Papers

Large Language Models (LLMs) offer a natural interface for translating human objectives into reward signals for cooperative multi-agent reinforcement learning (MARL), yet the training-time dynamics of this integration remain poorly understood. We show that dynamically updating LLM-generated reward weights during off-policy MARL violates the stationarity assumption of Potential-Based Reward Shaping (PBRS) and contaminates the experience replay buffer, whose stored transitions carry reward labels computed under stale shaping weights. We characterise the result as a regime-dependent failure whose severity depends on how competent the unshaped baseline already is. To control it we propose two stabilisation strategies: a Phase-Based Freeze Schedule that enforces strict stationarity within training phases, and Exponential Moving Average (EMA) smoothing that bounds per-episode weight drift. We evaluate across three cooperative environments and five random seeds with QMIX, complemented by an exploratory VDN extension, yielding a three-regime taxonomy. In the augmentative regime (Simple Spread), where the baseline is functional (74.4 %), EMA significantly improves success to 86.7 % ($+12.3$ pp, $p<0.01$) while naive dynamic updates collapse it to 15.2 %. In the essential regime (Level-Based Foraging), where the baseline is broken (0.1 %), any shaping unlocks the task (95.9 % under EMA). In the supplementary regime (SMAC 3m), where the baseline is near-saturated (98.8 %), stabilised shaping preserves performance (99.9 %) while unstabilised shaping adds variance without gain. These findings establish reward-signal stationarity as a necessary design constraint and indicate that regime placement is a practical predictor of whether dynamic LLM shaping helps or harms.

We propose and analyse a mathematical model of evolutionary adaptation for non-degenerate (permanent) replicator systems, in which the fitness landscape matrix evolves on a slow timescale – the evolutionary time – while the species dynamics unfold on a fast timescale. Under a two-timescale separation justified by Tikhonov's theorem, the adaptation problem reduces to maximising the mean fitness at steady state over a convex admissible set of fitness landscape matrices. We derive a fitness variation formula and establish necessary and sufficient conditions for a fitness maximum, showing that the optimisation reduces at each step to a linear programming problem. The algorithm is applied to four canonical replicator systems: the hypercycle, the bi-hypercycle, the anthill system, and the RNA molecule network. In all cases the evolutionary process follows a universal three-phase pattern: an initial phase of fitness growth without equilibrium shift, during which purely altruistic replication gives way to mixed altruistic-selfish behaviour; a second phase of dominant species emergence; and a stabilisation phase analogous to the error catastrophe threshold in quasispecies models. A key consequence is that all evolved systems acquire resistance to parasitic species. We further prove that without non-degeneracy constraints the process leads to sequential species annihilation, with a provable spectral lower bound on fitness increase by dimension reduction.

Volterra series feedback linearizes a class of nonlinear hyperbolic PDEs but produces a controller that, even after truncation, demands solving a tower of plant-specific kernel PDEs and evaluating nested integrals. We prove the truncated controller is jointly Lipschitz in plant and state, and learn it as a single neural operator from plant nonlinearity and state to boundary control. Once trained, no kernel is ever solved again, for any plant in the trained class. The closed loop is practically stable in class-$\mathcal{KL}$ form, with a residual ball scaling linearly with training accuracy.

Backstepping for nonlinear PDEs yields exact feedback linearizing laws in the form of infinite Volterra series – elegant in theory, but with challenges for implementation. This paper shows that even very low-order truncations of such controllers, no longer exactly linearizing, retain the stabilizing power. The key insight is that higher-order terms become negligible near the origin, so stability is recovered for any fixed truncation order by restricting the initial condition size. We establish spatial sup-norm results: finite-time practical stability and asymptotic stability characterized by a class-$\mathcal{KL}$ estimate. The region-of-attraction estimate grows with the truncation order and shrinks with the growth rate of the nonlinearity. The analysis overcomes the lack of pointwise kernel bounds and resolves well-posedness of the nonlinear closed loop, showing that surprisingly simple approximations already capture the essence of nonlinear PDE feedback linearization.

