Papers

For given $m \times n$ data matrices $X, Y$, we investigate the Normal Procrustes Problem—the least squares optimization problem that aims to minimize $\|AX-Y\|_F^2$, where $A$ is constrained to be a normal $m \times m$ matrix. As far as the author of this article is aware, no other method that attempts to solve the Normal Procrustes Problem exists in the literature; we thus propose what is, to our knowledge, the first such method. We, furthermore, adapt our approach to address the Real Normal Procrustes Problem, where $A$ must be real. In our treatment of these problems, we first reduce our complex and real objective functions to be purely optimizable over the Riemannian manifolds of the unitary and real orthogonal matrices, respectively. This reduction enables us to apply techniques in Riemannian manifold optimization to approximate solutions to both. The Closest Normal Matrix and Real Closest Normal Matrix Problems are both special cases of their respective Procrustes Problems (by setting $m = n$ and $X = I_m$) and have been previously studied in the literature. Our approach thus recovers a novel Riemannian optimization method for approximating solutions to both these problems. We further numerically test the performance of our method across all such problems (including against previously developed algorithms on the Closest Normal Matrix Problems) and obtain competitive residuals and favorable scaling in wall-clock time.

Most modern optimizers form their momentum as an exponential moving average (EMA) of past gradients, forgetting every direction at one fixed rate. However, the inputs a deep network sees during training can be highly anisotropic, with a few directions queried frequently while most are seen rarely. Preconditioning methods address this anisotropy by wrapping extra processing around this buffer and leave the momentum update itself unchanged. We propose Activation-Keyed Momentum (AK-Momentum), which builds direction-awareness into the momentum update rule. The gradient of a linear layer splits into an input activation that acts as a key and an output-side error that acts as a value. Keying on that activation, AK-Momentum updates the momentum buffer by the canonical delta rule, so each direction is forgotten at a rate set by how often it appears. We prove that it is a valid momentum, that it applies the input-side curvature correction without matrix inversion, and that it clears stale directions faster than EMA under both a fixed and a drifting optimum. It is a drop-in replacement for the momentum buffer of any optimizer, its coefficient transfers across widths under $\mu$P, and its extra compute stays between $22.2\%$ and $25.0\%$ of a gated-MLP block's linear cost with no persistent memory. In FineWeb-Edu pretraining, AdamW with AK-Momentum (AK-AdamW) reaches AdamW's validation loss in up to $46.39 \pm 4.32\%$ fewer steps at 67M and $22.12 \pm 0.80\%$ at 370M over three seeds, and the gain persists at 1B on a Chinchilla-optimal budget. A Muon baseline tuned under the same protocol sits above AK-AdamW at both language-model scales, and the gain holds for SGD, ResNet-18, and ViT-Tiny on CIFAR-10. Training-time diagnostics confirm the predicted mechanism, better gradient tracking and healthier input directions.

Quadratic manifold-based model order reduction offers a viable pathway to circumvent the limitations of linear subspaces for linear control systems characterized by slow Kolmogorov $n$-width decay. However, a system-theoretic framework for constructing such quadratic approximations remains absent from the literature. This paper presents a system-agnostic, optimization-free framework for the direct construction of quadratic projection matrices. We prove that the synthesized reduced-order model matches the nonlinear moments of the full-order system and preserves its exact center manifold mapping, thereby ensuring asymptotic tracking of steady-state outputs under specific input classes. Numerical results on transport-dominated benchmark problems, namely, the one-dimensional damped wave and advection equations, show that the proposed framework achieves high-fidelity trajectory reconstruction within a significantly reduced-dimensional state space, yielding substantial online computational savings.

