arXiv subjects

Martin Schmidt

Publications and source records attributed to Martin Schmidt.

At least 19 recordsLinked to original sources

arXiv2026

Stability of Flow Models for Graph Signals

Generating signals on graphs requires permutation-equivariant models that exhibit stability with respect to relative structural perturbations. While favorable stability properties of Graph Neural Networks (GNNs) have been well documented, it is unclear how structural errors propagate through the dynamics of continuous generative flow models that are gaining traction for graph signal generation. In this paper, we analyze continuous normalized flow models parameterized by GNNs and show that permutation equivariance is preserved for both the resulting continuous-time ordinary differential equations and their discrete numerical approximations used as graph signal samplers. Our primary contribution is to derive explicit stability bounds on the generated probability distributions, which quantify how relative graph perturbations affect the final sampled signals. Motivated by these theoretical bounds, we introduce a stability-promoting regularized flow matching strategy that actively penalizes the spatial Lipschitz constant of the vector field during model training. Experiments using synthetic smooth signals on stochastic block model graphs and real-world fMRI signals on brain connectomes demonstrate that this bound-oriented approach yields generative models that are more robust to structural noise, without sacrificing output quality.

eess.SP↗

arXiv2026

When do Mixed-Integer Games Admit Rational Equilibria?

We consider mixed-integer linear-quadratic generalized Nash equilibrium problems, i.e., games in which each player solves a mixed-integer program subject to linear constraints in her own and rivals' strategies as well as an objective which is quadratic in her own strategies and bilinear in her own and rivals' strategies. For this class of games, we study the question of the existence of rational equilibria assuming rational input data. We distinguish four subclasses according to the presence of player-quadratic terms in the objective and rival-dependent constraints. As our main result, we completely settle the rationality question for all four subclasses, i.e., we show that only player-linear games without player-quadratic terms and without rival-dependent constraints admit rational equilibria -- if the game admits equilibria at all. All other three classes contain instances with irrational equilibria only.

cs.GT↗

arXiv2026

Optical Stability and Photophysics of NV Centers in Diamond up to 120 GPa

The nitrogen vacancy (NV) center has emerged as a powerful quantum sensor in high-pressure research, with the observation of optically detected magnetic resonance at megabar pressures. However, some aspects of NV physics require further investigation to optimize the development of NV-based sensing under pressure. Here, we study both experimentally and theoretically the optical properties of the NV center under hydrostatic pressure. We investigate the evolution of the zero-phonon line (ZPL) position, radiative lifetimes, optical lineshapes, and photoionization thresholds of the NV center under pressures up to ~120 GPa. We also provide spectroscopic guidelines for performing high-pressure optical experiments. Our results confirm that the NV center remains a robust quantum sensor under extreme hydrostatic pressures, especially for magnetic characterization.

quant-ph↗

arXiv2026

Widefield NV Magnetic Field Reconstruction for Probing the Meissner Effect and Critical Current Density under Pressure

The spatial distribution of a magnetic field can be determined with micrometer resolution using widefield nitrogen vacancy (NV) center magnetic imaging. Nevertheless, reconstructing the magnetic field from the raw data can be challenging due to the degeneracy of the four possible NV axes and the tremendous amount of data. While a qualitative approach is sufficient for most analyses, a quantitative analysis offers deeper insight into the physical system. Here, we apply NV widefield magnetic imaging to a HgBa$_{2}$Ca$_{2}$Cu$_{3}$O$_{8+\delta}$ (Hg-1223) superconducting microcrystal at a pressure of 4 GPa. We fit the results with solutions from the Hamiltonian describing the NV center ground state and take into account the relative intensities of the resonances to determine the local magnetic field magnitude and angle. Thus, we reconstruct the temperature-dependent expulsion of the magnetic field due to the Meissner effect around the superconductor. By comparing the resulting parameters to Brandt's model, which describes the magnetic behavior of a type-II superconductor, we extract the critical current density $j_c$. Overall, this work showcases the first widefield quantitative reconstruction of the Meissner effect under pressure and an optical method to study critical current density. Thus, it provides new insights into the application of NV magnetometry to superconductivity research at high pressures.

