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Optimization problem

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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:

Continuous optimization problem[edit]

The standard form of a continuous optimization problem is[1] where

  • f : n 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[edit]

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

  • I is a set of instances;
  • given an instance xI, 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

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.[2]

See also[edit]

  • 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 problems
  • 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 relation
  • 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 constraints


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

External links[edit]