Papers

We present anyakrakusuma, an open-source Python library that solves the discrete static Schrödinger bridge problem, the entropically regularized counterpart of optimal transport, through a log-domain Sinkhorn–Knopp iteration and reconstructs the entropic interpolation between two empirical point clouds. The solver is paired with a diagnostic pipeline that characterizes the optimal coupling and the intermediate distributions through information-theoretic and geometric measures. We exercise the library on four idealized planar cases spanning a circle-to-circle dilation, a spiral-to-mixture fragmentation, a rigid reorientation of two moons, and a Lissajous-to-trefoil deformation. The log-domain formulation is necessary rather than merely convenient at the parameters studied, where the cost-to-regularization ratio reaches four hundred and the Gibbs kernel underflows double precision across most of its range; the iteration nonetheless attains a marginal residual of $10^{-9}$ and unit marginal fidelity in every case. Residual histories decay geometrically over approximately eight decades at per-iteration contraction factors between $0.966$ and $0.976$, which are local rates near the fixed point that lie many orders of magnitude below the worst-case Hilbert-metric bound. The covariance analysis recovers an imposed ninety-degree reorientation to within $0.07^\circ$, roughly forty times smaller than its uncertainty, across a masked interval of near-isotropy on which the principal axis is unobservable. The diagnostics are reported with explicit attention to the regimes in which each is well defined, including the differential entropy, which is meaningful only on the open interpolation interval. The presented cases are constructed rather than measured; quantitative application to empirical point clouds requires further study.

For smooth nonconvex stochastic multi-objective optimization, stochastic multi-gradient descent (SMG) computes an approximate steepest common descent direction of the objectives from stochastic gradients. Under standard assumptions, this note establishes a new convergence rate for SMG in terms of the squared Pareto-stationarity (PS) measure. With a constant stepsize and linearly growing mini-batches, the expected squared empirical PS measure at the algorithm's output is $\tilde{O}(T^{-1})$ after $T$ iterations. This improves on the $\tilde{O}(T^{-1/4})$ bound obtained by Chen et al. (JMLR, 2024) for linearly growing batches, without requiring bounded gradients. Here, $\tilde{O}(\cdot)$ suppresses logarithmic factors. The key to the rate improvement is to exploit the Lipschitz continuity of the PS measure, defined by the norm of the multi-gradient descent algorithm (MGDA) direction, rather than the $(1/2)$-Hölder continuity of the MGDA direction used by Chen et al. (2024). For variants of MoCo and MoCo+ with exact MGDA updates and constant batch sizes, the analysis gives expected squared empirical PS rates of $O(T^{-1/2})$ and $\tilde{O}(T^{-2/3})$, respectively. These bounds match the corresponding single-objective momentum rates up to logarithmic factors. The proof was accidentally discovered while the author was preparing homework for a graduate course: ChatGPT 5.4 Thinking Extended generated the initial proof strategy for the main SMG result in response to an author-written homework-solution prompt; the author then verified and reorganized the resulting argument. The appendices document the prompt, summarize the student submissions with LLM disclosures, and present additional proofs and results on momentum-based stochastic MGDA.

Entropy regularization smooths equilibrium policies in time-inconsistent stochastic control. At low temperature, the same Gibbs response can strongly amplify errors in learned rewards and dynamics. We show that the derivative of an exploratory equilibrium is governed by a backward Volterra-parabolic resolvent. Along an aligned positive mode, a lower bound has the same exponential order. A block decomposition identifies the source of the amplification: causal paths contribute powers of 1/tau, whereas a positive feedback cycle can produce exponential growth. At fixed temperature, a local equilibrium branch is twice differentiable with respect to finite-dimensional model parameters, which yields a function-valued delta method. A bounded uniformly elliptic diffusion realizes this path-cycle distinction in every finite dimension. Closing one positive cycle changes the root-n linear-response boundary from a power law to order 1/log n; along the cyclic Perron mode, right-endpoint discretization is relatively consistent exactly when N tau^2 -> infinity. An affine model also gives an exact nonlinear transition at the Lambert-W temperature beta T / W(beta T sqrt(n)). Numerical calculations illustrate these rates.

