Papers

In this article we show that the projected dynamical system obtained by restricting the state of an incrementally scattering passive system to a closed and convex subset K of the state space (a real Hilbert space), is also an incrementally scattering passive system. First we show that the projection of a maximal dissipative operator to the tangent cones of K is again maximal dissipative, hence, it determines a contraction semigroup.

Background and Objective: Quantitative analysis of cell dynamics is central to modern biological research, providing critical insights into immune cell interactions, disease progression, and drug mechanisms. Automated cell tracking in time-lapse microscopy remains challenging due to noise, morphological variations, overlapping cells, and dynamic events such as divisions and fusions. Methods: We present ARGUS, a framework for Accelerated, Robust, General, and Unsupervised Cell Tracking Solutions. ARGUS combines adaptive cell detection, dense Farneback optical-flow prediction, frame-to-frame linear assignment, and a sequence-level tracklet-refinement step that reconnects trajectory fragments across short temporal gaps. Results: On publicly available Cell Tracking Challenge datasets, ARGUS achieved detection accuracy of 0.905-0.971 and tracking accuracy of 0.897-0.964, with runtimes within 1 minute (5-6 seconds for 3 frames). Conclusions: ARGUS is a modular, interpretable framework that can be adapted to different imaging modalities and biological applications without training data or GPU infrastructure. The implementation is publicly available at https://github.com/Gitinc/argus

In this paper, we investigate a class of so-called sub-infinite horizon stochastic linear-quadratic optimal control problems, in which the initial time $t$ is arbitrarily taken from $[0,\infty)$ and the running cost is defined over $[t,t+T]$ for a given $T>0$. The optimal control of this type of problem can be obtained by standard methods; however, it is shown that the resulting optimal control is generally time-inconsistent. Thus, instead of seeking an optimal control, which is time-inconsistent, we aim to find a time-consistent, locally optimal, and time-invariant equilibrium strategy, by introducing a new and very interesting type of Riccati equation. Its main feature is that the generator depends on a delay term of the unknown. In other words, this Riccati equation is a backward ordinary differential equation (ODE) with delay, which is equivalent to a forward ODE with advanced terms. Such an equation is essentially a Fredholm integral equation, whose solvability is challenging. We overcome the difficulty by deriving a sharp a priori estimate and applying the Leray–Schauder fixed point theorem. To this end, we establish a comparison theorem between two matrix-valued nonlinear algebraic equations. The convergence behavior of the solution to the Riccati equation as $T\to\infty$ is also provided.

The proximal operator is a fundamental tool in variational analysis and optimization. In the setting of a Hilbert space, given a proper, lower semicontinuous convex functional, its proximal operator is non-expansive, that is, 1-Lipschitz continuous. In the Wasserstein setting, the contraction properties of this operator have been investigated from different perspectives by Carlen and Craig and by Adve and Mészáros, among others, and are not completely understood. In this paper, we study the stability properties of proximal maps, with a particular focus on non-expansivity, under various notions of convexity of the functional that can be considered in the Wasserstein space.

This paper revisits the adaptive extended golden-ratio primal–dual algorithm (aEGRPDA) proposed by Soe et al. (2026) for structured convex optimisation problems involving a differentiable term that is only locally smooth. We prove that the artificial upper bound imposed on the primal step-size in aEGRPDA is redundant, since the adaptive rule itself keeps the step-sizes bounded above. As a consequence, the ergodic $\mathcal O(1/N)$ estimates for the objective residual and feasibility violation, where $N\ge1$ denotes the number of iterations, are independent of this hyperparameter. Consequently, the resulting adaptive golden-ratio primal–dual method, therefore, requires neither a step-size cap, nor a linesearch procedure, nor a known global Lipschitz constant. We establish linear convergence of the algorithm when both the primal and dual functions are strongly convex. Furthermore, we develop two accelerated variants, in addition to the local smoothness assumption: one for the case where the nonsmooth primal component is strongly convex, and another for the case where the differentiable term is globally strongly convex. For these accelerated methods, we prove an ergodic $\mathcal O(1/N^2)$ convergence rate. Preliminary numerical experiments on a Poisson imaging problem illustrate the efficiency and robustness of the proposed approaches.

