Numerical Polynomial Homotopy Continuation Research Articles

Solution to the Economic Emission Dispatch Problem Using Numerical Polynomial Homotopy Continuation

The economic emission dispatch (EED) is a highly constrained nonlinear multiobjective optimization problem with a convex (or nonconvex) solution space. These characteristics and constraints make the EED a difficult problem to solve. Several approaches for a solution have been proposed, such as deterministic techniques, stochastic techniques, or a combination of both. This work presents the use of an algebraic (deterministic) technique, the numerical polynomial homotopy continuation (NPHC) method, to solve the EED problem. A comparison with the sequential quadratic programming (SQP) algorithm and the nondominated sorting genetic algorithm II (NSGA-II) is also presented. Results show that the NPHC algorithm finds all the roots (solutions) of the problem starting from any initial point and assures an accurate solution with a good convergence time. In addition, the NPHC algorithm provides a more accurate solution than the SQP algorithm and the NSGA-II.

Optimal Configurations in Coverage Control with Polynomial Costs

We revisit the static coverage control problem for placement of vehicles with simple motion on the real line, under the assumption that the cost is a polynomial function of the locations of the vehicles. The main contribution of this paper is to demonstrate the use of tools from numerical algebraic geometry, in particular, a numerical polynomial homotopy continuation method that guarantees to find all solutions of polynomial equations, in order to characterize the global minima for the coverage control problem. The results are then compared against a classic distributed approach involving the use of Lloyd descent, which is known to converge only to a local minimum under certain technical conditions.

Computing the Feasible Spaces of Optimal Power Flow Problems

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 algorithm that is guaranteed to compute the entire feasible spaces of small OPF problems to within a specified discretization tolerance. Specifically, the feasible space is computed by discretizing certain of the OPF problem's inequality constraints to obtain a set of power flow equations. All solutions to the power flow equations at each discretization point are obtained using the Numerical Polynomial Homotopy Continuation algorithm. To improve computational tractability, “bound tightening” and “grid pruning” algorithms use convex relaxations to preclude consideration of many discretization points that are infeasible for the OPF problem. The proposed algorithm is used to generate the feasible spaces of two small test cases.

Fixed Points of Belief Propagation-An Analysis via Polynomial Homotopy Continuation.

Belief propagation (BP) is an iterative method to perform approximate inference on arbitrary graphical models. Whether BP converges and if the solution is a unique fixed point depends on both the structure and the parametrization of the model. To understand this dependence it is interesting to find all fixed points. In this work, we formulate a set of polynomial equations, the solutions of which correspond to BP fixed points. To solve such a nonlinear system we present the numerical polynomial-homotopy-continuation (NPHC) method. Experiments on binary Ising models and on error-correcting codes show how our method is capable of obtaining all BP fixed points. On Ising models with fixed parameters we show how the structure influences both the number of fixed points and the convergence properties. We further asses the accuracy of the marginals and weighted combinations thereof. Weighting marginals with their respective partition function increases the accuracy in all experiments. Contrary to the conjecture that uniqueness of BP fixed points implies convergence, we find graphs for which BP fails to converge, even though a unique fixed point exists. Moreover, we show that this fixed point gives a good approximation, and the NPHC method is able to obtain this fixed point.

Numerical polynomial homotopy continuation method to locate all the power flow solutions

The manuscript addresses the problem of finding all solutions of power flow equations or other similar non-linear system of algebraic equations. This problem arises naturally in a number of power systems contexts, most importantly the direct methods for transient stability analysis and voltage stability assessment. Here, the authors introduce a novel form of homotopy continuation method called the numerical polynomial homotopy continuation method that is mathematically guaranteed to find all the solutions without ever encountering a bifurcation. Since finding real solutions is much more challenging, first the authors embed the real form of power flow equation in complex space, and then track the generally unphysical solutions with complex values of real and imaginary parts of the voltages. The solutions converge to physical real form in the end of the homotopy. The so-called gamma-trick mathematically rigorously ensures that all the paths are well-behaved along the paths, so unlike other continuation approaches, no special handling of bifurcations is necessary. The method is embarrassingly parallelisable. The authors demonstrate the technique performance by solving several test cases up to the 14 buses. Finally, they discuss possible strategies for scaling the method to large size systems, and propose several applications for security assessments.

