Papers

Recently, rectified flow has emerged as a fundamental framework for large-scale image generation, powering state-of-the-art systems such as FLUX.1 and Stable Diffusion 3. Despite its remarkable empirical success, the computational and statistical guarantees of iterative rectified flow have remained largely unexplored. We address this problem by studying c-rectified flow, a cost-aware class of rectified flow that projects velocity fields onto a gradient class while preserving endpoint marginals. The ordinary rectified flow can fail to recover the optimal transport coupling: in a Gaussian case study, the iteration converges to the optimal coupling if and only if the source and target covariance matrices commute. In contrast, under suitable compactness and uniform-integrability assumptions, iterative c-rectified flow always converges to the optimal transport coupling. We further establish quantitative one-step contraction and exponential convergence guarantees under projection-stability assumptions for both quadratic and strongly convex displacement costs. Finally, under a Hölder ball assumption, we develop new minimax-optimal score estimation rates and show that, when combined with iterative c-rectified flow, they yield a rate-optimal estimator of the optimal transport for the dimension \(d \ge 3\) and a nearly parametric rate for \(d=1,2\).

Timescale separation is a powerful tool for analyzing interconnected dynamical systems. Meanwhile, operator theory provides a general framework for studying the convergence of iterative methods formulated as fixed-point iterations, including algorithms arising in optimization, learning, and control. In this paper, we bridge these two areas by establishing timescale separation results for fixed-point iterations induced by both deterministic and stochastic operators. As customary in timescale separation, our results involve auxiliary systems that arise from the original interconnection in the limit as the timescale parameter tends to zero and separately capture the dynamics induced by the slow and fast operators. The proposed operator-theoretic framework yields explicit and readily checkable bounds on this tunable parameter, expressed in terms of standard operator constants. To illustrate the applicability of our results, we employ them to prove the convergence properties of a feedback optimization scheme in both deterministic and stochastic settings.

This paper studies the distributed online control problem over a network of linear time-invariant (LTI) systems in the presence of adversarial disturbances and time-varying convex costs. The network cost is characterized by the summation of local cost functions, where each local function is sequentially revealed only to the corresponding agent. The goal of each agent is to generate a control sequence, using only local observations and neighbor communication, that competes with the best centralized linear policy in hindsight. We extend the recently proposed Online Spectral Control framework from the centralized setting to the distributed setting. In particular, each agent applies a spectral controller obtained by convolving past disturbances with the leading eigenvectors of a Hankel matrix, while the controller parameters are updated through a distributed online gradient descent step over the local surrogate costs. We formulate this problem this problem as a regret minimization problem based on the spectral parameterization, and under standard assumptions, we establish a sublinear regret bound of $O(\frac{\sqrt{T}\text{poly}(\log T)}{\gamma^3})$, where $T$ is the time horizon and $\gamma$ denotes the stability margin. The resulting bound also captures the dependence on the network size and connectivity.

Creator economy platforms face a strategic dilemma: allocating traffic to established stars for immediate ad revenue versus nurturing emerging creators to build a follower base for future monetization. We develop a continuous-time dynamic optimization model to characterize the optimal traffic allocation policy for a platform managing heterogeneous creators with dual revenue streams (advertising and direct follower contributions). We characterize the optimal policy analytically, revealing a “most-valuable-creator-first" rule driven by a forward-looking activation set. Under Bass diffusion dynamics, this policy exhibits a sophisticated “conditional reversal" strategy, where the platform temporarily prioritizes lagging creators to capitalize on word-of-mouth effects. Regarding the ecosystem structure, we find the optimal policy acts as a selective gatekeeper. Unlike myopic policies that lead to a harsh “winner-take-all" market, or naive fairness-driven heuristics that foster inefficient “indiscriminate growth," the optimal policy imposes a strict capability threshold that screens which creators receive platform traffic. Furthermore, it enforces disciplined growth for successful entrants, capping their follower bases at an optimal ceiling to prevent over-investment. Finally, we demonstrate that simple heuristics can lead to significant revenue losses (up to 25%) and propose a practical “follower-growth adjusted" heuristic that achieves near-optimal performance by leveraging observed growth momentum.

