# 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 according to some criteria. Typically, a set of bound and more general constraints is also present, and the decision variables are optimized considering also the constraints.

## General

A common (standard) model form is the minimization of one real-valued function $f$ in the parameter-space $\vec{x}\in P$, or its specified subset $\vec{x}\in D$: here $D$ denotes the set defined by the constraints.

(The maximization of a real-valued function $g(x)$ is equivalent to the minimization of the function $f(x):=(-1)\cdot g(x)$.)

In many nonlinear optimization problems, the objective function $f$ has a large number of local minima and maxima. Finding an arbitrary local optimum is relatively straightforward by using classical local optimization methods. Finding the global minimum (or maximum) of a function is far more difficult: symbolic (analytical) methods are frequently not applicable, and the use of numerical solution strategies often leads to very hard challenges.

## Applications of global optimization

Typical examples of global optimization applications include:

## Approaches

### Deterministic methods

The most successful general strategies are:

### Stochastic methods

Main page: Stochastic optimization

Several Monte-Carlo-based algorithms exist:

### Heuristics and metaheuristics

Main page: Metaheuristic

Other approaches include heuristic strategies to search the search space in a more or less intelligent way, including:

### Response surface methodology based approaches

• IOSO Indirect Optimization based on Self-Organization

## Global optimization software

1. Free and opensource:

2. Commercial:

## References

Deterministic global optimization:

• R. Horst, H. Tuy, Global Optimization: Deterministic Approaches, Springer, 1996.
• R. Horst, P.M. Pardalos and N.V. Thoai, Introduction to Global Optimization, Second Edition. Kluwer Academic Publishers, 2000.
• A.Neumaier, Complete Search in Continuous Global Optimization and Constraint Satisfaction, pp. 271-369 in: Acta Numerica 2004 (A. Iserles, ed.), Cambridge University Press 2004.
• M. Mongeau, H. Karsenty, V. Rouzé and J.-B. Hiriart-Urruty, Comparison of public-domain software for black box global optimization. Optimization Methods & Software 13(3), pp. 203–226, 2000.
• J.D. Pintér, Global Optimization in Action - Continuous and Lipschitz Optimization: Algorithms, Implementations and Applications. Kluwer Academic Publishers, Dordrecht, 1996. Now distributed by Springer Science and Business Media, New York. This book also discusses stochastic global optimization methods.
• L. Jaulin, M. Kieffer, O. Didrit, E. Walter (2001). Applied Interval Analysis. Berlin: Springer.
• E.R. Hansen (1992), Global Optimization using Interval Analysis, Marcel Dekker, New York.
• R.G. Strongin, Ya.D. Sergeyev (2000) Global optimization with non-convex constraints: Sequential and parallel algorithms, Kluwer Academic Publishers, Dordrecht.
• Ya.D. Sergeyev, R.G. Strongin, D. Lera (2013) Introduction to global optimization exploiting space-filling curves, Springer, NY.

For simulated annealing:

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

For reactive search optimization:

• Roberto Battiti, M. Brunato and F. Mascia, Reactive Search and Intelligent Optimization, Operations Research/Computer Science Interfaces Series, Vol. 45, Springer, November 2008. ISBN 978-0-387-09623-0

For stochastic methods:

• A. Zhigljavsky. Theory of Global Random Search. Mathematics and its applications. Kluwer Academic Publishers. 1991.
• K. Hamacher. Adaptation in Stochastic Tunneling Global Optimization of Complex Potential Energy Landscapes, Europhys.Lett. 74(6):944, 2006.
• 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.

For continuation methods:

• Zhijun Wu. The effective energy transformation scheme as a special continuation approach to global optimization with application to molecular conformation. Technical Report, Argonne National Lab., IL (United States), November 1996.

For general considerations on the dimensionality of the domain of definition of the objective function:

• K. Hamacher. On Stochastic Global Optimization of one-dimensional functions. Physica A 354:547-557, 2005.