Papers

The growth of large language model (LLM) inference and search services increases the scale of online linear programming problems, motivating computationally efficient algorithms. We develop resource-adaptive stochastic gradient descent (RASGD) for stochastic online linear programming. The algorithm uses one request and current inventory to update resource prices, requiring O(m) operations for m resources and memory per arrival and no LP or sample-average optimization. The central idea is to express the current-resource pricing logic of re-solving through a first-order SGD update: each arrival refreshes the remaining-inventory allowance in the dual objective, while the stepsize decreases for early learning and increases later to match the speed of inventory adjustment. Under standard non-degeneracy conditions, our algorithm is feasible on every sample path and achieves O(\log T) expected regret against the realized fractional hindsight optimum, which matches the lower bound, even for policies that know the distribution and have unrestricted computation. The analysis converts curvature around the fixed reference price into inventory stability without tracking optimal prices at changing resource levels. Numerical experiments show that RASGD achieves regret competitive with per-arrival LP re-solving and improves upon the tested first-order baselines, while retaining the computational efficiency of first-order methods. These results establish RASGD as a computationally efficient approach to achieving high allocation quality in large-scale OLP.

For smooth convex–concave minimax optimization, the higher-order lower bound of Chen et al. (2026) applies to a restricted tensor-algorithm class with prescribed regularized Taylor-model updates. We establish the same bound for arbitrary adaptive deterministic and randomized algorithms, matching, up to logarithmic factors, the upper bound of Zhang et al. (2026). Fix an integer $p\ge 2$ and let $L_p>0$ bound the Lipschitz constant of the objective's $p$-th derivative on a compact convex product domain of diameter at most $D_Z>0$. Each feasible query returns the objective value and all derivatives through order $p$. For accuracy $\epsilon>0$, set $Q_{\mathrm{tan}}=L_pD_Z^p/\epsilon$ for tangent residual and $Q_{\mathrm{gap}}=L_pD_Z^{p+1}/\epsilon$ for saddle gap. Let $T_E^{\mathrm{det}}(\epsilon)$ and $T_E^{\mathrm{rand}}(\epsilon)$ denote the high-dimensional minimax query complexities for criterion $E\in\{\mathrm{tan},\mathrm{gap}\}$, with randomized success probability at least $2/3$ on every instance. Our lower bounds and the existing upper bound give $c_pQ_E^{2/(3p-1)}\le T_E^{\mathrm{rand}}(\epsilon)\le T_E^{\mathrm{det}}(\epsilon)\le C_pQ_E^{2/(3p-1)}[1+\log(3+Q_E)]^{6(p-1)}$ for sufficiently large $Q_E$, where $c_p,C_p>0$ depend only on $p$. Thus the same accuracy exponent holds beyond tensor update rules, even for randomized queries and arbitrary feasible outputs. The proof constructs a scalar convex–concave chain with exactly flat gates that hide complete derivative information. Direct product-domain error witnesses and adaptive transcript arguments establish the lower bounds for both criteria.

We establish tight polynomial tolerance bounds for stochastic second-order stationarity when each fresh oracle response is a derivative of one common random scalar loss. For a population objective $F$ with Lipschitz gradient and Hessian, the target is $\|\nabla F(x)\|\le \epsilon$ and $\lambda_{\min}(\nabla^2 F(x))\ge -\gamma$, with independent tolerances $\epsilon,\gamma>0$. Under bounded gradient variance and almost-surely bounded Hessian error, the minimax number of fresh gradient or Hessian-vector-product calls is $\widetilde{\Theta}\!\left(\epsilon^{-3}+\gamma^{-5}\right)$. The characterization fixes positive gap, smoothness, and noise parameters, suppresses logarithmic factors, and allows dimension to grow within an explicit polynomial envelope. The upper bound removes the mixed term $\epsilon^{-2}\gamma^{-2}$ from the earlier fresh-HVP guarantee. Direct random-line Hessian estimates and a dyadic gradient tracker separate gradient drift from randomly signed curvature motion. The lower bound realizes the endpoint costs through globally defined smooth random losses: a smooth partition localizes scalar noise without a chain-length penalty, while exact cancellation limits the information in the entire response. Consequently, the same tolerance exponents hold even for joint value, gradient, and full-Hessian responses with bounded value variance. For a population Hessian with Hölder exponent $\nu\in(0,1]$, fresh gradient/HVP complexity becomes $\widetilde{\Theta}\!\left(\epsilon^{-3}+\gamma^{-(3+2/\nu)}\right)$ under the corresponding dimension envelope.

