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We study convergence rates of the classic proximal bundle method for a variety of nonsmooth convex optimization problems. We show that, without any modification, this algorithm adapts to converge faster in the presence of smoothness or a Holder growth condition. Our analysis reveals that with a constant stepsize, the bundle method is adaptive, yet it exhibits suboptimal convergence rates. We overcome this shortcoming by proposing nonconstant stepsize schemes with optimal rates. These schemes use function information such as growth constants, which might be prohibitive in practice. We complete the paper with a new parallelizable variant of the bundle method that attains near-optimal rates without prior knowledge of function parameters. These results improve on the limited existing convergence rates and provide a unified analysis approach across problem settings and algorithmic details. Numerical experiments support our findings and illustrate the effectiveness of the parallel bundle method.
This work studies a class of non-smooth decentralized multi-agent optimization problems where the agents aim at minimizing a sum of local strongly-convex smooth components plus a common non-smooth term. We propose a general primal-dual algorithmic fr
The spectral bundle method proposed by Helmberg and Rendl is well established for solving large scale semidefinite programs (SDP) thanks to its low per iteration computational complexity and strong practical performance. In this paper, we revisit thi
We consider minimization of functions that are compositions of convex or prox-regular functions (possibly extended-valued) with smooth vector functions. A wide variety of important optimization problems fall into this framework. We describe an algori
We develop new perturbation techniques for conducting convergence analysis of various first-order algorithms for a class of nonsmooth optimization problems. We consider the iteration scheme of an algorithm to construct a perturbed stationary point se
We consider the problem of optimizing the sum of a smooth convex function and a non-smooth convex function using proximal-gradient methods, where an error is present in the calculation of the gradient of the smooth term or in the proximity operator w