Editing Nonlinear programming (section)

== Definition and discussion ==

Let ''n'', ''m'', and ''p'' be positive integers. Let ''X'' be a subset of ''R<sup>n</sup>'' (usually a box-constrained one), let ''f'', ''g<sub>i</sub>'', and ''h<sub>j</sub>'' be [[real-valued function]]s on ''X'' for each ''i'' in {''1'', ..., ''m''} and each ''j'' in {''1'', ..., ''p''}, with at least one of ''f'', ''g<sub>i</sub>'', and ''h<sub>j</sub>'' being nonlinear.

A nonlinear programming problem is an [[optimization problem]] of the form

:<math>
\begin{align}
  \text{minimize }   & f(x) \\
  \text{subject to } & g_i(x) \leq 0 \text{ for each } i \in \{1, \dotsc, m\} \\
                    & h_j(x) = 0 \text{ for each } j \in \{1, \dotsc, p\} \\
                    & x \in X.
\end{align}
</math>

Depending on the constraint set, there are several possibilities:

* ''feasible'' problem is one for which there exists at least one set of values for the choice variables satisfying all the constraints.
* an ''infeasible'' problem is one for which no set of values for the choice variables satisfies all the constraints. That is, the constraints are mutually contradictory, and no solution exists; the feasible set is the [[empty set]].
* ''unbounded'' problem is a feasible problem for which the objective function can be made to be better than any given finite value. Thus there is no optimal solution, because there is always a feasible solution that gives a better objective function value than does any given proposed solution.

Most realistic applications feature feasible problems, with infeasible or unbounded problems seen as a failure of an underlying model. In some cases, infeasible problems are handled by minimizing a sum of feasibility violations.

Some special cases of nonlinear programming have specialized solution methods:

* If the objective function is [[Concave function|concave]] (maximization problem), or [[Convex function|convex]] (minimization problem) and the constraint set is [[Convex set|convex]], then the program is called convex and general methods from [[convex optimization]] can be used in most cases.
* If the objective function is [[quadratic function|quadratic]] and the constraints are linear, [[quadratic programming]] techniques are used.
* If the objective function is a ratio of a concave and a convex function (in the maximization case) and the constraints are convex, then the problem can be transformed to a convex optimization problem using [[fractional programming]] techniques.