We investigate the asymptotic behavior and turnpike properties of graphon mean field control (GMFC) problems in the linear-quadratic setting. We consider both a finite-horizon GMFC problem and its associated ergodic counterpart, in which the controlled dynamics are governed by a graphon mean field stochastic differential equation with heterogeneous interactions. The optimal controls and state trajectories for both problems are characterized by systems of Riccati equations together with systems of generalized differential and algebraic equations on suitable Hilbert spaces. Under a stabilizability condition and appropriate positivity assumptions on the graphon-induced operators, we establish the unique solvability of the ergodic control problem and derive exponential convergence estimates for the finite-horizon system to its stationary limit. As a consequence, we establish an exponential turnpike property for the optimal pair and prove the convergence of the time-averaged value function for the finite-horizon GMFC problem.

Prediction markets are attracting growing attention as trading volumes rise and their practical relevance increases. To ensure efficient price discovery, liquidity provision becomes ever more important. Due to the binary settlement structure in prediction markets, optimal market making leads to an optimization problem that is fundamentally different from the ones studied in classical settings. In this paper, we develop a stochastic control framework for prediction markets in which the market price is modeled as a conditional probability of the outcome that is generated by a transformed latent belief diffusion. A market maker selects bid and ask quotes to maximize expected terminal wealth while controlling both mark-to-market inventory risk and the settlement risk of remaining positions at resolution. We derive the associated Hamilton–Jacobi–Bellman equation and characterize the unique optimal bid and ask quotes. By transforming the equation to the latent belief space and using a fixed-point argument, we prove existence and uniqueness of a classical solution and verify the resulting optimal quoting strategy. In addition, we provide a numerical analysis, which reveals how optimal liquidity provision in prediction markets depends on inventory, market beliefs, time to resolution, and risk aversion. Further, we demonstrate that the optimal quoting strategy substantially improves downside protection while preserving most of its expected profit relative to a myopic benchmark that maximizes the instantaneous expected mark-to-market profit.

We present CARDAL (Curvature-Aware Rank-Adaptive Distributed Augmented Lagrangian), a distributed multi-GPU solver for large-scale semidefinite programs (SDPs) based on a rank-adaptive Burer-Monteiro factorization and an augmented Lagrangian method. At fixed ranks, a matrix-free L-BFGS method with negative-curvature corrections targets an approximate Euclidean second-order stationary point of the factored augmented Lagrangian. A reverse multiplier shift turns a negative dual-slack direction into exact negative curvature after rank expansion, and a small joint rank-lift problem selects a batched low-rank correction. A verified slack lower bound provides an a posteriori approximate KKT certificate. Our analysis establishes generic global-optimality guarantees for heterogeneous products of PSD cones at per-block ranks near the Barvinok-Pataki scale, together with a finite-accuracy counterpart under blockwise cost smoothing. For scalable execution, CARDAL distributes constraint rows, factor columns, and PSD blocks over a Constraint x Rank x Cone device mesh. The primal residual, gradient, Hessian-vector products, and slack matrix-vector products are evaluated using device-local operations and axis-wise collectives. On the Mittelmann benchmark, CARDAL exhibits stronger robustness than existing low-rank GPU approaches under a uniform accuracy standard. Experiments on large-scale SDP relaxations from robotics, electronic structure, and Max-Cut demonstrate the complementary scaling regimes of the three distribution axes, with observed wall-clock speedups of up to 4x on four H100 GPUs.

We formulate and solve the reflected sub-Riemannian Schrödinger bridge (SB) problem: minimum-energy transport of probability distributions on a bounded domain for underactuated dynamics with degenerate diffusion and hard state constraints. The main difficulties are geometric and degeneracy: Euclidean normal reflection is generally incompatible with the horizontal subbundle, since it may push the process in directions not generated by the admissible control and noise fields. Also, the reference measure may not have full support. We address these by introducing an intrinsic oblique reflection mechanism that is compatible to the sub-Riemannian structure. Under Hörmander and non-characteristic boundary assumptions, we prove that the reflected degenerate reference process admits a smooth, strictly positive transition density. The resulting optimal control is characterized by a forward–backward PDE system with asymmetric boundary conditions: an oblique Neumann condition for the backward factor and a normal no-flux condition for the forward factor. Since explicit transition densities are generally not available in this setting, we develop a PDE-based Sinkhorn iteration that enforces these boundary conditions directly. Our examples reveal that SB depends on the topology of the domain and the geometry of the diffusion.

