Papers

Condat's algorithm is an efficient dynamic-threshold method for projection onto the simplex, but its extension to weighted equality constraints and the algorithmic roles of resetting and removal have received limited analysis. We develop a dynamic-threshold algorithm (DTA) for a continuous quadratic knapsack problem with a weighted equality constraint. DTA maintains a threshold invariant through three operations–addition, reset, and removal–and we establish its finite termination and correctness. A sufficient condition under which reset cannot occur motivates a simpler no-reset variant, NDTA. We construct instances for which DTA runs in ${\Theta} (n)$ time whereas NDTA requires $\Theta (n^2)$ time, although both algorithms have quadratic worst-case complexity. We further show that, when the weight ratio and the number of deletions per removal pass are bounded, a linear number of passes with positive threshold increments requires the minimum nonzero gap between input values, normalized by the data range, to be at most $\exp [- \Theta (n \log n)]$. Numerical experiments with up to 10^7 variables demonstrate that DTA and NDTA achieve approximately linear empirical scaling, and outperform Secant, WMVA, Variable Fixing, Newton, Median Search, Heap, and Sort in running time.

We estimate convergence rates of value functions for particle approximations of a controlled nonlinear filtering problem. The state is a McKean–Vlasov diffusion on the flat torus, driven by hidden idiosyncratic noise and observed common noise. The filter—the conditional law of the state given the observations—serves as the state variable of the control problem, and the associated value function solves a second-order Hamilton–Jacobi–Bellman equation on the Wasserstein space. We approximate this problem by a centralized \(N\)-particle control problem with independent idiosyncratic noises and a common observation noise. The framework accommodates nonseparable rewards and controlled drifts. Since a single control is applied to the entire population, the Hamiltonian is defined by an optimization performed after integration over the population. Under smoothness of the data, uniform ellipticity, and regularity of this Hamiltonian, we establish uniform value-function error bounds of order \(N^{-1/6}\) for \(d=1\), \(N^{-1/6}(\log N)^{1/3}\) for \(d=2\), and \(N^{-1/(3d)}\) for \(d>2\). The proof combines a translation lift in the common-noise direction, Fourier–Wasserstein inf- and sup-convolutions, viscosity comparison, and particle derivative estimates uniform in \(N\).

A study of the minimizers of one-dimensional Rudin-Osher-Fatemi-type functionals with linear or nonlinear total variation and fidelity term is undertaken. Conditions on the input datum and the parameters of the functional are found that force the input itself to be the minimizer or not. In the linear setting the discriminating conditions on the parameters are complementary highlighting the sharpness of the results. The nonlinear setting is substantially different: while the non-minimality of the datum is treated in analogy with the linear case, for the minimality of the input only partial answers are found. The results in the nonlinear setting hinge on auxiliary constrained or penalized minimization problems investigating the behavior of optimal transitions with prescribed height. Additionally, they are complemented by some numerical examples.

Mainstream plan-then-track approaches to multifingered grasp execution entail selecting a grasp, planning a collision-free trajectory, and tracking the resulting trajectory via a feedback controller. Pose-estimation error during execution or scene motion can invalidate this open-loop commitment and trigger replanning. We thus present Grasp Distance Fields (GDFs), smooth softmin distances to finite sets of arm-hand grasp configurations. Using their negative gradients as feedback, we jointly select and execute grasps without planning a trajectory. A CBF-CLF quadratic program (QP) enforces self-collision, workspace, object, and obstacle-clearance constraints, while its CLF slack quantifies obstruction of task progress. We bound the softmin approximation error by $\log N/\rho$ and prove forward invariance of the filtered safe set. To handle changes in contact topology, we combine a hysteretic contact-mode transition with a wrench-quality CBF that limits degradation of the realized force-closure margin relative to hold onset. Using our method, a fixed-base manipulator and a Unitree G1 equipped with the same underactuated hand grasp and lift 46 of 50 test objects amid clutter and moving obstacles. The realized grasps also retain a median 94% of their synthesized quality margin, and each QP solve requires 0.09 ms within a 20 ms control interval. Project page: www.clintonenwerem.com/gdf.