We analytically determine the ultimate limits of state preparation in two-level open quantum systems driven by coherent control. For a dissipative qubit governed by a GKSL master equation, we give an exact characterization of the reachable set in the Bloch ball. Dissipation excludes a region of states in the Bloch ball which cannot be approached even under arbitrarily strong coherent driving, and we prove that this region has a nontrivial geometry whose boundary is a surface of revolution around the $x$-axis which is analytic except for two conical singularities. We derive a closed-form control protocol for moving on this boundary, and construct an explicit protocol that steers the system arbitrarily close to any prescribed boundary state. These results provide a complete geometric constructive description of reachable qubit states in the standard dissipative environment, establishing fundamental bounds on controllability and state-preparation fidelity for open two-level quantum systems.

Presolve for mixed integer programming (MIP) problems aims to eliminate redundant information, strengthen the formulation, and extract useful structural information for the subsequent branch-and-cut process. An important type of such structural information is the variable implications (VIs), which describe how a bound on a variable depends on a bound of a binary variable. In this paper, we develop two new presolve techniques that exploit VIs to derive reductions for MIP problems. The first technique, called VI aggregation, aggregates multiple VIs into a single inequality by using implications between a variable and a set of binary variables that form a clique. This aggregation can reduce the number of constraints and tighten the linear programming relaxation. The second technique, called VI-aware linear constraint propagation (LCP), builds on the standard LCP but incorporates VIs associated with the variable being tightened to derive more reductions and can derive tighter variable bounds. We show that although VI information is additionally considered, the tightest lower or upper bound of a variable can still be derived in linear time. Moreover, compared with a state-of-the-art approach in the literature, the proposed VI-aware LCP can derive tighter variable bounds. Computational results on MIPLIB 2017 benchmark instances demonstrate the effectiveness of VI aggregation and VI-aware LCP in improving the performance of the open-source MIP solver HiGHS. In particular, using the two proposed presolve techniques, a reduction of 4% in solving time and 6% in node number on HiGHS can be achieved.

We study information design when a designer with commitment shapes the information of strategically interacting, far-sighted agents whose actions drive a persistent, controlled Markov state. We introduce the Markov Bayes correlated equilibrium (Markov BCE), the controlled-Markov generalisation of the BCE of Bergemann and Morris (2016), characterised by a dynamic obedience condition that adds a continuation-value term to the static one and reduces to it when actions cannot move the state. Recommending actions is without loss; the designer's problem is recursive in the agents' promised continuation utilities and is solved by a set-valued backward-induction algorithm whose optimum exists and lies between the no-disclosure and first-best values. For linear-quadratic-Gaussian payoffs the obedience condition becomes a covariance condition with a modified interaction matrix, and the stationary case reduces to an algebraic Riccati equation. When agents instead learn the transition, we identify the rent an agent earns from a model of the dynamics sharper than the designer anticipates: it is non-negative, zero at the known-dynamics benchmark, and deterred only by building slack into obedience. Under persistent excitation the cumulative rent grows logarithmically as heterogeneous agents' estimates converge. Two worked examples, in congestion and resource coordination, together with a numerical study illustrate the theory.

We propose the first deep learning algorithm, the Certainty Equivalent Learning (CEL) algorithm, for solving high-dimensional discrete-time dynamic programming problems with recursive utility. Dynamic programming with recursive utility is numerically challenging because the recursive utility does not have an explicit representation and the Bellman equation contains a certainty equivalent that is difficult to evaluate. The CEL algorithm learns this certainty-equivalent value directly with neural networks and jointly approximates value functions, policy functions, and certainty-equivalent functions. The CEL algorithm is mesh-free and simulation-based, allowing high-dimensional state and control spaces, and does not rely on Euler equations, first-order conditions, or differentiability of the state transition function. The CEL algorithm also works for dynamic programming problems with expected utility as expected utility is a special case of recursive utility. We apply the CEL to discounted linear exponential quadratic Gaussian control, small-noise robust control, Epstein-Zin DSGE, and multivariate strategic asset allocation problems. Compared with closed-form and VFI-based benchmarks, the CEL delivers accurate value and policy approximations, remains effective in high-dimensional problems, achieves accuracy comparable to VFI in the small-noise robust-control case, and produces out-of-sample Bellman errors and Euler or first-order residuals that are in the range from 1.0e-4 to 1.0e-3 for most problems.