We extend the trace–determinant framework of Nayak, Sharma, and Mishra~\cite{nayak2026} for bounding the H-eigenvalues of symmetric positive definite tensors. First, we replace the Arithmetic–Geometric Mean (AM–GM) relaxation underlying previous bounds by the exact solution of the associated constrained optimization problem, yielding sharp upper and lower bounds that are attained on the admissible spectral variety. Second, we incorporate higher-order power sums as additional spectral invariants and prove a structural theorem showing that any extremizer over a $K$-invariant feasibility region has at most $K$ distinct spectral values. This reduces the problem to a finite collection of low-dimensional polynomial systems and yields a hierarchy of increasingly tight bounds. For the four-invariant case $(T,S,p_3,D)$, we develop a complete theory including solution-count estimates, a multistart Newton algorithm, and sharpness conditions. We also derive closed-form bounds in small dimensions, establish perturbation estimates, and obtain refined Lyapunov region-of-attraction bounds. Numerical experiments for dimensions up to $d=100$ show that the sharp three-invariant bound reduces the median relative overestimation gap from $53\%$ to $6\%$ while maintaining low computational cost. The framework is validated on tensors with real H-spectrum.

Primal-dual first-order methods are widely used for large-scale semidefinite programming (SDP), but their ability to compute highly accurate solutions is not well explained by global convergence theory alone. We study the local convergence of the primal-dual hybrid gradient (PDHG) method applied to a standard primal–dual SDP pair. We show that PDHG converges eventually (R-)linearly whenever the limiting KKT point satisfies either strict complementarity or primal–dual nondegeneracy. The proof views PDHG as a preconditioned proximal point method for the KKT inclusion and combines its descent inequality with a local error bound. Under strict complementarity, the error bound follows from the local spectral geometry of the positive semidefinite cone; under primal-dual nondegeneracy, it follows from strong regularity of the KKT mapping. We also give a simple SDP instance where both regularity conditions fail and PDHG can converge only sublinearly. This contrasts with linear programming, where PDHG admits a local linear convergence regime even for degenerate instances. Numerical experiments support the theory and identify difficult SDP instances where PDHG struggles to reach high accuracy.

Diffusion-based policies have recently emerged as powerful policy parameterizations for reinforcement learning, representing state-conditioned action distributions as terminal laws of diffusion processes with parameterized drifts. This terminal-law representation has shown substantial expressive flexibility in practice, enabling diffusion policies to model complex, multimodal, and highly non-Gaussian action distributions; however, it remains unclear what mathematically drives this expressivity and how to fully exploit it when the policy is learned from finite data. In this paper, we identify the drift Lipschitz budget $K$ as a central quantity governing the expressivity and statistical behavior of diffusion policies. We quantify expressivity through approximation: diffusion policies with $K$-Lipschitz drifts can concentrate near optimal deterministic policies and achieve value approximation error of order $1/K$; moreover, we prove a matching lower bound under nondegenerate diffusion noise. This increased expressivity comes with a statistical cost. When the drift is parameterized by neural networks, increasing $K$ improves approximation but increases statistical complexity. Balancing these two terms yields a finite-sample performance gap of order $\tilde{O}(n^{-2/(m+6)})$ for generic neural-network drifts, and a sharper rate $\tilde{O}(n^{-2/(m+4)})$ for one-sided dissipative drift classes, where $n$ is the sample size and $m$ is the dimension of the state space. Numerical experiments provide empirical evidence for the sample-dependent trade-off in $K$, supporting both theoretical regimes. Our framework also suggests a practical implementation principle: choose the diffusion budget $K$ according to the available sample size, and then select a neural-network architecture with the corresponding fixed Lipschitz coefficient.

Pairwise comparisons are fundamental in the analytic hierarchy process. Various consistency indices have been proposed to assess inconsistencies in these comparisons. Since Saaty first proposed his consistency index, the assessment of the degree of consistency in pairwise comparison matrices has remained an open and hot topic in the study of the analytic hierarchy process. The consistency indices CI and CR proposed by Saaty are defined using the principal eigenvalue of the pairwise comparison matrix. In our previous study, we introduced an alternative index derived from the relationship between the coefficient of the characteristic polynomial and the consistency of comparisons. Saaty proposed a fixed threshold of 0.1 for CI or CR as a guideline for an acceptable level of consistency, regardless of the matrix size. However, whether this threshold represents an equivalent level of consistency across different matrix sizes, that is, across different numbers of evaluation items, remains unclear. This study analysed the relationship between consistency and matrix size by examining pairwise comparison matrices constructed from subsets of evaluation items. Based on this analysis, we propose the fundamental property to be satisfied by a size-independent consistency index. Furthermore, we refine our previously proposed index to ensure that it satisfies this property, demonstrating that it coincides with the existing consistency index. Finally, we visualise the relationship between the matrix size and consistency index values using randomly generated pairwise comparison matrices, thereby providing insights into the impact of matrix size on consistency evaluation.