Motivated by the use of modern high-capacity models in real-time control problems, this paper studies the adaptive control of high-dimensional discrete-time nonlinear systems with an unknown matrix-valued parameter. We focus on regimes where the number of unknown parameter entries is large, but the parameter matrix possesses exploitable structure, such as entrywise sparsity, group sparsity, low rank, and row-stochastic or density-matrix structure. To quantify transient performance, we consider a regret criterion relative to a nominal controller with full knowledge of the true parameter. We show that standard Euclidean update schemes, including recursive least squares and gradient descent, are ill-suited to this setting: their transient performance deteriorates as the dimension increases, even when the true parameter has low intrinsic complexity. To address this limitation, we propose a novel class of mirror-descent-type adaptive laws equipped with a non-Euclidean Polyak-type step size that exploit the geometry induced by the parameter structure. For the proposed update laws, we establish asymptotic state convergence and derive regret bounds with at most logarithmic dependence on the ambient dimension. Numerical experiments demonstrate the effectiveness of the proposed schemes.

A solution procedure is formulated and solved for treating the nonlinear Dissipation Inequality as a constraint equation within continuum mechanics, and allowing for incomplete knowledge of constitutive behavior. The scheme is demonstrated in the context of the rate-dependent, elastoplastic response of a bar, resulting in a nonlinear problem of constrained optimization. Both closed form and computational results are developed. The computational solutions utilize a sequence of convex optimization problems, and are shown to be accurate. In the example considered, the approach is shown to automatically correct an (intentionally) faulty constitutive specification, resulting in the solution to be in accord with the fundamental postulates of continuum mechanics.

In this paper, we propose a proximal gradient method with adaptive linesearch for multiobjective optimization problems whose objective functions are weakly smooth, i.e., they have Hölder continuous gradients. The proposed method is parameter-free as we do not require prior knowledge of parameters related to the weak smoothness of the objective function; the proposed linesearch finds an appropriate step-size that adapts to the weak smoothness. The complexity guarantee analyzed in this paper for the non-convex case is compatible with related works and our algorithm accepts coercer stationarity measure compared to existing methods. We also establish a novel complexity result for the convex case which improves the one in non-convex case.

We investigate Optimization under Input Uncertainty (OIU), in which the input to the objective function, rather than the objective function itself, is subject to uncertainty. OIU appears in manufacturing processes with production tolerance, control of physical systems with actuation noise, Mixture of Experts, and Reinforcement Learning (RL). Most of the existing approaches solve OIU by using the value of the objective function but discard the information of the realized input, even though the realized input is observable in various applications. The question here is whether the discarded information of the realized input is useful to accelerate the optimization process. We affirmatively answer this question for Evolutionary Strategy (ES) by theoretically showing that the information of the realized input can reduce the variance of the gradient estimator via Rao-Blackwellization. Using the Rao-Blackwellized gradient estimator, we propose Phenotype-Accelerated Evolutionary Strategy (PAES), which is a refinement of ES for OIU. Numerical experiments show that PAES converges faster than the usual ES from simple continuous optimization problems to RL benchmarks.

This paper investigates the problems of stability analysis and nonsingular sliding mode stabilization for exact prescribed-time control. By exploiting the isochronism of simple harmonic motion, this paper establishes a novel exact prescribed-time control framework. First, a novel Lyapunov analysis method for exact prescribed-time control is proposed, based on which the exact prescribed-time stabilization of scalar systems is achieved. It is theoretically established that for arbitrary non-zero initial conditions, the settling time is exactly equal to the prescribed time. Next, the framework is extended to a novel sliding mode control law. It guarantees exact prescribed-time convergence for almost all initial conditions even in the presence of external disturbances. Notably, the proposed control law is nonsingular across the entire state space and maintains uniformly bounded gains. Finally, simulation results validate the effectiveness of the proposed methods.

