DIRECT using local search on surrogates

The solution of noisy nonlinear optimization problems with nonlinear constraints and derivative information is becoming increasingly important, as many practical applications can be described by this type of problem in e.g., engineering applications. Existing local optimization methods show good convergence properties. However, they often depend on sufficiently good starting points and/or the approximation of gradients. In turn, global derivative free methods, which need no starting values to be initialized, require many evaluations of the objective function, particularly in the vicinity of the solution. A derivative free optimization algorithm is developed that combines advantages of both local and global methods. The DIRECT algorithm, which is often used for problems where no prior knowledge is available as kind of a brute force start, is extended by an inner loop using a surrogate based optimization method. The local search on the surrogate function determines better candidates for sampling than the hypercube center points chosen by DIRECT, especially if constraints are arising. This inner loop needs no additional evaluation of the original problem. Standard test problems and a computational more expensive test problem are chosen to show the performance of the new algorithm