Optimization problem


In mathematics, 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:
The standard form of a continuous optimization problem is
where
If, 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 is a quadruple, where
The goal is then to find for some instance an optimal solution, that is, a feasible solution with
For each combinatorial optimization problem, there is a corresponding decision problem that asks whether there is a feasible solution for some particular measure. For example, if there is a graph which contains vertices and, an optimization problem might be "find a path from to 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 to 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.