Gradient based optimization methods are nowadays the methods of choice for training deep neural networks (DNNs) in artificial intelligence (AI) systems. In practically relevant DNN training problems, one does usually not apply the standard gradient descent (GD) optimization method but instead one employs suitable sophisticated GD optimization methods, which incorporate adaptivity and/or acceleration techniques, such as the famous Adam optimizer. It is a key contribution of this work to provide a general unified convergence analysis for GD optimization methods in the training of DNNs with analytic activations such as the softplus and the popular Gaussian error linear unit (GeLU) activation. Our general unified convergence result applies to a large class of gradient based optimization methods such as the standard GD, the momentum, the Nesterov accelerated gradient (NAG), the RMSprop, the Adam, the Adamax, the Nadam, the Nadamax, the Adan, the AdaBelief, the AMSGrad, and the Yogi optimizers. Our analysis employs the theory of Kurdyka-Łojasiewicz (KL) inequalities to establish convergence to critical points in the training of DNNs. To the best of our knowledge, the generality of our convergence analysis is also just in the special situation of the Adam optimizer a new contribution to the literature on the analysis of AI optimization algorithms.

A central challenge in many mobile multi-robot applications is that communication topologies are inherently time-varying. Agents may enter or exit the network and such changes cannot generally be restricted a priori. This work introduces a distributed multi-agent control algorithm based on local communication that supports anytime agent joining and leaving the communication network without centralized coordination. The method scales efficiently with the number of agents by relying on a distance-based neighbor definition and on contracts derived from predicted trajectories. The resulting contract constraints guarantee collision avoidance and constraint satisfaction. We validate the proposed method in an autonomous multi-agent driving scenario, demonstrating effective collision avoidance in high-speed, dynamic environments with agents moving in opposite directions, in both simulated and real-world experiments.

We study a receding horizon game in which multiple agents drive linear systems subject to additive disturbances, private state and input constraints, and shared coupling constraints. We propose a robust game-theoretic control framework that combines tube-based constraint tightening with a finite-horizon generalized Nash equilibrium problem (GNEP), equipped with a discrete algebraic Riccati equation (DARE)-based terminal cost and a decoupled positively invariant terminal set. The framework guarantees recursive feasibility for every bounded disturbance realization. Exploiting the potential-game structure induced by tracking costs, we further establish asymptotic convergence of each agent's nominal state to a steady-state variational generalized Nash equilibrium (vGNE), and show that each agent's actual state converges to a neighborhood of the vGNE determined by the minimal robust positively invariant set.

We study the fundamental classification problem of computing a separating hyperplane for a binary-labeled dataset of size $n$ with normalized $d$-dimensional features. Letting $\Phi \in \mathbb{R}^{n \times d}$ denote the feature matrix and $\gamma$ the margin of the maximum-margin separating hyperplane, we present a randomized algorithm that solves this problem in $\tilde{O}(\gamma^{-2/3}\, \operatorname{nnz}(\Phi) + \gamma^{-2(\omega+1)/3})$-sequential running time (work), $\tilde{O}(\gamma^{-2/3})$-parallel (computational) depth, and accesses $\Phi$ only through $\tilde{O}(\gamma^{-2/3})$-matrix-vector queries (matvecs). We also present a second, faster randomized algorithm with a $\tilde{O}(\gamma^{-2/3}\, \operatorname{nnz}(\Phi) + \gamma^{-2})$-sequential running time that uses $\tilde{O}(\gamma^{-2/3})$-matvecs to $\Phi$, but achieves only $\tilde{O}(\gamma^{-4/3})$-parallel depth. Both algorithms match the near-optimal deterministic matvec complexity recently established by Kornowski and Shamir [2025], Karmarkar et al. [2026] and achieve improved sequential runtime and parallel depth, albeit at the expense of using randomness.

