Optimization problem

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 $$\begin{align} &\underset{x}{\operatorname{minimize}}& & f(x) \\ &\operatorname{subject\;to} & &g_i(x) \leq 0, \quad i = 1,\dots,m \\ &&&h_j(x) = 0, \quad j = 1, \dots,p \end{align}$$ where
 * $f : ℝ^{n} → ℝ$ is the objective function to be minimized over the $n$-variable vector $x$,
 * $g_{i}(x) ≤ 0$ are called inequality constraints
 * $h_{j}(x) = 0$ 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 $$m(x, y) = g\left\{ m(x, y') : y' \in f(x) \right\}.$$

For each combinatorial optimization problem, there is a corresponding decision problem that asks whether there is a feasible solution for some particular measure $m_{0}$. 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.