cond-mat.supr-con↗

arXiv2025

Minimal Regret Walras Equilibria for Combinatorial Markets

We consider combinatorial multi-item markets and propose the notion of a $\Delta$-regret Walras equilibrium, which is an allocation of items to players and a set of item prices that achieve the following goals: prices clear the market, the allocation is capacity-feasible, and the players' strategies lead to a total regret of $\Delta$. The regret is defined as the sum of individual player regrets measured by the utility gap with respect to the optimal item bundle given the prices. We derive a complete characterization for the existence of $\Delta$-regret equilibria by introducing the concept of a parameterized social welfare problem, where the right-hand side of the original social welfare problem is changed. Our characterization then relates the achievable regret value with the associated duality/integrality gap of the parameterized social welfare problem. For the special case of monotone valuations this translates to regret bounds recovering the duality/integrality gap of the original social welfare problem. We further establish an interesting connection to the area of sensitivity theory in linear optimization. We show that the sensitivity gap of the optimal-value function of two (configuration) linear programs with changed right-hand side can be used to establish a bound on the achievable regret. Finally, we use these general structural results to translate known approximation algorithms for the social welfare optimization problem into algorithms computing low-regret Walras equilibria. We also demonstrate how to derive strong lower bounds based on integrality and duality gaps but also based on NP-complexity theory.

cs.GT↗

arXiv2025

Branch-and-Cut for Computing Approximate Equilibria of Mixed-Integer Generalized Nash Games

Generalized Nash equilibrium problems with mixed-integer variables constitute an important class of games in which each player solves a mixed-integer optimization problem, where both the objective and the feasible set is parameterized by the rivals' strategies. However, such games are known for failing to admit exact equilibria and also the assumption of all players being able to solve nonconvex problems to global optimality is questionable. This motivates the study of approximate equilibria. In this work, we consider an approximation concept that incorporates both multiplicative and additive relaxations of optimality. We propose a branch-and-cut (B&C) method that computes such approximate equilibria or proves its non-existence. For this, we adopt the idea of intersection cuts and show the existence of such cuts under the condition that the constraints are linear and each player's cost function is either convex in the entire strategy profile, or, concave in the entire strategy profile and linear in the rivals' strategies. For the special case of standard Nash equilibrium problems, we introduce an alternative type of cut and show that the method terminates finitely, provided that each player has only finitely many distinct best-response sets. Finally, on the basis of the B&C method, we introduce a single-tree binary-search method to compute best-approximate equilibria under some simplifying assumptions. We implemented these methods and present numerical results for a class of mixed-integer flow games.

cs.GT↗

arXiv2025

Branch-and-Cut for Mixed-Integer Nash Equilibrium Problems

We study Nash equilibrium problems with mixed-integer variables in which each player solves a mixed-integer optimization problem parameterized by the rivals' strategies. We distinguish between standard Nash equilibrium problems (NEPs), where parameterization affects only the objective functions, and generalized Nash equilibrium problems (GNEPs), where strategy sets may additionally depend on rivals' strategies. We introduce a branch-and-cut (B&C) algorithm for such mixed-integer games that, upon termination, either computes a pure Nash equilibrium or decides their non-existence. Our approach reformulates the game as a bilevel problem using the Nikaido--Isoda function. We then use bilevel-optimization techniques to get a computationally tractable relaxation of this reformulation and embed it into a B&C framework. We derive sufficient conditions for the existence of suitable cuts and finite termination of our method depending on the setting. For GNEPs, we adapt the idea of intersection cuts from bilevel optimization and mixed-integer linear optimization. We can guarantee the existence of such cuts under suitable assumptions, which are particularly fulfilled for pure-integer GNEPs with decoupled concave objectives and linear coupling constraints. For NEPs, we show that suitable cuts always exist via best-response inequalities and prove that our B&C method terminates in finite time whenever the set of best-response sets is finite. We show that this condition is fulfilled for the important special cases of (i) players' cost functions being concave in their own continuous strategies and (ii) the players' cost functions only depending on their own strategy and the rivals' integer strategy components. Finally, we present preliminary numerical results for two different types of knapsack games, a game based on capacitated flow problems, and integer NEPs with quadratic objectives.

