Complexity Results and Active-Set Identification of a Derivative-Free Method for Bound-Constrained Problems
Year: 2026
Type: article
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 $$\mathcal {O}{(n\epsilon ^{-2})}$$ O ( n ϵ - 2 ) iterations are needed to drive a criticality measure below a predefined threshold $$\epsilon $$ ϵ , requiring at most $$\mathcal {O}{(n^2\epsilon ^{-2})}$$ O ( n 2 ϵ - 2 ) function evaluations. We also show that the total number of iterations where the criticality measure is not below $$\epsilon $$ ϵ is upper bounded by $$\mathcal {O}{(n^2\epsilon ^{-2})}$$ O ( n 2 ϵ - 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.
Authors: Andrea Brilli, Andrea Cristofari, Giampaolo Liuzzi, Stefano Lucidi
Institutions: Sapienza University of Rome, University of Rome Tor Vergata
Language: English
Subfield: Numerical Analysis
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,190
Funder: Sapienza Università di Roma
OpenAlex ID: W7134932508