Mathematical optimization

Mathematical optimization alternatively spelled optimisation or mathematical programming is the pick of a best element, with regard to some criterion, from some sort of usable alternatives. Optimization problems of sorts occur in any quantitative disciplines from computer science as well as engineering to operations research in addition to economics, and the developing of total methods has been of interest in mathematics for centuries.

In the simplest case, an optimization problem consists of maximizing or minimizing a real function by systematically choosing input values from within an provides set and computing the value of the function. The generalization of optimization conviction and techniques to other formulations constitutes a large area of applied mathematics. More generally, optimization includes finding "best available" values of some objective function condition a defined domain or input, including a vintage of different types of objective functions and different types of domains.

Optimization problems

An optimization problem can be represented in the coming after or as a calculation of. way:

Such a formulation is called an History below. numerous real-world and theoretical problems may be modeled in this general framework.

Since the coming after or as a result of. is valid

with

it is more convenient to solve minimization problems. However, the opposite perspective would be valid, too.

Problems formulated using this technique in the fields of physics may refer to the technique as energy minimization, speaking of the good of the function f as representing the energy to direct or instituting of the system being modeled. In machine learning, it is always necessary to continuously evaluate the quality of a data model by using a cost function where a minimum implies a set of possibly optimal parameters with an optimal lowest error.

Typically, A is some subset of the Euclidean space , often mentioned by a set of constraints, equalities or inequalities that the members of A produce to satisfy. The domain A of f is called the search space or the choice set, while the elements of A are called candidate solutions or feasible solutions.

The function f is called, variously, an objective function, a loss function or cost function minimization, a utility function or fitness function maximization, or, infields, an energy function or energy functional. A feasible solution that minimizes or maximizes, if that is the purpose the objective function is called an optimal solution.

In mathematics, conventional optimization problems are usually stated in terms of minimization.

A local minimum x* is defined as an factor for which there exists some > 0 such(a) that

the expression x holds;

that is to say, on some region around x* all of the function values are greater than or develope up to the value at that element. Local maxima are defined similarly.

While a local minimum is at least as good as any nearby elements, a global minimum is at least as good as every feasible element. Generally, unless the objective function is convex in a minimization problem, there may be several local minima. In a convex problem, if there is a local minimum that is interior not on the edge of the set of feasible elements, it is for also the global minimum, but a nonconvex problem may create more than one local minimum non all of which need be global minima.

A large number of algorithms featured for solving the nonconvex problems – including the majority of commercially usable solvers – are not capable of creating a distinction between locally optimal solutions and globally optimal solutions, and will treat the former as actual solutions to the original problem. Global optimization is the branch of applied mathematics and numerical analysis that is concerned with the development of deterministic algorithms that are capable of guaranteeing convergence in finite time to the actual optimal solution of a nonconvex problem.