cs.GT↗

arXiv2025

Non-invasive maturity assessment of iPSC-CMs based on optical maturity characteristics using interpretable AI

Human induced pluripotent stem cell-derived cardiomyocytes (iPSC-CMs) are an important resource for the identification of new therapeutic targets and cardioprotective drugs. After differentiation iPSC-CMs show an immature, fetal-like phenotype. Cultivation of iPSC-CMs in lipid-supplemented maturation medium (MM) strongly enhances their structural, metabolic and functional phenotype. Nevertheless, assessing iPSC-CM maturation state remains challenging as most methods are time consuming and go in line with cell damage or loss of the sample. To address this issue, we developed a non-invasive approach for automated classification of iPSC-CM maturity through interpretable artificial intelligence (AI)-based analysis of beat characteristics derived from video-based motion analysis. In a prospective study, we evaluated 230 video recordings of early-state, immature iPSC-CMs on day 21 after differentiation (d21) and more mature iPSC-CMs cultured in MM (d42, MM). For each recording, 10 features were extracted using Maia motion analysis software and entered into a support vector machine (SVM). The hyperparameters of the SVM were optimized in a grid search on 80 % of the data using 5-fold cross-validation. The optimized model achieved an accuracy of 99.5 $\pm$ 1.1 % on a hold-out test set. Shapley Additive Explanations (SHAP) identified displacement, relaxation-rise time and beating duration as the most relevant features for assessing maturity level. Our results suggest the use of non-invasive, optical motion analysis combined with AI-based methods as a tool to assess iPSC-CMs maturity and could be applied before performing functional readouts or drug testing. This may potentially reduce the variability and improve the reproducibility of experimental studies.

cs.LG↗

arXiv2025

Field-effect detected magnetic resonance of NV centers in diamond based on all-carbon Schottky contacts

The nitrogen vacancy (NV) center is a defect in diamond whose spin state can be read optically by exploiting its photoluminescence or electrically by exploiting its charge generation rate under illumination, both of which being spin-dependent. The latter method offers numerous opportunities in terms of integration and performance compared to conventional optical reading. Here, we investigate the physical properties of a graphitic-diamond-graphitic structure under illumination. We show how, for a type IIa diamond material, electron-hole pairs generated by an ensemble of NV centers lead to a p-type material upon illumination, making this all-carbon structure equivalent to two back-to-back Schottky diodes. We analyze how the reverse current flowing upon illumination changes as a function of bias voltage and radiofrequency-induced excitation of the NV ensemble spin resonances. Furthermore, we demonstrate how an additional field effect arising from the illumination scheme affects the reverse current, resulting in a photoelectrical signal that can exceed the optical signal under the same illumination conditions.

quant-ph↗

arXiv2025

A Computational Study for Solving Decision-Dependent Robust Problems as Bilevel Optimization Problems

Both bilevel and robust optimization are established fields of mathematical optimization and operations research. However, only until recently, the similarities in their mathematical structure has neither been studied theoretically nor exploited computationally. Based on the recent results by Goerigk et al. (2025), this paper is the first one that provides an extensive computational study for solving strictly robust optimization problems with decision-dependent uncertainty sets as equivalent bilevel optimization problems. If the uncertainty set can be dualized, the respective bilevel techniques to obtain a single-level reformulation are very similar compared with the classic dualization techniques used in robust optimization but lead to larger single-level problems to be solved. Our numerical study shows that this usually leads to larger computation times. For the more challenging case of decision-dependent uncertainty sets represented by mixed-integer linear models, one cannot apply classic dualization techniques from robust optimization. Thus, we compare the presented bilevel approach with an established method from the literature, which is based on quantified mixed-integer linear programs. Our numerical results indicate that, for the problem class of decision-dependent robust optimization problems with mixed-integer linear uncertainty sets, the bilevel approach performs better in terms of computation times.