In an incomplete round robin tournament, teams play against different sets of opponents. Given varying strengths of the teams, this raises a fairness issue. We establish the complexity of determining whether a fair draw exists under various scenarios involving the presence of pots (i.e., strength based groups of teams), teams coming from the same association, connectivity requirements, and others. We also show experimentally how fair incomplete round robin tournaments can be generated.

This paper addresses the problem of energy-efficient thermal management in Battery Electric Vehicles (BEVs), with the goal of extending driving range while ensuring passenger comfort and battery thermal safety. A hierarchical Nonlinear Model Predictive Control (NMPC) framework is proposed, in which the supervisory layer explicitly balances energy consumption and cabin temperature reference tracking over a long horizon and translates this trade-off into optimal actuator command sequences for the integrated heating, ventilation, and air conditioning (HVAC) system and the battery thermal management (BTM) system. These command sequences are then enforced by a lower-layer NMPC, which introduces local corrections to compensate disturbances and modeling uncertainty while tracking an externally specified cabin temperature reference trajectory. To enable predictive control of the highly nonlinear and strongly coupled thermal dynamics, data-driven neural network models are employed as prediction models within both control layers. The proposed approach is developed and validated using a high-fidelity BEV thermal simulator developed at Politecnico di Torino in collaboration with Stellantis N.V., and benchmarked against an industrial rule-based (RB) control strategy over the WLTC driving cycle. Simulation results demonstrate a significant reduction in energy consumption compared to the baseline strategy, while satisfying comfort and battery temperature constraints.

Nonlinear state estimation under structural model uncertainty remains a fundamental challenge in autonomous Guidance, Navigation, and Control (GNC) systems. Conventional Bayesian filtering separates state propagation from measurement correction, allowing model mismatch to accumulate during propagation, resulting in proposal–likelihood inconsistency, particle degeneracy, and degraded estimation accuracy. Existing approaches primarily improve proposal distributions or weighting strategies without explicitly incorporating measurement geometry into state propagation. This paper introduces a geometry-consistent Bayesian filtering framework that incorporates measurement geometry directly into the propagation process. The nominal drift is projected onto the measurement-consistent subspace, yielding a geometry-consistent proposal while preserving the Bayesian posterior through a rigorous change-of-measure formulation. A Geometric Projection Particle Filter (GPF) is developed together with a geometric co-state that quantifies instantaneous dynamics–measurement inconsistency. Theoretical analysis establishes existence and uniqueness of the projected dynamics, posterior preservation, standard Monte Carlo convergence of the particle approximation, and robustness under structural model uncertainty. The framework is validated using lunar descent navigation under partial observability and persistent model uncertainty. Compared with the bootstrap particle filter and conventional Gaussian filtering methods, GPF consistently achieves higher effective sample size and lower estimation error. These results demonstrate that geometry-consistent propagation provides a principled, computationally efficient, and theoretically grounded framework for robust nonlinear Bayesian filtering.

Value-function-type reformulations have generated a broad class of methods for bilevel optimization. However, the corresponding value-function-type constraints are inherently degenerate and generally fail to satisfy standard constraint qualifications, so the associated multiplier sequences may be unbounded and bounded-multiplier convergence analyses become inapplicable. We study this issue for the regularized gap-function reformulation of bilevel problems with constrained convex lower-level programs. We prove that accumulation points of approximate stationary sequences are C-stationary for the corresponding Karush-Kuhn-Tucker-based mathematical program with complementarity constraints (MPCC), even when the multiplier sequence associated with the regularized gap-function constraint is unbounded. The result holds under Mangasarian-Fromovitz constraint qualification (MFCQ) for the upper- and lower-level constraint systems and MPCC-MFCQ at the limiting MPCC point, without any constraint qualification on the regularized gap-function constraint itself. We further provide an example showing that approximate stationary points of the standard regularized gap-function reformulation may converge to a point that is C-stationary but not M-stationary. To guarantee M-stationarity, we introduce a slack-based two-parameter penalty formulation preserving exact multiplier-slack complementarity and establish M-stationarity under a domination condition on the penalty parameters. We develop an inexact slack-penalty method with adaptive penalty updates and feasibility correction, whose accumulation points are M-stationary under the stated assumptions.

