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In this paper, a new variant of accelerated gradient descent is proposed. Thepro-posed method does not require any information about the objective function,usesexact line search for the practical accelerations of convergence, convergesaccordingto the well-known lower bounds for both convex and non-convexobjective functions,possesses primal-dual properties and can be applied in thenon-euclidian set-up. Asfar as we know this is the rst such method possessingall of the above properties atthe same time. We also present a universalversion of the method which is applicableto non-smooth problems. We demonstratehow in practice one can efficiently use thecombination of line-search andprimal-duality by considering a convex optimizationproblem with a simplestructure (for example, linearly constrained).

Primal-dual accelerated gradient methods with small-dimensional relaxation oracle