Homotopy Path Research Articles

A constrained homotopy technique for excluding unwanted solutions from polynomial equations arising in kinematics problems

This paper presents a constrained technique which enables classical homotopy continuation solvers to exclude unwanted solutions from a system of polynomials. It is well known that extraneous and degenerate solutions often appear in polynomial equations derived from kinematics problems. These solutions usually do not have physical meanings and prevent us from understanding the intrinsic feature of the studied problem. Traditionally to exclude these solutions, we have to rely on a tedious post-processing technique which often suffers from numerical instability especially when the number of unwanted solutions is large. In the proposed constrained homotopy technique, the unwanted solutions are mapped to the solutions at infinity of a higher dimensional system. We show that the extra polynomials do not increase the homotopy path count when they are used with a classical homotopy solver. This convenient and user friendly technique does not require any changes to the homotopy codes or extra steps to post-process out unwanted solutions. Compared with the regeneration procedure, our approach offers more freedom in eliminating positive dimensional solutions. In addition, our numerical experiments show that the solutions obtained with the constrained technique seem to have a better accuracy than the post processing approach. Kinematics analysis and synthesis problems are employed to demonstrate that the technique is capable of eliminating unwanted isolated and positive dimensional solutions. In the revisit of the classical nine-point path generation of four-bar, we verify that the maximum number of distinct mechanisms is 1442 triplets.

Convergence of the homotopy path for a full-Newton step infeasible interior-point method

Roos [C. Roos, A full-Newton step O ( n ) infeasible interior-point algorithm for linear optimization. SIAM J. Optim. 16 (4) (2006) 1110–1136 (electronic)] proposed a new primal–dual infeasible interior-point method for linear optimization. This new method can be viewed as a homotopy method. In this work, we show that the homotopy path has precisely one accumulation point in the optimal set. Moreover, this accumulation point is the analytic center of a subset of the optimal set and depends on the starting point of the infeasible interior-point method.

Application of numerical continuation to compute all solutions of semilinear elliptic equations

Abstract We adapt numerical continuation methods to compute all solutions of finite difference discretizations of nonlinear boundary value problems involving the Laplacian in two dimensions. New solutions on finer meshes are obtained from solutions on coarser meshes using a complex homotopy deformation. Two difficulties arise. First, the number of solutions typically grows with the number of mesh points and some form of filtering becomes necessary. Secondly, bifurcations may occur along homotopy paths of solutions and efficient methods to swap branches are developed when the mappings are analytic. For polynomial nonlinearities we generalize an earlier strategy for finding all solutions of two-point boundary value problems in one dimension and then introduce exclusion algorithms to extend the method to general nonlinearities.

Solving generalized Nash equilibrium problem with equality and inequality constraints

In this paper, a homotopy mapping for a generalized Nash equilibrium problem (GNEP) with equality and inequality constraints is constructed. From proving the existence of a homotopy path, a globally convergent homotopy method is developed to find a solution to the GNEP. Several numerical examples are presented.

Path Deformation Roadmaps: Compact Graphs with Useful Cycles for Motion Planning

In this paper we describe a new approach to sampling-based motion planning with Probabilistic Roadmap Planner (PRM) methods. Our aim is to compute good quality roadmaps which encode the multiple connectedness of the configuration space inside small but yet representative graphs which capture the different varieties of free paths well. The proposed Path Deformation Roadmaps (PDRs) rely on a notion of path deformability indicating whether or not a given path can be continuously deformed into another existing path. By considering a simpler form of deformation than that allowed between homotopic paths, we propose a method that extends the Visibility-PRM technique to construct compact roadmaps which encode richer and more suitable information than representative paths of the homotopy classes. PDRs contain additional useful cycles between paths in the same homotopy class that can be hardly deformed into each other. Experimental results show that the technique enables small roadmaps to reliably capture the multiple connectedness of complex spaces in various problems involving free-flying and articulated robots in both two- and three-dimensional environments.

