279196 Nonconvex Generalized Benders Decomposition and Piecewise Convex Relaxations for Optimal Process Design and Operation Under Uncertainty
Integrated process design and operation problems are often cast as mathematical programming problems for systematic solution. If a process involves uncertain factors that could impact the process performance significantly, the uncertainties need to be explicitly addressed in the mathematical programming, which results in a stochastic programming problem.
Recently, a nonconvex generalized Benders decomposition (NGBD)  has been developed to solve a class of stochastic nonconvex mixed-integer nonlinear programming (MINLP) problems arising from integrated process design and operation under uncertainty. By exploiting the decomposable structure of the problem, NGBD can solve the problem to global optimality finitely, and it exhibits dramatic computational advantages over traditional branch-and-bound based global optimization methods. As 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 significantly improved by generating tighter convex relaxations.
It has been recognized in the process systems engineering literature that piecewise linear relaxation enables much tighter relaxation of bilinear programs and can expedite global optimization significantly in the branch-and-bound framework (e.g.    ). This paper generalizes the notion of piecewise relaxation to factorable functions in the context of McCormick relaxation   and integrates the so-obtained piecewise McCormick relaxation technique into the NGBD algorithm to reduce the gap between the original problem and its convex relaxation. In addition, most subproblems in a NGBD iteration can be solved without exchanging information among them, and therefore they can be solved in parallel to reduce the solution time. The parallelization for NGBD will be briefly discussed in the paper as well.
Case studies of a pump network configuration problem  and an energy polygeneration problem  show that, while NGBD can solve problems that are intractable for a state-of-the-art global optimization solver (BARON ), integrating the proposed piecewise convex relaxation into NGBD helps to reduce the solution time by up to an order of magnitude. The case study results also show that parallel computation can reduce the NGBD solution time significantly.
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