We prove sequential convergence results for the AdaGrad algorithm family optimizing convex differentiable objectives. Specifically, we provide necessary and sufficient conditions for the convergence of iterates for the three main AdaGrad variants (AdaNorm, AdaDiag, AdaFull) when the objective is convex and locally Lipschitz-smooth, closing the question left open from the literature. We harness this general result to study the three variants under the generalized $(L_0,L_1)$-smoothness condition and show sequential convergence for sufficiently small constant step size. Moreover, under the so-called $(L_0,L_1)$-polynomially modifiable smoothness assumption, which is a relaxation of the $(L_0,L_1)$ generalized smoothness property and is satisfied by many function classes such as $L$-smooth functions or univariate polynomials, sequential convergence for these AdaGrad variants is proved for arbitrary learning rates. This result provides conditions under which AdaGrad presents adaptivity, i.e., does not require tuning the parameters based on the instance. Finally, we provide numerical illustrations of the behavior of AdaGrad on convex and nonconvex functions. In particular, we construct a counterexample empirically showing that smoothness alone is not sufficient for the sequential convergence of AdaGrad-type algorithms, and suggesting that additional geometric hypotheses (e.g., convexity as in this paper, or the Kurdyka-Łojasiewicz inequality) are indispensable for sequential convergence results.

We study the fixed-reference entropy-regularized Wasserstein barycenter problem, in which a probability measure $\eta$ on the barycenter space is chosen before optimization and the $i$th transport plan is regularized relative to $\nu_i\otimes\eta$. Earlier work established a dual formulation for this model and equality between the primal and dual optimal values. However, equality of values does not by itself guarantee that the dual supremum is achieved. Our main result shows that, on compact metric spaces with continuous transport costs, the dual problem admits maximizing potentials in the original class of continuous functions. For completeness, we also establish existence and uniqueness of the optimal tuple of transport plans and of the barycenter. The maximizing potentials recover the optimal plans through exponential primal–dual relations and give an explicit density of the barycenter with respect to $\eta$. This density is continuous and strictly positive. We further show that the potentials inherit regularity from the costs and are unique up to the natural additive gauge transformations. These results strengthen the known value-duality theory and provide a complete continuous primal–dual description of the fixed-reference regularized barycenter problem.

State estimation in open client server networks is challenging because the set of active observation sources changes over time, and active sources need not be valid sensors of the latent state. We study a client-server estimation problem in which a fixed but unknown state-consistent sensing class measures the latent state when active, while nuisance sources may arrive in the active set and generate observations from a different signal class. The server observes active sources, but it does not know which agents observe valid state measurements. We propose a source-reliability-weighted observer. The state estimate is a standard fixed-prior weighted least-squares update, but the information assigned to each active observation is determined by a source-level reliability score learned from repeated innovation consistency. Numerical results show that the observer, when all active sources are considered, has a finite bias, while the proposed observer initially learns source reliability and then tracks the desired latent state after a finite learning period.

We study persistent-agent identification and mean tracking in an open client-server network where agents participate intermittently. The network consists of a core of highly regular persistent agents and a subset of transient/non-persistent agents that appear only sporadically. Since the server observes only binary activity indicators and scalar updates from active agents, naive averaging cannot recover the active persistent mean because transient-agent updates contaminate the aggregate, and heterogeneous participation introduces bias. To address this, we propose a two-phase estimation framework. In Phase I, we develop a two-layer window-based identification procedure: the first layer forms single-window activity decisions, while the second aggregates these decisions across windows to create a high-probability separation between persistent and transient agents. Under heterogeneous Bernoulli participation, we derive explicit finite-window bounds that specify how many windows the server must wait for this separation to occur with high probability. In Phase II, we use the resulting structural estimate to track the active persistent mean via a smoothing filter, and show that the tracking error contracts geometrically to a neighborhood determined by the target drift and the filter gain. We provide numerical simulations to illustrate the effectiveness of our theoretical results.