A double homotopy method for decentralised control design

An iterative algorithm that is based on the idea of two homotopy paths is proposed for output feedback decentralised ℋ∞ control design. The approach follows a bilinear matrix inequality (BMI) formulation of the decentralised control problem. Along one homotopy path the BMI problem, which is non-convex, is locally linearised and solved. Along the second homotopy path the controller is deformed at each step so that in the end it attains a decentralised structure. The proposed computational algorithm can also be applied to obtain reduced-order decentralised controllers. Numerical examples are used to demonstrate the efficacy of the proposed algorithm.

A new homotopy method for solving non-linear complementarity problems

In this article, a new homotopy method is proposed for solving the non-linear complementarity problem. An existence condition for a finite homotopy path is derived from the limiting behaviours of the complementarity mapping at infinity. This condition is different from any existing ones in the literature and can be easily verified. The method guarantees the existence of a solution to the non-linear complementarity problem without the mapping being P 0 or monotone. Numerical results show that the method is effective and efficient.

A quantum Goldman bracket in (2 + 1) quantum gravity**Dedicated to John Charap, in gratitude for his inspiration and vision.

In the context of quantum gravity for spacetimes of dimension (2 + 1), we describe progress in the construction of a quantum Goldman bracket for intersecting loops on surfaces. Using piecewise linear paths in (representing loops on the spatial manifold, i.e. the torus) and a quantum connection with noncommuting components, we review how holonomies and Wilson loops for two homotopic paths are related by phases in terms of the signed area between them. Paths rerouted at intersection points with other paths occur on the rhs of the Goldman bracket. To better understand their nature we introduce the concept of integer points inside the parallelogram spanned by two intersecting paths, and show that the rerouted paths must necessarily pass through these integer points.

A globally convergent interior point algorithm for non-convex nonlinear programming

In this paper, a new algorithm for tracing the combined homotopy path of the non-convex nonlinear programming problem is proposed. The algorithm is based on the techniques of β -cone neighborhood and a combined homotopy interior point method. The residual control criteria, which ensures that the obtained iterative points are interior points, is given by the condition that ensures the β -cone neighborhood to be included in the interior part of the feasible region. The global convergence and polynomial complexity are established under some hypotheses.

Modified bounded homotopies to enable a narrow bounding zone

The aim of this paper is to introduce modifications that enhance the usability of the bounded homotopy methods proposed by Paloschi [1995. Bounded homotopies to solve systems of algebraic nonlinear equations. Computers and Chemical Engineering 19, 1243–1254; 1997. Bounded homotopies to solve systems of sparse algebraic nonlinear equations. Computers and Chemical Engineering 21, 531–541], especially in the area of chemical engineering. In modified bounded homotopies, the homotopy path is tracked by exploiting mapped variables instead of unmapped ones. Path tracking based on mapped variables makes it significantly easier to track the bounded homotopy path even though the bounding zone has to be narrow. Mapping also improves the bounding effect of bounded homotopies and makes it possible to avoid unreasonable variable values in homotopy path tracking. The performance of the modifications is illustrated with test cases. These examples clearly show that the modifications enlarge the capability and accuracy of bounded homotopies when solving both small- and large-scale sets of nonlinear equations describing chemical engineering problems.

Solving the Karush–Kuhn–Tucker system of a nonconvex programming problem on an unbounded set

In the papers [G.C. Feng, B. Yu, Combined homotopy interior point method for nonlinear programming problems, in: H. Fujita, M. Yamaguti (Eds.), Advances in Numerical Mathematics; Proceedings of the Second Japan–China Seminar on Numerical Mathematics, in: Lecture Notes in Numerical and Applied Analysis, vol. 14, Kinokuniya, Tokyo, 1995, pp. 9–16; G.C. Feng, Z.H. Lin, B. Yu, Existence of an interior pathway to a Karush–Kuhn–Tucker point of a nonconvex programming problem, Nonlinear Analysis 32 (1998) 761–768; Z.H. Lin, B. Yu, G.C. Feng, A combined homotopy interior point method for convex programming problem, Applied Mathematics and Computation 84 (1997) 193–211], a combined homotopy interior method was presented and global convergence results obtained for nonconvex nonlinear programming when the feasible set is bounded and satisfies the so called normal cone condition. However, for when the feasible set is not bounded, no result has so far been obtained. In this paper, a combined homotopy interior method for nonconvex programming problems on the unbounded feasible set is considered. Under suitable additional assumptions, boundedness of the homotopy path, and hence global convergence, is proven.