We study one-dimensional screening when the Spence-Mirrlees condition fails along a monotone dividing curve, on which all types value marginal output equally. A single ratio, marginal distortion over marginal rent, governs the problem. When the relaxed solution stays in the positive region, non-decreasing optima in the positive single-crossing region are continuous if the ratio increases in the decision. When it crosses a flat curve and marginal rents are separable, every continuous contract is improved by a jump. When it crosses a strictly monotone curve, a contract that pools and jumps continues along the mirror of the pool if the ratio does not increase along it. Optimality among all implementable allocations, deterministic or random, is proved by a measure on the binding constraints whose weights are the same ratio; with constant curvatures and no crossing, it is also necessary. As applications, we solve problems of procurement and nonlinear pricing.

High Bandwidth Memory (HBM) systems can exhibit partial service rather than only full service or complete isolation: a controller-visible service unit may deliver full, reduced, or zero bandwidth because of sub-channel isolation, lane remapping, or protection overhead. This paper develops an exact multistate reliability framework in which each service unit carries an arbitrary finite set of bandwidth states and reliability is the probability that aggregate delivered bandwidth meets a demand. The binary k-out-of-n model is recovered as a special case, while closed-form binary-mapping error relations quantify mean-bandwidth distortion and provide a screening test for the simpler abstraction. For exact single-threshold evaluation a threshold-pruned multistate Binary-Addition-Tree (TP-mBAT) algorithm is proposed. It is deliberately regime-specific: fixed-grid dynamic programming is preferable on a compact common grid, where a 16-unit commensurate control required 273 pruned-DP updates versus 362,506 TP-mBAT node visits. On a reproducible 14-unit incommensurate benchmark, TP-mBAT is compared against a dynamic program carrying the same threshold rules, so that no baseline is weakened. Both expand the same state space, 6,862 nodes against 6,861 updates, and the separation lies entirely in retained state: 17 traversal entries against 412,121 probability states at central demand, reducing measured peak storage from 25.02 MB to 1,152 B. An exact probability-transfer sensitivity identifies when moving mass from a degraded state to a higher-bandwidth state changes system success, and a reserved-unit floor model admits a third exact pruning rule that is vacuous without such floors. A latent package-state mixture captures shared stress, where ignoring dependence overstates reliability by 8.73 percentage points.

We introduce Deep-MKV-TS, a path-dependent McKean-Vlasov framework for financial scenario generation. The stochastic dynamics are chosen by matching selected path and volatility features of generated scenarios to those observed in the data. Starting from an interpretable reference model, Deep-MKV-TS preserves the reference drift and adjusts its volatility, while a regularization penalty limits unnecessary departures from the calibrated dynamics. We solve the resulting control problem using a neural, sample-based implementation of the stochastic maximum principle. We validate the method against an exactly computable oracle. On Heston and Heston-mixture models, Deep-MKV-TS substantially reduces path-dependent and volatility-related deficiencies of the reference model. In delayed-volatility experiments, the correction remains effective as the forecasting horizon increases, while direct training becomes less reliable. On held-out intraday equity-index futures, the corrected model improves conditional forecasts relative to the reference and reaches a level of performance comparable to flexible generative and historical baselines. The resulting scenarios also support greater exposure than the reference under a fixed drawdown-risk target. These results show that path-dependent McKean-Vlasov control can enrich an interpretable reference model without replacing it.

We study a long-range perturbation of the perimeter functional by a nonlocal repulsive term defined through the Yukawa kernel, minimized under a volume constraint. Because the kernel decays exponentially, the competition between surface tension and repulsion is governed by two independent quantities, the screening rate and the volume. Specifically we prove that (i) above an explicit critical screening rate, minimizers exist at every volume, whereas earlier work required the volume to be large; (ii) below an explicit volume threshold minimizers exist for every screening rate; (iii) at small volume, and uniformly in the screening rate, the ball is the unique minimizer up to translation; and (iv) there exists a sharp volume threshold, depending on the screening rate, for the ball to be a stable volume-constrained critical point. The threshold is given in closed form for every screening rate and reduces to the known unscreened value when $\alpha=0$. Our proofs overcome new technical changes due to the lack of homogeneity in the Yukawa kernel.

