Mathematical optimization

Mathematical optimization, also called mathematical programming, is a way to find the best solution from many choices. It helps people make good decisions by looking at different options and choosing the best one based on certain rules or goals. This idea is useful in many areas, like solving problems with computers and machines or helping businesses and economies work better.

Optimization problems can be divided into two main types: discrete optimization, where the choices are separate and distinct, and continuous optimization, where the choices can be any value within a range. These problems appear in many fields that use numbers, such as computer science, engineering, operations research, and economics. For many years, mathematicians have worked on ways to solve these problems.

In general, an optimization problem means trying to make a number as big or as small as possible. This is done by choosing values to put into a formula and then seeing what number comes out. The study of these methods and ideas is a big part of applied mathematics, helping us understand and solve many real-world challenges.

Optimization problems

Main article: Optimization problem

Optimization problems are tasks where we try to find the best solution from a group of options. These problems can be split into two types based on the kind of values we can use: discrete and continuous. In discrete optimization, the values we can pick from are separate and distinct, like whole numbers or different arrangements. In continuous optimization, the values can change smoothly, like temperatures or speeds.

We can describe an optimization problem by stating a rule (called a function) that tells us how good each option is, and then looking for the option that makes this rule as small or as large as possible. For example, we might want to find the fastest route or the cheapest way to build something. Sometimes, there are extra rules (called constraints) that tell us which options are allowed. The best option we find is called the optimal solution.

Notation

Optimization problems use special symbols to show what we are trying to find. For example, we might want to find the smallest value of a math rule, like x² + 1, when x can be any real number. The smallest value here is 1, which happens when x is 0.

We can also look for the input values that give us the smallest or biggest result. For instance, we might search for the value of x that makes x² + 1 the smallest, but only when x is less than or equal to -1. In this case, the answer is x = -1.

Main article: Arg max

History

Mathematicians like Fermat and Lagrange found ways to get the best results using calculus. Later, Newton and Gauss made methods to move step-by-step toward the best answer.

The idea of "linear programming" was named by George B. Dantzig. Some of its theories were first shared by Leonid Kantorovich in 1939. Dantzig told people about his Simplex algorithm in 1947. Many other researchers have helped this field grow since then.

Major subfields

Mathematical optimization has two main types: discrete optimization and continuous optimization.

Discrete optimization solves problems where answers can only be specific values, like whole numbers. Continuous optimization solves problems where answers can change smoothly, like any value on a scale.

Important areas in optimization include:

• Linear programming: Both the goal and limits are straight-line equations.
• Integer programming: Some answers must be whole numbers.
• Nonlinear programming: Handles more complex curves and shapes in goals or limits.

There are also special methods for uncertainty:

• Stochastic programming: Handles random changes.
• Robust optimization: Finds solutions that work well even when data is unsure.

Classification of critical points and extrema

The feasibility problem is about finding any solution that works, without worrying about how good it is. Sometimes, we can start from any point and change it until it becomes a good solution.

There are special points in math where the best solution can be found. These points are where certain values stop changing. To tell if these points are the best solutions, we can look at how the values change around them. If the values get smaller, we have a minimum; if they get larger, we have a maximum. These ideas help us solve many kinds of problems.

Computational optimization techniques

To solve problems, researchers use special steps called algorithms. Some algorithms finish in a set number of steps. Others keep going until they get closer to the right answer.

There are also methods called heuristics. These might not always give the perfect answer but can be helpful in real situations.

One group of methods is called iterative methods. These are used for problems where the rules change in complex ways. These methods can be different depending on how much detail they need. Some need a lot of detail and can take more time. Others need less and might work faster for big problems. There are many types of these methods, like Newton's method and gradient descent. Each has its own uses and benefits.

Applications

Mathematical optimization is used in many different fields to find the best solution to a problem.

In mechanics, it helps solve problems about how objects move and interact, like making sure parts of a machine fit together properly.

In economics and finance, optimization helps people and businesses make the best decisions with limited resources. For example, consumers try to get the most satisfaction from what they buy, while companies aim to make the most profit. Optimization is also used in electrical engineering to design better circuits and antennas, and in civil engineering to plan roads and manage resources efficiently.

Operations research uses optimization to improve decision-making in business and industry. Control engineering applies optimization to design systems that can adjust themselves automatically. In geophysics, optimization helps scientists understand the properties of rocks and fluids from measurements like seismic waves. Optimization is also important in molecular modeling and machine learning, where it helps predict how molecules behave and improve computer algorithms.

Solvers

Main article: List of optimization software

Optimization problems are solved using special tools called solvers. These solvers help find the best answer from many choices. There are many types of optimization software. Each type is made to solve different kinds of problems. These tools are important in subjects like computer science, engineering, and economics. They help people make the best decisions.