The α BB Global Optimization Algorithm for Nonconvex Problems: An Overview

The αBB algorithm, a deterministic global optimization algorithm for constrained twice-differentiable NLPs, is presented. It is based on a branch-and-bound approach in which a convex relaxation of the original nonconvex problem is obtained through a dual procedure. First, all nonconvex terms of special structure (i.e., bilinear, trilinear, fractional, fractional trilinear, univariate concave) are replaced by customized tight convex lower bounding functions. Second, valid convex underestirnators are generated for general nonconvex terms by using a diagonal shift matrix A. A is related to the Hessian matrix of the nonconvex term and four rigorous methods, based on the interval Hessian matrix, are proposed for its computation. Branching variable selection strategies that improve the quality of the convex relaxations are presented. Variable bound updates are also found to affect the convergence rate positively. An implementation of the algorithm is used to solve three examples: a small but highly nonconvex problem, a stability problem and a large-scale problem involving the design of batch plant under uncertainty.