Among the many types of acceleration techniques designed to improve the performance of Benders decomposition, unified cut generation schemes have recently attracted a keen interest. Unified cuts aim for a better balance between the generation of optimality and feasibility cuts, while also providing a way to compare the strengths of different feasibility cuts. Our goal is to assess the experimental performance of a broad selection of unified and distinct Benders cuts in the context of the multi-commodity fixed-charge network design problem (MCFNDP). We express under a common mathematical structure and notation the construction of each unified or distinct Benders cut considered. We also explain how the generic formulations of the Benders cuts can be specialized to conform to the specifications of the MCFNDP. In addition, we suggest bespoke methods for comparing the performance of several solution methods when the benchmark is made up of heterogeneous problem instances. We report the results of a systematic empirical analysis comparing the performances of 50 Benders methods involving unified or distinct cuts in applications to a common testing bench made up of standardized MCFNDP instances. The analysis identifies a small number of leading Benders methods, namely those featuring the static Brandenberg-Stursberg cuts and the Hosseini-Turner l1-deepest cuts. In addition, we also report results obtained by using both Gurobi and CPLEX as the supporting solver to the SMS++ computation library.

We study approximation and structural simplification of optimal control policies for controlled regime-switching diffusion processes for discounted, ergodic, finite-horizon, and exit-time criteria. We first establish continuity of the cost functionals over classes of Markov and stationary Markov policies by exploiting elliptic and parabolic regularity of the corresponding Hamilton–Jacobi–Bellman and Poisson equations. Using density results of policies with finite-action, piecewise-constant, and Lipschitz continuous, we show that each control problem admits $\varepsilon$-optimal policies within these structured subclasses. We then construct an Euler–Maruyama approximation of the controlled regime-switching diffusion under piecewise-constant controls. We prove strong convergence of the controlled state process and establish convergence of the associated finite-horizon value functions with rate $O(h^{\gamma/2})$. Building on this discretization, we develop a finite-state approximation of the induced discrete-time Markov chain via state-space quantization. We show that the value functions of the finite models converge uniformly on compact sets to the value function of the original problem, and that optimal policies of the approximating models are asymptotically optimal. These results provide a systematic framework for approximating regime-switching diffusion control problems and justify the use of structured policies and finite-state models for numerical implementation.

We investigate quadratic bilinear systems by developing novel tree-based representations of their solutions. The proposed framework decomposes the solution into a sequence of coupled bilinear subsystems whose components admit explicit expansions indexed by full binary trees. These representations yield sufficient conditions for the existence of global solutions and lead to new output bounds in terms of reachability Gramians. Motivated by these estimates, we introduce time-limited and infinite-horizon reachability and observability Gramians, establish sufficient conditions for their existence, and characterize them through nonlinear matrix equations. The associated Gramians are employed to identify dominant state-spaces and to derive exact reduced-order models obtained by removing Gramian kernels. Building on these results, we develop a balanced truncation method for quadratic bilinear systems and prove an error bound for the reduced-order approximation. The proposed framework provides a unified connection between tree-based solution representations, nonlinear Gramian theory, and balanced truncation for quadratic bilinear systems, closing several theoretical gaps in the analysis of Gramian-based model reduction for this class of systems.

Modern machine learning (ML) increasingly relies on complex models whose behavior is difficult to characterize beyond empirical performance metrics. Across a wide range of tasks, including prediction, generation, and decision-making, models with similar empirical performance can exhibit markedly different properties in terms of their transparency, interpretability, robustness, fairness, privacy, and certifiability. This survey highlights how optimization- and certification-oriented reasoning can provide a useful framework for reasoning about such differences, supporting tasks ranging from model training and selection to auditing and certification. We review and synthesize recent advances at the intersection of combinatorial optimization (CO) and trustworthy ML, covering both training and post-training tasks, including interpretable model learning, explanation generation, robustness analysis, fairness auditing, model compression, and privacy attacks and protections. Across these domains, CO formulations offer additional capabilities over purely heuristic approaches, e.g., gradient-based ones, notably global guarantees, formal certificates, and explicit treatment of trade-offs. While scalability remains an important challenge, continued progress in solvers and hybrid algorithms suggests a growing role for CO in the design and deployment of trustworthy ML systems.

