Global algorithmic optimization
Introduction
Global optimization is a branch of applied mathematics and
numerical analysis that deals with the optimization of a function "Function (mathematical)") or a set of functions according to different criteria.
Typically, a set of more general bounds and constraints is also present, with the decision variables being optimized according to the constraints.
General
A model in its standard form is the minimization of a real function in the parameter space, or in the subspace defined by the constraints.
(The problem of maximizing the function
is equivalent to minimizing a function.).
In many non-linear optimization problems, the objective function has a large number of local minima and maxima, finding a local optimum is a simple task, using classical local optimization methods. In these cases, finding a global minimum or maximum is a task of greater complexity, since analytical methods cannot be used in the vast majority of cases and numerical strategies bring with them a large number of problems.
Applications
Some of the cases in which a global optimization approach should be used are:.
Deterministic methods
Contenido
Las estrategias más usadas son:.
Inner and outer approximation
In both strategies, the set on which the functions are going to be optimized is a polyhedron. In the interior approximation the polyhedron is contained in the set, while in the exterior approximation the polyhedron contains the set.
Cutting plans
Cutting plans is an umbrella term for optimization methods that iteratively refine a feasible set or objective function by adding linear inequalities, called cuts. These procedures are widely used to find integer solutions and mixed integer solutions of linear programming problems, as well as solving general convex optimization problems, not necessarily differentiable.