We consider an insurance risk model with a random walk structure in which the underlying probability law is unknown. A decision maker observes a statistic $Q_n$ computed from $n$ historical observations and then needs to choose a capital buffer of the form $C_n=n f(c,Q_n)$, where $c=\log(1/\delta)/n$ balances the amount of data and the tolerated ruin probability $\delta$. The classical safe capital buffer is inversely proportional to the adjustment coefficient $\gamma$, the exponential rate at which the ruin probability decays; when the law is unknown, this coefficient must be inferred from the data. The profile $f$ couples the statistical cost of observing an atypical historical statistic with the future ruin exponent induced by the resulting decision. We study the joint ex ante probability (over both the historical sample and an independent future risk process) that the future maximum exceeds the data-dependent buffer. We show that the naive plug-in rule fails to achieve the prescribed logarithmic decay exponent $c$, illustrating the adverse impact of model uncertainty when making decisions under rare-event constraints. We then identify a profile $f^*=f^*(c,Q_n)$ with the following appealing properties: its lower semicontinuous majorants are safe, while regular continuous rules that fall strictly below $f^*$ are, under mild conditions, unsafe. We illustrate the potential applicability of our framework by developing three parametric examples. In nonparametric settings, we show that a single exponential envelope leads to degenerate capital buffers, whereas a two-level exponential envelope yields a nondegenerate capital buffer which we prove to be safe.

We develop an adjoint-sensitivity framework for positional influence in causal residual Transformers and separate unconditional analytic results from conditional boundary-shape conclusions. The principal unconditional theorem is the residual-to-depth-flow estimate for layer controls converging in $L^1$, complemented by a finite-token-to-Volterra attention estimate that explicitly controls the first cells near the causal endpoint. We define a normalized adjoint-energy influence density and derive its exact evolution along full-batch gradient flow. The adjoint admits an exact generator-term decomposition into residual transmission, nonlocal Volterra, and local channels, including all covariance cross terms. Causal masking can amplify early-position sensitivity and residual identity paths can transmit a right-localized terminal bias, but neither mechanism alone forces a U-shaped profile. We therefore state boundary advantages under independently checkable energy, correlation, and local-channel bounds; these conditions are sufficient rather than necessary. Finite-token influence balancing, positional reweighting, and task-aligned observability are presented as diagnostics or regularizers with explicit differentiation requirements, computational costs, and limitations. Controlled simulations illustrate that each intervention controls its designated surrogate, while observability balance or outer-loop reweighting need not monotonically reduce the influence-based Lost-in-the-Middle diagnostic.

We study atomic splittable congestion games with $n\ge2$ players and nondecreasing $C^2$ resource costs $f$ with convex $x\mapsto xf(x)$. We characterize the largest resource cost class for which the Nash equilibrium is unique. This characterization combines a curvature inequality involving the first two derivatives and the number of players with strict increase of the scalar marginal costs $x\mapsto f(x)+xf'(x)$. The proof rests on first characterizing the largest resource cost classes for which the associated variational inequality operator is monotone, strictly monotone, or strongly monotone. We also draw a perhaps surprising connection to learning dynamics: the same curvature inequality characterizes universal local and global stability of Euclidean-regularized Frank–Wolfe dynamics on arbitrary compact convex strategy spaces. Both the uniqueness and stability characterizations assume closure under positive affine transformations. Both characterizations remain exact for network games; the necessity constructions use acyclic directed graphs whose arc costs can be chosen strictly positive. Finally, we show that on parallel-link networks, every interior equilibrium of the regularized Frank–Wolfe dynamics is locally exponentially stable even without the curvature condition.