Wasserstein gradient flows are intimately connected with evolution partial differential equations and diffusion processes. We take the first step in developing such connections for inner product Gromov–Wasserstein (IGW) gradient flows by studying the IGW gradient flow of the relative entropy $\mathsf{H}(\cdot\|\gamma)$ with respect to the standard Gaussian measure $\gamma$. We first show that $\mathsf{H}(\cdot\|\gamma)$ fails to be $\lambda$-convex along generalized or modified generalized IGW geodesics for any $\lambda \in \mathbb{R}$, and therefore falls outside the scope of the existing IGW gradient flow theory from Zhang et al. (2026). We bridge this gap by establishing a suitable local convexity estimate that enables the construction of the gradient flow and its extension to the infinite time horizon. We then obtain increasingly explicit representations of the resulting dynamics. Starting from a partial integro-differential equation, we derive a nonlinear Fokker–Planck equation and show that its second-moment dynamics decouple from the law as they satisfy an autonomous matrix ODE. This reduces the IGW dynamics to a linear, time-inhomogeneous Fokker–Planck equation, yielding a probabilistic representation as the time-marginal flow of a linear stochastic differential equation resembling the Ornstein–Uhlenbeck process. Finally, we study its asymptotic behavior by establishing exponential convergence of the flow to $\gamma$ in relative entropy.

This article introduces a numerical optimal control framework for minmax constrained density control for a class of noisy linear parabolic partial differential equations (PDEs), in particular the noisy heat equation. The goal is to transport an initial density to a target density while minimizing a specified cost with respect to control actions and maximizing it with respect to disturbances, all within a fixed time horizon while satisfying given convex path constraints. To address this, the spatial derivatives in the PDE are discretized using finite-difference approximations, transforming the problem into a system of ordinary differential equations in time. The admissible space of control and disturbance trajectories is then finitely parametrized, and the resulting optimal control problem is formulated as a convex semi-infinite program (SIP) under mild assumptions. By leveraging new numerical tools from convex SIP theory, we establish guarantees for exact solutions that account for constraint satisfaction under an infinite family of disturbance realizations, and we establish an optimization-based computationally efficient algorithm to recover these solutions. Comprehensive numerical examples to demonstrate and validate our findings are included.

We study stochastic control of multivariate Hawkes-driven stochastic differential equations with machine learning algorithms in a non-Markovian setting. Due to the path dependence of the memory of the Hawkes intensity, this problem does not fall within classical stochastic control theory outside particular Markovian kernels. We first develop a finite-dimensional Markovianization procedure and algorithm to approximate multivariate Hawkes processes with mixtures of exponential kernels. We prove the convergence of the Markovianized approximation of the Hawkes process, its intensity, and the value of the problem to the original non-Markovian processes and the value of the primal problem. We then formulate continuous-time deterministic policy gradient learning on the Markovianized approximation of the problem, called Hawkes-CT DDPG. We propose a model-free algorithm to solve the non-Markovian Hawkes-driven optimization by observing only the event times of the process, the realization of the solution to the SDE, and a chosen set of decay filters, while the Hawkes kernel coefficients remain unknown. We compare our continuous time reinforcement learning Hawkes-CT DDPG method with discrete time reinforcement learning techniques under three different types of kernels: simple exponential, Erlang, and power-law kernels.