This paper studies smoothed matrix-polar spectral gradient flows for unconstrained matrix-valued optimization.The canonical polar-factor map loses smoothness at rank-deficient matrices and becomes ill-conditioned as singular values approach zero, creating analytical difficulties.We therefore introduce a spectral feedback law generated by a smooth spectral potential and establish the regularity, monotonicity, boundedness, and dissipation properties of the feedback.Based on this feedback law, we propose a smoothed spectral gradient flow and prove well-posedness and global convergence of the flow.We derive convergence-rate results for the spectral gradient flow in nonconvex, convex, and Polyak–Lojasiewicz (PL) settings and analyze the Lyapunov structure and convergence of a momentum-augmented system under the same spectral feedback law. Furthermore, we provide a local descent-rate comparison between the smoothed spectral-gradient direction and the standard Frobenius-gradient direction using a general Hessian-based quadratic model. This analysis yields a verifiable normalized descent-rate advantage condition, showing that the local benefit of the spectral direction depends on both first-order alignment with the gradient matrix and the directional curvature induced by the Hessian.

In portfolio optimization, a cardinality constraint, which limits the number of assets held, plays a key role in cutting down monitoring and transaction costs. However, the resulting problem is NP-hard and becomes computationally difficult to solve globally as the number of candidate assets grows. Safe screening addresses this difficulty by fixing decision variables before optimization without excluding any globally optimal solution, thereby reducing the problem size while preserving optimality guarantees. We propose safe screening rules for cardinality-constrained portfolio optimization with a convex quadratic objective function and linear constraints. Using a perspective relaxation of the L2-regularization term and Fenchel duality, we derive asset-specific scores that incorporate the Lagrange multipliers of the linear constraints. Combined with a relaxation-based lower bound and a feasible-solution upper bound, these scores safely fix binary asset-selection variables to zero or one. Experiments on S&P 500 and Russell 2000 datasets show substantial computational improvements on challenging cases, particularly under moderate or strong regularization and less stringent return requirements. These results demonstrate the effectiveness of safe screening as an optimality-preserving preprocessing technique that greatly boosts computational efficiency in large-scale cardinality-constrained portfolio optimization.

We establish exponential mixing for the randomly forced and weakly damped KdV equation in $L^2(\mathbb{T})$. The noise is bounded, localized, and degenerate in high frequencies. Our proof relies on a general probabilistic framework in [11,33], nonlinear smoothing for KdV and its linearization via normal form transformation, and stabilization of the system by localized force. This paper continues a series of works connecting asymptotic compactness, control theory, and ergodicity and mixing for randomly forced dispersive PDEs.

To address the demand for inherently interpretable optimization methods, we introduce novel linear surrogates for budget allocation problems. These surrogates consist of sparse linear rules that map instances to feature-based representations of solutions. We present an exact approach based on mixed-integer programming as well as a heuristic for their computation. The performance of both approaches is analyzed through computational experiments.

We study null controllability of backward stochastic heat systems with constant couplings in both the state and the martingale integrand, common scalar diffusion, and a localized drift control. We prove that null controllability is equivalent to a noncommutative word-rank condition on the two coupling matrices and the control matrix. The proof combines a factorization of the dual dynamics, positivity and small-time coercivity of a finite-dimensional stochastic Gramian, and a Lebeau–Robbiano spectral argument. Failure of the rank condition yields an invariant unobservable subspace.

For mirror descent generated by a Legendre kernel, perhaps one of the most basic question in optimization is this: must every accumulation point of a bounded mirror descent sequence be Karush–Kuhn–Tucker (KKT) stationary under proper stepsizes? We show that the answer is no. A longstanding obstacle to resolving this question is the boundary blow-up of the Legendre gradient: it keeps every mirror step in the interior, while at a boundary limit, the inverse entropy metric vanishes on active coordinates and can erase the dual-feasibility in the KKT system. We construct $C^\infty$ objectives and bounded sequences generated by the Shannon-entropic mirror descent on the nonnegative orthant $\R_+^n$, for every $n\geq 3$, and on the probability simplex $\Delta_n$, for every $n\geq 4$, such that, in each case, the set of accumulation points is a smooth boundary circle containing a nonempty relatively open arc of non-KKT points. The steps satisfy $\alpha_k\asymp k^{-\beta}$ with $\beta\in(1/2,1)$, the objective values are nonincreasing, and the objectives are entropy-relatively smooth. Hence the pathology stems from the degeneracy of the Bregman geometry at the boundary, rather than from failure of descent, or improper stepsizes. To the best of our knowledge, these provide the first counterexamples to KKT accumulation for bounded mirror descent sequences with nonincreasing objective values.

