A Parallel, Asynchronous Method for Derivative-Free Nonlinear Programs

Derivative-free optimization algorithms are needed to solve real-world engineering problems that have computationally expensive and noisy objective function and constraint evaluations. In particular, we are focused on problems that involve running cumbersome simulation codes with run times measured in hours. In such cases, attempts to compute derivatives can prove futile because analytical derivatives are typically unavailable and noise limits the accuracy of numerical approximations. Furthermore, the objective and constraint functions may be inherently nonsmooth, i.e., because the underlying model is nonsmooth. Generating Set Search (GSS) methods [7] are particularly well-suited to such unwieldy optimization problems. GSS methods are a generalization of pattern search that derives its search directions from the generators of the -tangent cone of the linear constraints, i.e., a generating set. GSS methods offer several advantages: • Because search direction are based upon the local geometry of the feasible region defined by the linear constraints, and not the objective or nonlinear constraint functions, they are well-suited for problems with noise. • The function evaluations can be performed asynchronously in parallel [5, 10, 6]. • If the underlying objective function and constraints are smooth, GSS methods can bound the first-order optimality conditions in terms of step size. • They can easily accommodate undefined points within the feasible region, i.e, points where the simulation unexpectedly fails. The focus of this talk will be on the addition of constraint-handling abilities to APPSPACK, which is a C++ implementation of an asynchronous parallel GSS algorithm [4]. APPSPACK is a publicly available derivative-free software package whose handling of both linear and nonlinear constraints is based upon rigorous convergence theory. Specifically, we have added the ability to handle linear constraints using conforming search directions and nonlinear constraints using an augmented Lagrangian algorithm. We are interested in solving a nonlinear program of the form