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Proceedings of GO 2005, pp. 109 – 113.Rigorous Affine Lower Bound Functions for MultivariatePolynomials and Their Use in Global OptimisationJuergen Garloff and Andrew P. SmithUniversity of Applied Sciences / FH Konstanz, Department of Computer Science, Postfach 100543, D-78405 Konstanz,Germany {garloff,smith}@fh-konstanz.deAbstractKeywords:We address the problem of finding tight affine lower bound functions for multivariate polynomials,which may be employed when global optimisation problems involving polynomials are solved witha branch and bound method. These bound functions are constructed by using the expansion of thegiven polynomial into Bernstein polynomials. The coefficients of this expansion over a given boxyield a control point structure whose convex hull contains the graph of the given polynomial overthe box. We introduce different methods for computing tight affine lower bound functions based onthese control points, using either a linear interpolation of certain specially chosen control points ora linear approximation of all the control points. We present a bound on the distance between thegiven polynomial and its affine lower bound function, which, at least in the univariate case, exhibitsquadratic convergence with respect to the width of the domain. We also address the problem of howto obtain a verified affine lower bound function in the presence of uncertainty and rounding errors.Some numerical examples are presented.Bernstein polynomials, relaxation, affine bound functions, constrained global optimisation1. IntroductionIn our talk we wish to contribute to the solution of the constrained global optimisation problemThe set of feasible solutions F is defined by⎧ ∣⎨ ∣∣∣∣∣ g i (x) ≤ 0 for i = 1, . . . , mF :=⎩ x ∈ S h j (x) = 0 for j = 1, . . . , lx ∈ Xmin f(x). (1)x∈Fwhere S ⊆ R n , X is a box contained in S, and f, g i , h j are real-valued functions defined on S.The optimisation problemmin f(x) (2)x∈Ris called a relaxation of (1) if the set of feasible solutions fulfils F ⊆ R and f(x) ≤ f(x) holdsfor all x ∈ F .To generate an affine relaxation for problem (1), the functions f, g i (i = 1, . . . , m), and h j(j = 1, . . . , l) are replaced by affine lower bound functions f, g i, and h j , respectively. Then therelaxed problem (2) with the respective set of feasible solutions yields a linear programmingproblem. Its solution provides a lower bound for the solution of (1). This relaxation may beused in a branch and bound framework for solving problem (1).⎫⎬⎭ ,