Global optimization - Definition and Overview

Global optimization is a branch of applied mathematics and numerics that deals with the optimization of a function or 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)
• Simulated annealing
• 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.
• Spin glasses

Approaches

Stochastic, thermodynamics

Several Monte-Carlo-based algorithms exist:

• Direct Monte-Carlo sampling
• Stochastic tunneling
• Parallel tempering
• Monte-Carlo with minimization

Other random algorithms

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