Mathematical optimization

Mathematical optimization (also known as mathematical programming, optimization theory, or simply optimization) is a branch of applied mathematics and operations research that focuses on finding the best possible solution to a problem within a given set of constraints.

Overview

Mathematical optimization involves selecting the best element from a set of available alternatives. In the simplest case, this involves maximizing or minimizing a real function by systematically choosing input values from within an allowed set and computing the value of the function. More generally, optimization includes finding "best available" values of some objective function given a defined domain, including a variety of different types of objective functions and different types of domains.

Types of Mathematical Optimization

There are several types of mathematical optimization, including:

• Linear programming: This is a method to achieve the best outcome in a mathematical model whose requirements are represented by linear relationships.

• Nonlinear programming: This is a process of solving optimization problems where the objective function or the constraints, or both, are nonlinear.

• Integer programming: This is a mathematical optimization or feasibility program in which some or all of the variables are restricted to be integers.

• Convex programming: This is a subfield of optimization that studies the problem of minimizing convex functions over convex sets.

• Stochastic programming: This is a framework for modeling optimization problems that involve uncertainty.

Applications

Mathematical optimization has wide applications in fields such as economics, engineering, computer science, statistics, finance, logistics, and machine learning. It is used to find the most efficient and effective way to use resources, maximize output, minimize cost, and solve complex problems.

See also

• Operations research
• Game theory
• Decision theory
• Control theory
• Combinatorial optimization