math.OC↗

arXiv2025

On Coupling Constraints in Pessimistic Linear Bilevel Optimization

The literature on pessimistic linear bilevel optimization with coupling constraints is rather scarce and it has been common sense that these problems are harder to tackle than pessimistic bilevel problems without coupling constraints. In this note, we show that this is not the case. To this end, given a pessimistic problem with coupling constraints, we derive a pessimistic problem without coupling constraints that has the same set of globally optimal solutions. Moreover, our results also show that one can equivalently replace a pessimistic problem with such constraints with an optimistic problem without coupling constraints. This paves the way of both transferring theory and solution techniques from any type of these problems to any other one.

math.OC↗

arXiv2025

Imaging the Meissner Effect and Flux Trapping of Superconductors under High Pressure using N-V Centers

Pressure is a key parameter for tuning or revealing superconductivity in materials and compounds. Many measurements of superconducting phase transition temperatures have been conducted using diamond anvil cells (DACs), which provide a wide pressure range and enable concomitant microscopic structural characterization of the sample. However, the inherently small sample volumes in DACs complicate the unambiguous detection of the Meissner effect, the hallmark of superconductivity. Recently, the Meissner effect in superconductors within a DAC was successfully demonstrated using diamond nitrogen-vacancy (N-V) widefield magnetometry, a non-invasive optical technique. In this work, we show that N-V magnetometry can also map superconductivity with micrometer resolution. We apply this technique to a microcrystal of HgBa$_2$Ca$_2$Cu$_3$O$_{8+\delta}$ (Hg-1223) mercury-based cuprate superconductor under 4 GPa of pressure. The method is capable to detect the magnetic field expulsion and heterogeneities in the sample, visible in a set of characteristic parameters as the local critical temperature $T_{c}$. Flux pinning zones are identified through flux trapping maps. This approach could enable detailed investigations of superconductivity of a broad range of materials under high-pressure conditions.

cond-mat.supr-con↗

arXiv2024

Adjustable Robust Nonlinear Network Design Without Controllable Elements under Load Scenario Uncertainties

We study network design problems for nonlinear and nonconvex flow models without controllable elements under load scenario uncertainties, i.e., under uncertain injections and withdrawals. To this end, we apply the concept of adjustable robust optimization to compute a network design that admits a feasible transport for all, possibly infinitely many, load scenarios within a given uncertainty set. For solving the corresponding adjustable robust mixed-integer nonlinear optimization problem, we show that a given network design is robust feasible, i.e., it admits a feasible transport for all load scenario uncertainties, if and only if a finite number of worst-case load scenarios can be routed through the network. We compute these worst-case scenarios by solving polynomially many nonlinear optimization problems. Embedding this result for robust feasibility in an adversarial approach leads to an exact algorithm that computes an optimal robust network design in a finite number of iterations. Since all of the results are valid for general potential-based flows, the approach can be applied to different utility networks such as gas, hydrogen, or water networks. We finally demonstrate the applicability of the method by computing robust gas networks that are protected from future load fluctuations.

