Threading Homotopies and DC Operating Points of Nonlinear Circuits
Ross Geoghegan, Jeffrey C. Lagarias, Robert C. Melville · SIAM Journal on Optimization · 1998
This paper studies continuation methods for finding isolated zeros of nonlinear functions. Given a nonlinear function $F : {\Bbb R}^n \rightarrow {\Bbb R}^n$, a {\em threading homotopy} is a function $H( \mathbf{x} , \lambda ) : {\Bbb R}^{n+1} \rightarrow {\Bbb R}^n$ with $H( \mathbf{x} , 0) \equiv F( \mathbf{x} )$, such that the zero set of H is a single connected curve containing all zeros of $F( \mathbf{x} )$. For a $C^1$ function F, a necessary condition for the existence of a nondegenerate $C^1$ threading homotopy is that the topological degree of $F(\mathbf{x} )$ be 1, 0, or $-1$. For $C^2$ mappings in all dimensions, except possibly $n = 2$, this condition is also a sufficient condition for existence of a $C^2$ threading homotopy which is weakly proper over 0. A homotopy H is {\em weakly proper over }0 if, for every interval $[a,b]$, the set $H^{-1} ({\bf 0}) \cap ( {\Bbb R}^n \times [a,b])$ is compact. This condition rules out any part of the zero set escaping to infinity at a finite value of the homotopy parameter. Threading homotopies are potentially applicable in continuation methods for finding all dc operating points of nonlinear circuits. We show that most transistor circuits have dc operating point equations $F(\mathbf{x} ) = {\bf 0}$ with $\deg (F) = \pm 1$, so that threading homotopies exist in principle for such operating point equations. The explicit construction of such threading homotopies remains an open problem.