Low-rank adaptation (LoRA) makes finetuning large language models cheaper by adding to each weight matrix a trainable low-rank update parameterized as the product of two matrices. These matrices are usually trained with Adam, which treats them as a single flat vector of parameters and ignores both the matrix and product structure of LoRA. Applying a matrix-aware optimizer such as Muon to each factor does not consistently improve over Adam, and neither do the product-aware Muon variants proposed in concurrent works. To realize consistent gains, we introduce PoLoRA, a Preconditioned Orthogonalized LoRA optimizer built from three ingredients: a product-aware spectral update direction, curvature preconditioning derived from controlling the per-sample loss change, and a magnitude rule that controls the sizes of both the factor and merged updates. We evaluate PoLoRA on instruction-tuning datasets for code and math across models from 1B to 8B parameters, and find that it reaches the final held-out loss achieved by tuned Adam in 1.2-1.7 times fewer steps, while adding at most 3% per-step overhead. Compared to Adam, PoLoRA is also less sensitive to the learning rate, and its optimal learning rate is stable across ranks.

Stochastic nonconvex optimization is central to training deep networks and LLMs in modern machine learning. We give a black-box reduction from stochastic nonconvex optimization to ordinary static regret minimization in online convex optimization (OCO), thereby resolving the open problem posed by Chen and Hazan (2024). Our reduction maintains a predictable gradient tracker, while a black-box online learner $\mathcal{A}$ selects a preconditioner that transforms this tracker into the update direction. Given a \(\beta\)-smooth function with a range bounded by $M$ and an unbiased gradient oracle with variance bounded by $\sigma^2$, we bound the expected average squared gradient norm by $O(\sigma\sqrt{M\beta/T}+\sqrt{M\beta}\mathrm{Reg}_T(\mathcal{A})/T+\frac{M\beta}{T})$, where $\mathrm{Reg}_T(\mathcal{A})$ is the static regret of $\mathcal{A}$. Thus, any OCO oracle with $O(\sqrt{T})$ regret recovers the classical $O(T^{-1/2})$ convergence rate. We further extend the framework to nonsmooth nonconvex objectives, still relying only on ordinary static regret, and attain the optimal convergence rate for Goldstein-type stationarity. Finally, we conduct numerical experiments on nonconvex objectives to illustrate how the reduction exploits online-selected preconditioners while using the same stochastic-oracle budget as stochastic gradient descent.

We establish mean-square and concentration bounds for stochastic approximation (SA) with arbitrary norm contractive mappings, under a multiplicative noise model where the noise may scale affinely with the norm of the iterates, and the iterates are potentially unbounded. These settings arise in reinforcement learning, where operators are often contractive in the $\ell_\infty$ norm and the noise scales with the iterates. To address the arbitrary norm, earlier works replace the non-smooth squared norm with a smooth Lyapunov function constructed via the generalized Moreau envelope. For concentration analysis, these works handle multiplicative noise and unbounded iterates through a multi-stage bootstrapping argument that starts from a time-varying worst-case bound and iteratively refines it. We instead present a unified and elementary analysis that yields both bounds. Using an averaged noise sequence and corresponding auxiliary iterates, we obtain a one-step Lyapunov drift inequality for the normed error directly, without smoothing the norm or constructing an envelope. For the mean-square bound, we combine this drift inequality with an induction argument showing that the iterates remain bounded in expectation. For the concentration bound, we develop a probabilistic induction over a sequence of "good" events on which the iterates are controlled, allowing the standard Azuma-Hoeffding bound to be applied. Our approach yields the first sub-Gaussian tailed maximal (all-time) concentration bound for SA under multiplicative noise, by allowing the stepsize to depend logarithmically on the confidence level. Beyond the specific setting considered here, we discuss the generalizability of these proof techniques to other noise models and iterative algorithms.

We solve the optimal consumption-investment problem in incomplete markets, where the investor aims to maximise an Epstein-Zin type stochastic differential utility from consumption over an infinite time horizon. We verify that the optimal strategies can be characterised by the limit of a sequence of solutions of the HJB equation in bounded domains with carefully designed boundary conditions. We also conduct numerical experiments for three market models. Explicit solutions are constructed under some parameter regimes.

