On Polytopal Branch and Bound with Monotonicity

In the field of Interval Arithmetic Branch and Bound (B&B) monotonicity detection and its use for dimension reduction and eliminations of partition sets has a long tradition. Recent investigation of extending ideas to simplicial B&B has shown that the relative success depends on the dimension of the simplex, i.e. the dimension of its affine hull. Linear constraints may provide a feasible area which is a polytope in a lower dimension due to equality constraints. We investigate the question how to deal with subsets defined by polytopes. We study how to construct a converging branch and bound, how to define bounds and how to refine and how to exploit monotonicity when bounds are provided by the interval hull.