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Global optimization

Global Optimization is a branch of applied mathematics and numerics that deals with the optimization of a function/a set of functions to some criteria.

General

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

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

Applications of Global Optimization

Typical examples of global optimization applications include:

• Protein structure prediction (minimize the energy/free energy function) and
• Traveling salesman problem and circuit desing (minimize the path length).
• The starting point of several Molecular dynamics-simulations consists of an initial optimization of the energy of the system to be simulated, too.
• Spin glasses

Approaches

Stochastic / Thermodynamics

There are several Monte-Carlo-based algorithms such as:

• direct Monte-Carlo sampling
• Simulated annealing
• Stochastic tunneling
• Parallel Tempering
• Monte-Carlo-with-Minimization

Other Random Algorithms

There are several other approaches including genetic algorithms due to Holland and others and evolutionary strategies due to Schwefel et al.

Deterministic

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.