## Quick Reference

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.

maximize *f*(**x**), **x** ∈ *S* ⊆ *R** ^{n}*,

**x** = (*x*_{1},*x*_{2},…,*x** _{n}*)

^{T}

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

*Subjects:*
Computing.

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