On a coupled fixed point

Question

Let $X$ be a Banach space.

An element $(x^*,y^*)X\times X$ is called a coupled fixed point of a mapping $T : X ×X \rightarrow X$ if $$T(x^*,y^*) = x^*$$ and $$T(y^*,x^*) = y^*$$.

While studying this type of fixed point, this question has come to mind: why we don't transform this problem into a classical problem of fixed points, by putting $F(x,y)=(T(x,y),T(y,x))$, so we will have $$F(x^*,y^*)=(x^*,y^*)$$

So, why add a whole fixed point theory - coupled fixed point - , whereas it can be considered as a classic fixed point: $F(x,y)=(x,y)$

Answer 1

Perhaps there is some nice structure in the study of the coupled fixed points?

I'll use optimization as an example: Sure, we could say that all unconstrained optimization can be written as minimizing a single function $g$. However, often times we can exploit the structure of $g$ to get nicer methods for solving the problem. For instance, if $g$ is the sum of two functions, then we can activate each summand independently (via methods such as the forward-backward algorithm). This yields a lighter computational load while still solving the problem. In this way, we have used the underlying structure of $g$ which may not have been visible if we just used the standard theory.

EDIT: Continuing the example, suppose that $g=f+h$ is the sum of two convex lower-semicontinuous functions where $f$ has a Lipschitzian gradient and $h$ might not be differentiable. We also suppose that the proximity operator $\textrm{prox}_h$ can be computed. The proximity operator is a minimization tool that is often similar in cost to a gradient step, but it requires fewer assumptions on $h$. Now, if we want to minimize $g$, then in general $g$ will not be differentiable. Standard methods for just the problem of "minimize $g$" would dictate that we need either $\nabla g$ (which may not exist) or the ability to compute $\textrm{prox}_g$ (which is only rarely computable given only $\nabla f$ and $\textrm{prox}_{h}$).

Nonetheless, thanks to the structure, we can solve this problem using only activations of $\textrm{prox}_h$ and $\nabla f$ using the Forward-backward algorithm. Sweeping some details under the rug, the Forward-Backward algorithm allows us to solve the minimization task by iterating two steps: at every iteration $n\in\mathbb{N}$,

1. First, we perform a gradient step $y_n=x_n-\gamma\nabla f(x_n)$ (where $\gamma$ is a parameter).

2. Second, compute the next iterate by performing the proximal step $x_{n+1}=\textrm{prox}_h y_n$.

In addition to making this problem tractable, notice that we do not need to compute both $\nabla f$ and $\textrm{prox}_h$ at the same time. This allows us to save some storage space.