# mathematical programming

**mathematical programming** A wide field of study that deals with the theory, applications, and computational methods for optimization problems. An abstract formulation of such problems is to maximize a function *f* (known as an *objective function*) over a constraint set *S*, i.e. maximize *f*(** x**),

**∈**

*x**S*⊆

*R*,

^{n}where

*R*denotes the space of real

^{n}*n*-component vectors

**,**

*x***= (**

*x**x*

_{1},

*x*

_{2},…,

*x*)

_{n}^{T}

and

*f*is a real-valued function defined on

*S*. If

*S*consists only of vectors whose elements are integers, then the problem is one of

*integer programming*. Linear programming treats the case of

*f*as a linear function with

*S*defined by linear equations and/or constraints. Nonlinear objective functions with or without constraints (defined by systems of nonlinear equations) give rise to problems generally referred to as optimization problems.

Mathematical-programming problems arise in engineering, business, and the physical and social sciences.

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**mathematical programming**