Reliable mixture critical point computation using polynomial homotopy continuation

The numerical computation of multicomponent mixture critical points has been the subject of much study due to their theoretical and practical importance. Both deterministic and stochastic methods have been applied with varying degrees of reliability and robustness. In this work, we utilize numerical polynomial homotopy continuation (NPHC) to reliably identify all mixture critical points. This method is unique due to its robustness, initialization‐free nature and ease of parallelization. For a given system of equations, all complex solutions are found. Computational times are also found to be invariant to mixture composition. We validate this technique against previous work and extend the method to mixtures of up to eight components. NPHC is shown to be a modern and powerful technique which offers mathematical reliability at a moderate computational cost. © 2016 American Institute of Chemical Engineers AIChE J, 62: 4497–4507, 2016

Statistics of stationary points of random finite polynomial potentials

.The stationary points (SPs) of the potential energy landscapes (PELs) of multivariate random potentials (RPs) have found many applications in many areas of Physics, Chemistry and Mathematical Biology. However, there are few reliable methods available which can find all the SPs accurately. Hence, one has to rely on indirect methods such as Random Matrix theory. With a combination of the numerical polynomial homotopy continuation method and a certification method, we obtain all the certified SPs of the most general polynomial RP for each sample chosen from the Gaussian distribution with mean 0 and variance 1. While obtaining many novel results for the finite size case of the RP, we also discuss the implications of our results on mathematics of random systems and string theory landscapes.

Experiments on the Zeros of Harmonic Polynomials Using Certified Counting

Motivated by Wilmshurst’s conjecture, we investigate the zeros of harmonic polynomials. We utilize a certified counting approach, which is a combination of two methods from numerical algebraic geometry: numerical polynomial homotopy continuation to compute a numerical approximation of each zero and Smale’s alpha theory to certify the results. We provide new examples of harmonic polynomials having the most extreme number of zeros known so far; we also study the mean and variance of the number of zeros of random harmonic polynomials.

Energy landscape of the finite-size mean-field 2-spin spherical model and topology trivialization.

Motivated by the recently observed phenomenon of topology trivialization of potential energy landscapes (PELs) for several statistical mechanics models, we perform a numerical study of the finite-size 2-spin spherical model using both numerical polynomial homotopy continuation and a reformulation via non-Hermitian matrices. The continuation approach computes all of the complex stationary points of this model while the matrix approach computes the real stationary points. Using these methods, we compute the average number of stationary points while changing the topology of the PEL as well as the variance. Histograms of these stationary points are presented along with an analysis regarding the complex stationary points. This work connects topology trivialization to two different branches of mathematics: algebraic geometry and catastrophe theory, which is fertile ground for further interdisciplinary research.

Energy landscape of the finite-size spherical three-spin glass model

We study the three-spin spherical model with mean-field interactions and Gaussian random couplings. For moderate system sizes of up to 20 spins, we obtain all stationary points of the energy landscape by means of the numerical polynomial homotopy continuation method. On the basis of these stationary points, we analyze the complexity and other quantities related to the glass transition of the model and compare these finite-system quantities to their exact counterparts in the thermodynamic limit.

Enumerating Gribov copies on the lattice

In the modern formulation of lattice gauge fixing, the gauge-fixing condition is written in terms of the minima or stationary points (collectively called solutions) of a gauge-fixing functional. Due to the non-linearity of this functional, it usually has many solutions, called Gribov copies. The dependence of the number of Gribov copies, n[U], on the different gauge orbits plays an important role in constructing the Faddeev–Popov procedure and hence in realising the BRST symmetry on the lattice. Here, we initiate a study of counting n[U] for different orbits using three complimentary methods: (1) analytical results in lower dimensions, and some lower bounds on n[U] in higher dimensions, (2) the numerical polynomial homotopy continuation method, which numerically finds all Gribov copies for a given orbit for small lattices, and (3) numerical minimisation (“brute force”), which finds many distinct Gribov copies, but not necessarily all. Because n for the coset SU(Nc)/U(1) of an SU(Nc) theory is orbit independent, we concentrate on the residual compact U(1) case in this article, and establish that n is orbit dependent for the minimal lattice Landau gauge and orbit independent for the absolute lattice Landau gauge. We also observe that, contrary to a previous claim, n is not exponentially suppressed for the recently proposed stereographic lattice Landau gauge compared to the naive gauge in more than one dimension.

