optimization problem

      Optimization problem GudangMovies21 Rebahinxxi LK21

      In mathematics, engineering, computer science and economics, an optimization problem is the problem of finding the best solution from all feasible solutions.
      Optimization problems can be divided into two categories, depending on whether the variables are continuous or discrete:

      An optimization problem with discrete variables is known as a discrete optimization, in which an object such as an integer, permutation or graph must be found from a countable set.
      A problem with continuous variables is known as a continuous optimization, in which an optimal value from a continuous function must be found. They can include constrained problems and multimodal problems.


      Continuous optimization problem


      The standard form of a continuous optimization problem is











      minimize
      x





      f
      (
      x
      )






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      t

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      g

      i


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      x
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      0
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      i
      =
      1
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      …
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      m








      h

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      {\displaystyle {\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 : ℝn → ℝ is the objective function to be minimized over the n-variable vector x,
      gi(x) ≤ 0 are called inequality constraints
      hj(x) = 0 are called equality constraints, and
      m ≥ 0 and p ≥ 0.
      If m = p = 0, the problem is an unconstrained optimization problem. By convention, the standard form defines a minimization problem. A maximization problem can be treated by negating the objective function.


      Combinatorial optimization problem



      Formally, a combinatorial optimization problem A is a quadruple (I, f, m, g), where

      I is a set of instances;
      given an instance x ∈ I, f(x) is the set of feasible solutions;
      given an instance x and a feasible solution y of x, m(x, y) denotes the measure of y, which is usually a positive real.
      g is the goal function, and is either min or max.
      The goal is then to find for some instance x an optimal solution, that is, a feasible solution y with




      m
      (
      x
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      )
      =
      g

      {

      m
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      .


      {\displaystyle m(x,y)=g\left\{m(x,y'):y'\in f(x)\right\}.}


      For each combinatorial optimization problem, there is a corresponding decision problem that asks whether there is a feasible solution for some particular measure m0. For example, if there is a graph 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 of approximation 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.


      See also


      Counting problem (complexity) – Type of computational problem
      Design Optimization
      Ekeland's variational principle – theorem that asserts that there exist nearly optimal solutions to some optimization problemsPages displaying wikidata descriptions as a fallback
      Function problem – Type of computational problem
      Glove problem
      Operations research – Discipline concerning the application of advanced analytical methods
      Satisficing – Cognitive heuristic of searching for an acceptable decision − the optimum need not be found, just a "good enough" solution.
      Search problem – type of computational problem represented by a binary relationPages displaying wikidata descriptions as a fallback
      Semi-infinite programming – optimization problem with a finite number of variables and an infinite number of constraints, or an infinite number of variables and a finite number of constraintsPages displaying wikidata descriptions as a fallback


      References




      External links


      "How Traffic Shaping Optimizes Network Bandwidth". IPC. 12 July 2016. Retrieved 13 February 2017.

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