This work presents a novel approach for adapting neural network architecture along the depth based on a posteriori error estimation. By formulating neural network training as a continuous-time optimal control problem, we derive rigorous error estimates that quantify how approximation error distributes across network layers. This error decomposition enables a principled depth adaptation strategy: new layers are inserted at locations of maximum estimated error, allowing the network to efficiently capture complex, nonlinear variations in the underlying problem. Our framework introduces a novel network architecture that treats weights and biases as piecewise linear functions varying across layers, with the error estimator bounding the discrepancy between this discrete representation and the true continuous optimal control solution. The approach leverages dual weighted residual methodology from finite element analysis to derive computable upper bounds on the functional error. A key theoretical contribution is the derivation of explicit error bounds that decompose the total approximation error into interval-wise contributions, providing a rigorous basis for targeted architecture refinement. We demonstrate the effectiveness of our method on scientific datasets, including learning the observable-to-parameter map for the Navier-Stokes equation. Numerical results reveal that our approach consistently outperforms existing architecture adaptation methods in terms of generalization performance.

In this paper, we introduce the moving-anchored extra-gradient (MAEG) method for solving monotone inclusion problems involving the sum of a continuous monotone operator and a maximal monotone operator. Notably, the distance from the anchor point to the solution set is designed to be monotonically non-increasing. Under Lipschitz continuity of the forward operator, MAEG attains an $\mathcal{O}(1/k)$ non-asymptotic iteration complexity, and when a positive anchor-update parameter is used, it further achieves an $o(1/k)$ asymptotic rate. Furthermore, leveraging the specific behavior of the anchor point, we propose a tailored restart strategy. We demonstrate that this strategy ensures convergence even in the absence of local Lipschitz continuity, while preserving the original iteration complexity guarantees whenever the Lipschitz condition holds.

In nonlinear optimal control, uncertainties in system dynamics may affect not only closed-loop stability but also the achieved optimality properties of the resulting solutions. This paper develops a systematic robustness analysis for nonlinear optimal control beyond the conventional focus on stability in robust control theory. First, we demonstrate that the optimal value function retains its Lyapunov property under a quantifiable criterion, thereby guaranteeing the preservation of closed-loop stability. Building upon this foundation, we establish explicit characterizations for optimality deviations induced by model mismatch in both closed-loop performance and optimal controllers, and further reveal their consistency with classical linear-quadratic regulator (LQR) results. In addition, the robustness analysis admits a unified computational formulation that gives rise to an iterative scheme with guaranteed convergence, enabling quantitative assessment of optimality robustness in nonlinear control systems. Numerical examples validate the theoretical analysis.

We study quantitative convergence rates of the vanishing viscosity approximation of first-order time-dependent Mean Field Games with regularizing coupling acting in the Hamilton-Jacobi equation. Under standard structural assumptions on the Hamiltonian ensuring convergence of the vanishing viscosity approximation, previous results provide either qualitative convergence for the two unknown of the system or quantitative estimates merely for solutions of the Hamilton-Jacobi equation. In this work, we improve these convergence rates under the same assumptions and establish, in addition, quantitative estimates for the convergence of the associated forward Fokker-Planck equation. As a consequence, we obtain quantitative convergence rates for the full Mean Field Game system, thus extending and strengthening the existing theory for the first-order limit.

Reaction–diffusion systems can produce spatial patterns such as stripes and spots by diffusion-driven instability. In synthetic developmental biology, such patterns are engineered in cell populations, and steering them toward a prescribed target pattern can be formulated as an optimal control problem. When the control acts on the entire spatial domain, the existence of optimal controls and the first-order optimality conditions are well established. In experiments, the control is applied with light by optogenetic tools, and the light illuminates a chosen region of the tissue, so the control acts on a subdomain. We consider the setting where the control is restricted to a subdomain fixed in advance, for a synthetic circuit in which the slowly diffusing activator Nodal and the fast-diffusing inhibitor Lefty form the pattern. We study the optimality loss, the difference between the optimal costs of the subdomain problem and of the full-domain problem, and we ask how much coverage keeps it within a desired precision. In simulations with one parameter set and two target patterns, nine squares give a smaller terminal error than one square at the same coverage. Moreover, the error depends on the average distance from the uncontrolled points to the control region. In particular, small errors occur when this distance is comparable to the length scale set by Nodal diffusion and degradation.