Complexity of Bezout’s Theorem VI: Geodesics in the Condition (Number) Metric

This paper investigates the cost of solving systems of sparse polynomial equations by homotopy continuation. First, a space of systems of $n$-variate polynomial equations is specified through $n$ monomial bases. The natural locus for the roots of those systems is known to be a certain toric variety. This variety is a compactification of $(\mathbb C\setminus\{0\})^n$, dependent on the monomial bases. A toric Newton operator is defined on that toric variety. Smale's alpha theory is generalized to provide criteria of quadratic convergence. Two condition numbers are defined and a higher derivative estimate is obtained in this setting. The Newton operator and related condition numbers turn out to be invariant through a group action related to the momentum map. A homotopy algorithm is given, and is proved to terminate after a number of Newton steps which is linear on the condition length of the lifted homotopy path. This generalizes a result from Shub (2009).

String Tightening as a Self-Organizing Phenomenon

The phenomenon of self-organization has been of special interest to the neural network community throughout the last couple of decades. In this paper, we study a variant of the self-organizing map (SOM) that models the phenomenon of self-organization of the particles forming a string when the string is tightened from one or both of its ends. The proposed variant, called the string tightening self-organizing neural network (STON), can be used to solve certain practical problems, such as computation of shortest homotopic paths, smoothing paths to avoid sharp turns, computation of convex hull, etc. These problems are of considerable interest in computational geometry, robotics path-planning, artificial intelligence (AI) (diagrammatic reasoning), very large scale integration (VLSI) routing, and geographical information systems. Given a set of obstacles and a string with two fixed terminal points in a 2-D space, the STON model continuously tightens the given string until the unique shortest configuration in terms of the Euclidean metric is reached. The STON minimizes the total length of a string on convergence by dynamically creating and selecting feature vectors in a competitive manner. Proof of correctness of this anytime algorithm and experimental results obtained by its deployment have been presented in the paper.

Condition for global convergence of a homotopy method for variational inequality problems on unbounded sets

A condition for global convergence of a homotopy method for a variational inequality problem (VIP) on an unbounded set is introduced. The condition is derived from a concept of a solution at infinity to VIP. By an argument of the existence of a homotopy path, we show that VIP has a solution if it has no solution at infinity. It is proved that if any of several well-known conditions given in the literature holds, there is no solution at infinity. Furthermore, a globally convergent homotopy method is developed to compute a solution to VIP. Several numerical examples illustrate how to follow the homotopy path starting at an arbitrary point in the unbounded set.

Algorithm 857

Globally convergent, probability-one homotopy methods have proven to be very effective for finding all the isolated solutions to polynomial systems of equations. After many years of development, homotopy path trackers based on probability-one homotopy methods are reliable and fast. Now, theoretical advances reducing the number of homotopy paths that must be tracked and handling singular solutions have made probability-one homotopy methods even more practical. POLSYS_GLP consists of Fortran 95 modules for finding all isolated solutions of a complex coefficient polynomial system of equations. The package is intended to be used on a distributed memory multiprocessor in conjunction with HOMPACK90 (Algorithm 777), and makes extensive use of Fortran 95-derived data types and MPI to support a general linear product (GLP) polynomial system structure. GLP structure is intermediate between the partitioned linear product structure used by POLSYS_PLP (Algorithm 801) and the BKK-based structure used by PHCPACK. The code requires a GLP structure as input, and although finding the optimal GLP structure is a difficult combinatorial problem, generally physical or engineering intuition about a problem yields a very good GLP structure. POLSYS_GLP employs a sophisticated power series end game for handling singular solutions, and provides support for problem definition both at a high level and via hand-crafted code. Different GLP structures and their corresponding Bezout numbers can be systematically explored before committing to root finding.