We study convex approximations of mixed-integer recourse functions in two-stage stochastic programming. For first-stage decisions, the relevant approximation error is the signed expected residual between the convex approximation and the integer-recourse value function. On bounded uncertainty boxes, we identify residual conditions that yield deterministic bounds for this expectation error. The central condition is coordinate slice-centering, under which integration by parts gives an anisotropic total-variation bound and a mixed-derivative bound whose all-coordinate case is expressed through Vitali variation of the density. Exact centering, however, may be incompatible with convexity, even for totally unimodular ceiling recourse. We therefore use commuting coordinate projectors to decompose an arbitrary residual into a centered component and an explicit slice-mean defect. The resulting defect-adjusted bounds provide selectable error certificates and uniform first-stage decision-quality guarantees. We further establish sharp constants, extend the mixed-derivative estimate to nonsmooth densities, and develop tensor-product and geometric extensions of the framework, alongside a max-affine convex fitting and audit methodology.

We study inverse potential mean-field games (MFGs), in which an unknown spatial inverse-cost (mobility) map is inferred from observed population densities. We solve the forward MFG with a preconditioned primal-dual hybrid gradient (PDHG) method and develop Jacobian-free backpropagation (JFB-$r$), which records only the final $r$ iterations from a detached warm start while retaining the full forward solve. To analyze this truncated differentiation method, we show that the exact-proximal dual-extrapolated PDHG map is a metric resolvent of the maximal monotone KKT operator. This resolvent view shows that JFB-$r$ exactly differentiates a finite-trajectory surrogate and, under a locally fixed active set at an exact equilibrium detach point, converges to the implicit gradient as the tracked depth increases. Across several inverse-MFG settings, numerical experiments show that JFB-r at moderate tracked depths can achieve recovery accuracy comparable to full unrolling while reducing memory and runtime.

SignMuon compresses the Muon update to one bit per parameter by taking its elementwise sign, providing the most direct way to run a matrix-aware optimizer under an extremely low communication budget. It outperforms SignSGD in practice, yet it can ascend even on a linear function. Signing the gradient before the Linear Minimization Oracle (LMO), rather than after, does not repair this: we construct a small explicit instance on which sign-before (MuonUSign) and sign-on-both-sides (MuonSign) ascend as well, so no placement of the sign around the oracle descends in general. Error feedback, the standard remedy for a biased compressor, does not rescue SignMuon: when applied to Muon's output, error feedback can fail for every smoothness constant, step size, and momentum. Applied to the gradient, error feedback does work, and EF21-MuonUSign and EF21-MuonSign attain the standard $\mathcal{O}(T^{-1/2})$ rate for the squared gradient norm on smooth nonconvex problems, the latter at one bit in each direction. Experiments then reverse the ordering: across centralized CIFAR-10, federated CIFAR-10, and the nanoGPT speedrun, the strongest compressed method is consistently sign-after-the-LMO, precisely the placement we prove divergent, with the provably convergent variants trailing it. Compressing after the LMO, a heuristic, matters more at these scales than the guarantee does.

This paper presents a Matlab implementation of an arc-search infeasible interior point algorithm for linear programming (LP), which has a proven polynomial bound of $\mathcal{O}(\sqrt{n}L)$, the best among all interior-point algorithms for LP. Software architecture and major functions are discussed. Its ease of use is described by a simple example. Crucial strategies are summarized. Quality of the software is assured because this software has been extensively tested on both PC and Linux for the widely used Netlib benchmark linear programming problems in standard form. Some benchmark test problems involve tens of thousands of constraints and hundreds of thousands of variables. For all tested problems, the code found the optimal solution. The numerical results have been compared to those obtained by the popular Mehrotra's predictor-corrector algorithm. We conclude that the implemented algorithm not only has the best polynomial bound but also is computationally competitive compared to the popular Mehrotra's predictor-corrector algorithm.

