GLOBAL OPTIMIZATION OF MINLP PROBLEMS IN-PROCESS SYNTHESIS AND DESIGN

Two new methodologies for the global optimization of MINLP models, the Special structure Mixed Integer Nonlinear alpha BB, SMIN-alpha BB, an d the General structure Mixed Integer Nonlinear alpha BB, GMIN-alpha B B, are presented. Their theoretical foundations provide guarantees tha t the global optimum solution of MINLPs involving twice-differentiable nonconvex functions in the continuous variables can be identified. Th e conditions imposed on the functionality of the binary variables diff er for each method : linear and mixed bilinear terms can be treated wi th the SMIN-alpha BB; mixed nonlinear terms whose continuous relaxatio n is twice-differentiable are handled by the GMIN-alpha BB. While both algorithms use the concept of a branch & bound tree, they rely on fun damentally different bounding and branching strategies. In the GMIN-al pha BB algorithm, lower (upper) bounds at each node result from the so lution of convex (nonconvex) MINLPs derived from the original problem. The construction of convex lower bounding MINLPs, using the technique s recently developed for the generation of valid convex underestimator s for twice-differentiable functions (Adjiman et al., 1996; Adjiman an d Floudas, 1996), is an essential task as it allows to solve the under estimating problems to global optimality using the GBD algorithm or th e OA algorithm, provided that the binary variables participate separab ly and linearly. Moreover, the inherent structure of the MINLP problem can be fully exploited as branching is performed on the binary and th e continuous variables. In the case of the SMIN-alpha BB algorithm, th e lower and upper bounds are obtained by solving continuous relaxation s of the original MINLP. Using the alpha BB algorithm, these nonconvex NLPs are solved as global optimization problems and hence valid lower bounds are generated. Since branching is performed exclusively on the binary variables, the maximum size of the branch-and-bound tree is sm aller than that for the SMIN-alpha BB. The two proposed approaches are used to generate computational results on various nonconvex MINLP pro blems that arise in the areas of Process Synthesis and Design.