Non-smooth and non-convex optimization problems are pervasive in machine learning, control, and signal processing, due to the need for sparse solutions and the inherently non-convex nature of many objective functions. In this paper, we study non-smooth and non-convex distributed optimization problems. We propose a novel bi-level Consensus Alternating Direction Method of Multipliers (ADMM) algorithm, termed CADMM-Prox. The proposed algorithm integrates classical Consensus ADMM with a proximal mechanism by introducing a sufficiently large proximal term associated with an outer-level variable. Under the mild assumption that the local objective functions are semi-convex, CADMM-Prox is guaranteed to converge globally to a neighborhood of a Clarke stationary point. Numerical experiments on a phase retrieval problem demonstrate that our proposed method exhibits more stable convergence behavior compared with baseline algorithm.

This study provides a quantitative framework for analysis of systemic demand uncertainty and risk propagation cascades across general supply chain networks. By leveraging properties derived from stochastic networks embedded within a Newsvendor paradigm, we model multi-echelon networks under equilibrium and transient operational regimes. We mathematically validate that the systemic volatility behavior commonly referred to as the Bullwhip effect persists entirely as an unavoidable, inherent topological property of coordinated logistics networks, independent of traditional operational noise or information visibility constraints. Extending this paradigm to transient environments, we model inventory drawdown horizons as a multi-dimensional Skorokhod reflection problem. Crucially, we endogenize market-clearing feedback loops by incorporating non-linear price elasticity mechanisms and dynamic trade relation rebalancing, demonstrating how decentralized rational actions co-evolve with physical capacity bottlenecks and can accelerate systemic network degradation. Finally, we operationalize the framework through a data-driven numerical experiment mapping global oil trade dynamics, showing how localized chokepoint disruptions, such as a capacity shock in the Strait of Hormuz, trigger non-linear cascading stockouts and systemic reallocation across sovereign buffers over time.

As is well known, both the numerical performance and the theoretical properties of the Dai–Liao conjugate gradient algorithm are highly dependent on the adjustment of its key parameter. Here, we first employ the well-known Harmonic–Geometric–Arithmetic–Quadratic mean inequality to reinforce the optimality of two previously proposed scalar adaptive settings of the Dai–Liao parameter. In other words, we show how these parameter choices are capable of enhancing the well-conditioning of the Dai–Liao search direction matrix by shrinking the intervals containing its singular values. A similar analysis is also carried out for scaled memoryless quasi–Newton updating formulas in order to further justify the optimality of two classical scaling parameters associated with these updates. Then, as the main contribution of this work, we move from the classical scalar setting of the Dai–Liao parameter to a matrix setting aimed at enhancing flexibility and diversity within optimization methods. In particular, we show that, under such a matrix formulation of the parameter, a well-known three-term conjugate gradient algorithm emerges as a member of the proposed extended Dai–Liao class of algorithms while enjoying several computationally attractive properties. Among these, the well-conditioning of the associated search direction matrix is especially noteworthy, as well as the ability to make more explicit use of the second-order information of the model. Finally, to provide practical evidence supporting the proposed matrix setting of the Dai–Liao parameter, we conduct a series of numerical experiments on standard benchmark test problems and report the results in detail. Generally speaking, the proposed framework is shown to retain both theoretical soundness and computational reliability.

Highly constrained multi-objective design and decision problems are difficult to solve because of strict feasibility requirements and conflicting stakeholder preferences. Evolutionary algorithms are widely used for these problems but typically separate constraint handling from preference optimisation. Pareto-based methods provide a set of trade-off solutions but require post-processing to select a preferred option, while scalarisation can distort heterogeneous stakeholder preferences. This paper introduces the Integrative Maximisation of Aggregated Preferences (IMAP) and its Inter-Generational Solver (IMAP-IGS), which embed preference aggregation directly into the evolutionary fitness function. A constraint-violation preference function guides the search toward feasibility, while final scores reflect only stakeholder preferences. The framework can be integrated into any evolutionary algorithm that uses scalar fitness evaluation. Two implementations are presented: IMAP-BRKGA for combinatorial optimisation and IMAP-GA-II for continuous optimisation. Testing on DAS-CMOP, MO-VRPTW, and HVASP benchmarks shows that IMAP consistently outperforms weighted-sum and Pareto-based approaches when stakeholder preferences conflict. These advantages largely disappear when preferences align, indicating that IMAP's strength lies in resolving preference conflicts in highly constrained decision environments.

