Optimization problem: Difference between revisions

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{{Broader|Mathematical optimisationoptimization}}

In [[mathematics]], [[computer science]] and [[economics]], an '''optimisationoptimization problem''' is the [[Computational problem|problem]] of finding the ''best'' solution from all [[feasible solution]]s.

OptimisationOptimization problems can be divided into two categories, depending on whether the [[Variable (mathematics)|variables]] are [[continuous variable|continuous]] or [[discrete variable|discrete]]:

* An optimisationoptimization problem with discrete variables is known as a ''[[discrete optimisationoptimization]]'', in which an [[Mathematical object|object]] such as an [[integer]], [[permutation]] or [[Graph (discrete mathematics)|graph]] must be found from a [[countable set]].

* A problem with continuous variables is known as a ''[[continuous optimisationoptimization]]'', in which an optimal value from a [[continuous function]] must be found. They can include [[Constrained optimisationoptimization|constrained problem]]s and multimodal problems.

==Continuous optimisationoptimization problem==

The ''[[Canonical form|standard form]]'' of a [[Continuity (mathematics)|continuous]] optimisationoptimization problem is<ref>{{cite book|title=Convex OptimisationOptimization|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>\begin{align}

&\underset{x}{\operatorname{minimize}}& & f(x) \\

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* {{math|''m'' ≥ 0}} and {{math|''p'' ≥ 0}}.

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

==Combinatorial optimisationoptimization problem==

{{Main|Combinatorial optimization}}

Formally, a [[combinatorial optimisationoptimization]] 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;

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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''₀}}. 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 characterisedcharacterized as an optimisationoptimization problem.<ref name=Ausiello03>{{citation

| last1 = Ausiello | first1 = Giorgio

| year = 2003

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==See also==

*[[Counting problem (complexity)]]

*[[Design OptimisationOptimization]]

*[[Function problem]]

*[[Glove problem]]