Analysis of Reactance Oscillators Having Multi-Mode Oscillations

We consider oscillators consisting of a reactance circuit and a negative resistor. They may happen to have multi-mode oscillations around the anti-resonant frequencies of the reactance circuit. This kind of oscillators can be easily synthesized by setting the resonant and anti-resonant frequencies of the reactance circuits. However, it is not easy to analyze the oscillation phenomena, because they have multiple oscillations whose oscillations depend on the initial guesses. In this paper, we propose a Spice-oriented solution algorithm combining the harmonic balance method with Newton homotopy method that can find out the multiple solutions on the homotopy paths. In our analysis, the determining equations from the harmonic balance method are given by modified equivalent circuit models of DC, Cosine and Sine circuits. The modified circuits can be solved by a simulator STC (solution curve tracing circuit), where the multiple oscillations are found by the transient analysis of Spice. Thus, we need not to derive the troublesome circuit equations, nor the mathematical transformations to get the determining equations. It makes the solution algorithms much simpler.

On the complexity of a combined homotopy interior method for convex programming

In [G.C. Feng, Z.H. Lin, B. Yu, Existence of an interior pathway to a Karush–Kuhn–Tucker point of a nonconvex programming problem, Nonlinear Anal. 32 (1998) 761–768; G.C. Feng, B. Yu, Combined homotopy interior point method for nonlinear programming problems, in: H. Fujita, M. Yamaguti (Eds.), Advances in Numerical Mathematics, Proceedings of the Second Japan–China Seminar on Numerical Mathematics, Lecture Notes in Numerical and Applied Analysis, vol. 14, Kinokuniya, Tokyo, 1995, pp. 9–16; Z.H. Lin, B. Yu, G.C. Feng, A combined homotopy interior point method for convex programming problem, Appl. Math. Comput. 84 (1997) 193–211.], a combined homotopy was constructed for solving non-convex programming and convex programming with weaker conditions, without assuming the logarithmic barrier function to be strictly convex and the solution set to be bounded. It was proven that a smooth interior path from an interior point of the feasible set to a K–K–T point of the problem exists. This shows that combined homotopy interior point methods can solve the problem that commonly used interior point methods cannot solve. However, so far, there is no result on its complexity, even for linear programming. The main difficulty is that the objective function is not monotonically decreasing on the combined homotopy path. In this paper, by taking a piecewise technique, under commonly used conditions, polynomiality of a combined homotopy interior point method is given for convex nonlinear programming.

A combined homotopy interior point method for the linear complementarity problem

A combined homotopy interior point method for the linear complementarity problem (LCP) is proposed in this paper. We construct a homotopy path for the linear complementarity problem and prove that the homotopy path determines a smooth interior path from an arbitrary given interior point to the solution of LCP. Furthermore, we modify a predictor corrector algorithm with the technique of the cone neighborhood to trace the homotopy path. Finally, a numerical example is provided to show the effectiveness and feasibility of this method.

Delivering global DC convergence for large mixed-signal circuits via homotopy/continuation methods

Homotopy/continuation methods are attractive for finding dc operating points of circuits because they offer theoretical guarantees of global convergence. Existing homotopy approaches for circuits are, however, often ineffective for large mixed-signal applications. In this paper, we describe a robust homotopy technique that is effective for solving large metal-oxide-semiconductor (MOS)-based mixed-signal circuits. We demonstrate how certain common circuit structures involving turning-point nesting can lead to extreme inefficiency, or failure, of conventional probability-one homotopy methods. We also find that such situations can lead to numerical ill-conditioning and homotopy paths that fold back upon themselves, leading to algorithm failure. Our new homotopy model for MOS devices, dubbed Arc-tangent Schichman-Hodges (ATANSH), features decoupled continuation parameters that are instrumental in avoiding these problems. ATANSH-based homotopy methods in production use have led to the routine solution of large previously hard-to-solve industrial circuits, several examples of which are presented.

A fully parallel method for the singular eigenvalue problem

In this paper, a fully parallel method for finding some or all finite eigenvalues of a real symmetric matrix pencil ( A, B) is presented, where A is a symmetric tridiagonal matrix and B is a diagonal matrix with b 1 > 0 and b i ≥ 0, i = 2,3,…, n. The method is based on the homotopy continuation with rank 2 perturbation. It is shown that there are exactly m disjoint, smooth homotopy paths connecting the trivial eigenvalues to the desired eigenvalues, where m is the number of finite eigenvalues of ( A, B). It is also shown that the homotopy curves are monotonic and easy to follow.