The paper studies product-space norm minimisation problems involving distances to convex sets. Using standard tools from convex and functional analysis, we establish complete dual necessary and sufficient optimality conditions and show that the entire solution set can be constructed from the dual vectors arising from the optimality conditions at a given solution. As a consequence, we study optimality conditions and solution set descriptions for generalised versions of the Fermat-Torricelli problem, the Chebyshev centre problem, and the p-Fermat-Torricelli problem. Comparisons with existing results are provided whenever applicable. Examples in finite and infinite dimensional spaces equipped with different norms are presented to illustrate the results.

We consider the uniform exponential stability analysis of infinite-dimensional impulsive systems defined on a Banach or Hilbert space, whose flow is governed by a fixed $C_0$-semigroup generator and whose jumps occur at a prescribed time sequence. While the flow and jump maps are themselves time-invariant, the time-triggered impulses render the propagator a genuinely time-varying evolution family, which is the source of the analysis difficulty addressed here. We combine ideas from hybrid systems theory and infinite-dimensional systems to produce operator-based stability conditions, which can be analytically or numerically checked via convex programming. Necessary and sufficient conditions for the uniform exponential stability of impulsive systems on Banach spaces are obtained in the context of a fixed impulse-times sequence but also of arbitrary, constant, minimum, and range dwell-times using both non-coercive and coercive Lyapunov functionals. Some of those results are then adapted to systems on a Hilbert space and quadratic Lyapunov functionals. As an application, linear switched systems are shown to be an exact special case: reformulated as impulsive systems with unit-norm selector jumps, they inherit non-coercive and clock-dependent dwell-time stability conditions on both Banach and Hilbert spaces. Theoretical and numerical examples are given for illustration, notably on the sampled-data control of time-delay systems.

We study an insurance contract-design problem under moral hazard, endogenous participation, and strategic risk interdependence. Because the resulting $N$-agent game suffers from the curse of dimensionality, we approximate the strategic interactions via a heterogeneous mean-field game. We rigorously establish the existence of a lower-level mean-field Nash equilibrium using measurable selection arguments and the Kakutani fixed-point theorem. By proving the $L^1$-Lipschitz continuity of the aggregate participation threshold, we further establish equilibrium uniqueness via a contraction mapping. We then embed this mean-field response into the insurer's upper-level Stackelberg optimization problem. We formulate the objective through general performance envelopes to accommodate potential equilibrium multiplicity, proving the existence of upper-level $\varepsilon$-optimal contracts, and demonstrating the existence of an exact Stackelberg equilibrium under the uniqueness regime. We conclude by extending the model to finite contract menus, providing numerical evidence that multi-contract screening improves the principal's expected payoff in interdependent risk environments.

The use of ratio variables in convex Data Envelopment Analysis (DEA) models has long been recognized as a methodological issue, as ratios may be incompatible with the convexity assumptions underlying standard DEA. However, in this paper, we prove that if all variables are ratio variables sharing a common denominator, then convex combinations of feasible activities are also feasible. Moreover, we establish a result demonstrating the equivalence between DEA models with common-denominator ratio variables under variable returns to scale and DEA models with volume (non-ratio) variables under constant returns to scale. These significant results enable the development of a framework for evaluating the efficiency of the Organisation for Economic Co-operation and Development (OECD) countries based on the results of the Programme for International Student Assessment (PISA) report, using mean performance scores as outputs. In this framework, we give some methodological extensions, such as the incorporation of the index of economic social and cultural status (ESCS) as an input, thereby enabling fairer comparisons with countries with a lower socio-economic level. Furthermore, we introduce different methods for estimating directions of improvement and calculating targets appropriate to the difficulty of improving each performance score. Finally, we review and introduce several novel contributions to emerging methodologies that can complement classical radial and directional models, such as efficient frontier estimation with adaptive constrained enveloping splines (ACES), stochastic chance-constrained models, and fuzzy models. All these methodologies can be used to analyse data from other PISA or similar reports, allowing non-specialists to implement DEA appropriately.

