Convergence of Hybrid Space Mapping Algorithms

The space mapping technique is intended for optimization of engineering models which involve very expensive function evaluations. It may be considered a preprocessing method which often provides a very efficient initial phase of an optimization procedure. However, the ultimate rate of convergence may be poor, or the method may even fail to converge to a stationary point. We consider a convex combination of the space mapping technique with a classical optimization technique. The function to be optimized has the form \$H \$\backslash\$circ f\$ where \$H: \$\backslash\$dR\^m \$\backslash\$mapsto \$\backslash\$dR\$ is convex and \$f: \$\backslash\$dR\^n \$\backslash\$mapsto \$\backslash\$dR\^m\$ is smooth. Experience indicates that the combined method maintains the initial efficiency of the space mapping technique. We prove that the global convergence property of the classical technique is also maintained: The combined method provides convergence to the set of stationary points of \$H \$\backslash\$circ f\$.