The TOMLAB /CGO toolbox aims at efficiently solving global non-convex (integer) problems where the function f(x) is very costly to compute.

The toolbox consists of three general solvers:

• rbfSolve - using a Radial Basis Interpolation (RBF) algorithm.

• ego - using the Efficient Global Optimization (EGO) algorithm.

Response surface methods are discussed in a paper by Donald R. Jones:

A Taxonomy of Global optimization Methods Based on Response Surfaces Journal of Global Optimization 21 (4), 345:383, 2001.

Jones draws the conclusion that methods based on EGO and RBF algorithms are the most promising. The TOMLAB /CGO toolbox is based on these promising methods, and is continuously developed based on the state-of-the-art research in the field.

One example that motivated the research was industrial design of trains, where one f(x) value consumed 30 minutes to compute. The function value was the result of a simulation of 30 seconds of train ride. This problem is discussed in our paper:

Mattias Björkman, Kenneth Holmström: Global Optimization of Costly Nonconvex Functions Using Radial Basis Functions, Optimization and Engineering 1, 373-397, 2000.

The train design included costly nonlinear constraints. They were assigned a weight and added to the objective function. This approach showed to be very successful. The choice of weights was not very crucial.

## Main features:

• rbfSolve, EGO and arbfMIP have shown good results in practice on industrial and financial problems.

• The solvers are completely integrated in the TOMLAB Optimization Environment, and easy to combine with other solvers in TOMLAB.

• It is easy to use warm starts, and combine the EGO and (A)RBF solvers. The EGO solver sometimes have problem with ill-conditioning of the inverse of the correlation matrix when the number of sampled points grows. rbfSolve and arbfMIP have been shown to work without problem for up to 1000 sample points.

• The initial set of points is important. The TOMLAB /CGO solvers have several ways to generate initial points. The user may also specify any number of initial points to include.

• Both TOMLAB /CGO solvers have several algorithmic options that may be tuned for the particular class of problems.

• It is recommended to combine this toolbox with TOMLAB /NPSOL or TOMLAB /SOL, using NPSOL as a fast and robust local solver on the response surface.