Kantorovich Theorem

The Kantorovich theorem, or Newton–Kantorovich theorem, is a mathematical statement on the semi-local convergence of Newton's method. It was first stated by Leonid Kantorovich in 1948. It is similar to the form of the Banach fixed-point theorem, although it states existence and uniqueness of a zero rather than a fixed point.

Newton's method constructs a sequence of points that under certain conditions will converge to a solution $x$ of an equation $f(x)=0$ or a vector solution of a system of equation $F(x)=0$. The Kantorovich theorem gives conditions on the initial point of this sequence. If those conditions are satisfied then a solution exists close to the initial point and the sequence converges to that point.

Assumptions

Let $X\subset\R^n$ be an open subset and $F:X \subset \R^n \to\R^n$ a differentiable function with a Jacobian $F^(\mathbf x)$ that is locally Lipschitz continuous (for instance if $F$ is twice differentiable). That is, it is assumed that for any $x \in X$ there is an open subset $U\subset X$ such that $x \in U$ and there exists a constant $L>0$ such that for any $\mathbf x,\mathbf y\in U$

:$\, F'(\mathbf x)-F'(\mathbf y)\, \le L\;\, \mathbf x-\mathbf y\,$

holds. The norm on the left is some operator norm that is compatible with the vector norm on the right. This inequality can be rewritten to only use the vector norm. Then for any vector $\mathbf v\in\R^n$ the inequality

:$\, F'(\mathbf x)(\mathbf v)-F'(\mathbf y)(\mathbf v)\, \le L\;\, \mathbf x-\mathbf y\, \,\, \mathbf v\,$

must hold.

Now choose any initial point $\mathbf x_0\in X$. Assume that $F'(\mathbf x_0)$ is invertible and construct the Newton step $\mathbf h_0=-F'(\mathbf x_0)^F(\mathbf x_0).$

The next assumption is that not only the next point $\mathbf x_1=\mathbf x_0+\mathbf h_0$ but the entire ball $B(\mathbf x_1,\, \mathbf h_0\, )$ is contained inside the set $X$. Let $M$ be the Lipschitz constant for the Jacobian over this ball (assuming it exists).

As a last preparation, construct recursively, as long as it is possible, the sequences $(\mathbf x_k)_k$, $(\mathbf h_k)_k$, $(\alpha_k)_k$ according to
:\begin
\mathbf h_k&=-F'(\mathbf x_k)^F(\mathbf x_k)\\ .4em\alpha_k&=M\,\, F'(\mathbf x_k)^\, \,\, \mathbf h_k\, \\ .4em\mathbf x_&=\mathbf x_k+\mathbf h_k.
\end

Statement

Now if $\alpha_0\le\tfrac12$ then
#a solution $\mathbf x^*$ of $F(\mathbf x^*)=0$ exists inside the closed ball $\bar B(\mathbf x_1,\, \mathbf h_0\, )$ and
#the Newton iteration starting in $\mathbf x_0$ converges to $\mathbf x^*$ with at least linear order of convergence.

A statement that is more precise but slightly more difficult to prove uses the roots $t^\ast\le t^$ of the quadratic polynomial
:$p(t) =\left(\tfrac12L\, F'(\mathbf x_0)^\, ^\right)t^2 -t+\, \mathbf h_0\,$,
:$t^=\frac$
and their ratio
:$\theta =\frac =\frac.$
Then
#a solution $\mathbf x^*$ exists inside the closed ball $\bar B(\mathbf x_1,\theta\, \mathbf h_0\, )\subset\bar B(\mathbf x_0,t^*)$
#it is unique inside the bigger ball $B(\mathbf x_0,t^)$
#and the convergence to the solution of $F$ is dominated by the convergence of the Newton iteration of the quadratic polynomial $p(t)$ towards its smallest root $t^\ast$, if $t_0=0,\,t_=t_k-\tfrac$, then
#:$\, \mathbf x_-\mathbf x_k\, \le t_-t_k.$
#The quadratic convergence is obtained from the error estimate
#:$\, \mathbf x_-\mathbf x^*\, \le \theta^\, \mathbf x_-\mathbf x_n\, \le\frac\, \mathbf h_0\, .$

Corollary

In 1986, Yamamoto proved that the error evaluations of the Newton method such as Doring (1969), Ostrowski (1971, 1973), Gragg-Tapia (1974), Potra-Ptak (1980), Miel (1981), Potra (1984), can be derived from the Kantorovich theorem.

Generalizations

There is a ''q''-analog for the Kantorovich theorem. For other generalizations/variations, see Ortega & Rheinboldt (1970).

Applications

Oishi and Tanabe claimed that the Kantorovich theorem can be applied to obtain reliable solutions of linear programming.

References

Further reading

* John H. Hubbard and Barbara Burke Hubbard: ''Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach'', Matrix Editions,
preview of 3. edition and sample material including Kant.-thm.

* {{cite book , first=Tetsuro , last=Yamamoto , chapter=Historical Developments in Convergence Analysis for Newton's and Newton-like Methods , pages=241–263 , editor-first=C. , editor-last=Brezinski , editor2-first=L. , editor2-last=Wuytack , title=Numerical Analysis : Historical Developments in the 20th Century , publisher=North-Holland , year=2001 , isbn=0-444-50617-9

Functional analysis
Numerical analysis
Theorems in analysis
Optimization in vector spaces
Optimization algorithms and methods