieee.org/​xpl/​articleDetails.jsp?​arnumber=4407760&​filter%3DAND%28p_IS_Number%3A4407756%29|signal recovery problem]],​[[http://www.sciencedirect.com/science/article/​pii/​S0925231214000964|ensemble learning]], ​ [[http://​link.springer.com/​chapter/​10.1007/​978-1-4419-9569-8_17|minimal antenna-subset selection problem]], and [[https://​ieeexplore.ieee.org/​document/​8584116|bandwidth allocation problem]]. ​+  - **Nonsmooth Convex Optimization Problem**:​\\ It assumes that $f^{(i)}$ $(i\in \mathcal{I})$ is nonsmooth and convex. The problem includes [[http://​ieeexplore.ieee.org/​xpl/​articleDetails.jsp?​arnumber=4407760&​filter%3DAND%28p_IS_Number%3A4407756%29|signal recovery problem]],​[[https://ieeexplore.ieee.org/document/8744480|ensemble learning]], ​ [[http://​link.springer.com/​chapter/​10.1007/​978-1-4419-9569-8_17|minimal antenna-subset selection problem]], and [[https://​ieeexplore.ieee.org/​document/​8584116|bandwidth allocation problem]]. ​
   - **Smooth Nonconvex Optimization Problem**\\ It assumes that $f^{(i)}$ $(i\in \mathcal{I})$ is smooth and nonconvex. The problem has practical problems such as  [[http://​link.springer.com/​article/​10.1007%2Fs10107-010-0427-x#​|power control]] and [[http://​epubs.siam.org/​doi/​abs/​10.1137/​110849456|bandwidth allocation]].   - **Smooth Nonconvex Optimization Problem**\\ It assumes that $f^{(i)}$ $(i\in \mathcal{I})$ is smooth and nonconvex. The problem has practical problems such as  [[http://​link.springer.com/​article/​10.1007%2Fs10107-010-0427-x#​|power control]] and [[http://​epubs.siam.org/​doi/​abs/​10.1137/​110849456|bandwidth allocation]].
 +  - **Nonsmooth Nonconvex Optimization Problem**\\ It assumes that $f^{(i)}$ $(i\in \mathcal{I})$ is nonsmooth and nonconvex. The problem has practical problems such as [[https://​www.sciencedirect.com/​science/​article/​abs/​pii/​S037722171400424X|fractional programming]].
  
 We focus on the following algorithms for solving the above problems.  ​ We focus on the following algorithms for solving the above problems.  ​
ライン 24: ライン 25:
 In 2001, [[http://​www.sp.ss.titech.ac.jp/​index.php?​Isao%20Yamada|Yamada]] proposed [[http://​www.sciencedirect.com/​science/​article/​pii/​S1570579X01800288|the hybrid steepest descent method]] for //smooth convex optimization over the fixed point sets of nonexpansive mappings//. We have presented algorithms based on the hybrid steepest descent method and their convergence analyses. In 2001, [[http://​www.sp.ss.titech.ac.jp/​index.php?​Isao%20Yamada|Yamada]] proposed [[http://​www.sciencedirect.com/​science/​article/​pii/​S1570579X01800288|the hybrid steepest descent method]] for //smooth convex optimization over the fixed point sets of nonexpansive mappings//. We have presented algorithms based on the hybrid steepest descent method and their convergence analyses.
 The following are our previously reported results: The following are our previously reported results:
-  * [[:​en:​iiduka:​|H. Iiduka]]: [[http://​link.springer.com/​article/​10.1007/​s10107-013-0741-1|Acceleration Method for Convex Optimization over the Fixed Point Set of a Nonexpansive Mapping]], ​//Mathematical Programming,​ Vol. 149, No. 1, pp. 131-165, 2015.// +  * [[:​en:​iiduka:​|H. Iiduka]]: [[http://​link.springer.com/​article/​10.1007/​s10107-013-0741-1|Acceleration Method for Convex Optimization over the Fixed Point Set of a Nonexpansive Mapping]], Mathematical Programming,​ Vol. 149, No. 1, pp. 131-165, 2015. 
