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Backtracking linesearch is the de facto approach for minimizing continuouslydifferentiable functions with locally Lipschitz gradient. In recent years, ithas been shown that in the convex setting it is possible to avoid linesearchaltogether, and to allow the stepsize to adapt based on a local smoothnessestimate without any backtracks or evaluations of the function value. In thiswork we propose an adaptive proximal gradient method, adaPG, that uses novelestimates of the local smoothness modulus which leads to less conservativestepsize updates and that can additionally cope with nonsmooth terms. This ideais extended to the primal-dual setting where an adaptive three-term primal-dualalgorithm, adaPD, is proposed which can be viewed as an extension of the PDHGmethod. Moreover, in this setting the ``essentially'' fully adaptive variantadaPD$^+$ is proposed that avoids evaluating the linear operator norm byinvoking a backtracking procedure, that, remarkably, does not require extragradient evaluations. Numerical simulations demonstrate the effectiveness ofthe proposed algorithms compared to the state of the art.

Adaptive proximal algorithms for convex optimization under local Lipschitz continuity of the gradient