Optimization problem

Inmathematics,engineering,computer scienceandeconomics,anoptimization problemis theproblemof finding thebestsolution from allfeasible solutions.

Optimization problems can be divided into two categories, depending on whether thevariablesarecontinuousordiscrete:

An optimization problem with discrete variables is known as adiscrete optimization,in which anobjectsuch as aninteger,permutationorgraphmust be found from acountable set.
A problem with continuous variables is known as acontinuous optimization,in which an optimal value from acontinuous functionmust be found. They can includeconstrained problemsand multimodal problems.

Continuous optimization problem

Thestandard formof acontinuousoptimization problem is^[1] ${\begin{aligned}&{\underset {x}{\operatorname {minimize} }}&&f(x)\\&\operatorname {subject\;to} &&g_{i}(x)\leq 0,\quad i=1,\dots,m\\&&&h_{j}(x)=0,\quad j=1,\dots,p\end{aligned}}$ where

$f : ℝ n \to ℝ$ is theobjective functionto be minimized over the $n$ -variable vector $x$ ,
$g i (x) \leq 0$ are calledinequalityconstraints
$h j (x) = 0$ are calledequality constraints,and
$m \geq 0$ and $p \geq 0$ .

If $m = p = 0$ ,the problem is an unconstrained optimization problem. By convention, the standard form defines aminimization problem.Amaximization problemcan be treated bynegatingthe objective function.

Combinatorial optimization problem

Formally, acombinatorial optimizationproblem $A$ is a quadruple^{[citation needed]} $(I, f, m, g)$ ,where

$I$ is asetof instances;
given an instance $x \in I$ , $f (x)$ is the set of feasible solutions;
given an instance $x$ and a feasible solution $y$ of $x$ , $m (x, y)$ denotes themeasureof $y$ ,which is usually apositive real.
$g$ is the goal function, and is either $min$ or $max$ .

The goal is then to find for some instance $x$ anoptimal solution,that is, a feasible solution $y$ with $m(x,y)=g\left\{m(x,y'):y'\in f(x)\right\}.$

For each combinatorial optimization problem, there is a correspondingdecision problemthat asks whether there is a feasible solution for some particular measure $m 0$ .For example, if there is agraph $G$ which contains vertices $u$ and $v$ ,an optimization problem might be "find a path from $u$ to $v$ 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 $u$ to $v$ that uses 10 or fewer edges? "This problem can be answered with a simple 'yes' or 'no'.

In the field ofapproximation algorithms,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.^[2]

References

^Boyd, Stephen P.; Vandenberghe, Lieven (2004).Convex Optimization(pdf).Cambridge University Press. p. 129.ISBN 978-0-521-83378-3.
^Ausiello, Giorgio; et al. (2003),Complexity and Approximation(Corrected ed.), Springer,ISBN 978-3-540-65431-5

External links

"How Traffic Shaping Optimizes Network Bandwidth".IPC.12 July 2016.Retrieved13 February2017.

[1] Boyd, Stephen P.; Vandenberghe, Lieven (2004).Convex Optimization(pdf).Cambridge University Press. p. 129.ISBN 978-0-521-83378-3.

[Ausiello03-2] Ausiello, Giorgio; et al. (2003),Complexity and Approximation(Corrected ed.), Springer,ISBN 978-3-540-65431-5

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Optimization problem

Contents

Continuous optimization problem

Combinatorial optimization problem

See also

References

External links