We give a counterexample to the convergence conjecture in Remark 12 of [Bolte & Pauwels, 2021] for mini-batch stochastic approximation with definable potentials. The construction uses two convex piecewise-affine, hence semialgebraic, summands on $\mathbb{R}$. We choose a deterministic nonincreasing block stepsize sequence satisfying $\alpha_k = o(1/\log k)$ and an admissible minimum-norm selection from each aggregate batch field. On successive blocks, the iterates form lazy reflected random walks on nested dyadic lattices. An explicit endpoint-cover-time estimate, Markov's inequality, and the first Borel-Cantelli lemma imply that almost surely every sufficiently late block's iterates visit their entire lattice. Consequently, the iterates remain in $[-1,1]$ but do not converge, and their accumulation set is exactly $[-1,1]$, on which the averaged objective is constant. Finally, the construction has $\sum_k \alpha_k^2 =\infty$. Both Chat-GPT 5.6 (Sol) and Gemini Pro 3.1 (DeepThink) were used in the development and drafting of this result.

This paper concerns the null control of quasi-linear parabolic systems where the diffusion coefficient depends on the gradient of the state variable. In our main theoretical result, with some assumptions on the regularity and growth of the diffusion coefficient and regular initial data, we prove that local null controllability holds. To this purpose, we consider the null controllability problem for the linearized system, we deduce new estimates on the control and the state and, then, we apply a Local Inversion Theorem. We also formulate an iterative algorithm of the quasi-Newton kind for the computation of a null control and an associated state. We apply this method to some numerical approximations of the problem and illustrate the results with several experiments.

Occupancy-targeted curb pricing, including San Francisco's SFpark pilot, often relies on local threshold rules that do not account for spatial substitution or uncertainty in demand response. We estimate parking-price elasticities from disaggregated SFpark data using a hierarchical Bayesian demand model and use the resulting posterior distribution in a spatially coupled constrained pricing model. Partial pooling across districts gives a posterior mean elasticity of -0.241 with a 95% credible interval of [-0.325, -0.176]. The pricing problem is formulated as a constrained nonlinear program with spatial coupling, temporal price-change limits, neighbor price gaps, and a soft revenue floor, and is solved with Ipopt. Expected-loss and CVaR objectives are evaluated over joint posterior draws rather than a fixed elasticity. On held-out posterior scenarios, the posterior-based CVaR policy has a lower objective than the SFpark threshold rule and historical tariff with posterior probability 1.00, and than a literature-calibrated robust policy with probability 0.85. Epsilon-constraint frontiers show that posterior calibration changes the efficient pricing set. Results from 60 district-windows across Fillmore, Mission, and Marina also report the associated tradeoffs in occupancy-band performance, revenue, and spatial disparity.

This paper studies the problem of robust stabilization of linear input-state systems in discrete time. We prove that for any compact set of stabilizable systems, there exists a dynamic state-feedback controller that globally asymptotically stabilizes all systems in the set. In addition, we show that for some compact sets of stabilizable systems, no nonlinear static or linear dynamic state-feedback law can achieve this task. This proves that, in general, robust stabilization requires a feedback law that is both nonlinear and dynamic. We extend our study to robust exponential stabilization with a given rate of decay. Finally, for polytopic sets of systems, we introduce an algorithm for the design of robust feedback laws.

In this paper, we study an optimal control problem related to an ODE model of glioblastoma growth influenced by the oxygen. The model considers a couple of controls describing the chemotherapy and antiangiogenic therapies. The cost functional aims to reduce the tumor growth, bring the oxygen concentration close to a desired value, and penalize the use of therapies. We solve the optimal control problem, proving the existence of a global optimal control and deriving first-order necessary optimality conditions through the Pontryagin Minimum Principle. For the numerical approximation, we use Physics-Informed Neural Networks (PINNs) to solve the state and adjoint systems, together with a gradient descent method with Armijo line search for the controls. To address the strategy of PINNs we consider the methodology proposed in [15], making a decomposition of the time domain into several subintervals, using different neural networks in each subinterval and enforcing continuity conditions between successive time subintervals. This strategy, called sequential PINN formulations in time, including soft and hard-constrained versions, is compared with the corresponding approximation results of classical solvers and traditional PINNs counterparts.