This paper presents a method for computing inner polytopic approximations of admissible sets for continuous-time linear control systems subject to multiple affine state constraints, with a particular concern on computational tractability for large dimensional problems. In place of globally computing the admissible set and the part of its boundary called the barrier, we compute the so-called individual admissible sets and the corresponding barriers for each single constraint. We then use the exact sampling of linear systems and generate polytopes in half-space representation that provide an approximation of the individual admissible sets, to finally intersect them. We provide a complexity analysis of the whole procedure to evaluate its efficiency. The approach is illustrated by two examples -a triple integrator and a mass-spring-damper chain considered in 4, 6, 8, and 10 dimensions- with corresponding runtimes evaluated for both.

This paper studies the policy gradient update for a multi-arm bandit problem in diffusion environment that is described by a stochastic differential equation (SDE) under the continuous-time reinforcement learning framework by Wang et al. (2020), Jia and Zhou (2022b). With the logit parameterization for the stochastic policy, we show that it converges almost surely to the optimal arm under an arbitrary constant learning rate. Furthermore, we derive the non-asymptotic regret upper bound when the constant learning rate is below a time-invariant threshold; and the regret bound has order $O(\log T)$. We improve the analysis in Lattimore (2026a) for the same SDE by constructing a novel Lyapunov function and demonstrate the transparency of analyzing policy gradient using the tools in SDEs. In addition, the same Lyapunov function is also helpful in analyzing the discrete-time policy gradient algorithm.

For a given real matrix $A\in\R^{n\times n}$, a Pareto eigenvalue of $A$ is a real number $\lambda\in\R$ for which there exists a nonzero vector $x\in\R^n\setminus\{0\}$ such that $$ 0\leq x\perp Ax-\lambda x\geq0. $$ We prove that every matrix $A\in\R^{4\times4}$ has at most $23$ distinct Pareto eigenvalues. We first prove the result for matrices of order $4$ satisfying two conditions: every real eigenvalue of a principal submatrix is simple, and two different principal submatrices have no real eigenvalue in common. For these matrices, a fixed point argument shows that the number of Pareto eigenvalues is odd. Previous known results show that the Pareto capacity of order $4$ is between $23$ and $26$. Thus only $25$ remains to exclude. We exclude this case by using the support profiles in order $3$ and identities involving eigenvectors of principal submatrices. A perturbation and fixed point index argument then extends the bound to all real matrices of order $4$. An exact symbolic computation certifies that an explicit matrix of order $4$ has exactly $23$ distinct regular Pareto eigenvalues.

Promoting cooperative behaviour amongst decision makers has key implications for the long term sustainability of social systems. Incentives can promote cooperation in situations where defection is more favourable. Previous research has identified optimal decentralised incentives in structured populations where the social environment is described by a generic single-layer networks. Optimality is here intended in the sense of cost minimisation. Here, we provide and optimal incentive strategy for structured populations in general network, a rather unexplored case. Further, we look at populations where, through the network structure, each player interact with one set of neighbours while contributing to an opinion diffusion dynamics from a second set of neighbours. Hence we cover the case of multilayer networks. To fill this gap, we provide a solution to the optimal control problem by solving the Hamilton-Jacobi-Bellman equation and derive an analytic solution for distributing incentives in multilayer networks. By implementing the dynamics of the networked Prisoner's Dilemma Game, we provide a feedback control loop that yields an optimal incentive distribution over time. The dynamic incentive depends on the current state of cooperation, the game-layer network, the strategy-diffusion layer, and the payoff matrix design. We found that the optimal incentive is node-wise, unique to each node on the network, influenced by its relative position according to the adjacency matrices of both layers. We also found that the optimal solution is not exclusively reward or punishment. While one node can receive a reward, the other node may receive punishment at the same time, depending on its current level of cooperation, relative to other nodes. We provide analytic solutions and numerical validations for the cases studied, comparing our results to the existing literature.

