A Contractor Based on Convex Interval Taylor

International audienceInterval Taylor has been proposed in the sixties by the interval analysis community for relaxing continuous non-convex 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 andby Lin and Stadtherr to handle n × n (square) systems of equations. This paper presents an interval Newton-like operator, called X-Newton, 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 X-Newton operator can be rapidly encoded, and produces good speedups in constrained global optimization and constraint satisfaction. First experiments compare X-Newton with affine arithmetic

Global Optimization based on Contractor Programming

International audienceIn this paper, we will present a general pattern based on contractor programmingfor designing a global optimization solver. This approach allows to solve problems with awide variety of constraints. The complexity and the performance of the algorithm rely on theconstruction of contractors which characterize the feasible region

A global method for mixed categorical optimization with catalogs

In this article, we propose an algorithmic framework for globally solving mixed problems with continuous variables and categorical variables whose properties are available from a catalog. It supports catalogs of arbitrary size and properties of arbitrary dimension, and does not require any modeling effort from the user. Our tree search approach, similar to spatial branch and bound methods, performs an exhaustive exploration of the range of the properties of the categorical variables ; branching, constraint programming and catalog lookup phases alternate to discard inconsistent values. A novel catalog-based contractor guarantees consistency between the categorical properties and the existing catalog items. This results in an intuitive generic approach that is exact and easy to implement. We demonstrate the validity of the approach on a numerical example in which a categorical variable is described by a two-dimensional property space