Kernekoncepter
This paper proposes a novel safe zeroth-order optimization method that iteratively constructs quadratic approximations of the constraint functions, builds local feasible sets, and optimizes over them. The method guarantees that all samples are feasible and returns an η-KKT pair within a polynomial number of iterations and samples.
Resumé
The paper addresses smooth constrained optimization problems where the objective and constraint functions are unknown but can be queried. The main goal is to generate a sequence of feasible points converging towards a KKT primal-dual pair.
The key highlights are:
The authors propose a zeroth-order method that iteratively computes quadratic approximations of the constraint functions, constructs local feasible sets, and optimizes over them. This ensures that all samples are feasible.
The method is proven to return an η-KKT pair within O(d/η^2) iterations and O(d^2/η^2) samples, where d is the problem dimension.
Numerical experiments show that the proposed method can achieve faster convergence compared to state-of-the-art zeroth-order safe approaches, such as LB-SGD and SafeOptSwarm. The effectiveness is also illustrated on nonconvex optimization problems in optimal control and power system operation.
The authors extend their previous work on convex problems to handle non-convex objective and constraint functions. They prove that the accumulation points of the iterates are KKT pairs under mild assumptions.
The termination conditions of the algorithm are designed to ensure that the final output is an η-KKT pair. The complexity analysis shows that the number of iterations and samples required scales polynomially with the problem dimension and inversely with the desired accuracy η.