This paper presents a robust tube model predictive control (MPC) framework for nonlinear systems represented by bilinear Koopman models identified from data. We derive discrete-time robust control contraction metrics (RCCMs) for bilinear Koopman models, which certify a contraction property that explicitly accounts for the mismatch between the Koopman model and the true dynamics. To synthesize such certificates for the typically high-dimensional lifted state, we propose a scalable learning approach that trains neural network parameterizations of the metric and of an associated feedback controller over a sparsely sampled subset of the lifted state space. The resulting certificates are used to construct a robust tube MPC scheme with tightened constraints, for which we establish recursive feasibility, robust constraint satisfaction and input-to-state stability (ISS) of the closed-loop system with respect to the mismatch between the Koopman model and the true dynamics. By exploiting the linear parameter-varying structure of bilinear Koopman models, the MPC problem becomes convex for a fixed scheduling sequence, while the geodesic computation underlying the contractive feedback is reformulated, via a Chebyshev pseudospectral discretization, as a sequence of quadratic programs. The complete framework is validated on a nonlinear pendulum benchmark with a state-dependent input gain.

We investigate the forward-backward-splitting version of the Plug and Play (PnP) method for linear ill-posed problems with MMSE estimators as denoisers. In contrast to existing literature, we consider estimators which are specialized for (degenerate) Gaussian noise with possibly non-diagonal covariance matrices. We further deviate from the classical iteration by replacing parts of the descent step with a linear operator that relates the observation noise to that of the MMSE estimator. Under mild assumptions, we derive several properties of the denoiser and prove recovery guarantees of the iteration both pointwise and in the Wasserstein distance of the underlying probability distributions. Crucially, our analysis shows that the denoiser cannot be chosen in a physics-agnostic way, that is, independently of the forward model. We extend our results to the case where the MMSE denoiser is parametrized by a neural network and derive the corresponding recovery bounds.

We provide a framework for robust mean field control problems that describe multi-dimensional optimal liquidation problems under uncertainty from both the underlying stochastic process and the deterministic model parameters. The verification results are established with Hamilton-Jacobi-Bellman-Isaacs (HJBI) equations where the variables are probability measures and the Hamiltonian nonlinearly involves the joint distribution of position and momentum. Using novel a priori estimates, we establish the well-posedness of the HJBI equations featuring general or quadratic Hamiltonians that are neither displacement convex nor concave in their momentum. The a priori estimates and well-posedness results are extended during their application to optimal liquidation problems, where we allow the Hamiltonian to have derivatives of linear growth and solve the constrained multi-dimensional linear quadratic optimal liquidation problem under composite uncertainty.

Many real-world systems rely on predictive models to inform decisions, and fairness concerns arise in both the prediction and decision stages. We introduce end-to-end fairness optimization (E2EFO) as a unifying framework that integrates fairness across the prediction-to-decision pipeline. We focus on resource allocation with group-based fairness: the prediction task estimates allocation impacts while limiting accuracy disparity across groups, and the decision task distributes those impacts equitably by optimizing a group-based alpha-fairness measure. Within this framework, we propose fair decision-focused learning (FDFL), a training paradigm that jointly accounts for prediction accuracy, prediction fairness, and decision regret – the loss in decision fairness due to imperfect predictions. FDFL trains the predictor by gradient descent, combining the objective gradients through multi-task learning techniques. The core computational challenge is the decision Jacobian with respect to the predictor parameters: we derive exact closed-form formulas for a tractable class of fair allocation and apply a differentiable optimization layer in the general case. We further establish a finite-sample generalization bound for the scalarized FDFL objective. Numerical experiments on a healthcare-based single resource allocation and a synthetic multiple resource allocation illustrate the value of jointly accounting for prediction fairness and decision fairness in prediction-informed decision-making.

Admission control is a widely used mechanism for regulating congestion in stochastic service systems, where restricting arrivals is expected to reduce workload and improve performance. In many settings, however, customers make decisions based on the observed system state, creating a feedback loop between control actions and future arrivals. This raises a fundamental question: does admission control necessarily reduce congestion under state-dependent behavior? We address this question using stochastic ordering of workload processes. We show that admission control may fail to reduce congestion due to endogenous feedback effects, and construct explicit counterexamples illustrating this phenomenon. We then identify a general condition under which admission control is effective over the initial busy period. In particular, independence between customers' types and service requirements eliminates the adverse feedback mechanism and restores stochastic dominance. These results highlight fundamental limitations of admission control and the importance of accounting for behavioral responses in the design of stochastic service systems.

