Showing papers on "Homotopy analysis method published in 1979"

Solving the Nonlinear Complementarity Problem by a Homotopy Method

Abstract: Let F be a $C^2 $ map from n-dimensional Euclidean space into itself. It is proved that, under some mild conditions on F, the complementarily problem $z \geqq 0$, $F(z) \geqq 0$, $zF(z) = 0$ can be solved by a homotopy algorithm developed by Chow, Mallet-Paret, Yorke, and Watson. The algorithm is globally convergent with probability one, and uses Mangasarian’s nonlinear system equivalent to the complementarity problem. Convergence theorems for the algorithm simultaneously prove existence of a solution, although existence is already well known. Some computational results are included.

Scalar labelings for homotopy paths

Abstract: Two scalar labelings are introduced for obtaining approximate solutions to systems of nonlinear equations by simplicial approximation. Under reasonable assumptions the new scalar-labeling algorithms are shown to follow, in a limiting sense, homotopy paths which can also be tracked by piecewise linear vector labeling algorithms. Though the new algorithms eliminate the need to pivot on a system of linear equations, the question of relative computational efficiency is unresolved.

A Homotopy based approach to unconstrained optimization

Abstract: The problem of finding a local minimum of a real differentiable function is considered from a homotopic point of view. Using a Davidenko embedding method with a particular homotopy, an ordinary differential equation is derived. Solution of this equation by Euler's rule gives rise to an iteration formula for the optimization problem. Convergence and termination properties of this formula are discussed.