# Mixed-Integer Nonlinear Optimization in Process Synthesis

## Abstract

The use of networks allows the representation of a variety of important engineering problems. The treatment of a particular class of network applications, the *process synthesis problem*, is exposed in this paper. Process Synthesis seeks to develop systematically process flowsheets that convert raw materials into desired products. In recent years, the optimization approach to process synthesis has shown promise in tackling this challenge. It requires the development of a network of interconnected units, the process superstructure, that represents the alternative process flowsheets. The mathematical modeling of the superstructure has a mixed set of binary and continuous variables and results in a Mixed-Integer optimization model. Due to the nonlinearity of chemical models, these problems are generally classified as Mixed-Integer Nonlinear Programming (MINLP) problems.

A number of local optimization algorithms, developed for the solution of this class of problems, are presented in this paper: Generalized Benders Decomposition (GBD), Outer Approximation (OA), Generalized Cross Decomposition (GCD), Branch and Bound (BB), Extended Cutting Plane (ECP), and Feasibility Approach (FA). Some recent developments for the global optimization of nonconvex MINLPs are then introduced. In particular, two branch-and-bound approaches are dis-cussed:the Special structure Mixed Integer Nonlinear *α*BB (SMIN-αBB), where the binary variables should participate linearly or in mixed-bilinear terms, and the General structure Mixed Integer Nonlinear αBB (GMIN- *α*BB), where the continuous relaxation of the binary variables must lead to a twice-differentiable problem. Both algorithms are based on the *α*BB global optimization algorithm for nonconvex continuous problems.

Once the theoretical issues behind local and global optimization algorithms for MINLPs have been exposed, attention is directed to their algorithmic development and implementation. The framework **MINOPT** is discussed as a computational tool for the solution of process synthesis problems. It is an implementation of a number of local optimization algorithms for the solution of MINLPs. The use of **MINOPT** is illustrated through the solution of a variety of process network problems. The synthesis problem for a heat exchanger
network is then presented to demonstrate the global optimization SMIN-*α*BB algorithm.

## Keywords

Master Problem Outer Approximation MINLP Problem Heat Exchanger Network Outer Approximation Algorithm## Preview

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