We establish tight randomized higher-order oracle complexity for finding first-order stationary points of nonconvex finite sums. Let $n$ be the number of components, $\Delta>0$ the initial objective-gap bound, $L_p>0$ an individual $p$-th derivative Lipschitz bound, and $\epsilon>0$ the target gradient norm. For every fixed integer $p\ge 2$, the minimax number of exact component queries returning the value and all derivatives through order $p$, with success probability at least $2/3$, is \[ \Theta_p\!\left( n+\Delta L_p^{1/p}n^{1-1/(2p)} \epsilon^{-(p+1)/p} \right), \] where the constants depend only on $p$ and the worst case ranges over all finite dimensions. The lower bound holds for unrestricted randomized adaptive algorithms and closes the $\sqrt{n}$ gap between the previously known general-order upper and lower bounds in their dependence on $n$. We extend dense weak hiding to complete higher-order replies while keeping each component's regularity independent of the chain length. The matching upper bound retains the known finite-sum exponent, requires only mean-squared $p$-th derivative increments, and removes the fixed-confidence logarithmic loss by verifying entire recursive-estimation epochs with exact function values. The characterization includes the additive $n$ term for every positive parameter regime; it counts oracle calls with unrestricted internal computation.

Personalised pricing maximises expected revenue from a market but requires detailed information about individual customers. How much of this revenue can be recovered using a simple posted price based on a finite number of samples from the underlying value distribution? We answer this question by maximising the worst-case ratio between the expected revenues of posted and personalised pricing over the fundamental class of $\lambda$-regular value distributions. Our results reveal a structural transition as a function of $\lambda$. For the class of monotone hazard rate (MHR) distributions, corresponding to $\lambda = 0$, the sample mean is an optimal statistic: the entire sample can be compressed into its average without any loss of revenue. Beyond the MHR class, corresponding to $\lambda > 0$, this property disappears. We show that the sample mean is no longer optimal, revealing that optimal sample-based pricing rules become substantially more intricate. Nevertheless, we show that a remarkably simple order-statistic based pricing rule is asymptotically optimal as the number of samples $n$ grows, achieving the optimal approximation ratio up to a tight error of order $1/n$. Our analysis combines techniques from probability, approximation theory and optimization, including doubly infinite linear programming, hypergeometric functions, and combinatorial identities involving incomplete Beta functions.

One item set evaluated at two capacities $R<D$ produces two concave hulls of optimal value against cardinality. We ask which prescribed pairs arise. Exchange arguments give a necessary system on vertex witnesses, exchange closure (EC), whose scalar consequences form the linear closure. We exhibit a pair satisfying every scalar test that fails EC, so the linear closure is strictly larger already at larger terminal count three; and a globally coherent EC witness admitting no common-size representation although its target pair is realizable. Under a cardinality cap, a four-band classification of one family gives exact thresholds for cap-four realizability, uncapped realizability and the vertex-only capped closure. Pairs whose larger terminal count is at most two are characterized. Under the explicit encoding the decision problem lies in $\Sigma_2^p$ and is polynomial-time for fixed terminal cardinalities. Exact finite certificates establish agreement of the scalar and witness conditions on six specified domains; sufficiency of uncapped EC remains open.

