2008, Garloff, Jürgen, Smith, Andrew Paul

This paper addresses the problem of finding tight affine lower bound functions for multivariate polynomials, which may be employed when global optimisation problems involving polynomials are solved with a branch and bound method. These bound functions are constructed by using the expansion of the given polynomial into Bernstein polynomials. The coefficients of this expansion over a given box yield a control point structute whose convex hull contains the graph of the given polynomial over the box. We introduce a new method for computing tight affine lower bound functions based on these control points, using a linear least squares approximation of the entire control point structure. This is demonstrated to have superior performance to previous methods based on a linear interpolation of certain specially chosen control points. The problem of how to obtain a verfied affine lower bound function in the presence of uncertainty and rounding errors is also considered. Numerical results with error bounds for a series of randomly-generated polynomials are given.

2001, Garloff, Jürgen

We consider the class of the totally nonnegative matrices, i.e., the matrices having all their minors nonnegative, and intervals of matrices with respect to the chequerboard partial ordering, which results from the usual entrywise partial ordering if we reverse the inequality sign in all components having odd index sum. For these intervals in 1982 we stated in this journal the following conjecture: If the left and right endpoints of an interval are nonsingular and totally nonnegative then all matrices taken from the interval are nonsingular and totally nonnegative. In this paper we show that this conjecture is true if we restrict ourselves to the subset of the almost totally positive matrices.

2003, Garloff, Jürgen, Jansson, Christian, Smith, Andrew Paul

Relaxation techniques for solving nonlinear systems and global optimisation problems require bounding from below the nonconvexities that occur in the constraints or in the objective function by affine or convex functions. In this paper we consider such lower bound functions in the case of problems involving multivariate polynomials. They are constructed by using Bernstein expansion. An error bound exhibiting quadratic convergence in the univariate case and some numerical examples are given.

2000, Smith, Andrew Paul, Garloff, Jürgen

A method for enclosing all solutions to a system of polynomial equations inside a given box is presented. This method relies on the expansion of a multivariate polynomial into Bernstein polynomials and is a domain-splitting approach. After a pruning step a collection of subboxes remain which undergo an existence test provided by Miranda's Theorem.

2003, Garloff, Jürgen, Smith, Andrew Paul

In this paper we present a method for construction such affine lower bound functions for polynomials which requires the computation of slopes, the solution of a system of linear equations, and a sequence of back substitutions. This method in general requires fewer arithmetic operations and has lower complexity than our previous approach.