-  * S. Iemoto, K. Hishinuma, and [[:​iiduka:​|H. Iiduka]]: [[http://​www.fixedpointtheoryandapplications.com/​content/​2014/​1/​51|Approximate Solutions to Variational Inequality over the Fixed Point Set of a Strongly Nonexpansive Mapping]], ​//Fixed Point Theory and Applications,​ 51, Vol. 2014, No. 1, 2014.// +  * S. Iemoto, K. Hishinuma, and [[:​iiduka:​|H. Iiduka]]: [[http://​www.fixedpointtheoryandapplications.com/​content/​2014/​1/​51|Approximate Solutions to Variational Inequality over the Fixed Point Set of a Strongly Nonexpansive Mapping]], Fixed Point Theory and Applications,​ 51, Vol. 2014, No. 1, 2014. 
-  * [[:​en:​iiduka:​|H. Iiduka]] and I. Yamada: [[http://​epubs.siam.org/​doi/​abs/​10.1137/​110850542|Computational Method for Solving a Stochastic Linear-Quadratic Control Problem Given an Unsolvable Stochastic Algebraic Riccati Equation]], ​//SIAM Journal on Control and Optimization,​ Vol. 50, No. 4, pp. 2173-2192, 2012.// +  * [[:​en:​iiduka:​|H. Iiduka]] and I. Yamada: [[http://​epubs.siam.org/​doi/​abs/​10.1137/​110850542|Computational Method for Solving a Stochastic Linear-Quadratic Control Problem Given an Unsolvable Stochastic Algebraic Riccati Equation]], SIAM Journal on Control and Optimization,​ Vol. 50, No. 4, pp. 2173-2192, 2012. 
-  * [[:​en:​iiduka:​|H. Iiduka]]: [[http://​www.sciencedirect.com/​science/​article/​pii/​S0377042711005358|Fixed Point Optimization Algorithm and Its Application to Network Bandwidth Allocation]], ​//Journal of Computational and Applied Mathematics,​ Vol. 236, No. 7, pp. 1733-1742, 2012.// +  * [[:​en:​iiduka:​|H. Iiduka]]: [[http://​www.sciencedirect.com/​science/​article/​pii/​S0377042711005358|Fixed Point Optimization Algorithm and Its Application to Network Bandwidth Allocation]],​ Journal of Computational and Applied Mathematics,​ Vol. 236, No. 7, pp. 1733-1742, 2012. 
-  * [[:​en:​iiduka:​|H. Iiduka]]: [[http://​link.springer.com/​article/​10.1007%2Fs10957-010-9769-z#​|Iterative Algorithm for Solving Triple-hierarchical Constrained Optimization Problem]], ​//Journal of Optimization Theory and Applications,​ Vol. 148, No. 3, pp. 580-592, 2011.//  +  * [[:​en:​iiduka:​|H. Iiduka]]: [[http://​link.springer.com/​article/​10.1007%2Fs10957-010-9769-z#​|Iterative Algorithm for Solving Triple-hierarchical Constrained Optimization Problem]], Journal of Optimization Theory and Applications,​ Vol. 148, No. 3, pp. 580-592, 2011.  
-  * [[:​en:​iiduka:​|H. Iiduka]]: [[http://​www.sciencedirect.com/​science/​article/​pii/​S0096300311000099|Three-term Conjugate Gradient Method for the Convex Optimization Problem over the Fixed Point Set of a Nonexpansive Mapping]], ​//Applied Mathematics and Computation,​ Vol. 217, No. 13, pp. 6315-6327, 2011.// +  * [[:​en:​iiduka:​|H. Iiduka]]: [[http://​www.sciencedirect.com/​science/​article/​pii/​S0096300311000099|Three-term Conjugate Gradient Method for the Convex Optimization Problem over the Fixed Point Set of a Nonexpansive Mapping]], Applied Mathematics and Computation,​ Vol. 217, No. 13, pp. 6315-6327, 2011. 