We introduce the Max-Min Covering Location Blocker Problem, a bilevel optimization problem in which a leader blocks a minimum-cost set of candidate locations so that the optimal coverage of a budget-constrained follower does not exceed a prescribed target. Each customer's coverage is determined by the least favorable open facility that can serve it. The induced follower coverage set function is nonmonotone and nonsubmodular, so the validity of interdiction cuts cannot be inherited from existing frameworks. We develop an exact nested decomposition that exploits the max-min coverage structure at both levels. At the outer level, we establish the validity of interdiction cuts from this structure and strengthen them with facility-specific coefficients bounding the coverage lost when a critical facility is removed. We solve the NP-hard follower problem by branch-and-Benders-cut after projecting out the customer coverage variables, and characterize an optimal dual solution of the separation problem in closed form, generating Benders optimality cuts in linear time without solving a linear program. Computational experiments on benchmark instances show that both ingredients substantially improve the performance of the method. With a moderate number of candidate locations, instances with 300,000 customers are solved to optimality within one hour. A comparison with two general-purpose bilevel solvers from the literature shows the proposed method to be one to three orders of magnitude faster, and to solve to optimality instances that neither of them closes. The model applies wherever the planner cannot control which facility serves a customer, as in public automated external defibrillator (AED) networks. A case study on a network using real AED data from Virginia Beach illustrates the model.

We study the distributed average consensus problem under persistent link-level disturbances modeled as a martingale difference sequence with uniformly bounded conditional second moments. Under such disturbances, the standard stochastic-approximation-based linear iteration with diminishing stepsizes drives the network to consensus on an unbiased random variable with non-vanishing variance instead of the exact initial average. To understand and resolve this limitation, we develop an anchoring-based mechanism derived from a decentralized gradient descent formulation and study the effect of incorporating a decaying anchoring term that continuously pulls each agent state toward its initial value. This perspective provides an intuitive interpretation of how state anchoring counteracts disturbance accumulation. Under standard summability conditions, we prove that the resulting algorithm achieves exact average consensus almost surely. Furthermore, this decentralized gradient perspective offers a unifying framework for several related methods and an interpretable design principle for exact average consensus under persistent disturbances.

We study the asymptotic behavior of the optimal stabilization prefactor for the free Schrödinger equation, optimized over all bounded stationary linear feedbacks. For a prescribed decay rate $\delta>0$, let $\widehat{C}_S(\delta)$ denote this optimal prefactor. Assuming that the uncontrolled region contains a nonempty open set and that the observability cost satisfies $C(T)\le C_0e^{C_0/T}$ for small $T>0$, we prove $$ e^{c\sqrt{\delta}}\le \widehat{C}_S(\delta)\le e^{C\sqrt{\delta}}, \qquad \delta\gg1. $$ The lower bound is based on the uncontrolled open hole and a quantitative cutoff construction adapted to the Laplacian, while the upper bound combines an exponentially weighted Gramian with the small-time observability cost $C(T)\lesssim e^{C/T}$. The results apply to controlled Schrödinger equations in three settings: flat tori, including a strip-control example beyond the geometric control condition; the whole space $\mathbb{R}^n$ with exterior-ball control; and bounded smooth domains satisfying the geometric control condition for generalized geodesics.

Large-scale and high-dimensional discrete optimal transport often involves a tension between scalability and exact assignment feasibility: many scalable approaches optimize relaxed, factorized, or sparse transport plans rather than maintaining a one-to-one assignment throughout optimization. We introduce SWAP, an iterative solver that operates directly in permutation space for equally weighted point clouds. Starting from any feasible assignment, SWAP uses geometry-aware proposals to identify cost-decreasing local rearrangements. Consequently, every iterate is a valid permutation, so the one-to-one constraints are satisfied exactly throughout the optimization without rounding or feasibility correction. SWAP avoids constructing an $N\times N$ cost or coupling matrix and requires only $O(Nd)$ memory. We establish monotone descent, introduce a family of cycle-stability conditions, prove that SWAP reaches pairwise stability almost surely under generic nonparallel conditions, and derive sufficient conditions under which pairwise stability implies global optimality. Experiments on an ImageNet assignment with 640,500 points in 2,048 dimensions, synthetic benchmarks, and MERFISH spatial alignment demonstrate that SWAP achieves lower transport costs on challenging large-scale and high-dimensional problems, with substantially reduced runtime and memory compared with