In this paper, we introduce the notion of adaptive strong monotonicity and establish a linear convergence of the recently proposed Generalized Proximal Point Algorithm (GPPA) for computing approximate solutions to nonmonotone inclusion problems in Hilbert spaces. We also investigate the Inexact Generalized Proximal Point Algorithm (IGPPA) in the presence of nondiminishing errors.

Quadratic Gradient (QG) is a Newton-type optimization framework that bridges first-order gradient descent and second-order optimization by incorporating curvature information into gradient updates. Simplified Quadratic Gradient (SQG) reduces the complexity of QG construction while preserving its optimization capability, whereas Quasi-Quadratic Gradient (QQG) extends the quadratic gradient principle to quasi-Newton methods such as BFGS. In this paper, we propose **Generalized Quadratic Gradient (GQG)**, a unified framework that extends the quadratic gradient principle to a broader class of Newton-type optimization algorithms. By abstracting the common structure of existing quadratic gradient methods, we show that the fundamental requirement of quadratic gradient construction is not limited to specific Hessian approximations, such as constant Hessian matrices, diagonal Hessian approximations, or BFGS-based Hessian surrogates. Instead, it can be generalized to any positive-definite curvature matrix satisfying the stationary condition of a local quadratic model. Based on this perspective, we investigate the construction of generalized quadratic gradients using various positive-definite Hessian surrogates beyond BFGS, providing a broader foundation for developing curvature-aware optimization algorithms.

The safety and reliability of intelligent-transportation-systems (ITS) can be greatly enhanced through adding redundancy in the information-sharing network of the vehicles. In this paper, we first model the mixed traffic of human-driven and autonomous vehicles as a distributed system observability problem over a network of communicating vehicles. We clearly show that a strongly-connected network with minimum n links (with n as network size) is sufficient for the observability of mixed traffic network. Then, we present graph-theoretic results on adding redundancy to the changing network of vehicles to make it resilient to the failure of certain number of vehicles/sensors or their data-sharing links. Finally, we employ distributed observer design to validate our results over a simple mixed traffic example.

In this paper, we present a general formulation to address the problems of covering curves and polygonal chains with triangle, and fitting these curves into triangle. These problems can be formulated as special cases of Bellman's lost-in-a-forest problem (escaping triangular forest) and Moser's worm problem (covered by triangle). We model and reformulate the problem by keeping the curve stationary while allowing the triangle to translate and rotate. Subsequently, we derive the functional minimization formulation with support function constraints to solve. We also prove the equivalence and convergence of the formulas. Finally, we employ numerical methods and present results for covering curves with arbitrary triangles of various angles. We also present some corollaries and variant results, including closed curves and closed polygonal chains.

We consider mixed-integer convex optimization problems in which binary indicators control continuous variables. We introduce the Coordinate Optimality Reformulation (CORe) framework, which augments standard indicator formulations by incorporating coordinate-wise optimality information. The resulting reformulations preserve global optimality while substantially improving branch-and-bound performance, particularly in sparse and structured settings where the coordinate-wise optimality conditions expose exploitable problem structure. We first develop the main components of CORe, including coordinate-wise optimality conditions, closed-form characterizations, and disjunctive reformulations. We then demonstrate the framework across multiple problem families, including quadratic problems and robust single-index models. Computational experiments show that CORe can substantially improve solver performance compared with standard big-$M$ formulations.

This paper addresses the geometric inverse problem of simultaneously recovering two unknown deterministic source supports in a stochastic parabolic equation, where one source appears in the drift term and the other in the diffusion term. We establish that partial boundary flux measurements alone uniquely determine both supports. Moreover, we prove the existence of minimizers for a perimeter-regularized objective functional. For smooth interfaces, we derive a Hadamard-type boundary representation of the shape derivative. To the best of our knowledge, this is the first such formula for the simultaneous recovery of a drift-source support and a diffusion-source support in an SPDE setting. The derivative exhibits a genuinely stochastic two-channel structure: the adjoint state governs the sensitivity of the drift-source interface, while the martingale component controls the sensitivity of the diffusion-source interface. Based on this formula, we develop a shape-gradient reconstruction method. Numerical experiments demonstrate its effectiveness and its capacity to distinguish between the two source channels under both full and partial boundary observations.

