Deterministic global derivative-free optimization of black-box problems with bounded Hessian

Abstract

Obtaining guaranteed lower bounds for problems with unknown algebraic form has been a major challenge in derivative-free optimization. In this work, we present a deterministic global optimization method for black-box problems where the derivatives are not available or it is computationally expensive to obtain. However, a global upper bound on the diagonal Hessian elements is known. An edge-concave underestimator (Hasan in J Glob Optim 71:735–752, 2018) can be then constructed with vertex polyhedral solution. Evaluating this underestimator only at the vertices leads to a valid lower bound on the original black-box problem. We have implemented this lower bounding technique within a branch-and-bound framework and assessed its computational performance in locating $$\epsilon $$-global optimal solution for several box-constrained, nonconvex black-box functions.