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Working paper

Near-optimal tensor methods for minimizing the gradient norm of convex function

Optimization and Control. Working papers by Cornell University., 2020
Dvurechensky P., Gasnikov A., Остроухов П., Uribe C., Ivanova A.
Motivated by convex problems with linear constraints and, in particular, by entropy-regularized optimal transport, we consider the problem of finding ε-approximate stationary points, i.e. points with the norm of the objective gradient less than ε, of convex functions with Lipschitz p-th order derivatives. Lower complexity bounds for this problem were recently proposed in [Grapiglia and Nesterov, arXiv:1907.07053]. However, the methods presented in the same paper do not have optimal complexity bounds. We propose two optimal up to logarithmic factors methods with complexity bounds O~(ε−2(p+1)/(3p+1)) and O~(ε−2/(3p+1)) with respect to the initial objective residual and the distance between the starting point and solution respectively.