Minimizing Higgs potentials via numerical polynomial homotopy continuation

The study of models with extended Higgs sectors requires to minimize the corresponding Higgs potentials, which is in general very difficult. Here, we apply a recently developed method, called numerical polynomial homotopy continuation (NPHC), which guarantees to find all the stationary points of the Higgs potentials with polynomial-like non-linearity. The detection of all stationary points reveals the structure of the potential with maxima, metastable minima, saddle points besides the global minimum. We apply the NPHC method to the most general Higgs potential having two complex Higgs-boson doublets and up to five real Higgs-boson singlets. Moreover the method is applicable to even more involved potentials. Hence the NPHC method allows to go far beyond the limits of the Gröbner basis approach.

Numerical Polynomial Homotopy Continuation Method and String Vacua

Finding vacua for the four-dimensional effective theories for supergravity which descend from flux compactifications and analyzing them according to their stability is one of the central problems in string phenomenology. Except for some simple toy models, it is, however, difficult to find all the vacua analytically. Recently developed algorithmic methods based on symbolic computer algebra can be of great help in the more realistic models. However, they suffer from serious algorithmic complexities and are limited to small system sizes. In this paper, we review a numerical method called the numerical polynomial homotopy continuation (NPHC) method, first used in the areas of lattice field theories, which by construction findsallof the vacua of a given potential that is known to have only isolated solutions. The NPHC method is known to suffer from no major algorithmic complexities and isembarrassingly parallelizable, and hence its applicability goes way beyond the existing symbolic methods. We first solve a simple toy model as a warm-up example to demonstrate the NPHC method at work. We then show that all the vacua of a more complicated model of a compactified M theory model, which has anSU(3)structure, can be obtained by using a desktop machine in just about an hour, a feat which was reported to be prohibitively difficult by the existing symbolic methods. Finally, we compare the various technicalities between the two methods.

Numerical algebraic geometry

Numerical algebraic geometry uses numerical data to describe algebraic varieties. It is based on numerical polynomial homotopy continuation, which is an alternative to the classical symbolic approaches of computational algebraic geometry. We present a package, whose primary purpose is to interlink the existing symbolic methods of Macaulay2 and the powerful engine of numerical approximate computations. The core procedures of the package exhibit performance competitive with the other homotopy continuation software. INTRODUCTION. Numerical algebraic geometry [SVW, SW] is a relatively young subarea of computational algebraic geometry that originated as a blend of the well-understood apparatus of classical algebraic geometry over the field of complex numbers and numerical polynomial homotopy continuation methods. Recently steps have been made to extend the reach of numerical algorithms making it possible not only for complex algebraic varieties, but also for schemes, to be represented numerically. What we present here is a description of “stage one” of a comprehensive project that will make the machinery of numerical algebraic geometry available to the wide community of users of Macaulay2 [M2], a dominantly symbolic computer algebra system. Our open-source package [L1, M2] was first released in Macaulay2 distribution version 1.3.1. Stage one has been limited to implementation of algorithms that solve the most basic problem: Given polynomials f1, . . . , fn ∈ C[x1, . . . ,xn] that generate a 0-dimensional ideal I = ( f1, . . . , fn), find numerical approximations of all points of the underlying variety V (I) = {x ∈ Cn | f(x) = 0}. This task is accomplished by using homotopy continuation. To solve a target polynomial system f =( f1, . . . , fn)= 0, one constructs a start polynomial system g=(g1, . . . ,gn) with a similar structure, but readily available solutions. One option for a start system is g = (x f1) 1 − 1, . . . ,x deg( fn) n − 1). Given such an auxiliary system, we consider the homotopy (♠) h= (1− t)g+ γ tf ∈ C[x, t], γ ∈ C∗ . By specializing the values of t to the real interval [0,1], we obtain a collection of paths leading from the known solutions of the start system g = h|t=0 to the unknown solutions of the target system f = h|t=1. The following statement provides the key to making effective numerical computations. Theorem. For all but finitely many values of γ in the homotopy (♠), the homotopy continuation paths have no singularities with a possible exception of the endpoints corresponding to t = 1. Moreover, every solution of the target system, provided there are finitely many, is an endpoint (at t = 1) of some continuation path. 2010 Mathematics Subject Classification. 14Q99; 68N01. NumericalAlgebraicGeometry version 1.3.1. 5 Leykin :::: NumericalAlgebraicGeometry 6