  We are interested in building low-dimensional surrogate models to reduce
optimization costs, while having theoretical guarantees that the optimum will
satisfy the constraints of the full-size model, by making conservative
approximations. The surrogate model is constructed using a Gaussian process
regression (GPR). To ensure conservativeness, two new approaches are proposed:
the first one using bootstrapping, and the second one using concentration
inequalities. Those two techniques are based on a stochastic argument and thus
will only enforce conservativeness up to a user-defined probability threshold.
The method has applications in the context of optimization using the active
subspace method for dimensionality reduction of the objective function and the
constraints, addressing recorded issues about constraint violations. The
resulting algorithms are tested on a toy optimization problem in thermal
design.
