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Concept# Global optimization

Summary

Global optimization is a branch of applied mathematics and numerical analysis that attempts to find the global minima or maxima of a function or a set of functions on a given set. It is usually described as a minimization problem because the maximization of the real-valued function is equivalent to the minimization of the function .
Given a possibly nonlinear and non-convex continuous function with the global minima and the set of all global minimizers in , the standard minimization problem can be given as
that is, finding and a global minimizer in ; where is a (not necessarily convex) compact set defined by inequalities .
Global optimization is distinguished from local optimization by its focus on finding the minimum or maximum over the given set, as opposed to finding local minima or maxima. Finding an arbitrary local minimum is relatively straightforward by using classical local optimization methods. Finding the global minimum of a function is far more difficult: analytical methods are frequently not applicable, and the use of numerical solution strategies often leads to very hard challenges.
Typical examples of global optimization applications include:
Protein structure prediction (minimize the energy/free energy function)
Computational phylogenetics (e.g., minimize the number of character transformations in the tree)
Traveling salesman problem and electrical circuit design (minimize the path length)
Chemical engineering (e.g., analyzing the Gibbs energy)
Safety verification, safety engineering (e.g., of mechanical structures, buildings)
Worst-case analysis
Mathematical problems (e.g., the Kepler conjecture)
Object packing (configuration design) problems
The starting point of several molecular dynamics simulations consists of an initial optimization of the energy of the system to be simulated.
Spin glasses
Calibration of radio propagation models and of many other models in the sciences and engineering
Curve fitting like non-linear least squares analysis and other generalizations, used in fitting model parameters to experimental data in chemistry, physics, biology, economics, finance, medicine, astronomy, engineering.

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Global optimization

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