-  * [[:​en:​iiduka:​|H. Iiduka]] and [[http://​www.sp.ss.titech.ac.jp/​index.php?​Isao%20Yamada|I. Yamada]]: [[http://​epubs.siam.org/​action/​showAbstract?​page=1881&​volume=19&​issue=4&​journalCode=sjope8|A Use of Conjugate Gradient Direction for the Convex Optimization Problem over the Fixed Point Set of a Nonexpansive Mapping]], ​//SIAM Journal on Optimization,​ Vol. 19, No. 4, pp. 1881-1893, 2009.// +  * [[:​en:​iiduka:​|H. Iiduka]] and [[http://​www.sp.ss.titech.ac.jp/​index.php?​Isao%20Yamada|I. Yamada]]: [[http://​epubs.siam.org/​action/​showAbstract?​page=1881&​volume=19&​issue=4&​journalCode=sjope8|A Use of Conjugate Gradient Direction for the Convex Optimization Problem over the Fixed Point Set of a Nonexpansive Mapping]], SIAM Journal on Optimization,​ Vol. 19, No. 4, pp. 1881-1893, 2009. 
  
 ==== Decentralized Optimization Algorithms ​ ==== ==== Decentralized Optimization Algorithms ​ ====
-[[http://​epubs.siam.org/​doi/​abs/​10.1137/​S1052623499362111|The incremental subgradient method]] and [[http://​iopscience.iop.org/​0266-5611/​24/​6/​065014|the broadcast optimization method]] are useful for decentralized convex optimization. However, they can be applied to only the case where $C^{(i)} = C$ $(i\in \mathcal{I})$ and $C$ is simple in the sense that the projection onto $C$ can be easily computed (e.g., $C$ is a half-space). We have proposed decentralized optimization algorithms for solving convex optimization problems with more complicated $C^{(i)}$ (e.g., $C^{(i)}$ is the intersection of simple, closed convex sets), which include [[http://​epubs.siam.org/​doi/​abs/​10.1137/​120866877|bandwidth allocation problem]] ​and [[http://​ieeexplore.ieee.org/​xpl/​login.jsp?​tp=&​arnumber=4604754&​url=http%3A%2F%2Fieeexplore.ieee.org%2Fiel5%2F49%2F4604726%2F04604754.pdf%3Farnumber%3D4604754|storage allocation problem]].+[[http://​epubs.siam.org/​doi/​abs/​10.1137/​S1052623499362111|The incremental subgradient method]] and [[http://​iopscience.iop.org/​0266-5611/​24/​6/​065014|the broadcast optimization method]] are useful for decentralized convex optimization. However, they can be applied to only the case where $C^{(i)} = C$ $(i\in \mathcal{I})$ and $C$ is simple in the sense that the projection onto $C$ can be easily computed (e.g., $C$ is a half-space). We have proposed decentralized optimization algorithms for solving convex optimization problems with more complicated $C^{(i)}$ (e.g., $C^{(i)}$ is the intersection of simple, closed convex sets), which include [[http://​epubs.siam.org/​doi/​abs/​10.1137/​120866877|bandwidth allocation problem]][[http://​ieeexplore.ieee.org/​xpl/​login.jsp?​tp=&​arnumber=4604754&​url=http%3A%2F%2Fieeexplore.ieee.org%2Fiel5%2F49%2F4604726%2F04604754.pdf%3Farnumber%3D4604754|storage allocation problem]], and [[https://​ieeexplore.ieee.org/​document/​8744480|ensemble learning]].
   * [[:​en:​iiduka:​|H. Iiduka]]: [[https://​ieeexplore.ieee.org/​document/​8744480|Stochastic Fixed Point Optimization Algorithm for Classifier Ensemble]], IEEE Transactions on Cybernetics (accepted) ​   * [[:​en:​iiduka:​|H. Iiduka]]: [[https://​ieeexplore.ieee.org/​document/​8744480|Stochastic Fixed Point Optimization Algorithm for Classifier Ensemble]], IEEE Transactions on Cybernetics (accepted) ​
-  * [[:​en:​iiduka:​|H. Iiduka]]: [[https://​link.springer.com/​article/​10.1007/​s11081-019-09440-7|Decentralized Hierarchical Constrained Convex Optimization]],​ Optimization and Engineering ​(accepted) +  * [[:​en:​iiduka:​|H. Iiduka]]: [[https://​link.springer.com/​article/​10.1007/​s11081-019-09440-7|Decentralized Hierarchical Constrained Convex Optimization]],​ Optimization and Engineering, Vol. 21, No. 1, pp. 181–213, 2020. 
