arXiv subjects

Julian Schwarz

Publications and source records attributed to Julian Schwarz.

11 recordsLinked to original sources

arXiv2026

Tolls for Dynamic Equilibrium Flows

We consider dynamic network flows and study the following question: Which dynamic edge flows can be implemented as tolled dynamic equilibrium flows? We study this question for the heterogeneous-user model, where the flow particles are partitioned into populations characterized by their own source,destination-pairs and a cost function associating with any walk and departure time some costs. As our two main results, we first provide a duality-based characterization of implementability of dynamic edge flows for the multi-source, multi-destination case. Secondly, we derive both, a combinatorial and duality-based characterization of implementability of dynamic edge flows for the multi-source, single-destination case. Both results are derived under a fairly general network loading model. For the proof, we make several technical contributions: We formulate a novel infinite dimensional optimization problem, where the goal is to minimize the aggregated costs of the particles with respect to the fixed network loading induced by the given edge flow. This requires the recently introduced concept of autonomous network loadings for which we show several new structural insights. In particular, we give an alternative (tighter) characterization of the existence of autonomous network loadings for our setting by deriving a generalization of a result of M.A. Zarecki\uı on the Lusin $N^{-1}$ property of absolutely continuous monotone functions which may also be of independent interest. These insights allow us to prove the stated characterizations under the assumption of strong duality. Finally, for the case of a single-destination, we are able to provide a non-trivial proof that this assumption is always fulfilled for finitely supported edge flows with costs representing weighted travel times.

cs.GT↗

arXiv2026

Complex Refractive Index Determination via Microspectroscopy Through Magnifying Optics: Challenges and Opportunities

For the design and optimization of optoelectronic devices, accurate knowledge of the complex refractive indices of the constituent materials is essential. Herein, we present a fast and non-destructive approach for the extraction of the refractive indices from reflectance and transmittance spectra of samples with lateral dimensions down to the micrometer scale. Microspectroscopy, based on the combination of a standard optical microscope and a spectrometer, enables the assessment of the optical response of multilayer stacks using high-magnification optics with correspondingly large numerical apertures. Employing a numerical formalism explicitly accounting for the influence of the numerical aperture, allows for precise retrieval of the refractive index without resorting to dispersion models. We demonstrate the applicability of the proposed method for large-area, homogeneous, optically incoherent samples such as transparent glasses and absorbing 4H-SiC, for a SixNy thin film on glass substrate, and for mechanically exfoliated flakes of highly oriented pyrolytic graphite and MoO3, as representatives of uniaxial and biaxial optical anisotropy. While the results prove excellent agreement with values reported in literature, the case of graphite highlights the limitation for probing the out-of-plane refractive indices due to reduced sensitivity. Finally, we discuss possible extensions towards retrieving the full anisotropic tensor of the refractive index, establishing the proposed approach as a methodologically sound alternative to spectroscopic ellipsometry.

physics.app-ph↗

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

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↗

arXiv2026

Minimal Regret Walras Equilibria for Combinatorial Markets

We consider combinatorial multi-item markets and propose the notion of a $Δ$-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 $Δ$. 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 $Δ$-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↗

arXiv2026

A Decomposition Theorem for Dynamic Flows

The famous flow decomposition theorem of Gallai (1985) states that any static edge $s$,$d$-flow in a directed graph can be decomposed into a nonnegative linear combination of incidence vectors of paths and cycles. In this paper, we study the decomposition problem for the setting of dynamic edge $s$,$d$-flows assuming a quite general dynamic flow propagation model. We prove the following decomposition theorem: For any integrable dynamic edge $s$,$d$-flow, there exists a decomposition into a nonnegative linear combination of $s$,$d$-walk inflows and cycles of zero transit time. We show that a variant of the classical algorithmic approach of iteratively subtracting walk inflows from the current dynamic edge flow converges to a dynamic circulation and that every such circulation can be induced by inflows into cycles of zero transit time. The algorithm terminates in finite time, if there is a lower bound on the minimum edge travel times and the flow is finitely supported. We further characterize those dynamic edge flows which can be decomposed purely into nonnegative linear combinations of $s$,$d$-walk inflows. The proofs rely on the new concept of autonomous network loadings which allows us to describe how particles of a different walk flow would hypothetically propagate throughout the network under the fixed travel times induced by the given edge flow. We show several technical properties of this type of network loading and, as a byproduct, we also derive some general results on dynamic flows which could be of interest outside the context of this paper as well.

cs.DS↗

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

MindBenchAI: An Actionable Platform to Evaluate the Profile and Performance of Large Language Models in a Mental Healthcare Context