The unit commitment (UC) problem is paramount for optimal operation of power systems, but it faces computational limitations in large-scale settings, especially in investment or stochastic models, because of the binary variables that it contains. A lot of research has attempted to improve the computational performance of UC models, either by reducing model size, resulting in lower fidelity and accuracy, or by improving the tightness of the formulation. Tightness and model size are the best a priori indicators of the computational performance of UC models, but there is no clear overview of what the best formulation is for different generators. In this research, we define models with different levels of detail, and present a formulation for each level that is based on the convex hull. We show new proofs on the tightness of well-known formulations for ramping, for start-up and shut-down costs and capabilities, and for UC with investment. These models, with a different level of detail, can be incorporated into large-scale problems to reduce the computational burden, as demonstrated in Part II.

Greenhouse fruit-production control aims to maximize the economic performance (fruit revenue minus operating costs) while operating within system constraints under external weather disturbances. Control methods need to balance the delayed economic benefit of fruit yield with current operating costs. For such problems, model predictive control (MPC) can explicitly handle system constraints under future weather disturbances, but can become computationally demanding when using sufficiently long prediction horizons for (relatively large) nonlinear greenhouse fruit production models. In contrast, reinforcement learning (RL) can learn control policies offline while considering longer-term economic performance, but struggles to enforce system constraints, and performance may degrade under unseen weather trajectories. This work proposes trajectory-selection RL-MPC, a framework that incorporates longer-term economic information of fruit yield into a short-horizon MPC optimization problem. The framework uses an RL rollout trajectory to define a terminal region constraint and terminal cost. Next, a nonlinear MPC solves a short-horizon optimization problem with these terminal ingredients to find a local optimum. Finally, the framework selects and executes the first input from the trajectory with the better objective value, either from the MPC-predicted or the RL rollout trajectory. The method is applied to GreenLight, a large-scale greenhouse tomato production model that exhibits stiff dynamics. The simulation results show that trajectory-selection RL-MPC with a one-hour prediction horizon matches the closed-loop performance of a high-performing guiding policy while significantly improving over standalone MPC with the same horizon.

Maximal monotone mappings that are piecewise polyhedral arise from subdifferentials of convex functions and saddle functions that are piecewise linear-quadratic and enter into algorithmic constructions of importance in linear-quadratic optimization and associated splitting methods. The question of whether those constructions preserve maximal monotonicity is then crucial. The usual answers to that invoke constraint qualifications involving the nonemptiness of intersections of certain relative interiors, but it is shown here that the piecewise polyhedral structure allows the relative interiors to be bypassed.

Online truckload bid acceptance is a closed-loop stochastic decision problem in which a carrier or broker must, in real time, accept or reject a tendered load subject to operational feasibility, fleet repositioning costs, and opportunity cost against future demand. Public, reproducible benchmarks for this problem are scarce: existing routing benchmarks are static, while dynamic-fleet studies typically rely on private operator data. We introduce FreightBidBench, a public-calibrated, dependency-free, closed-loop benchmark in which feasibility (pickup reach, appointment windows, simplified hours-of-service, stochastic yard delays) and economics (service-failure penalty, terminal fleet value, daily price-premium window) are explicit, versioned, and reproducible from public Freight Analysis Framework and U.S. Department of Agriculture truck rate data. We develop two full-horizon hindsight ceilings: a simple LP style relaxation and a tighter Lagrangian-per-truck information relaxation that retains per-truck hours-of-service and sequencing structure and is 20.7% tighter than the LP relaxation on a tight-capacity scenario and 39.3% tighter on a scarce-capacity scenario. We introduce a parametric surrogate-rollout cascade with boundary-band and scarcity-pressure escalation triggers. On ten-seed tight and scarce scenarios, the best simple policy retains 91.0% and 86.5% of rollout profit and the standard-library surrogate 94.2% and 89.3%; a cascade at a single escalation band recovers roughly 98% on both at 40-56% of rollout's mean decision latency, and on the tight scenario is statistically indistinguishable from the rollout teacher (paired-bootstrap 95% CI on the profit delta spans zero).

