Nonlinear Programming

Nonlinear Programming

Xiaofeng Zhao (University of Mary Washington, USA)
Copyright: © 2014 |Pages: 10
DOI: 10.4018/978-1-4666-5202-6.ch147

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Introduction

Nonlinear programming (NLP) deals with the problem of optimizing an objective function in the presence of equality and inequality constraints, where some of the constraints or the objective functions are nonlinear. A general optimization problem is to select n decision variables 978-1-4666-5202-6.ch147.m01 from a given feasible region in such a way as to optimize (minimize or maximize) a given objective function978-1-4666-5202-6.ch147.m02of the decision variables. The problem is called a nonlinear programming problem (NLP) if the objective function is nonlinear and/or the feasible region is determined by nonlinear constraints.

For notational convenience, we usually let x denote the vector of n decision variables 978-1-4666-5202-6.ch147.m03that is, x = (978-1-4666-5202-6.ch147.m04) and write the problem more concisely. Throughout the article, we shall consider the following NLP problem:Minimize f (x),Subject to

978-1-4666-5202-6.ch147.m05
978-1-4666-5202-6.ch147.m06
978-1-4666-5202-6.ch147.m07
where X is a subset of978-1-4666-5202-6.ch147.m08, x is a vector of 978-1-4666-5202-6.ch147.m09 components978-1-4666-5202-6.ch147.m10, and f: X 978-1-4666-5202-6.ch147.m11R, g: X 978-1-4666-5202-6.ch147.m12978-1-4666-5202-6.ch147.m13 and h: X978-1-4666-5202-6.ch147.m14978-1-4666-5202-6.ch147.m15 are defined on X.

The function f is usually called the objective function. Each of the constraints 978-1-4666-5202-6.ch147.m16 is called an inequality constraint, and each of the constraints 978-1-4666-5202-6.ch147.m17, is called an equality constraint. Note also that the set X typically includes lower and upper bounds on the variables. The reason for separating variable bounds from the other inequality constraints is that they can play a useful role in some algorithms, i.e., they are handled in a specific way. A vector 978-1-4666-5202-6.ch147.m18satisfying all the constraints is called a feasible solution to the problem; the collection of all such points forms the feasible region. The NLP problem, then, is to find a feasible point 978-1-4666-5202-6.ch147.m19 such that 978-1-4666-5202-6.ch147.m20for each feasible point x. Needless to say, a nonlinear programming problem can be stated as a maximization problem, and the inequality constraints can be written in the form978-1-4666-5202-6.ch147.m21.

Key Terms in this Chapter

Infeasible Solution: A decision alternative or solution that does not satisfy one or more constraints.

Global Optimum: A feasible solution is a global optimal optimum if there are no other feasible points with better objective value in the entire feasible region. A global optimum may be either a global maximum or a global minimum.

Constraints: Restrictions or limitations imposed on a problem.

Convex Function: A function that is bowl-shaped up. For example, are convex functions.

Optimal Solution: The specific decision variable values or value that provide the best output for the model.

Concave Function: A function that is bowl-shaped down. For example, are convex functions.

Sensitivity analysis: The study of how changes in the coefficients of a nonlinear programming problem affect the optimal solution.

Local Optimization: A feasible solution is a local optimum if there are no other feasible solutions with a better function value in the immediate neighborhood. A local optimum may be either maximum or local minimum.

Objective Function: A mathematical expressions that describes the problem’s objective.

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