Modern supply chains span diverse operational environments, ranging from e-commerce distribution networks to customized production-to-order manufacturing lines. Across these settings, operational efficiency depends on coordinating two highly interdependent stages: order preparation and downstream delivery. Although these stages are traditionally managed in isolation, real-world fulfillment systems must satisfy stringent delivery expectations under dynamic stochastic order arrivals. To bridge this gap, we introduce the Dynamic Order Fulfillment Problem (DOFP), a new problem class unifying logistical challenges previously studied separately. We model DOFP as a Markov decision process whose state and decision spaces are partitioned into preparation and delivery sub-spaces, linked by synchronization constraints. While recent approaches attempt to optimize both fulfillment stages simultaneously over myopic rolling horizons, our framework isolates and optimizes the downstream delivery policy, treating preparation strictly as a state-level constraint filter. To solve this, we develop the Decomposition-Driven Framework with Value Function Approximation (DDF-VFA), which utilizes a novel policy-level decomposition. This design partitions the search into a delivery-stage master problem and a preparation-stage compatibility subproblem, iteratively refined via feedback loops. DDF-VFA executes this strategy by combining a large-neighborhood search over partial delivery decisions with a neural-network value function approximation for the cost-to-go. Numerical illustrations on two example variants using real-world datasets show that DDF-VFA consistently outperforms benchmarks that optimize the two stages independently or jointly without decomposition. Finally, the framework naturally scales to accommodate additional real-world complexities such as batched or multi-stage preparation.

We study sequential posted pricing for selling a single item under an exogenous deadline: a seller makes take-it-or-leave-it price offers to buyers one at a time, stopping when some buyer accepts or the selling opportunity expires. We consider both deterministic deadlines and random deadlines drawn from a known distribution, and allow buyer valuations to be either independent or arbitrarily correlated. Despite its practical relevance, this collection of deadline-constrained sequential pricing models has received limited algorithmic study beyond special cases. We develop the first approximation algorithms for revenue maximization across these settings.

Physical sensing and actuation noise floors should inform how much belief resolution a decision-making system can reliably use. We introduce Finite Reliability Representations (FRR), a framework for covering belief spaces by reliability cells: regions within which the optimal action-value function Q*(b,u) varies by at most a tolerance epsilon, uniformly over actions. The framework is formulated on beliefs rather than states and uses a cover rather than an equivalence quotient, because approximate decision-closeness is not transitive in general. A central technical point is that noisy Bayesian updates should not be treated as globally contractive on arbitrary beliefs. We therefore separate three objects: the fixed-observation filter map, the predictive observation law, and the controlled belief-transition kernel. For nonlinear continuous-state systems, FRR is obtained under a reachable-set Lipschitz modulus for the belief-transition kernel. For finite-state POMDPs, the same construction becomes exact on the belief simplex: prediction is linear, Bayesian correction is a normalized positive linear map, sensor noise enters through observation-distribution distinguishability, and actuation uncertainty enters through an action-execution channel. Under the corresponding action-value Lipschitz condition, an FRR cover supports a cell-constant policy whose suboptimality is bounded by 2 epsilon/(1 - gamma). We also introduce reliability entropy, the logarithm of the minimal number of reliability cells, as a measure of certified decision-relevant belief complexity. The framework distinguishes representation sufficiency from fundamental performance floors imposed by sensing, process, and actuation noise. It applies to finite POMDPs, linear-Gaussian filters, locally linearized nonlinear filters, and particle-filter implementations through analytic or empirical certification of reliability cells.

We establish finite-sample closed-loop stability guarantees for Model Predictive Path Integral (MPPI) control applied to discrete-time Linear Time-Invariant (LTI) systems with additive Gaussian process disturbances. The key observation is that, for unconstrained LTI/quadratic systems with the DARE terminal cost, the exact finite-horizon MPC law has the same first control action as the infinite-horizon LQR law for every planning horizon. Thus, finite-sample MPPI can be analyzed as a stochastic perturbation of LQR. First, we show that the MPPI control law approximates the LQR feedback with high probability. The approximation error decomposes into a Monte Carlo term that decreases with the sample count and an infinite-sample temperature bias that persists at finite temperature but vanishes as the temperature is reduced. The resulting constants are written in terms of the horizon-dependent stacked cost matrices, making explicit that the finite-sample certificate is parametrized by the selected planning horizon. Second, we use a Lyapunov perturbation argument to prove practical exponential stability in expectation. On sample paths that remain in a compact Lyapunov sublevel set over a finite operating horizon, the expected state norm decays exponentially up to three residual floors: a process-noise floor, an MPPI approximation floor, and a confidence floor from the per-step sampling failure probability. The sufficient sample threshold is explicit and computable from the DARE solution, LQR stability margin, MPPI sampling parameters, temperature, and planning horizon. In the joint limit of infinite samples and vanishing temperature bias, the result recovers the stochastic LQR stability bound.

