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A numerical irreducible decomposition for a polynomial system provides representations for the irreducible factors of all positive dimensional solution sets of the system, separated from its isolated solutions. Homotopy continuation methods…
Many combinatorial optimization problems can be formulated as the search for a subgraph that satisfies certain properties and minimizes the total weight. We assume here that the vertices correspond to points in a metric space and can take…
Recent advancements in quantum computing and quantum-inspired algorithms have sparked renewed interest in binary optimization. These hardware and software innovations promise to revolutionize solution times for complex problems. In this…
We provide an algorithmic method for constructing projective resolutions of modules over quotients of path algebras. This algorithm is modified to construct minimal projective resolutions of linear modules over Koszul algebras.
A unified approach to parametrization of the mixing matrix for $N$ generations is developed. This approach not only has a clear geometrical underpinning but also has the advantage of being economical and recursive and leads in a natural way…
An algorithm and associated strategy for solving polynomial systems within the optimization framework is presented. The algorithm and strategy are named, respectively, the penetrating gradient algorithm and the deepest descent strategy. The…
In this paper, we propose a class of super-schemes for efficiently solving nonlinear unconstrained optimization problems. The proposed approach introduces two novel choices of step-size parameters, leading to efficient descent directions…
A special homotopy continuation method, as a combination of the polyhedral homotopy and the linear product homotopy, is proposed for computing all the isolated solutions to a special class of polynomial systems. The root number bound of…
For many linear and nonlinear systems that arise from the discretization of partial differential equations the construction of an efficient multigrid solver is a challenging task. Here we present a novel approach for the optimization of…
Given an approximation to a multiple isolated solution of a polynomial system of equations, we have provided a symbolic-numeric deflation algorithm to restore the quadratic convergence of Newton's method. Using first-order derivatives of…
The usual methods for root finding of polynomials are based on the iteration of a numerical formula for improvement of successive estimations. The unpredictable nature of the iterations prevents to search roots inside a pre-specified region…
In this paper, we tackle the parametric complete multiplicity problem for a univariate polynomial. Our approach to the parametric complete multiplicity problem has a significant difference from the classical method, which relies on repeated…
Numerical continuation methods track a solution path defined by a homotopy. The systems we consider are defined by polynomials in several variables with complex coefficients. For larger dimensions and degrees, the numerical conditioning…
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…
A new algorithm for deciding the satisfiability of polynomial formulas over the reals is proposed. The key point of the algorithm is a new projection operator, called sample-cell projection operator, custom-made for Conflict-Driven Clause…
We propose a new algorithm to the problem of polygonal curve approximation based on a multiresolution approach. This algorithm is suboptimal but still maintains some optimality between successive levels of resolution using dynamic…
The tensor rank decomposition, also known as canonical polyadic(CP) or simply tensor decomposition, has a long history in multilinear algebra. However, computing a rank decomposition becomes particularly challenging when the rank lies…
In this article, we describe an implementation of a polynomial system solver to compute the approximate solutions of a 0-dimensional polynomial system with finite precision p-adic arithmetic. We also describe an improvement to an algorithm…
In this paper, we propose an algebraic approach to upgrade a projective reconstruction to a Euclidean one, and aim at computing the rectifying homography from a minimal number of 9 segments of known length. Constraints are derived from…
We present a general method for constructing real solutions to some problems in enumerative geometry which gives lower bounds on the maximum number of real solutions. We apply this method to show that two new classes of enumerative…