Potential transformation methods for large-scale constrained global optimization
Several techniques for global optimization treat the objective functionf as a force-field potential. In the simplest case, trajectories of the differential equationmx=−Δf sample regions of low potential while retaining the energy to surmount passes which might block the way to regions of even lower local minima. Apotential transformation is an increasing functionV:ℝ→ℝ. It determines a new potentialg=V(f), with the same minimizers asf and new trajectories satisfying\(m\ddot x = - \nabla g = - (dV/df)\nabla f\). We discuss a class of potential transformations that greatly increase the attractiveness of low local minima.
These methods can be applied to constrained problems through the use of Lagrange multipliers. We discuss several methods for efficiently computing approximate Lagrange multipliers, making this approach practical.
KeywordsConstrained optimization global optimization molecular dynamics Newtonian dynamics potential transformation methods PT methods
AMS subject classification49D10 90C30 70D05
Unable to display preview. Download preview PDF.
- R.L. Burden and J.D. Faires,Numerical Analysis (PWS-Kent, Boston, 4th ed., 1989).Google Scholar
- R.A. Donnelly, Generalized descent for geometry optimization, Unpublished manuscript (1987).Google Scholar
- H. Goldstein,Classical Mechanics (Addison-Wesley, Reading, 2nd ed., 1980).Google Scholar
- A.O. Griewank, Generalized descent for global optimization, J. Optim. Theory Appl. 34 (1981) 11–39.Google Scholar
- D.G. Luenberger,Linear and Nonlinear Programming, (Addison-Wesley, Reading, 2nd ed., 1973).Google Scholar
- J.W. Rogers, Jr. and R.A. Donnelly, Potential transformation methods for large-scale global optimization (1992) submitted.Google Scholar
- J.P. Ryckaert, G. Ciccotti and H.J.C. Berendsen, Numerical integration of the Cartesian equations of motion of a system with constraints: Molecular dynamics of n-alkanes, J. Comp. Phys. 23 (1977) 327–341.Google Scholar
- M. Yoneya and H.J.C. Berendsen, Non-iterative SHAKE: A new algorithm for constraint molecular dynamics simulation (1993) submitted.Google Scholar