Optimization - Optimization Problems

Optimization Problems

An optimization problem can be represented in the following way

Given: a function f : A R from some set A to the real numbers

Sought: an element x₀ in A such that f(x₀) ≤ f(x) for all x in A ("minimization") or such that f(x₀) ≥ f(x) for all x in A ("maximization").

Such a formulation is called an optimization problem or a mathematical programming problem (a term not directly related to computer programming, but still in use for example in linear programming – see History below). Many real-world and theoretical problems may be modeled in this general framework. Problems formulated using this technique in the fields of physics and computer vision may refer to the technique as energy minimization, speaking of the value of the function f as representing the energy of the system being modeled.

Typically, A is some subset of the Euclidean space Rn, often specified by a set of constraints, equalities or inequalities that the members of A have 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, cost function (minimization), indirect utility function (minimization), utility function (maximization), or, in certain fields, energy function, or energy functional. A feasible solution that minimizes (or maximizes, if that is the goal) the objective function is called an optimal solution.

By convention, the standard form of an optimization problem is stated in terms of minimization. Generally, unless both the objective function and the feasible region are convex in a minimization problem, there may be several local minima, where a local minimum x* is defined as a point for which there exists some δ > 0 so that for all x such that

the expression

holds; that is to say, on some region around x* all of the function values are greater than or equal to the value at that point. Local maxima are defined similarly.

A large number of algorithms proposed for solving non-convex problems – including the majority of commercially available solvers – are not capable of making a distinction between local optimal solutions and rigorous optimal solutions, and will treat the former as actual solutions to the original problem. 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 non-convex problem is called global optimization.