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Geovani Grapiglia (grapigliaufpr.br) Abstract: In this paper we consider the problem of finding \epsilonapproximate stationary points of convex functions that are ptimes differentiable with \nuHölder continuous pth derivatives. We present tensor methods with and without acceleration. Specifically, we show that the nonaccelerated schemes take at most O(\epsilon^{1/(p+\nu1)}) iterations to reduce the norm of the gradient of the objective below a given \epsilon\in (0,1). For accelerated tensor schemes we establish improved complexity bounds of O(\epsilon^{(p+\nu)/[(p+\nu1)(p+\nu+1)]}) and O(\log(\epsilon)\epsilon^{1/(p+\nu)}), when the Hölder parameter \nu\in [0,1] is known. For the case in which \nu is unknown, we obtain a bound of O(\epsilon^{(p+1)/[(p+\nu1)(p+2)]}) for a universal accelerated scheme. Finally, we also obtain a lower complexity bound of O(\epsilon^{2/[3(p+\nu)2]}) for finding \epsilonapproximate stationary points using porder tensor methods. Keywords: unconstrained minimization, highorder methods, tensor methods, Hölder condition, worstcase complexity Category 1: Convex and Nonsmooth Optimization (Convex Optimization ) Citation: Optimization Methods and Software (2020), DOI: 10.1080/10556788.2020.1818082 Download: [PDF] Entry Submitted: 07/13/2019 Modify/Update this entry  
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