We study the output regulator problem for an impedance passive linear plant, using the classical resonant internal model based controller.

Large-scale combinatorial optimization remains demanding for classical heuristics, particularly when dense Quadratic Unconstrained Binary Optimization (QUBO) formulations induce large memory footprints, high CPU utilization, and long execution times. While near-term quantum processors cannot yet deliver unconditional quantum advantage, hybrid architectures can provide practical value by reducing the resource burden. This paper presents a resource-efficiency study of Hybrid Quantum Neighborhood Selection (HQNS), a framework that decomposes large dense QUBO instances into bounded-width quantum subproblems via stochastic frontier selection. We evaluate HQNS on the Maximum Diversity Subset Selection Problem (MDSSP), focusing on the trade-off between solution quality retention and resource consumption. Benchmarks up to N=1000 candidates show that HQNS preserves 99.9908% of the mean diversity score of an 11-restart parallel Simulated Annealing baseline, while reducing wall-clock time by 94.91%, peak CPU utilization by 64.68%, and peak memory usage by 88.61%. The QPU execution time remains bounded within a 6-7 second envelope across scales, indicating that the quantum component is decoupled from the global QUBO dimension when the frontier size is fixed. These results suggest that HQNS provides a resource-aware pathway for deploying hybrid quantum optimization in practical large-scale settings, serving as an efficient architecture for incorporating near-term quantum processors into classical optimization pipelines.

Pairwise comparison (PC) via pairwise reciprocal matrices (PRMs) is central to the Analytic Hierarchy Process (AHP). Although the traditional eigenvector method is widely applied to derive priorities, its theoretical robustness in reflecting true priority vectors remains debated. Building upon a previous iteration of this study, this research develops the revised Least Penalty-Squared Prioritization (LPSP) optimization models, including the revised Least Product of Penalty and Direct Squares (LPPDS) and revised Weighted Squares (LPPWS), to minimize the revised Root Mean Penalty-Squared Variance (RMPSV) and the revised Root Mean Penalty-Weighted Square Variance (RMPSWV). However, solving these non-linear formulations is computationally complex for decision-makers. To overcome these limitations, this study proposes the Parallel Osprey Optimized Least Penalty-Squared Prioritization (POO-LPSP) method. By integrating an improved bio-inspired metaheuristic Parallel Osprey Optimization Algorithm (POOA), this framework efficiently solves complex LPSP models to minimize RMPSV and RMPSWV, thereby enhancing prioritization reliability. The practical utility and computational efficiency of the POO-LPSP method are validated through a numerical application focusing on a Generative AI (GAI) vendor selection problem. To extend, POO-LPSP can serve as a robust alternative to Saaty's Eigen system method for AHP applications.

This study presents a multi-fidelity framework for the systematic optimization of genetic algorithm (GA) hyperparameters. The framework integrates three fidelity levels: high-fidelity Fast Fourier Transform (FFT) homogenization for validation, a medium-fidelity 3D convolutional neural network surrogate for rapid property evaluation, and a low-fidelity Gaussian process (GP) surrogate within a Bayesian optimization (BO) framework to guide the hyperparameter search. Various acquisition functions are evaluated, with logNEI achieving the best performance by effectively accounting for the noise inherent in GA evaluations. The proposed framework identifies hyperparameter configurations that enable a 25-generation GA run to achieve elastic modulus values comparable to those obtained in a full 75-generation optimization. Furthermore, introducing a penalized BO objective significantly reduces the number of required lattices with only minor decreases in absolute achieved elastic modulus, revealing a practical trade-off between performance and the number of structures that must be evaluated. High-fidelity FFT validation verifies the effectiveness of the surrogate-driven optimization strategy. The optimized hyperparameters allow for rapid convergence, eliminate the need for lattice mutation, and reduce the overall computational cost by 24% (from 225 to 171 hours) while preserving mechanical performance. These results demonstrate the potential of multi-fidelity optimization as an efficient and practical approach for GA hyperparameter tuning and future experimental lattice design studies.

