A Robust Numerical Path Tracking Algorithm for Polynomial Homotopy Continuation

Quality Up in Polynomial Homotopy Continuation by Multithreaded Path Tracking

Speedup measures how much faster we can solve the same problem using many cores. If we can afford to keep the execution time fixed, then quality up measures how much better the solution will be computed using many cores. In this paper we describe our multithreaded implementation to track one solution path defined by a polynomial homotopy. Limiting quality to accuracy and confusing accuracy with...

Numerical solution of multivariate polynomial systems by homotopy continuation methods

pn(xi,...,xn) = 0 for x = (x\,... ,xn). This problem is very common in many fields of science and engineering, such as formula construction, geometric intersection problems, inverse kinematics, power flow problems with PQ-specified bases, computation of equilibrium states, etc. Elimination theory-based methods, most notably the Buchberger algorithm (Buchberger 1985) for constructing Grobner bas...

Numerical Stablity of Path Tracing in Polyhedral Homotopy Continuation Methods

The reliability of polyhedral homotopy continuation methods for solving a polynomial system becomes increasingly important as the dimension of the polynomial system increases. High powers of the homotopy continuation parameter t and ill-conditioned Jacobian matrices encountered in tracing of homotopy paths affect the numerical stability. We present modified homotopy functions with a new homotop...

Outlier Path: A Homotopy Algorithm for Robust SVM

In recent applications with massive but less reliable data (e.g., labels obtained by a semisupervised learning method or crowdsourcing), non-robustness of the support vector machine (SVM) often causes considerable performance deterioration. Although improving the robustness of SVM has been investigated for long time, robust SVM (RSVM) learning still poses two major challenges: obtaining a good ...

A review on Homotopy Continuation Methods for Polynomial Systems