arXiv subjects

Leo Lehmann

Publications and source records attributed to Leo Lehmann.

3 recordsLinked to original sources

arXiv2026

Projected Subgradient Methods for a Class of Nonsmooth and Nonconvex Optimization Problems

We investigate the optimization problem of minimizing a nonsmooth function that satisfies a nonsmooth version of the descent lemma over a nonempty and closed but not necessarily convex set. The objective function belongs to the class of upper-$\mathcal{C}^2$ functions, whereas the constraints may promote a sparse or low-rank structure. We propose a projected subgradient method with two different globalization strategies: (a) a nonmonotone linesearch and, under additional assumptions, (b) an auto-conditioned method, where the stepsize is given by a formula depending on data from past iterations. We show that both methods converge to solutions that satisfy a stronger stationarity concept than one would expect from the subdifferential sum-rule, which is particularly important since the optimization problems of interest are inherently nonconvex. Finally, we present promising numerical results when applying the algorithm to an MPEC-style problem as well as the matrix optimization problems MAXCUT and Robust PCA.

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arXiv2026

A Nonmonotone Descent Method for Optimization Problems Defined by Upper-$\mathcal{C}^2 $ Functions over Submanifolds

We consider the optimization problem of minimizing a nonsmooth function characterized by a nonsmooth formulation of the descent lemma over a manifold. In the unconstrained case over a Euclidean space, this class of functions is called upper-$\mathcal{C}^2$. Using the recent notion of projectional subdifferentials, we show that their descent property carries over to submanifolds. We propose a nonmonotone subgradient method to solve these problems and prove stationarity of accumulation points of the generated sequence as well as convergence and rate-of-convergence results under the Kurdyka-Lojasiewicz property. We also perform numerical experiments and show how our approach can be applied to a certain type of difference of convex functions as well as clustering problems on manifolds.

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arXiv2024

Convergence of Nonmonotone Proximal Gradient Methods under the Kurdyka-Lojasiewicz Property without a Global Lipschitz Assumption

We consider the composite minimization problem with the objective function being the sum of a continuously differentiable and a merely lower semicontinuous and extended-valued function. The proximal gradient method is probably the most popular solver for this class of problems. Its convergence theory typically requires that either the gradient of the smooth part of the objective function is globally Lipschitz continuous or the (implicit or explicit) a priori assumption that the iterates generated by this method are bounded. Some recent results show that, without these assumptions, the proximal gradient method, combined with a monotone stepsize strategy, is still globally convergent with a suitable rate-of-convergence under the Kurdyka-Lojasiewicz property. For a nonmonotone stepsize strategy, there exist some attempts to verify similar convergence results, but, so far, they need stronger assumptions. This paper is the first which shows that nonmonotone proximal gradient methods for composite optimization problems share essentially the same nice global and rate-of-convergence properties as its monotone counterparts, still without assuming a global Lipschitz assumption and without an a priori knowledge of the boundedness of the iterates.

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