Complexity results and active-set identification of a derivative-free method for bound-constrained problems

Abstract

In this paper, we analyze a derivative-free line search method designed for bound-constrained problems. Our analysis demonstrates that this method exhibits a worst-case complexity comparable to other derivative-free methods for unconstrained and linearly constrained problems. In particular, when minimizing a function with $n$ variables, we prove that at most ${\cal O(n\epsilon^{-2})}$ iterations are needed to drive a criticality measure below a predefined threshold $\epsilon$, requiring at most ${\cal O(n^2\epsilon^{-2})}$ function evaluations. We also show that the total number of iterations where the criticality measure is not below $\epsilon$ is upper bounded by ${\cal O(n^2\epsilon^{-2})}$. Moreover, we investigate the method capability to identify active constraints at the final solutions. We show that, after a finite number of iterations, all the active constraints satisfying the strict complementarity condition are correctly identified.