Editing Optimization problem (section)

==Continuous optimization problem==

The ''[[Canonical form|standard form]]'' of a [[Continuity (mathematics)|continuous]] optimization problem is<ref>{{cite book|title=Convex Optimization|first1=Stephen P.|last1=Boyd|first2=Lieven|last2=Vandenberghe|page=129|year=2004|publisher=Cambridge University Press|isbn=978-0-521-83378-3|url=https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf#page=143|format=pdf}}</ref>
<math display=block>\begin{align}
&\underset{x}{\operatorname{minimize}}& & f(x) \\
&\operatorname{subject\;to}
& &g_i(x) \leq 0, \quad i = 1,\dots,m \\
&&&h_j(x) = 0, \quad j = 1, \dots,p 
\end{align}</math>
where
* {{math|''f'' : [[Euclidean space|ℝ<sup>''n''</sup>]] → [[Real numbers|ℝ]]}} is the ''[[objective function]]'' to be minimized over the {{mvar|n}}-variable vector {{mvar|x}},
* {{math|''g<sub>i</sub>''(''x'') ≤ 0}} are called ''inequality [[Constraint (mathematics)|constraints]]''
* {{math|''h<sub>j</sub>''(''x'') {{=}} 0}} are called ''equality constraints'', and
* {{math|''m'' ≥ 0}} and {{math|''p'' ≥ 0}}.

If {{math|''m'' {{=}} ''p'' {{=}} 0}}, the problem is an unconstrained optimization problem. By convention, the standard form defines a '''minimization problem'''. A '''maximization problem''' can be treated by [[Additive inverse|negating]] the objective function.