We solve in semi-explicit form a class of non-Markovian stochastic optimal control problems with path-dependent rewards, using path signatures. We reformulate the control problem as the computation of Laplace transforms of signature functionals thanks to the Boué-Dupuis representation. Exploiting recent signature representations of such transforms on tensor algebras, we determine the value process and the optimal control through an infinite-dimensional system of Riccati equations on the extended tensor algebra. We establish an explicit feedback representation of the optimal control and the value process as an infinite linear combination of the time-extended signature of the controlled process, with time-dependent coefficients. The expansions being intrinsically local, we propose a dynamic recentering algorithm to ensure a global representation over the entire time horizon. We illustrate the approach on genuinely path-dependent, non-linear examples that go beyond the tractable linear-quadratic setting, including the tracking of linear functionals of the signature and signature lifts of Volterra control problems.

We develop a static mean field game formulation for optimal wind energy capacity siting in Alberta, Canada. The revenue model for agents is based on the expected wind resource at a location and the covariance of this wind resource with other locations. Agents choose locations to maximize long-run revenue while accounting for spatial correlations in wind resource availability. Spatial wind dependence is incorporated through an empirically calibrated covariance structure, combining a PCA-based component estimated from historical weather station data with a parametric residual kernel, to capture variability across locations. The equilibrium investment problem can be formulated as a quadratic program, which we solve for four policy scenarios that progressively restrict the feasible siting area, incorporating viewscape and transmission constraints beyond a baseline of minimal siting restrictions. These policy and infrastructure-driven land use restrictions are represented as constraints on the agents' action space, allowing us to examine their economic implications through a comparison of the MFG equilibria across policy scenarios.Our findings highlight the trade-off between regulatory land use objectives and economic efficiency in the use of renewable energy resources in the short and long term, and provide quantitative insights into how policy design shapes the spatial distribution and financial performance of wind investments. The static MFG formulation could be used to inform electricity system planning including, transmission planning and renewable energy development planning.

We present an L1-optimal control problem class with linear nonnegative costs subject to multiplicative Itô diffusion processes with elementwise linear input constraints. Forward invariance of the positive orthant is established for the considered stochastic dynamics, and a simulation method consistent with this invariance property is proposed. Both finite-horizon and discounted infinite-horizon stochastic L1-optimal control problems are considered. These problems admit explicit solutions characterized by a vector-valued ordinary differential equation in the finite-horizon case and by an algebraic equation in the infinite-horizon case. Notably, the optimal value function and feedback policy coincide with those of the corresponding deterministic problem, demonstrating robustness to multiplicative stochastic uncertainty. A portfolio example illustrates our results.

We study the online routing problem in large language model serving, where requests arrive sequentially and must be dispatched to parallel decode workers under tight batch-size and KV-cache constraints. Unlike widely used routing heuristics that are not tied to explicit service-level objectives (SLOs) and offer limited control over latency-throughput trade-offs, we introduce a multi-objective optimization framework that formulates routing as an online linear programming with interpretable decision rewards. We apply an efficient bid-price control policy based on the online linear programming that admits requests when their SLO-weighted benefit exceeds their shadow prices. To meet millisecond decision requirements, we develop a warm-started, projected first-order updates that track the evolving dual shadow prices online with predictable runtime. We integrate our router into the Vidur simulator and demonstrate substantial improvements over standard baselines across multiple SLO regimes, including end-to-end latency, time-to-first-token, throughput, and tail performance. A big picture from our result: a science-based approach outperforms others based on heuristics.

