Direct search nonsmooth constrained optimization via rounded ℓ1 penalty functions. (2nd January 2022) - Price, C. J.
ABSTRACT: A class of direct search methods for locally minimizing a Lipschitz continuous black-box function f subject to locally Lipschitz constraints is presented. A sequence of smoothed ℓ 1 penalty functions is used. Each smoothed penalty function is approximately minimized in turn. The smoothing is reduced after each minimization, exposing the ℓ 1 exact penalty function in the limit. Convergence to a constrained Clarke stationary point is shown under appropriate regularity conditions. Convergence to one or more KKT points is shown under similar conditions when f and all active constraints are strictly differentiable at each limit point. An implementation of one method in this class is numerically tested and shown to be effective in practice. The implementation uses a discrete quasi-Newton step when possible. Otherwise a global direction search is used to locate a descent direction. Theoretical convergence properties are independent of the quasi-Newton step.