[...]... integer variables is nonempty, and there exist nonlinear terms in the objective function and constraints, then (1.1) is a mixed- integer nonlinear programming MINLP problem The fundamentals of MINLP optimization are discussed in Chapter 6 The last class of MINLP problems features many applications in engineering and applied science, and a sample of these are discussed in Part 3 of this book It should... are discussed in Part 1 of this book If the set of integer variables is nonempty, the integer variables participate linearly and separably from the continuous, and the objective function and constraints are linear, then (1.1) becomes a mixed- integer linear programming MILP problem The basics of mixed- integer linear optimization are discussed in Part 2, Chapter 5, of this book If the set of integer variables... constraints 1.2 Structure of Nonlinear and Mixed- Integer Optimization Models In this book we will focus our studies on nonlinear and mixed integer optimization models and present the fundamental theoretical aspects, the algorithmic issues, and their applications in the area of Process Synthesis in chemical engineering Furthermore, we will restrict our attention to algebraic models with a single objective The... (1.1) includes the pure integer linear and nonlinear optimization problems which are not the subject of study of this book The interested reader in pure integer optimization problems is referred to the books by Nemhauser and Wolsey (1988), Parker and Rardin (1988), and Schrijver (1986) 1.3 Illustrative Applications Mixed- integer nonlinear optimization problems of the form (1.1) are encountered in a... 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 10, = tjj/T^ and including the bilinear constraints iw;TLi = tij The continuous variables n;,5i,T[,;,ii;j appear nonlinearly In particular, we have bilinear terms of n^Bi in the objective and constraints, bilinear terms of niTu and w^Tu in the... Bibliography, 435 Index, 453 xv This page intentionally left blank Nonlinear and Mixed- Integer Optimization This page intentionally left blank Chapter 1 Introduction 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... consideration or elimination of its elements If the set of integer variables is empty, and the objective function and constraints are linear, then (1.1) becomes a linear programming LP problem If the set of integer variables is empty, and there exist nonlinear terms in the objective function and/ or constraints, then (1.1) becomes a nonlinear programming NLP problem The fundamentals of nonlinear optimization. .. 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... Part 2 Part 2 introduces first the elementary notions of mixed- integer linear optimization and focuses subsequently on the theoretical and algorithmic developments in mixed- integer nonlinear optimization Part 3 introduces first the generic problems in the area of Process Synthesis, discusses key ideas in the mathematical modeling of process systems, and concentrates on the important application areas of... objective The structure of such nonlinear and mixed integer optimization models takes the following form: where x is a vector of n continuous variables, y is a vector of integer variables, h(x,y) = 0 are m equality constraints, g(jt,.y) < 0 are p inequality constraints, and f ( x , y ) is the objective function Introduction 5 Formulation (1.1) contains a number of classes of optimization problems, by appropriate . integer variables is nonempty, and there exist nonlinear terms in the objective function and constraints, then (1.1) is a mixed- integer nonlinear programming MINLP problem. The fundamentals . special topics in the mixed- integer nonlinear optimization part and the presented application areas. Material in this book has been used in graduate level courses in Optimization and . function and constraints are linear, then (1.1) becomes a mixed- integer linear programming MILP problem. The basics of mixed- integer linear optimization are discussed in Part