We propose and analyze a stochastic direct-search method for unconstrained zeroth-order minimization of locally Lipschitz, possibly nonsmooth, objectives. The method combines random polling directions with a stochastic extrapolating line search based on a sufficient-decrease test of order $p$. Under conditional accuracy assumptions on the stochastic estimates, we prove almost-sure convergence to Clarke stationary points. We further establish an expected iteration complexity bound. Specifically, using a supermartingale stopping-time argument, we prove that $\mathcal O\left( \max\left\{ r^{-p}, \varepsilon^{-p/(p-1)} \right\} \right) $ iterations are sufficient in expectation to reach an $(r,\varepsilon)$-Goldstein stationary point. Moreover, we derive a corresponding expected tested-point complexity bound of order $\mathcal O\bigl(\varepsilon^{1-n} \max\{r^{-p},\varepsilon^{-p/(p-1)}\}\bigr)$. To the best of our knowledge, this is the first convergence and expected-complexity analysis for an extrapolation-based direct-search method in a nonsmooth stochastic setting. Numerical experiments on a DFO benchmark suite highlight competitive performance against well-established stochastic direct-search methods.

We present an algorithm that finds an epsilon-approximate solution to a mixed integer quadratic programming (MIQP) problem, and that runs on a Turing machine in time polynomial in the size of the instance and in 1/epsilon, provided that the number of integer variables and the number of negative eigenvalues of the Hessian of the objective function are fixed. Unless P=NP, both restrictions are necessary, so this completes the characterization of the complexity of approximating MIQP in terms of the number of integer variables and the inertia of the Hessian; the result is new already in the purely continuous case. The main ingredient is a polynomial-time simultaneous diagonalization algorithm: it computes a rational change of basis that maps a given ellipsoid, presented in factored form, exactly to a ball, while making the objective function separable up to an arbitrarily small perturbation and preserving the inertia of its Hessian. The classical construction, in which the objective function is made exactly separable, requires a change of basis that is in general irrational, and cannot be carried out on a Turing machine; ours rests instead on rational Jacobi rotations, which we believe to be of independent interest.

We consider the problem of computing the Lebesgue volume of the unit sublevel set of a positive quasi-homogeneous polynomial. Pushing the Lebesgue measure of an ambient bounding box forward through the polynomial reduces this high-dimensional volume to a one-dimensional moment problem. This removes the ambient dimension from the optimization and confines the dimension to a single preprocessing stage, computing the moments of the polynomial over the box, which is polynomial in the ambient dimension for sparse or separable polynomials. We propose two deterministic algorithms for the resulting univariate relaxations, each returning certified upper and lower bounds on the volume. Both bypass semidefinite optimization entirely and rely only on standard numerical linear algebra. The first approximates a piecewise-constant function by a Chebyshev polynomial, so that each relaxation reduces to a fast cosine transform, and converges at a polynomial rate in the relaxation order. The second extracts the volume bounds from a single generalized eigenvalue problem involving moment and localizing matrices whose size grows linearly with the relaxation order, and converges at an exponential rate; the ratio governing this rate is determined by an a priori upper bound on the polynomial over the bounding box. Finally, the univariate polynomials produced by either algorithm are feasible for the multivariate moment-SOS volume hierarchy. The algebraic and geometric rates therefore transfer to the hierarchy itself, improving on its best known convergence rates.

We propose GLORO, a bilevel solution methodology for robust optimization involving expensive function evaluations. The methodology is carefully designed to achieve satisfactory optimization results in a computationally efficient manner. It is based on Gaussian Process surrogates in the parameter space for approximate computations of the objective / constraints of the robust optimization problem via Bayesian optimization (lower level), and local models constructed in the variable space using these approximate evaluations (upper level). Both the use of Bayesian optimization (in parameter space) and the use of local models (in variable space) are motivated by the idea of guiding the expensive function evaluations to the regions of interest for the optimization process. The methodological work is motivated by and tested on a robust airfoil design optimization problem. This application focuses on the RAE2822 airfoil, optimizing its shape to ensure a robust lift-to-drag ratio under operational uncertainties in Mach number and angle of attack.

