====== Fixed Point Algorithms ====== ===== Fixed Point Problem and Its Applications ===== Consider the following fixed point problem on a real Hilbert space with inner product $\langle \cdot, \cdot \rangle$ and norm $\| \cdot \|$: \begin{align*} \text{Find } x \in \mathrm{Fix}\left(T\right) := \left\{ x\in H \colon T \left( x \right) = x \right\}, \end{align*} where $T \colon H \to H$ is nonexpansive (i.e., $\|T(x) - T(y) \| \leq \| x-y \|$ $(x,y\in H)$). A number of [[http://en.wikipedia.org/wiki/Fixed-point_theorem|fixed point theorems]] have been studied by renowned mathematicians, such as [[http://en.wikipedia.org/wiki/Stefan_Banach|Banach]], [[http://en.wikipedia.org/wiki/Luitzen_Egbertus_Jan_Brouwer|Brouwer]], [[http://en.wikipedia.org/wiki/James_Caristi|Caristi]], [[http://en.wikipedia.org/wiki/Ky_Fan|Fan]], [[http://en.wikipedia.org/wiki/Shizuo_Kakutani|Kakutani]], [[http://en.wikipedia.org/wiki/William_Arthur_Kirk|Kirk]], [[http://en.wikipedia.org/wiki/Juliusz_Schauder|Schauder]], [[http://www.sci.keio.ac.jp/en/member/detail.php?eid=00119&katagaki=8&status=1|Takahashi]], and so on. We give the following two examples of the fixed point problem. - Convex Feasibility Problem:\\ [[http://dx.doi.org/10.1137/S0036144593251710|The problem]] is to find \begin{align*}x^\star \in C:= \bigcap_{i\in \mathcal{I}} C_i,\end{align*} where $C_i$ $(\subset H)$ $(i\in \mathcal{I} := \{1,2,\ldots,I \})$ is nonempty, closed, and convex. It includes practical problems, such as [[http://dx.doi.org/10.1109/78.782189|signal recovery problem]].\\ Let us define $T := P_1 P_2 \cdots P_I$((We may define $T$ by $T := P_I P_{I-1} \cdots P_1$ or $T:= \sum_{i\in \mathcal{I}} w_i P_i$, where $(w_i)_{i\in \mathcal{I}} \subset (0,1)$ satisfies $\sum_{i\in\mathcal{I}} w_i =1$.)), where $P_i$ $(i\in \mathcal{I})$ stands for the metric projection onto $C_i$.(($P_i$ is defined by $P_i(x) \in C_i$ and $\|P_i(x) - x \| = \inf_{y\in C_i}\| y-x \|$ $(x\in H)$, and it satisfies the nonexpansivity condition.)) Accordingly, $T$ is nonexpansive and $\mathrm{Fix}(T) = C = \bigcap_{i\in \mathcal{I}} C_i$. - Constrained Convex Optimization Problem:\\ Suppose that $C$ $(\subset H)$ is nonempty, closed, and convex, $f\colon H \to \mathbb{R}$ is Fréchet differentiable and convex, and its gradient, denoted by $\nabla f$, is Lipschitz continuous with a constant $L$ $(>0)$. The constrained convex optimization problem is to find \begin{align*}x^\star \in C \text{ such that } f(x^\star) \leq f(x) \text{ for all } x\in C.\end{align*} See [[en:intro:researches:optimization|Optimization Algorithms]] page for examples of the convex optimization problem. \\ Here, we define $T := P_C (\mathrm{Id} - \alpha \nabla f)$, where $\mathrm{Id}$ is the identity mapping on $H$,((It is defined by $\mathrm{Id}(x) := x$ $(x\in H)$.)) $P_C$ is the metric projection onto $C$, and $\alpha \in (0,2/L]$. Then, $T$ is nonexpansive and a fixed point of $T$ coincides with a minimizer of $f$ over $C$. ===== Conventional Fixed Point Algorithms ===== The following three algorithms are good ways to solve the fixed point problem. * The [[http://mi.mathnet.ru/eng/umn7954|Krasnosel'skii]]--[[http://dx.doi.org/10.1090/S0002-9939-1953-0054846-3|Mann]] Algorithm * The [[http://dx.doi.org/10.1090/S0002-9904-1967-11864-0|Halpern]] Algorithm * The [[http://dx.doi.org/10.1016/S0022-247X(02)00458-4|Hybrid]] Algorithm ===== Acceleration Algorithms for Solving Fixed Point Problems ===== ==== Acceleration of the Krasnosel'skii-Mann Algorithm ==== We numerically and theoretically compared our algorithm with the Krasnosel’skii-Mann algorithm and showed that it reduces the running time and iterations needed to find a fixed point compared with that algorithm. It is summarized as the following papers (You can get our papers from [[en:intro:publications|Publications]] page). * K. Fujiwara and [[:en:iiduka:|H. Iiduka]]: [[http://www.ybook.co.jp/online2/oplna/vol3/p189.html|Modification of the Krasnosel'skii-Mann Fixed Point Algorithm by Using Three-term Conjugate Gradients]], Linear and Nonlinear Analysis, Vol. 3, No. 2, pp.189-202, 2017. * [[:en:iiduka:|H. Iiduka]]: [[http://fixedpointtheoryandapplications.springeropen.com/articles/10.1186/s13663-016-0567-7| Line Search Fixed Point Algorithms Based on Nonlinear Conjugate Gradient Directions: Application to Constrained Smooth Convex Optimization]], Fixed Point Theory and Applications, Vol. 2016, No. 77, 2016. * K. Hishinuma and [[:en:iiduka:|H. Iiduka]]: [[http://www.ybook.co.jp/online-p/JNCA/Open/16/jncav16n11p2243-oa/FLASH/index.html|On Acceleration of the Krasnosel'skii-Mann Fixed Point Algorithm Based on Conjugate Gradient Method for Smooth Optimization]], 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. 2243-2254, 2015. ==== Acceleration of the Halpern Algorithm ==== We presented an algorithm to accelerate the Halpern algorithm and proved that, under certain assumptions, our algorithm strongly converges to a fixed point of $T$. We numerically compared our algorithm with the Halpern algorithm and showed that it reduces the running time and iterations needed to find a fixed point compared with that algorithm. It is summarized as the following paper (You can get our papers from [[en:intro:publications|Publications]] page). * [[en:kaito:|K. Sakurai]] and [[en:iiduka:|H. Iiduka]]: [[http://www.fixedpointtheoryandapplications.com/content/2014/1/202|Acceleration of the Halpern Algorithm to Search for a Fixed Point of a Nonexpansive Mapping]], Fixed Point Theory and Applications, Vol. 2014, 202, 2014. ==== Related Acceleration Methods ==== We have presented the following acceleration methods for convex optimization over the fixed point sets of nonexpansive mappings. If you are interested, please see the following papers (You can get our papers from [[en:intro:publications|Publications]] page). * [[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://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.