Editing Mathematical optimization (section)

== Notation ==
Optimization problems are often expressed with special notation. Here are some examples:

=== Minimum and maximum value of a function ===
Consider the following notation:
:<math>\min_{x\in\mathbb R}\; \left(x^2 + 1\right)</math>

This denotes the minimum [[Value (mathematics)|value]] of the objective function {{math|''x''<sup>2</sup> + 1}}, when choosing {{mvar|x}} from the set of [[real number]]s <math>\mathbb{R}</math>. The minimum value in this case is 1, occurring at {{math|1=''x'' = 0}}.

Similarly, the notation

:<math>\max_{x\in\mathbb R}\; 2x</math>

asks for the maximum value of the objective function {{math|2''x''}}, where {{mvar|x}} may be any real number. In this case, there is no such maximum as the objective function is unbounded, so the answer is "[[infinity]]" or "[[Undefined (mathematics)|undefined]]".

=== Optimal input arguments ===
{{Main|Arg max}}
Consider the following notation:
:<math>\underset{x\in(-\infty,-1]}{\operatorname{arg\,min}} \; x^2 + 1,</math>
or equivalently
:<math>\underset{x}{\operatorname{arg\,min}} \; x^2 + 1, \; \text{subject to:} \; x\in(-\infty,-1].</math>

This represents the value (or values) of the [[Argument of a function|argument]] {{mvar|x}} in the [[interval (mathematics)|interval]] {{math|(−∞,−1]}} that minimizes (or minimize) the objective function {{math|''x''<sup>2</sup> + 1}} (the actual minimum value of that function is not what the problem asks for). In this case, the answer is {{math|''x'' {{=}} −1}}, since {{math|''x'' {{=}} 0}} is infeasible, that is, it does not belong to the [[feasible set]].

Similarly,
:<math>\underset{x\in[-5,5], \; y\in\mathbb R}{\operatorname{arg\,max}} \; x\cos y,</math>
or equivalently
:<math>\underset{x, \; y}{\operatorname{arg\,max}} \; x\cos y, \; \text{subject to:} \; x\in[-5,5], \; y\in\mathbb R,</math>

represents the {{math|{''x'', ''y''<nowiki>}</nowiki>}} pair (or pairs) that maximizes (or maximize) the value of the objective function {{math|''x'' cos ''y''}}, with the added constraint that {{mvar|x}} lie in the interval {{math|[−5,5]}} (again, the actual maximum value of the expression does not matter). In this case, the solutions are the pairs of the form {{math|{5, 2''k''{{pi}}<nowiki>}</nowiki>}} and {{math|{−5, (2''k'' + 1){{pi}}<nowiki>}</nowiki>}}, where {{mvar|k}} ranges over all [[integer]]s.

Operators {{math|arg min}} and {{math|arg max}} are sometimes also written as {{math|argmin}} and {{math|argmax}}, and stand for ''argument of the minimum'' and ''argument of the maximum''.