Optimization

Optimization or mathematical optimization is a branch of mathematics that deals with finding the best solution from a set of available alternatives. It involves selecting the best element from some set of available alternatives. In the simplest case, this involves linearly ordering the elements from best to worst and choosing the best one. More generally, it involves the study of mathematical structures that have some notion of a best element.

Overview

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, while an optimization problem with continuous variables is known as a continuous optimization.

In a discrete optimization problem, we are looking for an object such as an integer, permutation or graph from a finite or countably infinite set. Examples of such problems are the traveling salesman problem and the knapsack problem.

In a continuous optimization problem, we are looking for a real number or a vector of real numbers. Examples of such problems are the linear programming and quadratic programming.

Applications

Optimization has wide applications in many fields, including economics, engineering, physics, and computer science. In economics, optimization techniques are used to find the best allocation of resources. In engineering, they are used to design systems that operate in the most efficient and effective way. In physics, they are used to predict the behavior of physical systems. In computer science, they are used to design algorithms that solve complex problems in the most efficient way.

See also

• Linear programming
• Quadratic programming
• Discrete optimization
• Continuous optimization
• Traveling salesman problem
• Knapsack problem

References

Optimization Resources
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