arXiv subjects

Meggie Marschner

Publications and source records attributed to Meggie Marschner.

4 recordsLinked to original sources

arXiv2025

Regularization methods for solving hierarchical variational inequalities with complexity guarantees

We consider hierarchical variational inequality problems, or more generally, variational inequalities defined over the set of zeros of a monotone operator. This framework includes convex optimization over equilibrium constraints and equilibrium selection problems. In a real Hilbert space setting, we combine a Tikhonov regularization and a proximal penalization to develop a flexible double-loop method for which we prove asymptotic convergence and provide rate statements in terms of gap functions. Our method is flexible, and effectively accommodates a large class of structured operator splitting formulations for which fixed-point encodings are available. Finally, we validate our findings numerically on various examples.

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arXiv2025

Extragradient methods with complexity guarantees for hierarchical variational inequalities

In the framework of a real Hilbert space we consider the problem of approaching solutions to a class of hierarchical variational inequality problems, subsuming several other problem classes including certain mathematical programs under equilibrium constraints, constrained min-max problems, hierarchical game problems, optimal control under VI constraints, and simple bilevel optimization problems. For this general problem formulation, we establish rates of convergence in terms of suitably constructed gap functions, measuring feasibility gaps and optimality gaps. We present worst-case iteration complexity results on both levels of the variational problem, as well as weak convergence under a geometric weak sharpness condition on the lower level solution set. Our results match and improve the state of the art in terms of their iteration complexity and the generality of the problem formulation.

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arXiv2025

Tikhonov regularized exterior penalty methods for hierarchical variational inequalities

We consider nested variational inequalities con- sisting in a (upper-level) variational inequality whose feasible set is given by the solution set of another (lower-level) variational inequality. This class of hierarchical equilibrium contains a wealth of important applications, including purely hierarchical convex bilevel optimization problems and certain multi-follower games. Working within a real Hilbert space setting, we develop a double loop prox-penalization algorithm with strong conver- gence guarantees towards a solution of the nested VI problem. We present various application that fit into our framework and present also some preliminary numerical results.

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arXiv2025

Complexity guarantees for risk-neutral generalized Nash equilibrium problems

In this paper, we address \ac{SGNEP} seeking with risk-neutral agents. Our main contribution lies the development of a stochastic variance-reduced gradient (SVRG) technique, modified to contend with general sample spaces, within a stochastic forward-backward-forward splitting scheme for resolving structured monotone inclusion problems. This stochastic scheme is a double-loop method, in which the mini-batch gradient estimator is computed periodically in the outer loop, while only cheap sampling is required in a frequently activated inner loop, thus achieving significant speed-ups when sampling costs cannot be overlooked. The algorithm is fully distributed and it guarantees almost sure convergence under appropriate batch size and strong monotonicity assumptions. Moreover, it exhibits a linear rate with possible biased estimators, which is rather mild and imposed in many simulation-based optimization schemes. Under monotone regimes, the expectation of the gap function of an averaged iterate diminishes at a suitable sublinear rate while the sample-complexity of computing an $\epsilon$-solution is provably $\mathcal{O}(\epsilon^{-3})$. A numerical study on a class of networked Cournot games reflects the performance of our proposed algorithm.

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