Scenario-tree model predictive control (MPC) represents future information by a rooted tree and optimizes a nonanticipative policy over that tree. Numerical methods for solving the resulting nonlinear program typically compute their search directions through a sequence of branched linear-quadratic regulator (LQR) subproblems. The standard tree Riccati recursion requires linear work but has a dependency chain proportional to tree height. We present an algebraically exact parallel solver based on rake-compress tree contraction. After independent local control condensation, its two operations act on node and edge data that represent conditional quadratic functions. A rake eliminates a leaf and its parent edge, adding their reduced contribution to the parent-node data. A compress eliminates a unary node and replaces its two adjacent edges by one edge, using the same conditional-value composition as parallel Riccati methods on a chain. Together they contract an arbitrary rooted tree to its root; reversing the contraction recovers every Riccati coefficient, state, control, and multiplier. Given a reusable topology plan, a solve with $N$ nodes and fixed state and control dimensions has $O(N)$ arithmetic work and storage and $O(\log N)$ span, independently of tree height, balance, and maximum out-degree. The formulation allows positive-semidefinite dual regularization, including the unregularized case, and an exact linear-size lifting covers the standard scenario-MPC convention of one control per information node. We prove the contraction identities and equivalence to the Karush-Kuhn-Tucker (KKT) system. Three MIT-licensed JAX packages implement the bidirectional contraction, the dual-regularized LQR solver, and a user-facing primal-dual interior-point solver for tree-structured optimal control.

Symmetric Numerical Three-Dimensional Matching (SN3DM) asks whether three disjoint labeled classes with identical weight multisets can be partitioned into class-transversal triples of one common target sum. Its theme is role recovery under marginal symmetry: identical numerical catalogues force the asymmetric source roles to be reconstructed from incidence structure alone. This tutorial develops three complementary hardness results for that symmetry restriction. Part I gives a unary-polynomial reduction from N3DM. Source roles become ports in one common occurrence set, a uniquely forced filler system reserves one main incidence per port, bipartite edge coloring restores the output-class labels, and a no-carry mixed-radix encoding packs four coordinates into positive integers. Hence SN3DM is strongly NP-complete. Part II studies Max-SN3DM, for which strong NP-hardness alone does not exclude a PTAS. Two numerical compilers lift Petrank's perfect-completeness gap for bounded 3DM to unary Max-N3DM, and a defect-stability lemma shows that a symmetric matching of size 13n - d yields a source matching of size at least n - 21d, where n is the multiset cardinality, and d is a symmetric defect. Hence, for some epsilon > 0, it is NP-hard to separate perfect instances from those of optimum at most (1- epsilon) times perfect, so no PTAS exists unless P = NP. Every maximal legal triple matching is a 3-approximation, placing the problem in APX. Part III supplies the approximation-preserving reduction Part II does not claim. An exact pair compiler and a one-live-port separation map degree-three Maximum 3DM to unary Max-SN3DM with OPT(Max-SN3DM) = Gamma + OPT(Max-3DM) for a fixed offset Gamma and one-for-one optimum-error transfer. The L-reduction has constants alpha = 764 and beta = 1, so Max-SN3DM is APX-complete. The two are incomparable; worked yes / no instances audit each construction.

