Nonlinear programming

In mathematics, nonlinear programming (NLP) is the process of minimization or maximization of a function of a set of real variables (termed objective function), while simultaneously satisfying a set of equalities and inequalities ( collectively termed constraints), where some of the constraints or the objective function are nonlinear.

Mathematical formulation

A nonlinear programming problem can be stated as:

$\min _{{\mathbf {x}}\in X}f({\mathbf {x}})$

or

$\max _{{\mathbf {x}}\in X}f({\mathbf {x}})$

where

$f:R^{n}\to R$

$X\subseteq R^{n}.$

In the above equations, the set X is also called the feasible set or feasible region of the problem. The function to be minimized is often called the objective function or cost function.

The feasible region is often defined in terms of a set of equalities and inequalities termed constraints. In this case, the NLP problem can be stated as

$\min f({\mathbf {x}})$

subject to:

$c_{i}({\mathbf {x}})\leq 0,i\in I$

and

$c_{j}({\mathbf {x}})=0,j\in E$

In the above problem statement, the second line defines the inequality constraints, and the third line defines the equality constraints. Constraints define the feasible set X, which is the set of points that satisfy all the constraints. I is the index set of inequality constraints and E is the index set of inequality constraints. Functions c_i(x) and c_i(x) are termed constraint functions.

See also

• Optimization

External links

• Nonlinear programming FAQ
• Mathematical Programming Glossary