Derivative-Free Optimization with Transformed Objective Functions (DFOTO) and the Algorithm Based on the Least Frobenius Norm Updating Quadratic Model

Abstract: Derivative-free optimization problems are optimization problems where derivative information is unavailable. The least Frobenius norm updating quadratic interpolation model function is one of the essential under-determined model functions for model-based derivative-free trust-region methods. This article proposes derivative-free optimization with transformed objective functions and gives a trust-region method with the least Frobenius norm model. The model updating formula is based on Powell's formula. The method shares the same framework with those for problems without transformations, and its query scheme is given. We propose the definitions related to optimality-preserving transformations to understand the interpolation model in our method. We prove the existence of model optimality-preserving transformations beyond translation transformation. The necessary and sufficient condition for such transformations is given. The affine transformation with a positive multiplication coefficient is not model optimality-preserving. We also analyze the corresponding least Frobenius norm updating model and its interpolation error when the objective function is affinely transformed. Convergence property of a provable algorithmic framework containing our model is given. Numerical results of solving test problems and a real-world problem with the implementation NEWUOA-Trans show that our method can successfully solve most problems with objective optimality-preserving transformations, even though such transformations will change the optimality of the model function. To our best knowledge, this is the first work providing the model-based derivative-free algorithm and analysis for transformed problems with the function evaluation oracle (not the function-value comparison oracle). This article also proposes the "moving-target" optimization problem.