# 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 fixed point theorems have been studied by renowned mathematicians, such as Banach, Brouwer, Caristi, Fan, Kakutani, Kirk, Schauder, Takahashi, and so on. We give the following two examples of the fixed point problem.

- Convex Feasibility Problem:

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 signal recovery problem.

Let us define $T := P_1 P_2 \cdots P_I$^{1)}, where $P_i$ $(i\in \mathcal{I})$ stands for the metric projection onto $C_i$.^{2)}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 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 $H$,^{3)}$P_C$ 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 Krasnosel'skii–Mann Algorithm
- The Halpern Algorithm
- The 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 Publications page).

- H. Iiduka: 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 H. Iiduka: 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 Publications page).

- K. Sakurai and H. Iiduka: 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 Publications page).

- H. Iiduka: Acceleration Method for Convex Optimization over the Fixed Point Set of a Nonexpansive Mapping, Mathematical Programming, Vol. 149, No. 1, pp. 131-165, 2015.
- H. Iiduka: 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.
- H. Iiduka and I. Yamada: 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.

^{1)}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 $\sum_{i\in\mathcal{I}} w_i =1$.

^{2)}$P_i$ 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.

^{3)}It is defined by $\mathrm{Id}(x) := x$ $(x\in H)$.