Conservative surrogate models for optimization with the active subspace method

Low-dimensional surrogate models are useful for reducing optimization costs. However, when using them to approximate constraints, additional conditions are needed to guarantee that the optimum will satisfy the constraints of the full-size model. One such way is to make the surrogate conservative. The surrogate model is constructed using a Gaussian process regression. To ensure conservativeness, two new approaches are proposed: the first one using a concentration inequality, and the second one using bootstrapping. Those two techniques are based on a stochastic argument and thus will only enforce conservativeness up to a user-defined probability threshold. The method is applied in the context of optimization using the active subspace method for dimensionality reduction. It addresses recorded issues about constraint violations when nonlinear constraints are replaced by low-dimensional surrogate models. The resulting algorithms are tested on a toy optimization problem in thermal design.