Polynomial Homotopy Continuation Research Articles

A short proof for the parameter continuation theorem

The Parameter Continuation Theorem is the theoretical foundation for polynomial homotopy continuation, which is one of the main tools in computational algebraic geometry. In this note, we give a short proof using Gröbner bases. Our approach gives a method for computing discriminants.

Complete Solutions for the Approximate Synthesis of Spherical Four-Bar Function Generators

Abstract Kinematic synthesis to meet an approximate motion specification naturally forms a constrained optimization problem. Instead of using local methods, we conduct global design searches by direct computation of all critical points. The idea is not new, but performed at scale is only possible through modern polynomial homotopy continuation solvers. Such a complete computation finds all minima, including the global, which allows for a full exploration of the design space, whereas local solvers can only find the minimum nearest to an initial guess. We form equality-constrained objective functions that pertain to the synthesis of spherical four-bar linkages for approximate function generation. We consider the general case where all mechanism dimensions may vary and a more specific case that enables the placement of ground pivots. The former optimization problem is shown to permit 268 sets of critical points, and the latter permits 61 sets. Critical points are classified as saddles or minima through a post-process eigenanalysis of the projected Hessian. The discretization points are contained within the coefficients of the stationarity polynomials, so the algebraic structure of the synthesis equations remains invariant to the number of points. The results of our computational work were used to design a mechanism that coordinates the folding wings. We also use this method to parameterize mechanism dimensions for a family of hyperbolic curves.

Polynomial Systems, Homotopy Continuation, and Applications

Polynomial Systems, Homotopy Continuation, and Applications

Numerical Construction of Lyapunov Functions Using Homotopy Continuation Method

Lyapunov functions are frequently used for investigating the stability of linear and nonlinear dynamical systems. Though there is no general method of constructing these functions, many authors use polynomials in $ p-forms $ as candidates in constructing Lyapunov functions while others restrict the construction to quadratic forms. By focussing on the positive and negative definiteness of the Lyapunov candidate and the Hessian of its derivative, and using the sum of square decomposition, we developed a method for constructing polynomial Lyapunov functions that are not necessarily in a form. The idea of Newton polytope was used to transform the problem into a system of algebraic equations that were solved using the polynomial homotopy continuation method. Our method can produce several possibilities of Lyapunov functions for a given candidate. The sample test conducted demonstrates that the method developed is promising

Computing saddle graphs via homotopy continuation for the approximate synthesis of mechanisms

An approach for approximate kinematic synthesis of mechanisms is proposed in this paper which computes a graph that identifies minima of an objective function as vertices and connections between them as edges. Such a graph is interactively presented to a designer, whereby edges are continuously traversed to navigate families of design candidates in between minima. Candidates are evaluated continuously according to auxiliary considerations for the exploration of design trade-offs. Relevant design specifications tend to be particular per application and are either unclear as how to incorporate into an objective, or clear but with great consequence to the complexity of function evaluation. Computing the aforementioned graphs begins with finding all minima and saddles of an objective function through polynomial homotopy continuation. Connections between minima that minimize their maximum objective value must pass through a saddle to do so. Therefore, after gathering saddles, each is perturbed both ways in its least eigendirection to seed gradient descent paths which connect two minima when pieced together. Discovered connections between minima are organized into a graph, where edges correspond to gradient descent paths.

Mixed precision path tracking for polynomial homotopy continuation

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 that builds on a local understanding of the region of convergence of Newton’s method and the distance to the closest singularity following Telen, Van Barel, and Verschelde. To handle numerically challenging situations, the algorithm uses mixed precision arithmetic. The efficiency and robustness are demonstrated in several numerical examples.

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.