We investigate the computation of optimal bias potentials for accelerating transitions between metastable states and for computation of equilibrium properties in molecular dynamics simulations. We formulate optimal biasing as an ergodic optimal control problem (OCP), which can be recast as a linear eigenvalue problem for the infinitesimal generator of the unbiased dynamics. We demonstrate that data-driven learning methods for the generator enable reliable solution of the OCP, computation of biasing potentials, extraction of equilibrium properties, and acceleration of state transitions. We also explore the relation of the control problem to coarse grained representations and learning of coarse grained dynamics.

In an $n$-dimensional cropped hypercube each of the $2^n$ cropping inequalities chops off a single corner of the $0$–$1$ hypercube by an $\ell_1$-distance $\rho$. The case $\rho = 1/2$ has been extensively studied in the literature. This paper shows that the Lasserre rank of the $n$-dimensional cropped hypercube where $\rho = 1/2$, $n \geq 2$, is the smallest integer $0\leq t \leq n$ such that $\Delta_t < 0$ in the recurrence $\Delta_{-1} = 1$, $\Delta_{0} = n-1$, $\Delta_t = (n-1)\Delta_{t-1} - t(n-t+1)\Delta_{t-2}$. It follows that the Lasserre rank can be computed in time $O(n^2 \log^2 n)$. Asymptotically, the rank is $\frac{n}{2} + c_{1/2}\sqrt{n} + o(\sqrt{n})$, where $c_{1/2}$ is the unique zero of a given function. Numerically, $c_{1/2} \approx 0.3825$. In fact, we prove such results for any fixed $0 < \rho < 1$.

Seasonal hydropower operation requires release decisions that account for uncertain future inflows and the opportunity cost of stored water. We assess the Modelling and Optimisation of Stochastic Seasonal Hydropower Storage (MOSSHOOS) framework for the Manantali reservoir using 660 monthly energy-inflow observations from 1961–2015. The framework combines multiscale inflow representation, blocked resampling and scenario reduction with stochastic dual dynamic programming, and uses a seasonal square-root diffusion and Hamilton–Jacobi–Bellman formulation as a continuous-time interpretation of marginal water value. Scenario reduction improves the energy score by 47.6% relative to independent monthly sampling, but the marginal distribution is rejected by a Kolmogorov–Smirnov test and the lower-tail p10 error remains 103.9%, indicating inadequate dry-tail calibration. Consistent with that limitation, 50 paired Monte Carlo evaluations show no detectable cost or deficit advantage of the stochastic policy over its deterministic-equivalent counterpart at the archived operating configuration. As an independent benchmark, the repaired open-source plan4res seasonal storage valuation chain yields a dominant marginal value of 81.66~USD/MWh, within 3% of the MOSSHOOS mean of 79.25~USD/MWh. These results support a coherent stochastic inflow-to-water-value methodology while showing that hydrological tail representation, rather than economic scaling, is the principal limitation to address before operational application.

In reinforcement learning, more accurate policy evaluation has been frequently shown to improve downstream policy optimization performance. For tractability, most value estimators rely on $n$-step truncation and thereby omit the long-horizon behavior of the transition dynamics. Methods that instead capture this behavior through the global structure of the dynamics can converge faster, but their computational and memory requirements limit their scalability to large or continuous state spaces. To reconcile these limitations, we introduce Limiting-Kernel Q($\lambda$) (LKQL), an off-policy value estimator that uses the limiting kernel to approximate the long-horizon terms omitted by truncation. LKQL has the same order of complexity as $n$-step estimators and integrates directly into both on- and off-policy actor-critic algorithms. We prove that, under aperiodicity and in the near-on-policy regime, the operator underlying LKQL improves the policy evaluation convergence rate over its truncated counterpart for sufficiently large $n$, and that LKQL itself converges almost surely to the optimal values in finite Markov decision processes (MDPs) under a fixed behavior policy. On the MuJoCo continuous-control benchmark, we show that LKQL improves over $n$-step baselines in most settings, particularly on long-horizon tasks.

