Mathematical optimization

We want to find the best choice.

Sometimes we want the most of something. Other times we want the least.

We look at many different options. Some options are whole numbers. Other options can be any amount.

We use rules to find the best answer. This can help us solve big puzzles. It works in science and math.

People have studied this for a long time. It helps us plan things well. We can find the best way to do a job.

This helps us pick the best path.

Imagine you want to find the very best choice from many options.

This is called mathematical optimization. You might want the most of something, like profit. Or you might want the least of something, like error or cost.

There are two main ways to look at these choices. One way uses discrete variables. These are whole things you can count, like a set of numbers. The other way uses continuous variables. These can be any amount, like a smooth line.

Sometimes, it is hard to find the absolute best answer. You might find a local minimum. This is a spot that is better than the choices right next to it. But it might not be the global minimum. The global minimum is the best choice in the whole set.

Many people have worked on this for a long time. Fermat and Lagrange used math rules to find these spots. Newton and Gauss used steps to move toward the best answer. In 1947, George B. Dantzig shared the Simplex algorithm. This helps solve a type of math called linear programming.

Imagine you are looking for the very best way to do something. You might want to find the most profit or the lowest cost. This process of picking the best choice from many options is called mathematical optimization.

In math, we use a special rule called an objective function to measure how good a choice is. If we want the lowest value, we call it minimization. If we want the highest value, we call it maximization. People use these tools in many fields, such as computer science, engineering, and economics.

There are two main ways to group these math problems. The first way uses discrete variables. These are things you can count, like whole numbers or specific shapes. The second way uses continuous variables. These can be any value along a smooth line or space.

Some problems have constraints, which are rules you must follow. A solution that follows all the rules is called a feasible solution. When a solution is the absolute best possible, it is called an optimal solution.

Finding the best answer can sometimes be a hard job. You might find a local minimum. This is a spot that is better than the choices right next to it. However, it might not be the best choice in the entire group. The global minimum is the best choice out of every possible option. In some math problems, called convex problems, a local minimum is also the global minimum. But in other problems, there might be many local spots that are not the true best.

Many smart people have studied these patterns for centuries. Fermat and Lagrange found math formulas to help identify these best spots. Newton and Gauss suggested ways to move step by step toward an answer. In 1939, Leonid Kantorovich introduced some of the ideas used in linear programming. Later, in 1947, George B. Dantzig published the Simplex algorithm. Dantzig studied these problems while working with the United States military on schedules. John von Neumann also worked on the theory of these math problems around that same time.

Optimization is used in many parts of our daily world. In machine learning, it helps find the lowest error in a data model. In physics, scientists use it to find the lowest energy in a system. It can even help with scheduling tasks or finding the best path. Whether it is splitting a pizza or running a huge company, math helps us find the best way. It turns a big list of choices into one clear, winning answer.

Mathematical optimization is the science of selecting the best possible element from a set of available alternatives. It is a vital part of applied mathematics used to make decisions based on specific criteria.

In practice, this means finding the highest or lowest value of a mathematical rule. This rule is called an objective function. Depending on the goal, the process is called maximization or minimization. If the goal is to find the lowest value, it is often called a cost function or loss function. If the goal is to find the highest value, it is called a utility or fitness function. Optimization helps experts in computer science, engineering, and economics solve complex problems.

To solve these problems, mathematicians work within a defined search space. This space contains all the possible choices, known as candidate solutions or feasible solutions. A solution is only considered feasible if it satisfies certain rules called constraints. These constraints can be equalities or inequalities that limit the choices. When a candidate solution perfectly satisfies the objective function by being the best, it is called an optimal solution. In many fields, such as physics, optimization is used to find energy minimization. This means finding the state where a system has the lowest possible energy.

Optimization problems are divided into two primary categories based on their variables. The first type is discrete optimization. In this category, the variables are countable, such as integers, permutations, or graphs. The second type is continuous optimization. This involves finding optimal arguments from a continuous set, such as any real number on a line. Problems can also be classified by their shape or structure. For example, convex problems are often easier to solve than nonconvex problems. In a convex minimization problem, any local minimum is also the global minimum. However, nonconvex problems may have many local minima that are not the absolute best.

A local minimum is an element that is better than all nearby elements. A global minimum is the absolute best element among every feasible choice in the entire set.

Many modern computer solvers struggle to distinguish between these two types of solutions. They may stop at a local minimum and treat it as the final answer. Global optimization is a specific branch of mathematics dedicated to this challenge. It develops deterministic algorithms that can guarantee they will find the true global solution in a finite amount of time.

The history of optimization is filled with significant mathematical discoveries. Early mathematicians like Fermat and Lagrange developed calculus-based formulas to identify optimal points. Later, Newton and Gauss proposed iterative methods to move step-by-step toward an optimum. In 1939, Leonid Kantorovich introduced important theories related to linear programming. In 1947, George B. Dantzig published the Simplex algorithm. Dantzig developed this while studying logistics and training schedules for the United States military. Around this same time, John von Neumann and other researchers worked on the theoretical aspects of linear programming, such as the theory of duality.

There are many specialized subfields within optimization that address different mathematical structures. Linear programming is a common type of convex programming where the objective function and constraints are linear. Integer programming is more difficult because it requires some variables to be whole numbers. Nonlinear programming handles cases where the functions or constraints are not straight lines. Other advanced methods include semidefinite programming, which uses semidefinite matrices, and stochastic programming, which accounts for random variables. Some techniques, known as heuristics, are used to find approximate solutions for very complicated problems when a perfect answer is too hard to find.

Optimization connects to many different scientific and technical systems. In machine learning, it is used to evaluate the quality of data models by minimizing error. In the field of artificial intelligence, constraint satisfaction is used for automated reasoning. Scientists also use the calculus of variations to find the best way to achieve a goal, such as finding a surface with the least possible area. Optimal control theory and dynamic programming are used to make decisions over time. By turning complex choices into mathematical models, optimization provides a clear path to the best possible results in a changing world.