| 両方とも前のリビジョン 前のリビジョン | 次のリビジョン両方とも次のリビジョン |
| en:intro:researches:optimization [2020/02/21 10:49] – [Decentralized Optimization Algorithms] Hideaki IIDUKA | en:intro:researches:optimization [2020/02/21 11:16] – Hideaki IIDUKA |
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| 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.\\ |
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| 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]],[[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]]. | - **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/S0377221719307945?via%3Dihub|fractional programming]]. |
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| We focus on the following algorithms for solving the above problems. | We focus on the following algorithms for solving the above problems. |