Introduction

This book is on modern continuous nonlinear optimization. Continuous nonlinear optimization problems have a simple mathematical model and always refer to a real physical system, the running of which we want to optimize. Firstly, a nonlinear optimization problem 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 which 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. The dynamic constraints always involve a Jacobian with a block-diagonal structure. Besides, these functional constraints are nonlinear functions which may be equalities or inequalities, or even range constraints. Another very important type of constraints is the simple bounds on variables. The real applications of optimization include simple bounds on variables which express the constructive engineering conditions and limits of the system to be optimized. The constraints of an optimization problem define the so-called feasible domain of the problem. Both the objective function and the constraints may depend on some parameters with known values which characterize the system under optimization. The process of identifying the variables, parameters, simple bounds on variables, the objective functions, and constraints is known as modeling, one of the finest intellectual activities. 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 continuously differentiable) with known first-order derivatives. When the number of variables and the number of constraints is large, then the optimization problem is quite challenging.