-  * [[:​en:​iiduka:​|H. Iiduka]]: [[https://​www.jstage.jst.go.jp/​article/​jorsj/​58/​4/​58_330/​_pdf|Parallel Optimization Algorithm for Smooth Convex Optimization over Fixed Point Sets of Quasi-nonexpansive Mappings]], Journal of the Operations Research of Japan, ​//Vol. 58, No. 4, pp. 330-352, 2015//+  * [[:​en:​iiduka:​|H. Iiduka]]: [[https://​www.jstage.jst.go.jp/​article/​jorsj/​58/​4/​58_330/​_pdf|Parallel Optimization Algorithm for Smooth Convex Optimization over Fixed Point Sets of Quasi-nonexpansive Mappings]], Journal of the Operations Research of Japan, Vol. 58, No. 4, pp. 330-352, 2015. 
-  * [[:​en:​iiduka:​|H. Iiduka]]: [[http://​www.ybook.co.jp/​online-p/​JNCA/​Open/​16/​jncav16n11p2159-oa/​FLASH/​index.html|Distributed Convex Optimization Algorithms and Their Application to Distributed Control in Peer-to-Peer Data Storage System]], Journal of Nonlinear and Convex Analysis: : Special Issue-Dedicated to Wataru Takahashi on the occasion of his 70th birth day, //Vol. 16, No. 11, pp. 2159-2179, 2015//+  * [[:​en:​iiduka:​|H. Iiduka]]: [[http://​www.ybook.co.jp/​online-p/​JNCA/​Open/​16/​jncav16n11p2159-oa/​FLASH/​index.html|Distributed Convex Optimization Algorithms and Their Application to Distributed Control in Peer-to-Peer Data Storage System]], Journal of Nonlinear and Convex Analysis: : Special Issue-Dedicated to Wataru Takahashi on the occasion of his 70th birth day, Vol. 16, No. 11, pp. 2159-2179, 2015. 
-  * [[:​en:​iiduka:​|H. Iiduka]] and K. Hishinuma: [[http://​epubs.siam.org/​doi/​abs/​10.1137/​130939560|Acceleration Method Combining Broadcast and Incremental Distributed Optimization Algorithms]], ​//SIAM Journal on Optimization,​ Vol. 24, No. 4, pp. 1840-1863, 2014.// +  * [[:​en:​iiduka:​|H. Iiduka]] and K. Hishinuma: [[http://​epubs.siam.org/​doi/​abs/​10.1137/​130939560|Acceleration Method Combining Broadcast and Incremental Distributed Optimization Algorithms]],​ SIAM Journal on Optimization,​ Vol. 24, No. 4, pp. 1840-1863, 2014. 
-  * [[:​en:​iiduka:​|H. Iiduka]]: [[http://​epubs.siam.org/​doi/​abs/​10.1137/​120866877|Fixed Point Optimization Algorithms for Distributed Optimization in Networked Systems]], ​//SIAM Journal on Optimization,​ Vol. 23, No. 1, pp. 1-26, 2013.// +  * [[:​en:​iiduka:​|H. Iiduka]]: [[http://​epubs.siam.org/​doi/​abs/​10.1137/​120866877|Fixed Point Optimization Algorithms for Distributed Optimization in Networked Systems]], SIAM Journal on Optimization,​ Vol. 23, No. 1, pp. 1-26, 2013.
  
 ===== Optimization Algorithms for Nonsmooth Convex Optimization ===== ===== Optimization Algorithms for Nonsmooth Convex Optimization =====
ライン 45: ライン 46:
 The following are the results of the algorithms based on the above methods. The following are the results of the algorithms based on the above methods.