We study finite-horizon distributionally robust linear-quadratic control with disturbances that can be arbitrarily correlated in time. The ambiguity set is modeled as a single 2-Wasserstein ball centered at a nominal elliptically contoured distribution of the disturbances. Despite the infinite-dimensionality of both the policy space and ambiguity set, we prove that the optimal policy is affine and that the worst-case distribution is an affine push-forward of the nominal distribution. These affine maps can be computed efficiently via a best-response algorithm based on the Frank–Wolfe algorithm. Experiments demonstrate improved out-of-sample performance over LQG and its distributionally robust extensions when the true disturbances are correlated, incurring only a small loss of performance under uncorrelated disturbances.

The design of profile extrusion dies remains a challenging task due to the complex rheological behavior of polymer melts and the geometric intricacies of flow channels. Traditional manual optimization approaches, which rely heavily on human experience, are inefficient and often employ unvalidated heuristics. To address these challenges, we present a deterministic and explainable framework for automatic die design based on adjoint-based shape optimization. This approach enables the computation of sensitivities that directly indicate beneficial modifications to the flow channel geometry. A major difficulty in such optimization processes lies in generating boundary-conforming meshes that evolve consistently with changing geometries. To overcome this issue, we employ non-boundary conforming geometry representation methods that eliminate the need for an explicit surface representation along physical boundaries. A dedicated reconstruction technique is developed to recover accurate sensitivity information at the virtual interface between fluid and solid regions. The proposed algorithm is demonstrated on 3D geometries with varying complexity, including realistic extrusion die flow channels. Several objective functionals relevant to industrial applications, such as flow balance at the outflow, are considered. The results highlight significant improvements in performance metrics while maintaining numerical robustness. This work showcases the potential of adjoint-based techniques for automated die design in a domain still largely governed by manual trial-and-error procedures, establishing a foundation for data-efficient, sustainable manufacturing workflows using computational rheology.

This paper investigates a stochastic linear-quadratic (SLQ) control problem for a regime-switching jump-diffusion system. Unlike traditional regime-switching diffusion systems that couple a diffusion process with a Markov chain, we incorporate the jumps of the Markov chain into the state equation. This modeling methodology effectively captures potential gains or losses of the system during the regime transitions. It should be noted that the introduction of Markov chain jumps into the state equation leads to increased complexity in the corresponding coupled differential Riccati equations (CDREs), thereby rendering the solvability of the control problem more challenging. Under the assumption that the cost functional is uniformly convex, we establish the unique solvability of the corresponding CDREs. Building upon this foundation, we derive a closed-loop representation for the unique open-loop optimal control. Finally, we apply our theoretical results to the mean-variance portfolio selection problem in a regime-switching financial market with Markov chain jumps and obtain its efficient frontier.

Magnetic navigation has emerged as a promising alternative for navigation in Global Positioning System (GPS)-denied environments, leveraging geomagnetic field maps in conjunction with onboard magnetometer measurements. However, its performance is highly sensitive to trajectory-dependent observability, which limits its practical effectiveness under conventional flight paths. This paper proposes an optimal trajectory design framework for magnetic navigation that maximizes information content along the flight path. The trajectory generation problem is formulated as an optimal control problem that minimizes the posterior Cramér–Rao lower bound on the position estimation error, subject to a penalty on path length. The resulting trajectories are non-intuitive and significantly enhance the observability of the navigation system. Simulation results demonstrate that the proposed optimal trajectories yield substantial reductions in estimation error compared to conventional straight-line trajectories, highlighting the critical role of trajectory design in enabling high-accuracy magnetic navigation. These findings suggest that trajectory optimization can substantially improve the viability of magnetic navigation as a robust alternative for aerospace applications in GPS-denied environments.

