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Derivative-free optimization problems are optimization problems wherederivative information is unavailable. The least Frobenius norm updatingquadratic interpolation model function is one of the essential under-determinedmodel functions for model-based derivative-free trust-region methods. Thisarticle proposes derivative-free optimization with transformed objectivefunctions 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 thesame framework with those for problems without transformations, and its queryscheme is given. We propose the definitions related to optimality-preservingtransformations to understand the interpolation model in our method. We provethe existence of model optimality-preserving transformations beyond translationtransformation. The necessary and sufficient condition for such transformationsis given. The affine transformation with a positive multiplication coefficientis not model optimality-preserving. We also analyze the corresponding leastFrobenius norm updating model and its interpolation error when the objectivefunction is affinely transformed. Convergence property of a provablealgorithmic framework containing our model is given. Numerical results ofsolving test problems and a real-world problem with the implementationNEWUOA-Trans show that our method can successfully solve most problems withobjective optimality-preserving transformations, even though suchtransformations will change the optimality of the model function. To our bestknowledge, this is the first work providing the model-based derivative-freealgorithm and analysis for transformed problems with the function evaluationoracle (not the function-value comparison oracle). This article also proposesthe "moving-target" optimization problem.

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