This book is an introduction to nonlinear programming. It deals with the theoretical foundations and solution methods, beginning with the classical procedures and reaching up to "modern" methods like trust region methods or procedures for nonlinear and global optimization. A comprehensive bibliography including diverse web sites with information about nonlinear programming, in particular software, is presented. Without sacrificing the necessary mathematical rigor, excessive formalisms are avoided. Several examples, exercises with detailed solutions, and applications are provided, making the text adequate for individual studies.
The book is written for students from the fields of applied mathematics, engineering, economy, and computation.
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Many problems of operations research, engineering and diverse other quantitative areas maybe formulated as an optimization problem. Basically, such a problem consists of maximizing or minimizing a function of one or several variables such that the variables satisfy certain constraints in the form of equations or inequalities.
Let us consider the following standard form of an optimization problem:
min ƒ(x)
subject to
(1.1)
where ƒ : ℝn → ℝ, gi : ℝn → ℝ(i = 1,..., m, see the Notations at the beginning of the book) are real-valued functions of n variables. In this book we will restrict ourselves to the case in which all these functions are continuous. We are searching for a point
such that
0 for i = 1,..., m and
for all x satisfying the inequalities in (1.1).
An example for (1.1) with n = 3 and m = 2 is the problem
subject to
In this case we have
. (Note that ƒ(x) consists of the total expression following the symbol “min”.)
Instead of (1.1) we may use one of the compact forms:
min ƒ(x)
subject to
(1.2)
with M := {x ∈ ℝn |gi(x) ≤ 0; i = 1,...,m} or
min ƒ(x)
subject to
(1.3)
where g1(x) := (g1(x),...,gm (x))T is a vector-valued function and 0 is the null vector of ℝm.
The function ƒ will be called the objective function, and the inequalities in (1.1) are called constraints or restrictions. The elements of the set M in (1.2) are called feasible solutions or feasible points and M is referred to as a feasible set or feasible region. A point
∈ M is called an optimal solution or optimal point of (1.1) if
holds for all x ∈ M. The value
of an optimal solution
is called the optimal value. Whenever we refer to “the solution” of an optimization problem, an optimal solution is meant.
Various optimization problems, which are not written in the form (1.1) can be reformulated as a problem in standard form. For example, in the case of maximization we can transform
(1.4)
into the equivalent problem
(1.5)
Of course, the optimal values of (1.4) and (1.5) differ in sign. An inequality gi (x) ≥ 0 can be written as – gi (x) ≤ 0, and the equation gi(x) = 0 can be substituted by the two inequalities
