Foundations of Optimization pp 209-250 | Cite as

# Nonlinear Programming

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## Abstract

A
where

*nonlinear program*, or a*mathematical program*, is a constrained optimization (say minimization) problem having the form min$$\begin{array}{ll}
\min & f(x)\\
{\rm{s.t.}} & g_i(x) \leq 0, \quad i = 1, \ldots, r, \quad (P)\\
{} & h_j(x) = 0, \quad j = 1,\ldots,m,\\
\end{array}$$

(9.1)

*f*, \(f, \{g_i\}^{r}_{1}\), and \(\{h_j\}^m_1\)are real-valued functions defined on some subsets of Rn. The function f is called the*objective function*of (*P*), and the inequalities and equalities involving*g*_{ i }and*h*_{ j }, respectively, are called the constraints of the problem. The*feasible region (or constraint set)*of (*P*) is the set of all points satisfying all the constraints,$$\mathcal{F}(P) = \{x \in \mathbb{R}^n\ :\ g_i(x) \leq 0, i = 1.,\ldots, r, h_j(x) = 0, j = 1, \ldots,m\}.$$

## Keywords

Nonlinear Programming Feasible Region Nonlinear Program Feasible Point Active Constraint
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer New York 2010