We introduce the concept of homotopy iterators for performing polynomial homotopy continuation tasks in a memory efficient manner. The main idea is to push forward an iterator for the start solutions of a homotopy via the function which tracks them along the homotopy. Doing so produces a representation of the target solutions, bypassing the need to hold all solutions in memory. We discuss several applications of this datatype ranging from solution counting to data compression.
We propose a new algorithm for numerical path tracking in polynomial homotopy continuation. The algorithm is `robust' in the sense that it is designed to prevent path jumping and in many cases, it can be used in (only) double precision arithmetic. It is based on an adaptive stepsize predictor that uses Padé techniques to detect local difficulties for function approximation and danger for path jumping. We show the potential of the new path tracking algorithm through several numerical examples and compare with existing implementations.
This article develops a new predictor-corrector algorithm for numerical path tracking in the context of polynomial homotopy continuation. In the corrector step it uses a newly developed Newton corrector algorithm which rejects an initial guess if it is not an approximate zero. The algorithm also uses an adaptive step size control which builds on a local understanding of the region of convergence of Newton's method and the distance to the closest singularity. To handle numerically challenging situations the algorithm uses mixed precision arithmetic. The efficiency and robustness are demonstrate…
A Special Homotopy Continuation Method For A Class of Polynomial Systems
LMAM & School of Mathematical Sciences, Peking University（北京大学数学科学学院）；Chongqing Inst. of Green and Intelligent Techn. Chinese Academy of Sciences（中国科学院重庆绿色智能技术研究院）
Galois groups of Schubert problems via homotopy computation
Institute for Mathematics；its Applications University of Minnesota Minneapolis MN \ 55455 USA；Department of Mathematics Texas A\&M University College Station Texas \ 77843 USA
Dipartimento di Matematica, Università di Trento（特伦托大学数学系）；Worcester State University（伍斯特州立大学）；Department of Applied and Computational Mathematics and Statistics, University of Notre Dame（圣母大学应用与计算数学及统计学系）；Inria Sophia Antipolis Méditerranée（索菲亚-安提波利斯地中海国家信息与自动化研究所）
In this paper we propose a method that uses Lagrange multipliers and numerical algebraic geometry to find all critical points, and therefore globally solve, polynomial optimization problems. We design a polyhedral homotopy algorithm that explicitly constructs an optimal start system, circumventing the typical bottleneck associated with polyhedral homotopy algorithms. The correctness of our algorithm follows from intersection theoretic computations of the algebraic degree of polynomial optimization programs and relies on explicitly solving the tropicalization of a corresponding Lagrange system.…
The generalized Nash equilibrium problem (GNEP) is a kind of game to find strategies for a group of players such that each player's objective function is optimized. Solutions for GNEPs are called generalized Nash equilibria (GNEs). In this paper, we propose a numerical method for finding GNEs of GNEPs of polynomials based on the polyhedral homotopy continuation and the Moment-SOS hierarchy of semidefinite relaxations. We show that our method can find all GNEs if they exist, or detect the nonexistence of GNEs, under some genericity assumptions. Some numerical experiments are made to demonstrate…
We design a homotopy continuation algorithm, that is based on numerically tracking Viro's patchworking method, for finding real zeros of sparse polynomial systems. The algorithm is targeted for polynomial systems with coefficients satisfying certain concavity conditions. It operates entirely over the real numbers and tracks the optimal number of solution paths. In more technical terms; we design an algorithm that correctly counts and finds the real zeros of polynomial systems that are located in the unbounded components of the complement of the underlying A-discriminant amoeba.
Locating Power Flow Solution Space Boundaries: A Numerical Polynomial Homotopy Approach
North Carolina State University（北卡罗来纳州立大学）；Schweitzer Engineering Laboratories（施维茨工程实验室）；University of Notre Dame（圣母大学）；University of Adelaide（阿德莱德大学）；United Technologies Research Center（联合技术研究中心）
