Potential transformation methods for large-scale constrained global optimization

Abstract

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.