We revisit Decomposition-Invariant Conditional Gradient methods, originally introduced by Garber and Meshi in 2016, for minimizing a convex and $\beta$-smooth function over a polytope in $\reals^n$, under an $\alpha$-quadratic growth condition. For 2-level polytopes we design a simple and parameter-free dyadic step-size rule that yields a linear convergence rate which scales with the dimension of the optimal face and not with the ambient dimension as in standard away-step-based conditional gradient methods for polytopes. For general polytopes, under a slightly stronger condition of $\alpha_{\mathrm{F}}$-facial quadratic growth, we introduce a method whose number of iterations to reach an $\epsilon$-approximate solution is of the order $n+\frac{\beta{}D^2}{\alpha{}r^{*2}} + \frac{(d^*+1)\beta{}D^2}{\alpha_{\mathrm{F}}}\log(1/\epsilon)$, where $d^*$ is the dimension of the optimal face, $r^*$ is a separation parameter between the optimal set and faces that do not contain an optimal solution, and $D$ is the diameter of the polytope. This method is also parameter-free and only relies on standard line-search computations. The second result improves upon previous conditional gradient methods, whose number of iterations to $\epsilon$-approximation scales with $\frac{\beta{}D^2n}{\alpha}\log(1/\epsilon)$, in a meaningful regime $\max\{\frac{\alpha}{\alpha_{\rm F}}(d^*+1), \frac{1}{r^{*2}}\} \ll n$

Distributed target tracking in multi-agent networks plays a critical role in cooperative sensing and autonomous navigation. However, it faces significant challenges in highly dynamic and adversarial setups. This study aims to enhance the resilience of decentralized target tracking algorithms against measurement faults and cyber-physical threats, especially false data injection attacks. We propose a consensus-based estimation algorithm that integrates a nearly-constant-velocity model with saturation-based filtering to suppress impulsive measurement variations and promote robust, distributed state estimation. To counteract adversarial conditions, we incorporate a dynamic false data injection detection and isolation mechanism that uses innovation thresholds to identify and disregard suspicious measurements before they can degrade the global estimate. The effectiveness of the proposed algorithms is demonstrated through a series of simulation-based case studies under both benign and adversarial conditions. The results show that increased network connectivity and higher consensus iteration rates improve estimation accuracy and convergence speed, while properly tuned saturation filters achieve a practical balance between fault suppression and accurate estimation. Furthermore, under localized, coordinated, and transient false data injection attacks, the detection mechanism successfully identifies compromised agents and prevents their data from corrupting the distributed global estimate. Overall, this study illustrates that the proposed algorithm provides a simplified fault-tolerant solution that significantly enhances the accuracy and resilience of distributed target tracking without imposing excessive communication or computational burdens.

In recent years, the prevalence of large-scale data-sets and the demand for sophisti-cated learning models have necessitated the development of efficient distributed ma-chine learning (ML) solutions. Convergence speed is a critical factor influencing the practicality and effectiveness of these distributed frameworks. Recently, non-Lipschitz continuous optimization algorithms have been proposed to improve the slow conver-gence rate of the existing linear solutions. The use of signum-based functions is previ-ously considered in consensus and control literature to reach fast convergence in the prescribed time and also to provide robust algorithms to noisy/outlier data. However, as shown in this work, these algorithms lead to an optimality gap and steady-state re-sidual of the objective function in discrete-time setup. This motivates us to investigate the distributed optimization and ML algorithms in terms of trade-off between conver-gence rate and optimality gap. In this direction, we specifically consider the distributed regression problem and check its convergence rate by applying both linear and non-Lipschitz signum-based functions. We check our distributed regression approach by extensive simulations. Our results show that although adopting signum-based func-tions may give faster convergence, it results in large optimality gaps. The findings pre-sented in this paper may contribute to and advance the ongoing discourse of similar distributed algorithms, e.g., for distributed constrained optimization and distributed estimation.

