Constrained, Global Optimization of Functions with Lipschitz Continuous Gradients.

We present two first-order, sequential optimization algorithms to solve constrained optimization problems. We consider a black-box setting with a priori unknown, non-convex objective and constraint functions that have Lipschitz continuous gradients. The proposed algorithms balance the exploration of the a priori unknown feasible space with the pursuit of global optimality within in a pre-specified finite number of first-order oracle calls. The first algorithm accommodates an infeasible start, and provides either a near-optimal global solution or establishes infeasibility. However, the algorithm may produce infeasible iterates during the search. For a strongly-convex constraint function and a feasible initial solution guess, the second algorithm returns a near-optimal global solution without any constraint violation. In contrast to existing methods, both of the algorithms also compute global suboptimality bounds at every iteration. We also show that the algorithms can satisfy user-specified tolerances in the computed solution with near-optimal complexity in oracle calls for a large class of optimization problems. We propose tractable implementations of the algorithms by exploiting the structure afforded by the Lipschitz continuous gradient property.