Introduction

This book is on nonlinear optimization in the GAMS technology. Continuous nonlinear optimization problems have a simple mathematical model and always refer to a system its running we want to optimize. Firstly, it contains an objective function which measures the performances or requirements of the system. Often, this function represents a profit, a time interval, a level, a sort of energy, or combination of different quantities which have a physical significance for the modeler. The objective function depends on some characteristics of the system, called variables or unknowns. The purpose of any optimization problem is to find the values of these variables that minimize (or maximize) the objective function, subject to some constraints the variables must satisfy. Constraints of an optimization problem may have different algebraic expressions. There are static and dynamic constraints called functional constraints. The difference between these types of constraints comes from the structure of their Jacobian. Another very important type of constraints is the simple bounds on variables. Both the objective function and the constraints may depend on some parameters with known values which represent the constructive characteristics of the system under optimization. The process of identifying the variables, parameters, the objective functions, and constraints is known as modeling, one of the finest intellectual activities. It is worth saying that in this book, we assume that the variables can take real values and the objective function and the constraints are smooth enough (at least twice differentiable) with known first-order derivatives. When the number of variables and the number of constraints are large, the optimization problem is quite challenging.