- A Trust Region Algorithm with a Worst-Case Iteration Complexity of ${\cal O}(\epsilon^{-3/2})$ for Nonconvex Optimization Frank E. Curtis (frank.e.curtisgmail.com) Daniel P. Robinson (daniel.p.robinsonjhu.edu) Mohammadreza Samadi (mohammadreza.samadilehigh.edu) Abstract: We propose a trust region algorithm for solving nonconvex smooth optimization problems. For any $\bar\epsilon \in (0,\infty)$, the algorithm requires at most $\mathcal{O}(\epsilon^{-3/2})$ iterations, function evaluations, and derivative evaluations to drive the norm of the gradient of the objective function below any $\epsilon \in (0,\bar\epsilon]$. This improves upon the $\mathcal{O}(\epsilon^{-2})$ bound known to hold for some other trust region algorithms and matches the $\mathcal{O}(\epsilon^{-3/2})$ bound for the recently proposed Adaptive Regularisation framework using Cubics, also known as the arc algorithm. Our algorithm, entitled trace, follows a trust region framework, but employs modified step acceptance criteria and a novel trust region update mechanism that allow the algorithm to achieve such a worst-case global complexity bound. Importantly, we prove that our algorithm also attains global and fast local convergence guarantees under similar assumptions as for other trust region algorithms. We also prove a worst-case upper bound on the number of iterations, function evaluations, and derivative evaluations that the algorithm requires to obtain an approximate second-order stationary point. Keywords: unconstrained optimization, nonlinear optimization, nonconvex optimization, trust region methods, global convergence, local convergence, worst-case iteration complexity, worst-case evaluation complexity Category 1: Nonlinear Optimization Category 2: Nonlinear Optimization (Unconstrained Optimization ) Citation: F. E. Curtis, D. P. Robinson, and M. Samadi. A Trust Region Algorithm with a Worst-Case Iteration Complexity of O(ε−3/2) for Nonconvex Optimization. Mathematical Programming, 162(1):1–32, 2017. Download: Entry Submitted: 10/21/2014Entry Accepted: 10/21/2014Entry Last Modified: 08/26/2019Modify/Update this entry Visitors Authors More about us Links Subscribe, Unsubscribe Digest Archive Search, Browse the Repository Submit Update Policies Coordinator's Board Classification Scheme Credits Give us feedback Optimization Journals, Sites, Societies Optimization Online is supported by the Mathematical Optmization Society.