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Title: Weak contraction mapping and optimization

Paper content:
The weak contraction mapping is a self mapping that the range is always a subset of the domain, which admits a unique fixed-point.
The iteration of weak contraction mapping is a Cauchy sequence that yields the unique fixed-point.
A gradient-free optimization method as an application of weak contraction mapping is proposed to achieve global minimum convergence.
The optimization method is robust to local minima and initial point position.
Many gradient-based optimization methods, such as gradient descent method, Newton's method and so on, face great challenges in finding the global minimum point of a function.
As is known, searching for the global minimum of a function with many local minima is difficult.
In principle, the information from the derivative of a single point is not sufficient for us to know the global geometry property of the function.
For a successful minimum point convergence, the initial point is required to be sufficiently good and the derivative calculation need to be accurate enough.
In the gradientbased methods, the domain of searching area will be divided into several subsets with regards to local minima.
And eventually it will converge to one local minimum depends on where the initial point locates at.Let (X,d) be a metric space and let T:X → X be a mapping.
For the inequality that, d(T
(x), T
(y)) ≤ qd(x,
y), ∀x, y ∈ X.(1)if
q ∈ [0, 1), T is called contractive; if q ∈ [0, 1], T is called nonexpansive; if q < ∞, T is called Lipschitz continuous(1; 2). The gradient-based
methods are usually nonexpansive mapping the solution exists but is not unique for general situation. For instance, if the
gradient descent method is written as a mapping T and the objective function has many local minima, then there are many fixed points accordingly. From the perspective
of spectra of bounded operator, for a nonexpansive mapping any minima of the objective function is an eigenvector of eigenvalue equation T (x) = λx ,in which λ
= 1. In the optimization
problem, nonexpansive mapping sometimes works but their disadvantages are obvious. Because both the existence
and uniqueness of solution are important so that the contractive mapping is more favored than the nonexpansive mapping(3; 4).Banach fixed-point theorem
is a very powerful method to solve linear or nonlinear system. But for optimization problems
, the condition of contraction mapping T : X → X that d(T (x), T (y)) ≤ qd(x, y) is usually
too strict
and luxury.
In the paper, we are trying to extend
the Banach fixedpoint theorem to an applicable method for optimization problem, which is called weak contraction mapping.In short, weak contraction mapping is a self mapping that always map to the subset of its domain. It is proven that weak contraction mapping
admits a fixed-point in the following section. How to apply the weak contraction mapping
to solve an optimization problem? Geometrically, given a point, we calculate
the height of this point and utilize a hyperplane at the same height to cut the objective function, where the intersection between the hyperplane and the objective function will form a contour or contours. And then map to a point insider a contour,
which the range of this mapping is always the subset of its domain. The iteration of the weak contraction mapping
yields a fixed-point, which coincides with the global minimum of the objective function.
The weak contraction mapping is a self mapping that always map to a subset of domain.
Intriguingly, as an extension of Banach fixed-point theorem, the iteration of weak contraction mapping is a Cauchy and yields a unique fixed-point, which fit perfectly with the task of optimization.
The global minimum convergence regardless of initial point position and local minima is very significant strength for optimization algorithm.
We hope that the advanced optimization with the development of the weak contraction mapping can contribute to empower the modern calculation.

assistant: text: A gradient-free method is proposed for non-convex optimization problem