A Contractor Based on Convex Interval Taylor

Abstract

Abstract:Interval Taylor has been proposed in the sixties by the interval analysis community for relaxing non-convex continuous constraint systems. However, it generally produces a non-convex relaxation of the solution set. A simple way to build a convex polyhedral relaxation is to select a corner of the studied domain/box as expansion point of the interval Taylor form, instead of the usual midpoint. The idea has been proposed by Neumaier to produce a sharp range of a single function and by Lin and Stadtherr to handle n × n (square) systems of equations. This paper presents an interval Newton-like operator, called ❳✲◆❡✇t♦♥, that iteratively calls this interval convexification based on an endpoint interval Taylor. This general-purpose contractor uses no preconditioning and can handle any system of equality and inequality constraints. It uses Hansen's variant to compute the interval Taylor form and uses two opposite corners of the domain for every constraint. The ❳✲◆❡✇t♦♥ operator can be rapidly encoded, and produces good speedups in constrained global optimization and non-convex constraint satisfaction. First experiments compare ❳✲◆❡✇t♦♥ with affine arithmetic.

Reliable non-linear state estimation involving time uncertainties

“…, X n } containing the x i 's. The variables x i can be symbols, real numbers [3] or vectors of R n . The constraints can be non-linear equations between the variables, such as x 3 = cos x 1 + exp(x 2 ) .…”

Bracketing the solutions of an ordinary differential equation with uncertain initial conditions

“…• A polygon in each box [p] of P that encloses all the possible trajectories inside the box, From the unique constraint, we can deduce two contractors that will be applied on doors and polygons [4,1,5]: • A flow contractor that contracts the polygons and doors consistently with f ,…”

“…Now, in the interval literature, only bounded intervals of R are bisected and the cut point is the center. For instance, the interval [−1, 3] is bisected into [−1, 1] and [1,3]. Here, to bisect unbounded intervals we propose to fix a support interval 5.…”

“…In this paper, we deal with a dynamical system S defined by the following state equation:ẋ (t) = f (x(t)) (1) where x(t) ∈ R n is the state vector and f : R n → R n is the evolution function of S. Denote by ϕ f the flow map of the system. This means that if at time t 0 , the initial state vector is x 0 , then the solution of the state equation is…”

“…This set can be interpreted as an approximation of all solutions of (1), except that we loose the dependency with respect to t.…”