We study an optimal control problem for a quasilinear parabolic equation modeling nonlinear heat conduction during heat treatment of metals. Under general assumptions, we prove existence of an optimal control and characterize the associated optimal control-state pair in a practically relevant framework. The main difficulty is the quadratic gradient term, which we handle through energy estimates and differentiability properties of the control-to-state map.

We study makespan minimisation on two identical parallel machines that share a single server for both loading and unloading. Each job must be loaded, processed without interruption on its assigned machine, and unloaded immediately after processing, with a common positive integer duration for all loading and unloading operations. We prove that the decision problem is NP-complete for every fixed server-operation duration and strongly NP-complete when this duration is part of the input. We then analyse ordinary list scheduling and the longest-processing-time rule in the non-unit setting. List scheduling has a tight supremum ratio of two. For the longest-processing-time rule, we obtain the exact worst-case ratio when all processing times are at least the server-operation duration, and derive new parameter-dependent lower and upper bounds for unrestricted instances. The results show that both processing-time granularity and blocking generated by short jobs shape the approximation behaviour of the common-server problem.

This paper addresses morning commute congestion caused by concentrated school-related trips in urban networks. We propose a bi-level optimization framework for regulating school start times in a multi-region urban network characterized by Macroscopic Fundamental Diagrams (MFDs), explicitly coupling system-level regulation with multi-class user-equilibrium-based departure-time choices. The Upper-Level problem jointly minimizes total time spent and deviations from current school schedules, while the Lower-Level problem models commuter behavior through a deterministic dynamic multi-class user equilibrium formulation incorporating alpha-beta-gamma preferences for travel time, earliness, and lateness costs. To address the computational challenges arising from the bilevel structure, non-convex traffic dynamics, and endogenous demand responses, an iterative algorithm alternating between the Upper- and Lower-Level problems is developed. The Upper-Level problem is approximated through a formulation solvable with standard mathematical programming solvers, while an iterative algorithm provides an approximate solution to the Lower-Level equilibrium problem. Numerical results demonstrate substantial congestion reductions and characterize the trade-off between school start-time flexibility and traffic efficiency. Sensitivity analyses further examine the effects of MFD uncertainty and scheduling preferences.

The Ramsey number $R(m,n)$ is the smallest order at which every red-blue edge coloring of a complete graph must contain a blue clique (a complete subgraph) of size $m$ or a red clique of size $n$. Determining these numbers exactly is extremely hard, and even certifying a lower bound requires exhibiting an explicit coloring that avoids both cliques. We develop an integer programming framework for certifying such lower bounds, restricting the search to circulant graphs, whose rotational symmetry lets us reformulate the problem in a projected distance space, reducing the number of binary variables from quadratic to linear in the graph order. We strengthen this projected model through coefficient reduction and solve it with a branch-and-cut algorithm whose separation routine exploits the common neighborhood structure of circulant graphs, combining heuristic and exact maximum-clique algorithms. In an extensive computational campaign on circulant graphs with up to 410 vertices, we improve the best lower bounds previously obtained by other methods by up to 11 points for 25 values of $R(3,n)$ with $24\le n\le49$ and $n\neq27$, each backed by an explicit graph certificate that can be independently verified with a stand-alone exact clique solver. To the best of our knowledge, our method also provides the first reproducible optimization-based procedure for certifying circulant Ramsey numbers $R_C(m,n)$, which we use to establish eight new values of $R_C(3,n)$ with $13\le n\le20$. Our framework, graph certificates, and stand-alone checker are provided as supplementary material to support independent verification and reuse.

