Editing Optimization problem (section)

==Combinatorial optimization problem==
{{Main|Combinatorial optimization}}

Formally, a [[combinatorial optimization]] problem {{mvar|A}} is a quadruple{{Citation needed|date=January 2018}} {{math|(''I'', ''f'', ''m'', ''g'')}}, where
* {{math|I}} is a [[Set (mathematics)|set]] of instances;
* given an instance {{math|''x'' ∈ ''I''}}, {{math|''f''(''x'')}} is the set of feasible solutions;
* given an instance {{mvar|x}} and a feasible solution {{mvar|y}} of {{mvar|x}}, {{math|''m''(''x'', ''y'')}} denotes the [[Measure (mathematics)|measure]] of {{mvar|y}}, which is usually a [[Positive (mathematics)|positive]] [[Real number|real]].
* {{mvar|g}} is the goal function, and is either {{math|[[Minimum (mathematics)|min]]}} or {{math|[[Maximum (mathematics)|max]]}}.

The goal is then to find for some instance {{mvar|x}} an ''optimal solution'', that is, a feasible solution {{mvar|y}} with
<math display=block>m(x, y) = g\left\{ m(x, y') : y' \in f(x) \right\}.</math>

For each combinatorial optimization problem, there is a corresponding [[decision problem]] that asks whether there is a feasible solution for some particular measure {{math|''m''<sub>0</sub>}}. For example, if there is a [[Graph (discrete mathematics)|graph]] {{mvar|G}} which contains vertices {{mvar|u}} and {{mvar|v}}, an optimization problem might be "find a path from {{mvar|u}} to {{mvar|v}} that uses the fewest edges". This problem might have an answer of, say, 4. A corresponding decision problem would be "is there a path from {{mvar|u}} to {{mvar|v}} that uses 10 or fewer edges?" This problem can be answered with a simple 'yes' or 'no'.

In the field of [[approximation algorithm]]s, algorithms are designed to find near-optimal solutions to hard problems. The usual decision version is then an inadequate definition of the problem since it only specifies acceptable solutions. Even though we could introduce suitable decision problems, the problem is more naturally characterized as an optimization problem.<ref name=Ausiello03>{{citation
 | last1 = Ausiello | first1 = Giorgio
 | year = 2003
 | edition = Corrected
 | title = Complexity and Approximation
 | publisher = Springer
 | isbn =  978-3-540-65431-5
|display-authors=etal}}</ref>