In data-driven contextual stochastic optimization, existing approaches are predominantly frequentist: they commit to a single predictive model and treat it as ground truth, yielding prescriptions that are fragile to model uncertainty and sampling variability in small- and moderate-sample regimes. We propose Bayesian contextual optimization (BCO), a framework that maintains a Gibbs posterior over the parameter space. This decision-focused posterior weights candidate models by their empirical decision quality rather than statistical fit, thereby avoiding commitment to a potentially misspecified likelihood while encoding the full distribution of plausible models consistent with the data. A Bayesian contextual policy is then derived by minimizing the expected decision cost under the posterior predictive distribution, hedging prescriptions against model uncertainty by aggregating over the posterior. We establish three theoretical guarantees: (i) the Gibbs posterior concentrates exponentially fast around the frequentist best-in-class parameter set; (ii) BCO can strictly improve over the frequentist best-in-class policy when model uncertainty is non-negligible; and (iii) BCO attains an $O(n^{-1/2})$ excess risk rate against the oracle over all probability measures up to a misspecification term and an oracle aggregate gap. Computationally, we tailor a variational inference scheme that has the same per-iteration cost as frequentist alternatives and a gradient-free Metropolis-Hastings algorithm that handles nondifferentiable problems. Numerical experiments on two-stage shipment planning, contextual newsvendor, and return-constrained portfolio problems confirm that BCO consistently reduces out-of-sample cost and variance relative to kernel estimators and decision-focused baselines, with the most pronounced gains in small- and moderate-sample regimes under substantial model uncertainty.

We investigate the optimal damping design and exponential stabilization for a class of higher-dimensional weakly degenerate damped wave equations. In weighted Sobolev spaces, we establish well-posedness and uniform exponential energy decay, and verify the existence of long-time optimal damping potentials. To overcome analytical difficulties arising from system non-self-adjointness, we use finite-time approximation via semigroup operator norms. Through linearization and adjoint-state analysis, we derive first-order necessary optimality conditions and show finite-time optimal dampings possess a bang-bang structure.

Algorithms in multi-agent systems such as federated learning, mobile robotic swarming, and consensus control can be designed and analyzed as distributed stochastic approximation algorithms. Such algorithms involve information exchanges between agents for various computations. The freshness of the information can be quantified using the Age of Information (AoI) metric. Consider robotic teams operating in highly obstructed geographical settings, such as subterranean or dense urban environments. Because of spatial disconnections, AoI has empirically been observed to be heavy-tailed with unbounded moments. However, most analyses assume AoI with bounded moments, creating a gap between theory and practice. To the best of our knowledge, ours is the first analysis under general heavy-tailed AoI with potentially infinite mean. We study the stability (almost sure boundedness of the distributed iterates) and convergence of multi-agent systems that are strictly dissipative in the scaling limit (system at “infinity”). Examples include most gradient-based and consensus algorithms under the Robbins-Monro step-size regime.

We prove that the Turán density of the tetrahedron $K_4^{(3)}$ satisfies $\pi(K_4^{(3)}) \le 14993367693127837/26880000000000000 < 0.557789$, improving Baber's upper bound of $0.5615$ and closing about $62\%$ of the gap to the conjectured value $5/9$. The proof uses an exact seven-vertex flag-algebra certificate incorporating degree-stationarity from Razborov's differential method. To find the certificate, we combine the established techniques of cutting planes and column generation to optimize jointly over flag families whose types have at most five vertices. We give a complete formal proof of this Turán density bound in Lean 4.