We give a self-contained triangular formulation of anisotropic higher-order covering and inverse estimates for smooth mappings and generalized equations. For a $C^p$ mapping into a finite-dimensional target, an indexed target decomposition is fixed, the corresponding triangular factor condition is imposed, and a direction satisfying the associated triangular $p$-kernel condition is chosen. Surjectivity of the resulting triangular $p$-factor operator yields a fixed-scale inclusion in which each nonzero block $Y_i$ is covered at order $t^i$, and hence an inverse estimate with exponent $1/i$ on that block. Zero blocks are allowed, and $p$ denotes the highest active grade. The smooth result is a derivative-level graded formulation: it verifies the model directly from the $C^p$ derivatives and records the blockwise fixed-scale inclusion needed for the set-valued analysis. For generalized equations, an auxiliary exact-model theorem isolates the range-transfer mechanism with acyclic correction fibres. The main sufficient criterion uses a closed convex-process inner approximation whose sum with the smooth triangular factor operator is surjective, allowing the set-valued term to supply missing directions.

We study the fixed-support doubly regularized Wasserstein barycenter problem. Using the semi-dual formulation of entropic optimal transport, we reformulate the problem as a smooth, unconstrained, convex optimization problem in the dual variables. We then derive explicit expressions for the gradient and Hessian and develop an exact Newton method for high-accuracy barycenter computation. To improve scalability, we propose a sparse Newton variant that sparsifies the transport probability matrices, thereby reducing the cost of Hessian-vector products. We establish theoretical results for the proposed methods, including Hessian approximation bounds and convergence results. Experiments on synthetic and real datasets show that the sparse Newton method converges faster than

We formulate a baseball plate appearance as a finite two-person, zero-sum, extensive-form game with chance moves and imperfect information. Successive pitch cycles are connected through the evolving count, while the Batter chooses an action without observing the Pitcher's current pitch selection. Terminal outcomes are valued through a run-expectancy utility that incorporates both runs scored during the plate appearance and the continuation value of the half-inning. We implement the sequence-form representation of the game, whose linear-programming formulation grows linearly with the game tree and permits exact minimax equilibrium computations for trees far beyond the practical range of the normal form. Nature's transition probabilities are estimated from MLB Statcast data. We also establish two complementary dynamic-programming interpretations. For general finite two-person zero-sum games with perfect recall, we show that the dual variables of the sequence-form best-response programs decompose into reach weights and conditional continuation values. Under the additional state-Markov assumptions of the baseball model, we prove that the full minimax equilibrium can be computed by backward induction over the twelve non-terminal counts. The resulting computations produce equilibrium strategies, continuation values, and conditional and reach-weighted measures of the strategic cost of non-optimal actions.

This paper studies a low-rank Kalman-Bucy filtering framework for linear time-varying systems through the tracking analysis of Oja's principal component flow. Under structured assumptions on the eigenspaces and their time variation, we show that the Oja flow can remain in a neighborhood of the time-varying dominant subspace by tuning a parameter of the flow, rather than tracking it exactly. These restrictive assumptions identify a tractable class of linear time-varying systems and provide a theoretical basis for low-rank filtering, as illustrated by a numerical experiment.

This paper is concerned with the non-Markovian stochastic optimal control problems in which the value function is a random field characterized by a stochastic Hamilton-Jacobi-Bellman (SHJB) equation. When the stochastic integration coefficients are not controlled, the SHJB equation takes a semilinear form, which is subject to computational challenges compared to the Markovian case due to the measurable randomness. We introduce a policy-iteration algorithm based on successive linearization that reduces the nonlinear SHJB equation to a sequence of linear ones. Furthermore, we prove that the resulting approximation sequence converges monotonically to the value function in the mean-square sense with an exponential rate.