Abstract. The search tree size of the spatial Branch-and-Bound algorithm for Mixed-Integer Nonlinear Programming depends on many factors, one of which is the width of the variable ranges at every tree node. A range reduction technique often employed is called Feasibility Based Bounds Tightening, which is known to be practically fast, and is thus deployed at every node of the search tree. From time to time, however, this technique fails to converge to its limit point in finite time, thereby slowing the whole Branch-and-Bound search considerably. In this paper we propose a polynomial time method, based on solving a linear program, for computing the limit point of the Feasibility Based Bounds Tightening algorithm applied to linear equality and inequality constraints. Keywords: global optimization, MINLP, spatial Branch-and-Bound, range reduction, constraint programming. 1

Interval-gradient cuts are (nonlinear) valid inequalities for nonconvex NLPs defined for constraints g(x) ≤ 0 with g being continuously differentiable in a box [x, ¯x]. In this paper we define intervalsubgradient cuts, a generalization to the case of nondifferentiable g, and show that no-good cuts (which have the form ‖x−ˆx ‖ ≥ ε for some norm and positive constant ε) are a special case of interval-subgradient cuts whenever the 1-norm is used. We then briefly discuss what happens if other norms are used.

One of the foremost difficulties in solving Mixed Integer Nonlinear Programs, either with exact or heuristic methods, is to find a feasible point. We address this issue with a new feasibility pump algorithm tailored for nonconvex Mixed Integer Nonlinear Programs. Feasibility pumps are successive projection algorithms that iterate between solving a continuous relaxation and a mixed-integer relaxation of the original problems; such approaches currently exist in the literature for Mixed-Integer Linear Programs and convex Mixed-Integer Nonlinear Programs. Both cases exhibit the distinctive property that the continuous relaxation can be solved in polynomial time. In nonconvex Mixed Integer Nonlinear Programming such a property does not hold and the main innovations in this paper are tailored algorithmic methods to overcome such a difficulty. We present extensive computational results on the MINLPLib, showing the effectiveness and efficiency of our algorithm.