Global Optimization of Nonconvex MINLP by a Hybrid Branch-and-Bound and Revised General Benders Decomposition Approach

Abstract

Mixed-integer nonlinear programming, MINLP, has played a crucial role in chemical process design via superstructures that always involve discrete and continuous variables. In this paper, a global optimization algorithm for nonconvex MINLP problems is developed by addressing the nonconvexity caused by the nonconvex continuous functions with a convex quadratic underestimator within a branch-and-bound framework, as well as the joint problem caused by the mixed natures of integer and continuous variables through a revised general Benders decomposition (GBD) method, where the latter is designed mainly for three favorable structures, i.e., separable, bilinear, and partly linear, between the two domains of continuous and binary variables. The convergence of the revised GBD method to the global solution of the relaxed MINLP subprobelm over each subregion generated in the above framework is guaranteed by the convex underestimation functions in terms of the twice-differentiable assumptions of the continuous functions and the above three favorable joint structures. Then, the convergence of the proposed hybrid algorithm can be established by the exhaustive partition of the constrained region, the monotonicity of the lower bound, and the reliability of the infeasibility detection. Finally, a very simple example for process design is used to verify the different implementation aspects of the proposed approach, especially the unique underestimator construction and the infeasibility detection in each lower-bounding problem.