 ==== Decentralized Optimization Algorithms ==== ==== Decentralized Optimization Algorithms ====
-  * K. Shimizu, K. Hishinuma, [[:​en:​iiduka:​|H. Iiduka]]: Parallel Computing Proximal Method for Nonsmooth Convex Optimization ​with Fixed Point Constraints of Quasi-nonexpansive Mappings, ​submitted ​ +  * K. Shimizu, K. Hishinuma, [[:​en:​iiduka:​|H. Iiduka]]: Parallel Computing Proximal Method for Nonsmooth Convex Optimization ​With Fixed Point Constraints of Quasi-nonexpansive Mappings, ​Applied Set-Valued Analysis and Optimization,​ accepted 
-  * H. Oishi, Y. Kobayashi, [[:​en:​iiduka:​|H. Iiduka]]: Incremental Proximal Method for Nonsmooth Convex Optimization ​with Fixed Point Constraints of Quasi-nonexpansive Mappings, Linear and Nonlinear Analysis ​(accepted) +  * H. Oishi, Y. Kobayashi, [[:​en:​iiduka:​|H. Iiduka]]: ​[[http://​www.ybook.co.jp/​online-p/​LNA/​Open/​vol5/​lnav5n3p477-oa/​index.html|Incremental Proximal Method for Nonsmooth Convex Optimization ​With Fixed Point Constraints of Quasi-nonexpansive Mappings]], Linear and Nonlinear Analysis, Vol. 5, No. 3, pp. 477-493, 2019. 
-  * [[:​en:​iiduka:​|H. Iiduka]]: [[https://​ieeexplore.ieee.org/​document/​8584116|Distributed Optimization for Network Resource Allocation with Nonsmooth Utility Functions]],​ IEEE Transactions on Control of Network Systems ​(accepted) +  * [[:​en:​iiduka:​|H. Iiduka]]: [[https://​ieeexplore.ieee.org/​document/​8584116|Distributed Optimization for Network Resource Allocation with Nonsmooth Utility Functions]],​ IEEE Transactions on Control of Network Systems, Vol. 6, No. 4, pp. 1354-1365, 2019. 
-  * K. Hishinuma and [[en:​iiduka:​|H. Iiduka]]: Convergence Analysis of Incremental and Parallel Line Search Subgradient Methods in Hilbert Space, Journal of Nonlinear and Convex Analysis: Special Issue-Dedicated to Wataru Takahashi on the occasion of his 75th birth day, Vol. 20, No. 9, pp.1937-1947,​ 2019.+  * K. Hishinuma and [[en:​iiduka:​|H. Iiduka]]: ​[[http://​www.ybook.co.jp/​online2/​opjnca/​vol20/​p1937.html|Convergence Analysis of Incremental and Parallel Line Search Subgradient Methods in Hilbert Space]], Journal of Nonlinear and Convex Analysis: Special Issue-Dedicated to Wataru Takahashi on the occasion of his 75th birth day, Vol. 20, No. 9, pp.1937-1947,​ 2019.
   * K. Hishinuma and [[:​en:​iiduka:​|H. Iiduka]]: [[https://​www.frontiersin.org/​articles/​10.3389/​frobt.2019.00077/​full|Incremental and Parallel Machine Learning Algorithms with Automated Learning Rate Adjustments]],​ Frontiers in Robotics and AI: Resolution of Limitations of Deep Learning to Develop New AI Paradigms, Vol. 6, Article 77, 2019.    * K. Hishinuma and [[:​en:​iiduka:​|H. Iiduka]]: [[https://​www.frontiersin.org/​articles/​10.3389/​frobt.2019.00077/​full|Incremental and Parallel Machine Learning Algorithms with Automated Learning Rate Adjustments]],​ Frontiers in Robotics and AI: Resolution of Limitations of Deep Learning to Develop New AI Paradigms, Vol. 6, Article 77, 2019. 
