Projection-based curve pattern search for black-box optimization over smooth convex sets

In this paper, we deal with the problem of optimizing a black-box smooth function over a full-dimensional smooth convex set. We study sets of feasible curves that allow us to properly characterize stationarity of a solution and possibly carry out sound backtracking curvilinear searches. We then propose a general pattern search algorithmic framework that exploits curves of this type to carry out poll steps and for which we prove properties of asymptotic convergence to stationary points. We particularly point out that the proposed framework covers the case where search curves are arcs induced by the Euclidean projection of coordinate directions. The method is finally proved to arguably be superior, on smooth problems, than other recent projection-based algorithms and is competitive with state-of-the-art methods from the literature on constrained black-box optimization.