Nonconvex Generalized Benders Decomposition with Piecewise Convex Relaxations for Global Optimization of Integrated Process Design and Operation Problems

This paper considers the global optimization of challenging stochastic or multiperiod mixed-integer nonconvex programs that arise from integrated process design and operation. The difficulties of the problems are large scale and the nonconvexity involved. Recently, a novel decomposition method, called nonconvex generalized Benders decomposition (NGBD), has been developed to solve this problem to global optimality finitely, and this method shows dramatic computational advantages over traditional branch-and-bound based global optimization methods because it can exploit well the decomposable structure of such problems. Since the convergence rate of NGBD is largely dependent on the tightness of the convex relaxations of the nonconvex functions, the efficiency of NGBD can be improved by generating tighter convex relaxations. Building on the success of piecewise linearization for bilinear programs in the process systems engineering literature, this paper develops a piecewise convex relaxation framework, which can yield tighter convex relaxations for factorable nonconvex programs, and integrates this framework into NGBD to expedite the solution. Case studies of a classical literature problem and an industry-level problem show that, while NGBD can solve problems that are intractable for a state-of-the-art global optimization solver, integrating the proposed piecewise convex relaxation into NGBD helps to reduce the solution time by up to an order of magnitude.