In this work, we address a real-life course timetabling problem that incorporates several practical academic requirements, including the chronological precedence of theoretical sessions over practical ones, the enforcement of rest periods for lecturers, and the simultaneous assignment of lecturers and teaching assistants. The formulation involves a trade-off between minimizing course conflicts and maximizing schedule compactness by reducing the number of dispersed working days for lecturers under strict daily working hour constraints. To handle this trade-off, we develop a bi-objective mixed-integer linear programming (MILP) model solved using a weighted-sum objective approach. Computational experiments using a real-world dataset demonstrate that the proposed model reduces timetable conflicts by 13.33%, improves lecturer schedule compactness by 26%, and completely eliminates violations of theoretical-practical session precedence compared to the real-life timetable. Overall, the findings highlight the efficiency and robustness of the proposed MILP model for complex university timetabling problems.

Stability is often assumed in learning and identification, yet it is rarely characterized directly from input–output data. We show that an input–output family admits a stable finite-dimensional state-linear realization iff it has finite Hankel-rank and its response decays uniformly with time; for state-linear realizable maps this decay is necessarily exponential. We extend these results to state-affine, LPV, and linear switched systems via suitable input-forgetting notions, and relate forgetting to decay of impulse responses (sub-Markov parameters). In all cases, the decay/forgetting rate determines the decay rate of every minimal realization.

Time delays are inherent in many physical and engineered systems and can significantly affect their stability and performance. In this work, we investigate the dissipativity of a class of nonlinear time-delay systems with multiple discrete delays and derive sufficient conditions for both delay-dependent and delay-independent dissipativity using Bessel-Legendre inequalities. For linear systems, the resulting dissipativity conditions are expressed in terms of linear matrix inequalities (LMIs) which can be solved numerically to obtain Lyapunov-Krasovskii-type storage functions.

Large-scale Capacitated Vehicle Routing Problems (CVRPs) are commonly solved by partitioning customers into smaller routing problems that can be optimized independently. While this substantially reduces computational complexity, independently constructed routing solutions may leave some customer demand unserved even when sufficient resources exist elsewhere in the fleet. We present Collaborative Routing Constructors (CoRC), a routing framework that enables independently solved subproblems to exchange customers and vehicles during optimization rather than relying solely on a fixed partition or a subsequent global re-optimization stage. Computational experiments on AGS benchmark instances and synthetic instances containing up to 200,000 customers compare CoRC against independent routing, post-routing global re-optimization, and state-of-the-art, end-to-end routing frameworks. Across all evaluated partitioning strategies, CoRC consistently constructs feasible routing solutions where competing partition-based methods do not. Furthermore, it remains effective on problem instances for which the evaluated end-to-end routing frameworks did not produce solutions under the same computational budget. These results demonstrate that collaboration between routing subproblems provides a robust and scalable approach for feasible large-scale route construction.

The Gaussian Error Linear Unit is usually motivated as the expected output of an input-dependent Bernoulli gate. This work gives an alternative interpretation: GELU is the expected output of a hard linear gate with a Gaussian random threshold. This view provides a generative interpretation for the Bernoulli gate: the gate opens once the input clears a latent Gaussian threshold. This interpretation stems from a decomposition based on well-known results in stochastic inventory theory and leads to a threshold-transmission family that includes ReLU, GELU, SiLU/Swish, and hard swish as special cases. By considering a latent uniform threshold, we recover a hard-swish-like piecewise-polynomial gate whose nonlinear transition is confined to a finite interval, yielding fixed- and learned-width variants. Controlled experiments on compact vision and language models show that calibrated or learned uniform-threshold gates are consistently competitive with GELU, ReLU, and SiLU/Swish, display architecture-dependent learned widths, and use the finite transition region nontrivially.

While traditional hub capacity planning models optimize effectively for quantitative inputs, they often fail to digest qualitative business context. We propose a novel framework where a large language model (LLM) agent iteratively proposes hub capacity decisions guided by natural-language business context descriptions. The key mechanism is a chain-of-thought reasoning protocol: the LLM constructs a structured decision table that maps each contextual item to specific capacity adjustments based on the implied direction and magnitude of changes. The new capacity decision is then validated through a feedback loop with an optimization model, which provides routing-based performance metrics to guide the agent's selection. On a real-world 13-hub freight network in the southeastern US, our framework achieves a 2.8% optimality gap relative to the hidden ground-truth, a significant improvement over the 11.0% gap produced by the traditional optimization model without textual business inputs. This demonstrates that LLMs can serve as a contextual bridge, integrating qualitative business insights into Operations Research workflows.

