Mixed-integer nonlinear optimization formulation

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. 

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). 

The problem of portfolio selection is easily expressed numerically as a constrained optimization maximize economic criterion subject to constraint on available capital. This is a form of the knapsack problem, which can be formulated as a mixed-integer linear program (MILP), as long as the project sizes are fixed. (If not, then it becomes a mixed-integer nonlinear program.) In practice, numerical methods are very rarely used for portfolio selection, as many of the strategic factors considered are difficult to quantify and relate to the economic objective function. 

In contrast with the use of objective functions such as observability or reliability that had been used, Bagajewicz (1997, 2000) formulated a mixed integer nonlinear programming to obtain sensor networks satisfying the constraints of residual precision, resilience, error detectability at minimal cost. A tree enumeration was proposed where at each node the optimization problem of the different characteristics are solved. 

Although, as explained in Chapter 9, many optimization problems can be naturally formulated as mixed-integer programming problems, in this chapter we will consider only steady-state nonlinear programming problems in which the variables are continuous. In some cases it may be feasible to use binary variables (on-off) to include or exclude specific stream flows, alternative flowsheet topography, or different parameters. In the economic evaluation of processes, in design, or in control, usually only a few (5-50) variables are decision, or independent, variables amid a multitude of dependent variables (hundreds or thousands). The number of dependent variables in principle (but not necessarily in practice) is equivalent to the number of independent equality constraints plus the active inequality constraints in a process. The number of independent (decision) variables comprises the remaining set of variables whose values are unknown. Introduction into the model of a specification of the value of a variable, such as T = 400°C, is equivalent to the solution of an independent equation and reduces the total number of variables whose values are unknown by one. 

This chapter introduces the reader to elementary concepts of modeling, generic formulations for nonlinear and mixed integer optimization models, and provides some illustrative applications. Section 1.1 presents the definition and key elements of mathematical models and discusses the characteristics of optimization models. Section 1.2 outlines the mathematical structure of nonlinear and mixed integer optimization problems which represent the primary focus in this book. Section 1.3 illustrates applications of nonlinear and mixed integer optimization that arise in chemical process design of separation systems, batch process operations, and facility location/allocation problems of operations research. Finally, section 1.4 provides an outline of the three main parts of this book. 

This book aims at presenting the fundamentals of nonlinear and mixed-integer optimization, and their applications in the important area of process synthesis and chemical engineering. The first chapter introduces the reader to the generic formulations of this class of optimization problems and presents a number of illustrative applications. For the remaining chapters, the book contains the following three main parts ... 

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