Mathematical Optimization

Mathematical optimization is a branch of applied mathematics which involves selecting the best element from a set of available alternatives.

In the simplest case, an optimization problem consists of maximizing or minimizing a real function by systematically choosing input values from within an allowed set and computing the value of the function. The generalization of optimization theory and techniques to other formulations constitutes a large area of applied mathematics.

Types of Mathematical Optimization

There are several types of optimization problems, including:

  • Linear Optimization: The function to be maximized or minimized is linear, and the set of allowable solutions is defined by linear equality and inequality constraints.

  • Nonlinear Optimization: The function to be maximized or minimized or the constraints include nonlinear components. This can be significantly more challenging than linear optimization.

  • Integer Optimization: Some or all of the decision variables are required to be integer-valued.

  • Stochastic Optimization: The function to be maximized or minimized is subject to random shocks.

  • Combinatorial Optimization: The goal is to find the best arrangement of a finite number of items, given some constraints. Examples include the traveling salesperson problem and the problem of factoring large numbers.

Uses of Mathematical Optimization

Optimization techniques are used in many areas of science, engineering, economics, and industry. For example, they may be used to minimize the cost of producing a product, to maximize the efficiency of an energy system, to design an optimal control strategy for a robotic system, or to determine the best investment strategy for a portfolio of financial assets.

However, optimization problems are often computationally challenging, and many real-world problems involve a combination of discrete and continuous variables, nonlinear relationships, and stochastic elements. Therefore, a wide variety of numerical optimization algorithms have been developed to solve different types of optimization problems.