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.
| Contents |
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>.
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 <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 design (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.
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.