Using Estimated Gradients in Bound-Constrained Global Optimization

A method for bound-constrained global optimization which uses estimated gradients is presented. The objective function is assumed to have a Lipschitz continuous gradient. This method alternates between a local trust region method using a quasi-Newton Hessian approximation and random search on the feasible region \(\Omega \) $\mathrm{\Omega }$ and selected subsets of \(\Omega \) $\mathrm{\Omega }$ . The local trust region subproblem is incompletely solved using Steihaug-Toint conjugate gradients. Convergence is shown almost surely, and the local trust region algorithm’s convergence properties are shown independently. The method was tested on 55 standard test problems in 2–60 dimensions, and further randomly generated Schoen functions. Results show the method is competitive and robust in the face of ill-conditioning and that using estimated or analytic gradients can be beneficial.