   * [[:​en:​iiduka:​|H. Iiduka]]: [[http://​www.tandfonline.com/​doi/​full/​10.1080/​10556788.2018.1425860|Two Stochastic Optimization Algorithms for Convex Optimization with Fixed Point Constraints]],​ Optimization Methods and Software, Vol. 34, No. 4, pp.731-757, 2019.   * [[:​en:​iiduka:​|H. Iiduka]]: [[http://​www.tandfonline.com/​doi/​full/​10.1080/​10556788.2018.1425860|Two Stochastic Optimization Algorithms for Convex Optimization with Fixed Point Constraints]],​ Optimization Methods and Software, Vol. 34, No. 4, pp.731-757, 2019.
ライン 65: ライン 66:
   * [[en:​iiduka:​|H. Iiduka]] and M. Uchida: [[http://​ieeexplore.ieee.org/​xpl/​login.jsp?​tp=&​arnumber=5752800&​url=http%3A%2F%2Fieeexplore.ieee.org%2Fiel5%2F4234%2F5895119%2F05752800.pdf%3Farnumber%3D5752800|Fixed Point Optimization Algorithms for Network Bandwidth Allocation Problems with Compoundable Constraints]],​ IEEE Communications Letters, Vol. 15, No. 6, pp. 596-598, 2011.   * [[en:​iiduka:​|H. Iiduka]] and M. Uchida: [[http://​ieeexplore.ieee.org/​xpl/​login.jsp?​tp=&​arnumber=5752800&​url=http%3A%2F%2Fieeexplore.ieee.org%2Fiel5%2F4234%2F5895119%2F05752800.pdf%3Farnumber%3D5752800|Fixed Point Optimization Algorithms for Network Bandwidth Allocation Problems with Compoundable Constraints]],​ IEEE Communications Letters, Vol. 15, No. 6, pp. 596-598, 2011.
   * [[en:​iiduka:​|H. Iiduka]] and [[http://​www.sp.ss.titech.ac.jp/​index.php?​Isao%20Yamada|I. Yamada]]: [[http://​www.tandfonline.com/​doi/​abs/​10.1080/​02331930701762829|A Subgradient-type Method for the Equilibrium Problem over the Fixed Point Set and its Applications]],​ Optimization,​ Vol. 58, No. 2, pp. 251-261, 2009.    * [[en:​iiduka:​|H. Iiduka]] and [[http://​www.sp.ss.titech.ac.jp/​index.php?​Isao%20Yamada|I. Yamada]]: [[http://​www.tandfonline.com/​doi/​abs/​10.1080/​02331930701762829|A Subgradient-type Method for the Equilibrium Problem over the Fixed Point Set and its Applications]],​ Optimization,​ Vol. 58, No. 2, pp. 251-261, 2009. 
- 
- 
- 
  
 ==== Decentralized Optimization Algorithms ==== ==== Decentralized Optimization Algorithms ====
-  * K. Hishinuma and [[en:​iiduka:​|H. Iiduka]]: [[https://​doi.org/​10.1016/​j.ejor.2019.09.037|Fixed Point Quasiconvex Subgradient Method]], European Journal of Operational Research, Vol. 282, No. 2, 428–437, 2020 
   * [[en:​iiduka:​|H. Iiduka]]: [[http://​www.ybook.co.jp/​|Distributed Iterative Methods for Solving Nonmonotone Variational Inequality over the Intersection of Fixed Point Sets of Nonexpansive Mappings]], Pacific Journal of Optimization,​ Vol. 10, No. 4, pp. 691-713, 2014.   * [[en:​iiduka:​|H. Iiduka]]: [[http://​www.ybook.co.jp/​|Distributed Iterative Methods for Solving Nonmonotone Variational Inequality over the Intersection of Fixed Point Sets of Nonexpansive Mappings]], Pacific Journal of Optimization,​ Vol. 10, No. 4, pp. 691-713, 2014.
  
 +===== Optimization Algorithms for Nonsmooth Nonconvex Optimization ===== 
 +==== Centralized Optimization Algorithms ==== 
 + * K. Hishinuma and [[en:​iiduka:​|H. Iiduka]]: [[https://​doi.org/​10.1016/​j.ejor.2019.09.037|Fixed Point Quasiconvex Subgradient Method]], European Journal of Operational Research, Vol. 282, No. 2, 428–437, 2020
  
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  • 最終更新: 2019/12/17 10:18
  • by Hideaki IIDUKA