Related papers: Numerical Polynomial Homotopy Continuation Method …
Three aspects of applying homotopy continuation, which is commonly used to solve parameterized systems of polynomial equations, are investigated. First, for parameterized systems which are homogeneous, we investigate options for performing…
The Holomorphic Embedding Load-Flow Method (HELM) was recently introduced as a novel technique to constructively solve the power-flow equations in power grids, based on advanced complex analysis. In this paper, the theoretical foundations…
We consider the problem of tracking one solution path defined by a polynomial homotopy on a parallel shared memory computer. Our robust path tracker applies Newton's method on power series to locate the closest singular parameter value. On…
Polynomial systems occur in many fields of science and engineering. Polynomial homotopy continuation methods apply symbolic-numeric algorithms to solve polynomial systems. We describe the design and implementation of our web interface and…
Numerical homotopy continuation methods for three classes of polynomial systems are presented. For a generic instance of the class, every path leads to a solution and the homotopy is optimal. The counting of the roots mirrors the resolution…
The solution to an optimal power flow (OPF) problem provides a minimum cost operating point for an electric power system. The performance of OPF solution techniques strongly depends on the problem's feasible space. This paper presents an…
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…
We use our recent implementation of a certified homotopy tracking algorithm to search for start systems that minimize the average complexity of finding all roots of a regular system of polynomial equations. While finding optimal start…
In this paper has been considered probability-one global convergence of NFPH (Newton-Fixed Point Homotopy) algorithm for system of nonlinear equations and has been proposed a probability-one homotopy algorithm to solve a regularized…
The power flow equations, which relate power injections and voltage phasors, are at the heart of many electric power system computations. While Newton-based methods typically find the "high-voltage" solution to the power flow equations,…
In this article, we introduce a new homotopy function to trace the trajectory by applying modified homotopy continuation method for finding the solution of the linear complementarity problem. Earlier several authors attempted to propose…
Recent advances have shown that the circuit simulation algorithms that allow for solving highly nonlinear circuits of over one billion variables can be applicable to power system simulation and optimization problems through the use of an…
While automatically generated polynomial elimination templates have sparked great progress in the field of 3D computer vision, there remain many problems for which the degree of the constraints or the number of unknowns leads to…
Alternating-Current Optimal Power Flow (AC-OPF) is framed as a NP-hard non-convex optimization problem that solves for the most economical dispatch of grid generation given the AC-network and device constraints. Although there are no…
The recently proposed non-iterative load flow method, called the holomorphic embedding method, may encounter the precision issue, i.e. nontrivial round-off errors caused by the limit of digits used in computation when calculating the…
This paper proposes a convex optimization based method that either locates all real roots of a set of power flow equations or declares no real solution exists in the given area. In the proposed method, solving the power flow equations is…
Homotopy methods have proven to be a powerful tool for understanding the multitude of solutions provided by the coupled-cluster polynomial equations. This endeavor has been pioneered by quantum chemists that have undertaken both elaborate…
We revisit the problem of certifying the correctness of approximate solution paths computed by numerical homotopy continuation methods. We propose a conceptually simple approach based on a parametric variant of the Krawczyk method from…
We describe, for the first time, a completely rigorous homotopy (path--following) algorithm (in the Turing machine model) to find approximate zeros of systems of polynomial equations. If the coordinates of the input systems and the initial…
Developing efficient and guaranteed nonconvex algorithms has been an important challenge in modern machine learning. Algorithms with good empirical performance such as stochastic gradient descent often lack theoretical guarantees. In this…