A continuation-based method for finding laminated composite stacking sequences

A method of recovering laminate ply stacking sequences from a set of up to twelve lamination parameters using polynomial homotopy continuation techniques is presented. The ply angles are treated as continuous variables, and are allowed to take any value between -90° and +90°. The individual plies are assumed to be orthotropic and have constant stiffness. The method is fully deterministic, and does not rely on an optimisation process to establish the stacking sequence. Polyhedral continuation methods are used to limit the solution space in which the stacking sequences are sought. The method can reliably find every stacking sequence solution that exists to achieve a precisely specified set of lamination parameter “targets”, with the number of real solutions to a feasible combination of target properties found to vary from 1 to over 100. The same method is also demonstrated to be able to find stacking sequences to satisfy a set of specified ABD stiffness matrix terms, as might be required following a direct-stiffness modelling design process.

A Robust Numerical Path Tracking Algorithm for Polynomial Homotopy Continuation

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 it with existing implementations.

Open quantum systems are harder to track than open classical systems

For a Markovian (in the strongest sense) open quantum system it is possible, by continuously monitoring the environment, to perfectly track the system; that is, to know the stochastically evolving pure state of the system without altering the master equation. In general, even for a system with a finite Hilbert space dimension D, the pure state trajectory will explore an infinite number of points in Hilbert space, meaning that the dimension K of the classical memory required for the tracking is infinite. However, Karasik and Wiseman [Phys. Rev. Lett., 106(2):020406, 2011] showed that tracking of a qubit (D=2) is always possible with a bit (K=2), and gave a heuristic argument implying that a finite K should be sufficient for any D, although beyond D=2 it would be necessary to have K>D. Our paper is concerned with rigorously investigating the relationship between D and Kmin, the smallest feasible K. We confirm the long-standing conjecture of Karasik and Wiseman that, for generic systems with D>2, Kmin>D, by a computational proof (via Hilbert Nullstellensatz certificates of infeasibility). That is, beyond D=2, D-dimensional open quantum systems are provably harder to track than D-dimensional open classical systems. We stress that this result allows complete freedom in choice of monitoring scheme, including adaptive monitoring which is, in general, necessary to implement a physically realizable ensemble (as it is known) of just K pure states. Moreover, we develop, and better justify, a new heuristic to guide our expectation of Kmin as a function of D, taking into account the number L of Lindblad operators as well as symmetries in the problem. The use of invariant subspace and Wigner symmetries (that we recently introduced elsewhere, [New J. Phys. https://doi.org/10.1088/1367-2630/ab14b2]) makes it tractable to conduct a numerical search, using the method of polynomial homotopy continuation, to find finite physically realizable ensembles in D=3. The results of this search support our heuristic. We thus have confidence in the most interesting feature of our heuristic: in the absence of symmetries, Kmin∼D2, implying a quadratic gap between the classical and quantum tracking problems. Explicit adaptive monitoring schemes that realize the discovered finite ensembles are obtained numerically, thus facilitating future experimental investigations.

Impact of vacuum stability constraints on the phenomenology of supersymmetric models

We present a fast and efficient method for studying vacuum stability constraints in multi-scalar theories beyond the Standard Model. This method is designed for a reliable use in large scale parameter scans. The minimization of the scalar potential is done with the well-known polynomial homotopy continuation, and the decay rate of a false vacuum in a multi-scalar theory is estimated by an exact solution of the bounce action in the one-field case. We compare to more precise calculations of the tunnelling path at the tree- and one-loop level and find good agreement for the resulting constraints on the parameter space. Numerical stability, runtime and reliability are significantly improved compared to approaches existing in the literature. This procedure is applied to several phenomenologically interesting benchmark scenarios defined in the Minimal Supersymmetric Standard Model. We utilize our efficient approach to study the impact of simultaneously varying multiple fields and illustrate the importance of correctly identifying the most dangerous minimum among the minima that are deeper than the electroweak vacuum.