Optimizer experiments observe responses to algorithmic configurations without uniquely revealing hidden mechanisms. We develop a causal optimizer interaction calculus that separates pathwise realization, Mobius decomposition, and experimental identification. Under a fixed innovation coupling, every finite-horizon innovation-driven optimizer admits a behaviorally minimal pathwise realization. For any finite effect support and intervention design, an incidence operator gives the complete observational gauge, exact identifiability, sharp quotient stability, held-out predictions, and exact noiseless configuration complexity. Smooth hidden relaxation generates interactions through inverse hidden-state stiffness. Building on this structural law, we prove an observable-readout transfer theorem: arbitrary smooth update or trace readouts inherit an explicit five-term interaction through first and second hidden responses. Unlike the reduced optimal value, a general readout has no universal interaction sign. Its Boolean effects remain exact integrals of continuous interaction curvature and can therefore be identified by factorial interventions. We also derive Gaussian quotient minimax risk, exact confidence sets and tests, misspecification decomposition, certified downstream decisions, and optimal replication. A controlled real-data experiment on a 65-dimensional strongly convex logistic model validates the complete reduced-value chain. Boolean effects and independently integrated curvature agree within 4.21e-11, while nine held-out continuous intensities agree within 8.88e-13. Gaussian campaigns attain the predicted coverage and power, and 4,500 real-minibatch observations reject an order-two interaction model. Neural trace audits provide complementary evidence that the declared response classes remain informative in nonconvex training.

Structured preconditioners restrict optimization to a small family of positive metrics, but endpoint condition-number reachability does not measure the geometric effort required to reach a useful metric. We formulate this effort as a path-space value problem. Restricted dynamic geometric complexity is the least affine-invariant length of an admissible metric path whose endpoint reaches a Hessian-relative generalized-eigenvalue condition target. Path elimination gives an exact min-plus semigroup and Bellman principle, while fixed-horizon kinetic energy is exactly squared complexity divided by twice the horizon. The main response result is global on a Hadamard state space: geodesic convexity produces a smooth intervention-cube path branch and a uniformly coercive Jacobi form, while one Green inverse generates the value Hessian, two-sided force-to-curvature bounds, exact Möbius effects, and arbitrary prescribed finite-order responses. For the hard condition target, a bordered Jacobi–KKT theorem differentiates the moving projection endpoint and multiplier on every regular active spectral stratum; its indefinite inverse also explains why hard-target interactions need not share the unconstrained sign. The theory specializes to affine-invariant positive-definite geometry. A determinant-one two-dimensional diagonal model has an exact target interval, a closed-form forced path, and a strictly negative-definite interaction matrix. A moving diagonal Hessian gives a closed-form hard-target projection, multiplier, and pair effects of either sign, while a coordinate-sequential three-dimensional protocol yields an exact path metric strictly larger than the ambient projection distance. Thus the global Green and bordered hard-target responses are explicit laws of restricted metric-path elimination built on Bellman composition.

Computationally efficient models for multimodal traffic flows with inter-modal interactions are foundational for coordinated traffic management. However, such models are lacking in the current literature. This study introduces a multimodal link transmission model (M-LTM) accommodating both continuous road traffic and discrete tramway traffic at the network level. M-LTM builds on a link model that captures the inter-modal interactions through the moving bottleneck theory. We further develop node models describing the flow transfer, tram operations, and mode interactions at tram stops and intersections. Specifically, the tram dwell process and dwell-induced congestion of road traffic are simulated at the stop node, and the intersection node model can reproduce queue spillback and tram priority. Case studies are conducted on two synthetic one-node networks, a synthetic arterial network, and the real network of Dresden to demonstrate the predictive power and the applicability of the proposed model in network traffic analysis and management. The GEH statistics of road traffic are below 4, and the average tram arrival time absolute errors are less than 10 seconds.

Data-Enabled Predictive Control (DeePC) has recently emerged as a framework for controlling unknown systems from data. However, its performance relies on the relevance of the collected data, and as such, disturbances lead to inevitable errors. This paper addresses this problem by proposing an augmentation of DeePC using Model-Free Disturbance Observer with Online Modification (MFDOOM). The method corrects output predictions based on previous prediction errors using a dedicated continuously updated Hankel matrix. We compare our method, both theoretically and through simulation, to other recent algorithms designed for time-varying systems in the DeePC framework. It is shown that for disturbances that can be modeled as the output of an autonomous linear time-invariant system, this approach can reduce tracking error and online-update burden compared with existing online DeePC variants.