Motivated by the need for reliable traffic management under fluctuating travel demand, we study the problem of determining the worst-case congestion in a multi-commodity traffic network subject to demand uncertainty. To this end, we stress-test a given network by identifying demand realizations and corresponding travelers' route choices that maximize congestion. The users of the traffic network are assumed to act according to one of the two Wardrop principles, the user equilibrium or the system optimum, so that the resulting congestion models can be seen as bilevel problems with a single leader and multiple followers. To address uncertain travel demand, we consider different models such as ellipsoidal or budgeted uncertainty sets and the hose polyhedron. We present single-level mixed-integer nonlinear reformulations of the congestion models that exploit binary variables and big-M constants, prove the existence of optimal solutions, derive valid big-Ms, and propose several enhancement techniques to further strengthen the formulations. An extensive computational study on instances of the Sioux Falls network and instances from the SNDlib demonstrates the computational effectiveness of the proposed techniques and provides insight into the impact of different congestion measures and uncertainty models on the resulting worst-case congestion.

We study a long-run risk-sensitive portfolio problem with proportional transaction costs in a continuous-time market whose log-prices are given as a Lévy process, and rebalancing is possible only at random moments of investment opportunities. Using a Schauder fixed-point argument, we solve the ergodic Bellman equation without any mixing assumptions. Under additional full-support and growth conditions, the solution to the Bellman equation is unique, with the unique continuous maximiser. We also prove vanishing risk-aversion asymptotics towards the risk-neutral (Kelly) problem and convergence of a dyadic-time-grid approximation of the random intervention moments. Numerical examples illustrate the results.

This paper presents a real-time control-constrained Differential Dynamic Programming (DDP) framework for underactuated legged robots. To address the limitation of classical DDP in handling control constraints, we propose an Accelerated Projected Gradient (APG)-based control-constrained DDP (ABC-DDP), which efficiently computes constrained solutions and identifies active sets without repeated Karush-Kuhn-Tucker (KKT) inversions. A virtual constraint is introduced to integrate control constraints within a feasibility-driven multiple-shooting framework, enabling stable optimization even from dynamically infeasible initializations. The proposed method supports real-time model predictive control (MPC) with short horizons under strong underactuation. Simulation results demonstrate static two-leg standing under external disturbances, along with diverse dynamic motions including slow catwalk, upright walking, and high-speed running within a unified MPC framework. To the best of our knowledge, this is the first demonstration of static two-leg standing of a quadruped robot achieved using real-time finite-horizon MPC.

Using the notion of data informativity, the existing results have given conditions under which the data are informative for controller designs for various control problems of unknown linear discrete-time systems, and have developed methods to compute such controllers from data. Nevertheless, the existing conditions for computing a dynamic output feedback control law based on the input and output data are somehow stringent. In this paper, we first focus on developing data informativity analysis and control for an unknown system with a known input matrix. It turns out that the informativity conditions for such a system are much milder and the methods for computing state feedback control laws for stabilization, deadbeat control, and the linear-quadratic regulator (LQR) for such a system are much more straightforward. Further, based on the parameterized observer, we show that the problem of computing a dynamic output feedback control law for an unknown system can be converted to the problem of designing a state feedback control law for an ancillary system whose input matrix is known. Therefore, the results of the first part of this paper can be directly used to compute a dynamic output feedback control law for stabilization and deadbeat control for unknown linear discrete-time systems based on the input and output data. Moreover, we present a dynamic output feedback control law which will asymptotically approach a state feedback LQR solution to the original unknown system.

Railcar switching, or shunting operations decisions play a significant role in the efficient operation of railyard systems, which are in turn critical to the fast and effective movement of goods. In flat yards, switching operations are primarily performed using locomotives to push and pull railcars in order to assemble and disassemble trains. In such settings, railcars with predefined destinations are located across multiple parallel rail tracks, and must be moved, or switched, in order to form desired outbound trains. This study addresses the Railcar Assignment Problem (RAP) in flat yards with an objective of minimizing the total number of switching movements. We present a novel mixed-integer programming (MIP) model for this problem that incorporates practical operational constraints in rail yards, and demonstrate its NP-hardness. To solve large-scale instances, we propose a comprehensive Zone-based Double Deep Q-Network (Zone-DDQN) heuristic method that integrates railway structure, yard-zone decomposition, and a Double Deep Q-Network (DDQN). The yard-zone decomposition strategy partitions the yard into multiple parallel yard zones, after which the DDQN is applied to solve the problem within each zone individually and sequentially. Computational experiments across small-, medium-, and large-scale yards were conducted on a series of RAP instances. Average results show that the Zone-DDQN heuristic achieves an average optimality gap of $5.71\%$ across small-scale yard instances. For large-scale yard instances containing more than 150 railcars and 30 tracks, the MIP model was not able to obtain solutions within 24 hours. In contrast, the Zone-DDQN heuristic was able to solve these instances with an average running time of 214.42 seconds.

