This chapter presents the fundamentals and algorithms for mixed-integer nonlinear optimization problems. Sections 6.1 and 6.2 outline the motivation, formulation, and algorithmic approaches. Section 6.3 discusses the Generalized Benders Decomposition and its variants. Sections 6.4, 6.5 and 6.6 presents the Outer Approximation and its variants with Equality Relaxation and Augmented Penalty. Section 6.7 discusses the Generalized Outer Approximation while section 6.8 compares the Generalized Benders Decomposition with the Outer Approximation. Finally, section 6.9 discusses the Generalized Cross Decomposition. 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 integer variables can be used to model, for instance, sequences of events, alternative candidates, existence or nonexistence of units (in their zero-one representation), while discrete variables can model, for instance, different equipment sizes. The continuous variables are used to model the input-output and interaction relationships among individual units/operations and different interconnected systems. The nonlinear nature of these mixed-integer optimization problems may arise from (i) nonlinear relations in the integer domain exclusively (e.g., products of binary variables in the quadratic assignment model), (ii) nonlinear relations in the continuous domain only (e.g., complex nonlinear input-output model in a distillation column or reactor unit), (iii) nonlinear relations in the joint integer-continuous domain (e.g., products of continuous and binary variables in the scheduling/ planning of batch processes, and retrofit of heat recovery systems). In this chapter, we will focus on nonlinearities due to relations (ii) and (iii). An excellent book that studies mixed-integer linear optimization, and nonlinear integer relationships in combinatorial optimization is the one by Nemhauser and Wolsey (1988). The coupling of the integer domain with the continuous domain along with their associated nonlinearities make the class of MINLP problems very challenging from the theoretical, algorithmic,and computational point of view. Apart from this challenge, however, there exists a broad spectrum of applications that can be modeled as mixed-integer nonlinear programming problems.
Extended Duality in Fuzzy Optimization Problems
