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Frank E. Curtis (frank.e.curtisgmail.com) 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 worstcase 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 worstcase upper bound on the number of iterations, function evaluations, and derivative evaluations that the algorithm requires to obtain an approximate secondorder stationary point. Keywords: unconstrained optimization, nonlinear optimization, nonconvex optimization, trust region methods, global convergence, local convergence, worstcase iteration complexity, worstcase 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 WorstCase Iteration Complexity of O(ε−3/2) for Nonconvex Optimization. Mathematical Programming, 162(1):1–32, 2017. Download: Entry Submitted: 10/21/2014 Modify/Update this entry  
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