This work presents a data-driven methodology for multi-objective optimization under uncertainty of process parameters in the fused filament fabrication (FFF) process. The proposed approach optimizes the process parameters with the objectives of minimizing the geometric inaccuracy and maximizing the filament bond quality of the manufactured part. First, experiments are conducted to collect data pertaining to the part quality. Then, Bayesian neural network (BNN) models are constructed to predict the geometric inaccuracy and bond quality as functions of the process parameters. The BNN model captures the model uncertainty caused by the lack of knowledge about model parameters (neuron weights) and the input variability due to the intrinsic randomness in the input parameters. Using the stochastic predictions from these models, different robustness-based design optimization formulations are investigated, wherein process parameters such as nozzle temperature, nozzle speed, and layer thickness are optimized under uncertainty for different multi-objective scenarios. Epistemic uncertainty in the prediction model and the aleatory uncertainty in the input is considered in the optimization. Finally, Pareto surfaces are constructed to estimate the tradeoffs between the objectives. Both the BNN models and the effectiveness of the proposed optimization methodology are validated using the actual manufacturing of the parts.

Optimal execution (OE) in multi-asset settings involves complex interactions across assets, particularly through shared capital constraints during portfolio rebalancing. While existing models capture cross-impact and portfolio-level dynamics, they largely overlook the role of explicit cash constraints along the execution trajectory. As a result, the feasibility of execution strategies under limited capital remains poorly understood. In this paper, we extend the classical Almgren-Chriss framework to incorporate intertemporal constraints on expected cash consumption, requiring that the expected cumulative cash spent does not exceed a prescribed budget at every trading period. We show that the resulting multi-asset OE problem can be equivalently formulated as a quadratically constrained quadratic program (QCQP), and further establish that it admits a convex representation under mild conditions. This provides a tractable framework for analyzing execution strategies under dynamic capital constraints. Through controlled synthetic experiments, we show that the proposed cash constraints qualitatively alter OE schedules toward cash-feasible sell-first executions as the constraints become tighter. Furthermore, evaluations in an out-of-sample agent-based market simulator demonstrate that our method substantially reduces peak cash drawdown while maintaining implementation shortfall comparable to existing execution strategies. Our results highlight the importance of explicitly modeling financial feasibility in multi-asset execution and provide a foundation for bridging theoretical OE models with practical capital constraints.

A unit-speed searcher starts at the origin of the Euclidean plane and must hit an unknown straight line whose direction and distance from the origin are both unknown. We prove that every deterministic search path has competitive ratio at least $C_{\log}\approx 12.5937096701246675$. The bound is unconditional: the path need not be cyclic, self-similar, spiral-like, or monotone in angle. For each projection direction, we compare the path with a zigzag obtained by sorting its alternating record turns. The resulting completion constraints are interpreted as jobs with scale-dependent deadlines and lower-bounded through a finite-window scheduling argument in logarithmic time. Averaging these directional bounds then uses the exact Euclidean velocity budget. In one dimension, the same method recovers the optimal cow-path constant $9$. Finally, Arb ball arithmetic provides a rigorous numerical enclosure of the constant.