Unified Selective Harmonic Elimination for Cascaded H-Bridge Asymmetric Multilevel Inverter

The unequal dc-link voltages in the cascaded H-bridge (CHB) asymmetric multilevel inverters (AMLIs) lead to asymmetry of the equations for selective harmonic elimination (SHE), which increases the difficulty in solving the switching angles. On the other hand, the unequal dc-link voltages provide a large amount of synthesized waveforms which can be used to optimize the performance of inverters. However, as most of the traditional SHE approaches are derived according to a given output waveform, it cannot handle multiple synthesized waveforms. In this paper, the unified SHE equations are derived for all the possible synthesized waveforms under a specific switching angles distribution among the cells. By using the polynomial homotopy continuation (PHC) algorithm, all the possible solutions of the unified SHE equations can be found without the selection of initial values. As the waveforms generated by H-bridge are limited to three levels, some of the solutions are not physically realizable and should be discarded, hence, the criterion is proposed to distinguish these specious solutions. The case of a CHB converter with two cells and six switching angles is studied; in total, there exist 86 groups of candidate solutions and 14 of them are physically realizable. Experiments are also shown to validate the correctness of this unified SHE approach for CHB AMLIs.

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.

The method of Gauss–Newton to compute power series solutions of polynomial homotopies

We consider the extension of the method of Gauss–Newton from complex floating-point arithmetic to the field of truncated power series with complex floating-point coefficients. With linearization we formulate a linear system where the coefficient matrix is a series with matrix coefficients, and provide a characterization for when the matrix series is regular based on the algebraic variety of an augmented system. The structure of the linear system leads to a block triangular system. In the regular case, solving the linear system is equivalent to solving a Hermite interpolation problem. We show that this solution has cost cubic in the problem size. In general, at singular points, we rely on methods of tropical algebraic geometry to compute Puiseux series. With a few illustrative examples, we demonstrate the application to polynomial homotopy continuation.

Determinantal representations of hyperbolic curves via polynomial homotopy continuation

A smooth curve in the real projective plane is hyperbolic if its ovals are maximally nested. By the Helton-Vinnikov Theorem, any such curve admits a definite symmetric determinantal representation. We use polynomial homotopy continuation to compute such representations numerically. Our method works by lifting paths from the space of hyperbolic polynomials to a branched cover in the space of pairs of symmetric matrices.

3D Indoor Building Environment Reconstruction using Least Square Adjustment, Polynomial Kernel, Interval Analysis and Homotopy Continuation

Abstract. Nowadays, municipalities intend to have 3D city models for facility management, disaster management and architectural planning. Indoor models can be reconstructed from construction plans but sometimes, they are not available or very often, they differ from ‘as-built’ plans. In this case, the buildings and their rooms must be surveyed. One of the most utilized methods of indoor surveying is laser scanning. The laser scanning method allows taking accurate and detailed measurements. However, Terrestrial Laser Scanner is costly and time consuming. In this paper, several techniques for indoor 3D building data acquisition have been investigated. For reducing the time and cost of indoor building data acquisition process, the Trimble LaserAce 1000 range finder is used. The proposed approache use relatively cheap equipment: a light Laser Rangefinder which appear to be feasible, but it needs to be tested to see if the observation accuracy is sufficient for the 3D building modelling. The accuracy of the rangefinder is evaluated and a simple spatial model is reconstructed from real data. This technique is rapid (it requires a shorter time as compared to others), but the results show inconsistencies in horizontal angles for short distances in indoor environments. The range finder horizontal angle sensor was calibrated using a least square adjustment algorithm, a polynomial kernel, interval analysis and homotopy continuation.

Numerical algebraic geometry for model selection and its application to the life sciences.

Researchers working with mathematical models are often confronted by the related problems of parameter estimation, model validation and model selection. These are all optimization problems, well known to be challenging due to nonlinearity, non-convexity and multiple local optima. Furthermore, the challenges are compounded when only partial data are available. Here, we consider polynomial models (e.g. mass-action chemical reaction networks at steady state) and describe a framework for their analysis based on optimization using numerical algebraic geometry. Specifically, we use probability-one polynomial homotopy continuation methods to compute all critical points of the objective function, then filter to recover the global optima. Our approach exploits the geometrical structures relating models and data, and we demonstrate its utility on examples from cell signalling, synthetic biology and epidemiology.

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