math.OC↗

arXiv2024

Mixed-Integer Linear Optimization for Cardinality-Constrained Random Forests

Random forests are among the most famous algorithms for solving classification problems, in particular for large-scale data sets. Considering a set of labeled points and several decision trees, the method takes the majority vote to classify a new given point. In some scenarios, however, labels are only accessible for a proper subset of the given points. Moreover, this subset can be non-representative, e.g., due to collection bias. Semi-supervised learning considers the setting of labeled and unlabeled data and often improves the reliability of the results. In addition, it can be possible to obtain additional information about class sizes from undisclosed sources. We propose a mixed-integer linear optimization model for computing a semi-supervised random forest that covers the setting of labeled and unlabeled data points as well as the overall number of points in each class for a binary classification. Since the solution time rapidly grows as the number of variables increases, we present some problem-tailored preprocessing techniques and an intuitive branching rule. Our numerical results show that our approach leads to a better accuracy and a better Matthews correlation coefficient for biased samples compared to random forests by majority vote, even if only few labeled points are available.

math.OC↗

arXiv2024

A Gauss-Seidel method for solving multi-leader-multi-follower games

We design a computational approach to find equilibria in a class of Nash games possessing a hierarchical structure. By using tools from mixed-integer optimization and the characterization of variational equilibria in terms of the Karush-Kuhn-Tucker conditions, we propose a mixed-integer game formulation for solving this challenging class of problems. Besides providing an equivalent reformulation, we design a proximal Gauss--Seidel method with global convergence guarantees in case the game enjoys a potential structure. We finally corroborate the numerical performance of the algorithm on a novel instance of the ride-hail market problem.

math.OC↗

arXiv2024

On Coupling Constraints in Linear Bilevel Optimization

It is well-known that coupling constraints in linear bilevel optimization can lead to disconnected feasible sets, which is not possible without coupling constraints. However, there is no difference between linear bilevel problems with and without coupling constraints w.r.t. their complexity-theoretical hardness. In this note, we prove that, although there is a clear difference between these two classes of problems in terms of their feasible sets, the classes are equivalent on the level of optimal solutions. To this end, given a general linear bilevel problem with coupling constraints, we derive a respective problem without coupling constraints and prove that it has the same optimal solutions (when projected back to the original variable space).

math.OC↗

arXiv2024

Mixed-Integer Linear Optimization for Semi-Supervised Optimal Classification Trees

Decision trees are one of the most popular methods for solving classification problems, mainly because of their good interpretability properties. Moreover, due to advances in recent years in mixed-integer optimization, several models have been proposed to formulate the problem of computing optimal classification trees. The goal is, given a set of labeled points, to split the feature spacewith hyperplanes and assign a class to each part of the resulting partition. In certain scenarios, however, labels are only available for a subset of the given points. Additionally, this subset may be non-representative, such as in the case of self-selection in a survey. Semi-supervised decision trees tackle the setting of labeled and unlabeled data and often contribute to enhancing the reliability of the results. Furthermore, undisclosed sources may provide extra information about the size of the classes. We propose a mixed-integer linear optimization model for computing semi-supervised optimal classification trees that cover the setting of labeled and unlabeled data points as well as the overall number of points in each class for a binary classification. Our numerical results show that our approach leads to a better accuracy and a better Matthews correlation coefficient for biased samples compared to other optimal classification trees, even if onlyfew labeled points are available.

math.OC↗

arXiv2023

Relax and penalize: a new bilevel approach to mixed-binary hyperparameter optimization

In recent years, bilevel approaches have become very popular to efficiently estimate high-dimensional hyperparameters of machine learning models. However, to date, binary parameters are handled by continuous relaxation and rounding strategies, which could lead to inconsistent solutions. In this context, we tackle the challenging optimization of mixed-binary hyperparameters by resorting to an equivalent continuous bilevel reformulation based on an appropriate penalty term. We propose an algorithmic framework that, under suitable assumptions, is guaranteed to provide mixed-binary solutions. Moreover, the generality of the method allows to safely use existing continuous bilevel solvers within the proposed framework. We evaluate the performance of our approach for two specific machine learning problems, i.e., the estimation of the group-sparsity structure in regression problems and the data distillation problem. The reported results show that our method is competitive with state-of-the-art approaches based on relaxation and rounding

cs.LG↗