Certified Parallelotope Continuation for One-Manifolds

Abstract

Starting from an initial solution, continuation methods efficiently produce a sequence of points on a manifold typically defined as the solution set of an underconstrained system of equations. They have a wide range of applications ranging from curve plotting to polynomial root-finding by homotopy. However, classical methods cannot guarantee that the returned points all belong to the same connected component of the manifold, i.e., they may jump from one component to another. Trying to overcome this issue has given birth to several sophisticated heuristics on the one hand and to guaranteed methods based on rigorous computations on the other hand. In this paper we introduce a new rigorous predictor corrector continuation method based on interval computations. Its novelty lies in the fact that it uses parallelotopes as defined in A. Goldsztejn and L. Granvilliers, A new framework for sharp and efficient resolution of NCSP with manifolds of solutions, Constraints, 15 (2010), pp. 190-212, to enclose consecutive portions of the followed manifold. Though computationally more expensive than regular interval computations, the fact that their orientation can be automatically adapted to the local topology of the followed curve makes parallelotopes a very suitable tool for a certified continuation method, as shown by reported experimental results. Introduction.Continuation methods [1] allow exploring step by step a solution manifold, usually defined as the solution set of an underconstrained system of equations F (x) = 0 with F : R n → R m and m < n. This paper focuses on systems where n − m = 1, whose solution manifolds are curves, and proposes a robust singleparameter continuation method. It has a wide range of applications, e.g., polynomial root finding via homotopy [3], nonlinear eigenvalue problems [4], biobjective optimization [18,33], and robot path planning [24]. The most simple and effective continuation method is the predictor corrector algorithm. It includes the simple embedding method and several of its improvements such as the parameter switching [28,29] and the pseudo-arclength methods that have the definite advantage over the simple embedding method that they naturally track folds [6]. They are all, however, subject to jumping between disconnected components without any prompt. For a given system, there exist some theoretical upper bounds on the step size that enforce the connectivity of the continuation, e.g., Theorem 5.2.1 in [1], but they are impractical. While this may be acceptable in some contexts since solutions are eventually computed, such jumps contradict the essence of continuation. Furthermore, connectivity is mandatory in some applications, e.g., robot command synthesis, where such jumps would result in a nonfeasible command path, or in homotopy methods, where such jumps would prevent computing all the solutions of the original system.

A robust and efficient method for solving point distance problems by homotopy

“…Notice that this simple path tracking algorithm does not avoid jumps between paths. Approaches using interval arithmetic (see [10,5,13]) could be used to certify the path tracking. Here we suppose that δ max is small enough to follow considered curves while avoiding such jumps.…”

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“…Then one of paths (and the corresponding solution) is lost. To avoid this, in [43], [44] robust techniques are proposed for path tracing. But, these methods are simply not available as mature public codes yet.…”

On continuation methods for non-linear bi-objective optimization: towards a certified interval-based approach

“…Using the certified continuation method [22] for performing the continuation on the first order conditions allows in particular certifying the feasibility, which can turn out to be critical e.g. when feasible vectors are to be used as bounds in a global algorithm.…”

“…Several certified continuation methods have been proposed: Some concentrate on deriving a guaranteed step size, like [2] based on Smale's α-theory [42], [8] based on interval analysis or [4] based on the reach of a manifold. Others perform rigorous computations using interval analysis and employ solution existence procedures to certify the enclosure of connected portions of the followed manifold [16,22]. Although more expensive than non certified methods, they can be used within a global optimization algorithm, since the price of rigorousness can be small with respect to the price of globality, while certification is a real advantage, e.g.…”

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