-- Leveraging on recent advancements on adaptive methods for convex minimization problems, this paper provides a linesearch-free proximal gradient framework for glob-alizing the convergence of popular stepsize choices such as Barzilai-Borwein and one-dimensional Anderson acceleration. This framework can cope with problems in which the gradient of the differentiable function is merely locally Hölder continuous. Our analysis not only encompasses but also refines existing results upon which it builds. The theory is corroborated by numerical evidence that showcases the synergetic interplay between fast stepsize selections and adaptive methods. Convex nonsmooth optimization problems are encountered in various engineering applications such as image denoising [4], signal processing and digital communication [16], machine learning [7], and control [15], to name a few. Traditional constant stepsizes require the gradient of the function f to satisfy global Lipschitz continuity [5, Prop.

We show that adaptive proximal gradient methods for convex problems are not restricted to traditional Lipschitzian assumptions. Our analysis reveals that a class of linesearch-free methods is still convergent under mere local H\"older gradient continuity, covering in particular continuously differentiable semi-algebraic functions. To mitigate the lack of local Lipschitz continuity, popular approaches revolve around $\varepsilon$-oracles and/or linesearch procedures. In contrast, we exploit plain H\"older inequalities not entailing any approximation, all while retaining the linesearch-free nature of adaptive schemes. Furthermore, we prove full sequence convergence without prior knowledge of local H\"older constants nor of the order of H\"older continuity. In numerical experiments we present comparisons to baseline methods on diverse tasks from machine learning covering both the locally and the globally H\"older setting.

Building upon recent works on linesearch-free adaptive proximal gradient methods, this paper proposes AdaPG$^{\pi,r}$, a framework that unifies and extends existing results by providing larger stepsize policies and improved lower bounds. Different choices of the parameters $\pi$ and $r$ are discussed and the efficacy of the resulting methods is demonstrated through numerical simulations. In an attempt to better understand the underlying theory, its convergence is established in a more general setting that allows for time-varying parameters. Finally, an adaptive alternating minimization algorithm is presented by exploring the dual setting. This algorithm not only incorporates additional adaptivity, but also expands its applicability beyond standard strongly convex settings.

Backtracking linesearch is the de facto approach for minimizing continuously differentiable functions with locally Lipschitz gradient. In recent years, it has been shown that in the convex setting it is possible to avoid linesearch altogether, and to allow the stepsize to adapt based on a local smoothness estimate without any backtracks or evaluations of the function value. In this work we propose an adaptive proximal gradient method, adaPG, that uses novel estimates of the local smoothness modulus which leads to less conservative stepsize updates and that can additionally cope with nonsmooth terms. This idea is extended to the primal-dual setting where an adaptive three-term primal-dual algorithm, adaPD, is proposed which can be viewed as an extension of the PDHG method. Moreover, in this setting the ``essentially'' fully adaptive variant adaPD$^+$ is proposed that avoids evaluating the linear operator norm by invoking a backtracking procedure, that, remarkably, does not require extra gradient evaluations. Numerical simulations demonstrate the effectiveness of the proposed algorithms compared to the state of the art.

In large-scale optimization, the presence of nonsmooth and nonconvex terms in a given problem typically makes it hard to solve. A popular approach to address nonsmooth terms in convex optimization is to approximate them with their respective Moreau envelopes. In this work, we study the use of Lasry-Lions double envelopes to approximate nonsmooth terms that are also not convex. These envelopes are an extension of the Moreau ones but exhibit an additional smoothness property that makes them amenable to fast optimization algorithms. Lasry-Lions envelopes can also be seen as an "intermediate" between a given function and its convex envelope, and we make use of this property to develop a method that builds a sequence of approximate subproblems that are easier to solve than the original problem. We discuss convergence properties of this method when used to address composite minimization problems; additionally, based on a number of experiments, we discuss settings where it may be more useful than classical alternatives in two domains: signal decoding and spectral unmixing.