# Global optimization

Global optimization is a branch of applied mathematics and numerical analysis that deals with the optimization of a function or a set of functions to some criteria.

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## General

The most common form is the minimization of one real-valued function [itex]f(\vec{x})[itex] in the parameter-space [itex]\vec{x}\in P[itex]. There may be several constraints on the solution vectors [itex]\vec{x}_{min}[itex].

In real-life problems, functions of many variables have a large number of local minima and maxima. Finding an arbitrary local optimum is relatively straightforward by using local optimisation methods. Finding the global maximum or minimum of a function is a lot more challenging and has been impossible for many problems so far.

The maximization of a real-valued function [itex]g(x)[itex] can be regarded as the minimization of the transformed function [itex]f(x):=(-1)\cdot g(x)[itex].

## Applications of global optimization

Typical examples of global optimization applications include:

## Approaches

### Stochastic, thermodynamics

Several Monte-Carlo-based algorithms exist:

### Other random algorithms

Several other approaches include genetic algorithms, developed by Holland and others, and evolutionary strategies, developed by Schwefel et al.

## References

For simulated annealing:

• S. Kirkpatrick, C.D. Gelatt, and M.P. Vecchi. Science, 220:671–680, 1983.

For stochastic tunneling:

• K. Hamacher and W. Wenzel. The Scaling Behaviour of Stochastic Minimization Algorithms in a Perfect Funnel Landscape. Phys. Rev. E, 59(1):938-941, 1999.
• W. Wenzel and K. Hamacher. A Stochastic tunneling approach for global minimization. Phys. Rev. Lett., 82(15):3003-3007, 1999.

For parallel tempering:

• U. H. E. Hansmann. Chem.Phys.Lett., 281:140, 1997.

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