Individuals are increasingly utilizing large language model (LLM)based tools for mental health guidance and crisis support in place of human experts. While AI technology has great potential to improve health outcomes, insufficient empirical evidence exists to suggest that AI technology can be deployed as a clinical replacement; thus, there is an urgent need to assess and regulate such tools. Regulatory efforts have been made and multiple evaluation frameworks have been proposed, however,field-wide assessment metrics have yet to be formally integrated. In this paper, we introduce a comprehensive online platform that aggregates evaluation approaches and serves as a dynamic online resource to simplify LLM and LLM-based tool assessment: MindBenchAI. At its core, MindBenchAI is designed to provide easily accessible/interpretable information for diverse stakeholders (patients, clinicians, developers, regulators, etc.). To create MindBenchAI, we built off our work developing MINDapps.org to support informed decision-making around smartphone app use for mental health, and expanded the technical MINDapps.org framework to encompass novel large language model (LLM) functionalities through benchmarking approaches. The MindBenchAI platform is designed as a partnership with the National Alliance on Mental Illness (NAMI) to provide assessment tools that systematically evaluate LLMs and LLM-based tools with objective and transparent criteria from a healthcare standpoint, assessing both profile (i.e. technical features, privacy protections, and conversational style) and performance characteristics (i.e. clinical reasoning skills).

cs.HC↗

arXiv2025

Are System Optimal Dynamic Flows Implementable by Tolls?

A seminal result of [Fleischer et al. and Karakostas and Kolliopulos, both FOCS 2004] states that system optimal multi-commodity static network flows are always implementable as tolled Wardrop equilibrium flows even if users have heterogeneous value-of-time sensitivities. Their proof uses LP-duality to characterize the general implementability of network flows by tolls. For the much more complex setting of $\textit{dynamic flows}$, [Graf et al., SODA 2025] identified necessary and sufficient conditions for a dynamic $s$-$d$ flow to be implementable as a tolled dynamic equilibrium. They used the machinery of (infinite-dimensional) strong duality to obtain their characterizations. Their work, however, does not answer the question of whether system optimal dynamic network flows are implementable by tolls. We consider this question for a general dynamic flow model involving multiple commodities with individual source-destination pairs, fixed inflow rates and heterogeneous valuations of travel time and money spent. We present both a positive and a, perhaps surprising, negative result: For the negative result, we provide a network with multiple source and destination pairs in which under the Vickrey queuing model no system optimal flow is implementable -- even if all users value travel times and spent money the same. Our counter-example even shows that the ratio of the achievable equilibrium travel times by using tolls and of the system optimal travel times can be unbounded. For the single-source, single-destination case, we show that if the traversal time functions are suitably well-behaved (as is the case, for example, in the Vickrey queuing model), any system optimal flow is implementable.

cs.GT↗

arXiv2024

Generalized Nash Equilibrium Problems with Mixed-Integer Variables

We consider generalized Nash equilibrium problems (GNEPs) with non-convex strategy spaces and non-convex cost functions. This general class of games includes the important case of games with mixed-integer variables for which only a few results are known in the literature. We present a new approach to characterize equilibria via a convexification technique using the Nikaido-Isoda function. To any given instance of the GNEP, we construct a set of convexified instances and show that a feasible strategy profile is an equilibrium for the original instance if and only if it is an equilibrium for any convexified instance and the convexified cost functions coincide with the initial ones. We develop this convexification approach along three dimensions: We first show that for quasi-linear models, where a convexified instance exists in which for fixed strategies of the opponent players, the cost function of every player is linear and the respective strategy space is polyhedral, the convexification reduces the GNEP to a standard (non-linear) optimization problem. Secondly, we derive two complete characterizations of those GNEPs for which the convexification leads to a jointly constrained or a jointly convex GNEP, respectively. These characterizations require new concepts related to the interplay of the convex hull operator applied to restricted subsets of feasible strategies and may be interesting on their own. Note that this characterization is also computationally relevant as jointly convex GNEPs have been extensively studied in the literature. Finally, we demonstrate the applicability of our results by presenting a numerical study regarding the computation of equilibria for three classes of GNEPs related to integral network flows and discrete market equilibria.

cs.GT↗

arXiv2020

Psychiatric Home Treatment for Inpatient Care -- Design, Implementation and Participation

The use of information and communication technologies (ICT) to support long-term care is gaining attention, also in the light of population ageing. Known in Scandinavian countries under the term of welfare technology, it aims to increase the quality of life and independence of people with physical, psychological or social impairments. In Germany, a new form of psychiatric home treatment, inpatient equivalent treatment (IET), is offered since 2018. It should allow service users with severe mental health issues to stay in their familiar environment during crisis, while being treated in the same complexity and flexibility like in an inpatient unit. However, this change in delivering healthcare services leads to sociotechnical challenges, such as coordination of work, integration into existing healthcare workflows and ensuring continuity of care. Hence, the objective of this exploratory study is to examine how information and communication technologies (ICT) interact in the new setting and how this process can be improved. Further, we also ask how service users can participate in designing home treatment services. Methodologically, this study follows a qualitative research approach. Different methods including participant observation, interviews and focus groups were conducted to answer the research questions. Data was collected during a field visit at the psychiatric department of a German clinic in summer 2019. Field notes and interviews were analyzed using the R package for qualitative data analysis RQDA. A list of socio-technical challenges and opportunities related to IET were identified. New forms of communication, gaps in documentation practices and continuity of care are seen to be highly relevant for designing and implementing home treatment services in psychiatric care. We also discuss how service users and health professionals can take pro-active part in designing these services.

cs.CY↗