One-way spatial marching methods separate upstream- from downstream-propagating disturbances using a rational approximation of a spectral projector. Among existing one-way Navier–Stokes (OWNS) formulations, the recursive variant, OWNS-R, is the most economical, evaluating the approximation as a product of $N$ resolvent factors. This product amplifies rounding errors multiplicatively, imposing a flow-dependent upper limit on the approximation order that cannot be known in advance. We reformulate the same approximation as an additive partial-fraction sum (OWNS-Summation, OWNS-S). In exact arithmetic, the recursive and summation evaluations are equivalent when supplied with identical weights and poles; in floating-point arithmetic, they are not. Each of the $N$ resolvent solves acts on the same input state and contributes independently to a weighted sum, preventing multiplicative error amplification. The approximation order $N$ therefore becomes a pure convergence parameter, and the solves can run in parallel. Using the same auxiliary poles as OWNS-R, OWNS-S remains accurate in every configuration tested, showing that the recursive evaluation, rather than the poles, is the dominant source of instability. A paired greedy parameter-selection procedure is also introduced, with candidates drawn from analytic estimates of the upstream and downstream spectral regions, avoiding eigen-decomposition during the numerical march. OWNS-S is validated on incompressible, hypersonic and transonic boundary layers. In the transonic case, the disturbance spectrum reorganises from a subsonic to a supersonic topology during the march. The one-way computation proceeds continuously through this transition at a streamwise resolution unattainable by the parabolised stability equations.

In this paper, we consider stochastic optimal control problems with infinite-horizon joint chance constraints. By means of an appropriate state augmentation, we reformulate the original problem as a constrained Markov decision process, in which both the cost and the constraint function exhibit an additive structure. We then prove that this formulation enjoys strong duality, thereby enabling us to reformulate the problem as an equivalent unconstrained one in the Lagrange dual framework. We propose a dual-ascent algorithm to solve the resulting problem and show that it converges to a deterministic Markov policy defined over the augmented state space that is both optimal and feasible. To accommodate continuous state-input spaces, we propose a dedicated learning algorithm to approximate the value function in an offline training setting, thereby significantly reducing the computational complexity of the online control phase. We then test our approach on a numerical example and demonstrate its effectiveness compared to online predictive control methods in terms of performance and computational complexity.

This study investigates a new approach for scheduling freight rail engineers based on call windows under the Hours of Service (HOS) regulations. Unlike planned trip assignments, call windows specify a time interval during which an engineer may be required to start work, thus providing greater flexibility to handle uncertain trip demand while offering drivers more predictable off-duty periods. Currently in the U.S., all major freight operators require drivers to be available 24/7 outside of mandatory rest periods, raising concerns over workforce fatigue and safety. We formalize the call-window scheduling problem and propose two solution approaches: a Set-covering-type optimization model and a Direct Algorithm. Both aim to maximize demand coverage while minimizing the number of engineers subject to HOS feasibility. Computational experiments for 2-city and 3-city instances with varying call window lengths, maximum delay allowances, and whether to allow deadhead trips show that the Set-covering-type model yields higher demand undercoverage (94.89% and 91.09% for 2-city and 3-city instances, respectively) than the Direct Algorithm (93.96% and 71.83%). It also offers greater rest opportunities and reduced delays. Sensitivity analysis reveals that the upper limit on engineer availability significantly affects all key performance metrics.

Generative AI tools are rapidly becoming standard for coding and other business operations. However, while these tools can drive productivity, their unpredictable, and often significant, costs are putting severe financial strain on organizations. The leadership of such organizations must split a fixed budget between human headcount and generative-AI token capacity, yet standard deterministic planning ignores both the heavy-tailed volatility of token consumption and the cognitive cost of auditing AI-generated output. We model the joint allocation by intersecting a cognitive-friction-adjusted production function with a chance-constrained stochastic budget frontier. Per-engineer token usage is treated as i.i.d., non-negative, and right-skewed; no parametric family is assumed, as only its mean, variance, and skewness enter the analysis. The chance constraint is reduced to a deterministic equivalent via the Central Limit Theorem with a Cornish–Fisher skewness correction. Solving the resulting Lagrangian yields a closed-form optimal headcount, a transcendental condition for optimal per-capita token intensity, and a bordered-Hessian second-order condition that holds for any reasonable volatility-to-headcount ratio. Comparative statics show that rising individual token volatility raises optimal per-capita token intensity while contracting headcount: although human labor and tokens are complements in production, the stochastic budget makes them substitutes at the margin. Individual usage skewness acts as a pure deadweight tax that shrinks the budget without altering the substitution dynamics. Both conclusions presuppose that usage dispersion is invariant to the planned mean allocation. In the case where dispersion instead scales proportionally with the mean, the substitution reverses, so which regime applies is a sharp, empirically testable question.