Multidimensional unconstrained global optimization arises in a vast spectrum of applications, ranging from financial and physical to biological. Local methods cannot adequately deal with the often numerous local minima and other challenging features of an objective functional, such as flat portions and oscillations. This paper exploits the strengths of n-dimensional B-spline approximations to the objective functional where convexity bounds necessary for the branch-and-bound approach used are easily found for the splines, which are then used on increasingly refined search domains. Good quality approximations are key to the approach, and to this end, Latin-hypercube sample points are used.

The presentation is well done, with detailed pseudocode and accompanying descriptions for the steps in the B-spline, and branch-and-bound implementations complete with data structure and search tree maintenance details. Color is used to effectively illustrate objective functional behavior and plots of experimental results. A matrix formulation (using normal equations but not expanding on numerical considerations) helps clarify the B-spline determination. One strength of the report is a well-presented set of experiments that examine a number of algorithm features, showing good results with a set of difficult examples from the literature. Competitive behavior, compared to a number of other methods, is demonstrated. As the development is multidimensional, some examples beyond the 2D ones used would have been a useful addition. This work is for unconstrained optimization and, as indicated, adding constraints is a topic for further study.