A Discussion on Variational Analysis in Derivative-Free Optimization

Variational Analysis studies mathematical objects under small variations. With regards to optimization, these objects are typified by representations of first-order or second-order information (gradients, subgradients, Hessians, etc). On the other hand, Derivative-Free Optimization studies algorithms for continuous optimization that do not use first-order information. As such, researchers might conclude that Variational Analysis plays a limited role in Derivative-Free Optimization research. In this paper we argue the contrary by showing that many successful DFO algorithms rely heavily on tools and results from Variational Analysis.