Optimization problem: Difference between revisions

Revision as of 10:04, 6 April 2021

Revision as of 10:05, 6 April 2021

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

==Combinatorial optimizationoptimisation problem==

{{Main|Combinatorial optimization}}

Formally, a [[combinatorial optimizationoptimisation]] 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 characterizedcharacterised as an optimizationoptimisation problem.<ref name=Ausiello03>{{citation

| last1 = Ausiello | first1 = Giorgio

| year = 2003