Finite-player nonzero-sum optimal stopping games typically lead to coupled equilibrium systems whose complexity grows rapidly with the number of players. We introduce an independently randomized formulation in which each stopping rule is represented by an adapted, nondecreasing cumulative stopping process. The canonical embedding preserves pure-profile payoffs, and a pure profile is a Nash equilibrium of the original game if and only if its embedding is a Nash equilibrium of the randomized game. We adopt the $\alpha$-potential approach to construct an $\alpha_N$-potential function, with the error $\alpha_N=O(N^{-1})$ under weak-interaction. We also identify an exact-potential subclass with a closed-form threshold equilibrium. For local stopped-status interactions, randomized payoffs admit a local stopped-mass representation, and potential maximization can be formulated as a multidimensional singular-control problem with local gradient constraints and a nonlocal condition for finite jumps. Under suitable regularity assumptions, we study the associated Hamilton-Jacobi-Bellman quasi-variational inequality and its regularity properties. For unknown model coefficients, we propose a bounded-intensity Potential-CT-DDPG learning algorithm. Numerical experiments closely match the analytical benchmark and yield estimated best-response improvements consistent with $N^{-1}$ scaling.

Anytime solvers return usable solutions before they terminate and improve them while time remains. Their progress is commonly summarized by a primal integral, a final gap, or convergence curves. Each summary leaves consequential choices open: how to score a run before its first feasible solution, whether poor incumbents are truncated, and whether the reference value is updated after the experiment or version-frozen beforehand. These choices become visible when runs produce no valid incumbent or improve a published best-known value. We study a normalized signed primal integral built from a bounded relative gap. It assigns an intrinsic worst value to an empty run, requires no acceptance threshold, and assigns negative instantaneous gaps to incumbents that beat a frozen reference. We compare the smooth difference-over-sum kernel with a signed version of Berthold's max-normalized gap and use the former as a working default. We also distinguish analysis-time from version-frozen reference policies and recommend reading the score with the raw final gap, mean convergence curve, and target-attainment curve. We evaluate these choices on a five-arm routing campaign and two model-fidelity ladders. The two signed kernels preserve every panel ordering in this study, whereas a common acceptance threshold compresses the distances between arms and reverses one panel ordering. Updating 54 of the 212 reference values changes score levels and removes all negative scores, while leaving the observed panel orderings unchanged. An exploratory screen also identifies 12 panel comparisons in which similar integral scores conceal materially different endpoints or attainment rates. The implementation, frozen inputs, and generators are openly released.

Stochastic mean-field control (MFC) provides a fundamental framework for coordinating large populations of interacting agents under uncertainty, with a wide range of applications. Existing numerical and deep-learning methods solve one MFC problem instance at a time and must be re-optimized whenever the task changes. In this work, we formulate stochastic MFC as an operator-learning problem and develop, to the best of our knowledge, the first mesh-free, self-supervised neural operator for stochastic MFC. The main challenge is that the diffusion term in the controlled Fokker–Planck equation precludes deterministic transport-map representations. We address this challenge by combining the probability-flow ODE with an invertible normalizing-flow-based transformer, which recasts the dynamics as a deterministic continuity equation and enables closed-form score evaluation through the exact inverse and analytical log-determinant of the normalizing flow, with $\mathcal{O}(d)$ cost per particle for networks of fixed size. Through transformer-based in-context learning, task prompts, represented by compact distribution parameters or raw particle clouds, condition the transport map, enabling a single pretrained operator to solve unseen tasks in one forward pass. The resulting Normalizing Flow Invertible Solution Transformer (NFIST) is trained end-to-end by minimizing the stochastic control objective directly, requiring no precomputed numerical solutions for training. We further prove the consistency of the proposed operator-learning formulation with task-by-task optimization. Numerical experiments on stochastic optimal control, Schrödinger bridge, systemic-risk control, and obstacle-avoiding path planning demonstrate effective zero-shot generalization while substantially reducing the computational cost of solving large families of stochastic MFC problems.