For a class of non-autonomous linear SPDEs, we establish the equivalence among asymptotic regularization, weak observability, and approximate null controllability for the associated deterministic control systems. This equivalence provides a deterministic control-theoretic characterization of stochastic smoothing and offers a systematic approach to studying SPDEs driven by spatially localized noise. We further establish a criterion for semilinear SPDEs based on weak observability of the linearized equations. Our approach combines methods from PDE control theory with Malliavin calculus. As applications, we consider the stochastic Oseen equation, non-autonomous uniformly parabolic equations, and the parabolic Sine–Gordon equation, all driven by finite-dimensional, spatially localized white-in-time noise.

Graph k-CUT problems include many important variants whose objectives combine cut value, volume, and cardinality terms in different ways. Most existing algorithms are designed for individual formulations, which limits their transferability across related models. In this paper, we organize a broad family of graph partitioning problems into two classes, MaxGCP and MinGCP, according to their optimization orientation and balance-related structure. Based on this classification, we propose a unified Parallel Evolutionary Algorithm Framework (PEAF). This framework combines structure-inheriting crossover operators, a hierarchical mutation mechanism based on the Multiple Mutation Heuristic (MMH) and the Auxiliary Cut Mutation Heuristic (ACMH), and a diversity-preserving selection strategy. Extensive experiments on G-set with k \in\{2, 3, 4, 5\} show that PEAF-ACMH consistently outperforms Gurobi on nine representative k-CUT problems. For MaxGCP, PEAF-ACMH improves several best-known solutions for Max-k-Cut with k \geq 3, and through numerical bounds derived from its relation to Max-k-Cut, verifies the high quality of the obtained solutions for Judicious-k-Partition and AntiCheeger-k-Cut. The results further indicate that Judicious-k-Partition usually yields more balanced partitions than AntiCheeger-k-Cut. For MinGCP, theoretical and computational comparisons show that Cheeger-k-Cut and Sparsest-k-Cut produce more balanced partitions than Normalized-k-Cut and Ratio-k-Cut, respectively. PEAF-ACMH also obtains highly similar partitions for Min-k-Cut and MinMax-k-Cut within short running times, providing numerical evidence for their structural affinity. These results demonstrate that PEAF is both an effective unified solver and a useful tool for revealing structural properties of graph k-CUT models.

We study a continuous-time dynamic pricing problem for a data platform that purchases raw data from privacy-sensitive providers and sells data products to consumers. The platform controls both the acquisition price offered to providers and the selling price charged to consumers. Provider and consumer arrivals are modeled by point processes whose intensities depend on the platform's current data stock, capturing feedback between data accumulation and market participation. We formulate the platform's problem as an infinite-horizon stochastic control problem with a jump-diffusion state process and derive the associated nonlinear integro-differential HJB equation. We prove that the value function is the unique viscosity solution, establish classical regularity under suitable conditions, and verify the optimal feedback pricing policy. Finally, we conduct numerical analyses to examine the influences of model parameters on the optimal pricing policies.

The exponential growth of AI is accelerating the deployment of data centers (DCs), placing unprecedented strain on power infrastructures. In response, major IT corporations are increasingly adopting on-site solar generation to reduce grid dependence and meet sustainability targets. However, the true impacts of this strategy remain ambiguous. While DCs are flexible assets capable of temporal load-shifting, anchoring them to self-generated power may inadvertently constrain their grid responsiveness. To evaluate these trade-offs, we propose a receding-horizon optimization (RHO) framework coordinating job scheduling, grid interactions, and on-site solar generation for a stand-alone DC. Our findings reveal a critical paradox: although solar integration increases energy self-sufficiency and reduces overall DC emissions, it inherently limits the facility's capacity to absorb low-cost, low-carbon electricity from the grid. This implies a fundamental tension between individual corporate sustainability goals and system-wide grid decarbonization.