Mixed-integer nonlinear programming MINLP

The specific problem characteristics are modeled most appropriately by a combination of concepts from various general modeling frameworks leading to a mixed-integer nonlinear programming (MINLP) model. 

Mixed-integer nonlinear programming MINLP (refer to Chapter 9). 

Flowsheet optimization is also regarded as a key task in the structural optimization of a flowsheet. As a described in the introduction, structural optimization for process design can be formulated as a mixed integer nonlinear program (MINLP). This then allows for addition or replacement of existing units, and consideration of a number of design options simultaneously. In these formulations individual units are turned on and off over the course of the optimization, as suggested by the MINLP master problem. 

The above formulation is a mixed-integer nonlinear programming MINLP model and has the following characteristics. The binary variables appear linearly and separably from the continuous variables in both the objective and constraints, by defining a new set of variables Wj = tij/Tu and including the bilinear constraints WiTu = Uj- The continuous variables nitBi,Tu,Wi appear nonlinearly. In particular, we have bilinear terms of riiBi in the objective and constraints, bilinear terms of niT i and WiTu in the constraints. The rest of the continuous variables Vj, (V fc)ra, Vfjk appear linearly in the objective function and constraints. 

A wide range of nonlinear optimization problems involve integer or discrete variables in addition to the continuous variables. These classes of optimization problems arise from a variety of applications and are denoted as Mixed-Integer Nonlinear Programming MINLP problems. 

The general mathematical model of the superstructure presented in step 2 of the outline, and indicated as (7.1), has a mixed set of 0 - 1 and continuous variables and as a result is a mixed-integer optimization model. If any of the objective function and constraints is nonlinear, then (7.1) is classified as mixed- integer nonlinear programming MINLP problem. 

A process-synthesis problem can be formulated as a combination of tasks whose goal is the optimization of an economic objective function subject to constraints. Two types of mathematical techniques are the most used mixed-integer linear programming (MILP), and mixed-integer nonlinear programming (MINLP). 

Kocis, G. R., and Grossmann, I. E. Global Optimization of Nonconvex Mixed-Integer Nonlinear Programming (MINLP) Problems in Process Synthesis, Ind. Eng. Chem. Res. 27,1407 (1988). 

One approach to making discrete design decisions is to pose a mixed-integer nonlinear program (MINLP) and apply a suitable optimization method to the selection of discrete design parameters. We have not adopted this approach for the following reasons. 

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