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A Trust Region Algorithm with a Worst-Case Iteration Complexity of ${\cal O}(\epsilon^{-3/2})$ for Nonconvex Optimization

Frank E. Curtis (frank.e.curtis***at***gmail.com)
Daniel P. Robinson (daniel.p.robinson***at***jhu.edu)
Mohammadreza Samadi (mohammadreza.samadi***at***lehigh.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 $O(\epsilon^{-3/2})$ for Nonconvex Optimization. Mathematical Programming, DOI: 10.1007/s10107-016-1026-2, 2016.

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Entry Submitted: 10/21/2014
Entry Accepted: 10/21/2014
Entry Last Modified: 01/02/2017

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