This paper presents an accelerated solver for the unit commitment problem in large-scale power systems. The approach is based on the concurrent execution of GPU- and CPU-based optimization solvers on a single machine, with the solver that converges first terminating the other to minimize overall runtime. This strategy effectively harnesses the complementary strengths of different solvers. Convergence is further accelerated through a systematic and aggressive presolve approach. Numerical experiments on a 6,049-bus system with millions of decision variables and constraints demonstrate speedups ranging from 2.14x to 5.61x, reducing the maximum runtime from 42.12 minutes to 5.77 minutes across 45 test cases. These results highlight the scalability and computational efficiency of the proposed GPU-CPU concurrent solver framework.

In emerging urban mobility and logistics applications, such as advanced air mobility, electric vehicle charging, and shared service systems, agents with heterogeneous valuations choose among multiple outcomes while sharing congested resources. Existing congestion game and resource-allocation models, however, typically restrict either the structure of heterogeneous outcome preferences or the form of congestion externalities. We introduce a nonatomic congestion game framework in which agents have multidimensional heterogeneous valuations over a finite set of outcomes, with the population represented by a measure over the valuation space. We show that, although there is a continuum of heterogeneous agents, both Nash equilibria and social optima admit finite-dimensional characterizations. Nash equilibria are characterized by aggregate flows and outcome-level threshold vectors, while social optima admit a semi-discrete optimal transport formulation characterized by aggregate flows and strategy-level Kantorovich dual potentials. For separable utilities and latency functions $\ell_e$ such that $x\mapsto x\ell_e(x)$ is convex for every resource $e$, the social welfare problem becomes a convex optimization problem, and we characterize social optima through marginal congestion costs and capacity-dual prices. Our results provide a geometric characterization of congestion games with multidimensional heterogeneous valuations and connect nonatomic congestion games with semi-discrete optimal transport.

Observer-quotient security studies interactive cryptographic systems whose security depends on what an admissible observer can distinguish across transcripts, leakage traces, and hidden implementation continuations. The paper defines observer-indexed experiments with session identifiers, adaptive schedulers, oracle forwarding, simulators, ideal quotient functionalities, and nonuniform environments, and proves a real/ideal emulation theorem in which sequential morphism defects add, parallel defects obey a product-TV bound, and adaptive observer choice is absorbed by an explicit wrapper construction. The resulting advantage bound is indexed by $\delta_{\mathrm{obs},t}$, $\delta_{K,t}$, $\delta_{\mathrm{post},t}$, $\delta_{\mathrm{sim},t}$, $\eta_t$, and the residual floor $\rho_T(\mathbb E Z_T)$. The framework is instantiated for IND-CPA encryption with timing leakage, deterministic encryption with entropy ledgers, and finite-state side-channel refinement under transcript, timing, cache, power, EM, and profiled observers. The optimization/control component identifies hidden continuations with observability kernels, treats sensor redesign as quotient refinement, and converts dissipativity, PL-type rates, and ISS residual bounds into concrete reductions in distinguishing advantage. Ancillary code and synthetic data reproduce the finite-state leakage audit and LTI observer-design benchmark.

We show that projected Adam for online optimization with arbitrary moment decay parameters $\beta_1,\beta_2\in[0,1)$ can have average regret bounded away from zero. A similar result of Reddi-Kale-Kumar from 2018 required $\beta_1<\sqrt{\beta_2}$. Similar to their result, we use a three-periodic sequence of linear functions on $[-1,1]$ with slopes $c,-1,-1$, though we use $c$ slightly larger than $2$. This nonzero average regret result extends to Adam variants such as AdamW, RMSProp, NAdam, Adan, AdaMax, Muon, and to an i.i.d. variant of the three-periodic sequence of slopes for Adam.