Homotopy methods. (English) Zbl 0847.65031

Forster, W.

Horst, Reiner et al., Handbook of global optimization. Dordrecht: Kluwer Academic Publishers. Nonconvex Optim. Appl. 2, 669-750 (1995).

This article surveys numerical homotopy methods up to ca 1994. It handles both the piecewise linear (simplicial) approach and the path following approach based on differential topology. The article has a discussion of applications, a history of the subject, and an extensive bibliography. It treats the relationship between classical fixed point theorems, antipodal point theorems and the combinatorical labelling lemmas such as those of Sperner and Kuhn. One of the applications discussed in detail is the solution of systems of polynomial equations in the complex context, e.g., Bezout’s theorem. A listing of available software is given at the conclusion. The article does not concentrate upon technical numerical issues or proofs.
For the entire collection see [Zbl 0805.00009].

Reviewer: E.L.Allgower (Fort Collins)

MSC:

65H20	Global methods, including homotopy approaches to the numerical solution of nonlinear equations
65H10	Numerical computation of solutions to systems of equations
65-02	Research exposition (monographs, survey articles) pertaining to numerical analysis

Keywords:

survey article; simplicial; numerical homotopy methods; piecewise linear; path following; differential topology; bibliography; fixed point theorems; antipodal point theorems; combinatorical labelling lemmas; systems of polynomial equations; Bezout’s theorem; software