Taylor Series Models in Deterministic Global Optimization

Deterministic global optimization requires a global search with rejection of subregions. To reject a subregion, bounds on the range of the constraints and objective function can be used. Although often effective, simple interval arithmetic sometimes gives impractically large bounds on the ranges. However, Taylor models as developed by Berz et al may be effective in this context. Efficient incorporation of such models in a general global optimization package is a significant project. Here, we use the system COSY-INFINITY by Berz et al to study the bounds on the range of various order Taylor models for certain difficult test problems we have previously encountered. Based on that, we conclude that Taylor models may be useful for some, but not all, problems in verified global optimization. Forthcoming improvements in the COSY-INFINITY interface will help us reach stronger conclusions. 1 Deterministic Global Optimization Deterministic global optimization involves exhaustive search over the domain. The domain is subdivided (“branching”), and those subdomains that cannot possibly contain global minimizers are rejected. For example, if the problem is the unconstrained problem Enclose the minimizers of φ(x) subject to x ∈ x, (1.1) then evaluating φ at a particular point x gives an upper bound for the global minimum of φ over the region x. Some method is then used to bound the range of φ over subregions x ⊂ x. If the lower bound φ, so obtained, for φ over x has φ > φ(x), then x may be rejected as not containing any global optima; see Figure 1 for the situation in one dimension. A related problem is that of finding all roots within a given region, that is, Enclose all x with f(x) = 0 subject to x ∈ x. (1.2) 2 R. Baker Kearfott, Alvard Arazyan In equation (1.2), bounds on f over a subregion x ⊂ x are obtained; denote the interval vector representing such bounds by f(x). If 0 6∈ f(x), that is, unless the lower bound for each component of f is less than zero and the upper bound is greater than zero, then there cannot be a solution of f(x) = 0 in x, and x can be rejected.