We investigate a class of differential inclusions governed by non-autonomous and autonomous maximal monotone operators, involving set-valued perturbations with a Volterra integral term and subject to a nonlocal condition. Under suitable assumptions on the governing operator, the set-valued perturbation, and the Volterra kernel, we establish existence results for solutions. In particular, the set-valued perturbation is assumed to satisfy a one-sided Lipschitz condition, while the Volterra kernel is required to be Lipschitz continuous. The existence of a trajectory is established by means of an iterative construction and an application of Zorn's lemma. Finally, several examples are presented to illustrate the applicability of the abstract results.

Proximal-Tracking is a novel algorithm proposed in [Falsone and Prandini, Automatica, 135 (2022), 109938] for nonsmooth distributed consensus optimization with local set constraints. The global convergence and numerical behavior of Proximal-Tracking have been well studied in [Falsone and Prandini, Automatica, 135 (2022), 109938]. As a complement, in this paper we provide a theoretical analysis regarding its linear convergence rate under additional strong convexity and calmness assumptions. In particular, this analysis does not require assumptions such as differentiability or smoothness, thus preserving the original advantages of the algorithm.

We study trust-region minimization of a smooth, possibly nonconvex functional plus a convex functional composed with a bounded linear operator. A relative primal–dual gap condition controls both the error in an approximate proximal-gradient step and its linear-model decrease. Together with a computable absolute stationarity test, it yields a finite Cauchy search, convergence of the proximal stationarity measure to zero, and an $O(\varepsilon^{-2})$ bound on outer trials. The outer analysis allows the linear operator to take values in a Banach space and does not require dual attainment. When the operator takes values in a Hilbert space and the regularizer is finite and Lipschitz, the dual proximal-gradient method produces finite gaps tending to zero, provided the required proximal maps and functional values can be evaluated. We prove $O(j^{-1})$ gap bounds for both recovered and averaged primal candidates and give a sharper bound on the primal error for exactly recovered points. A semilinear elliptic control problem with unsmoothed total-variation regularization and an $L^2$ control cost illustrates the method in the full $H^1$ metric. Across five meshes, outer and state Newton counts remain constant, while interior-point iteration counts vary mildly.

In this paper, we study sampled-data stabilization of general nonlinear stochastic systems under a local exponential-type Lyapunov condition for the continuous-time closed-loop system. The coefficients are assumed only locally Lipschitz, with neither linear-growth nor Khasminskii-type conditions. Two counterexamples show that almost-sure stabilization is generally unattainable. Instead, given an initial state, we establish exponential stability with arbitrarily high probability under sufficiently fast sampling. Separate methods are developed for \(p\ge 2\) and \(0<p<2\), where \(p\) is the growth order in the Lyapunov condition. Numerical examples illustrate the results.

In this paper, we propose and analyze an augmented Lagrangian framework for solving stochastic nonconvex optimization problems with expectation-based equality constraints over a closed and convex constraint set. The framework generates a sequence of nonconvex primal subproblems, which are solved inexactly using stochastic second-order methods. We establish iteration complexity results for obtaining approximate second-order stationary points, both in expectation and with prescribed probability, under corresponding conditions on the accuracy of the primal subproblem solutions. We further establish sample complexity results for the framework with general stochastic second-order subproblem solvers under reasonable assumptions on their theoretical guarantees. Moreover, we incorporate three existing stochastic second-order solvers and derive the corresponding deterministic and probabilistic sample complexity results for obtaining approximate second-order stationary points. To the best of our knowledge, such sample complexity guarantees for obtaining approximate second-order stationary points in nonconvex expectation-constrained optimization have not been established previously. Finally, we demonstrate the empirical performance of the proposed framework on two nonconvex machine learning problems.

A high dimensional bi-directionally coupled $N$-loop thermosyphon system in a tandem is modeled by $3N$ Lorenz type of differential equations. The fluid flow in each loop is driven by the heat source (Rayleigh numbers) as well as the moment and heat exchanges between the adjacent loops. The flow becomes chaotic when the Rayleigh numbers are large. A decentralized controller design via proportional local state feedback is employed to stabilize the chaotic flows in each loop. As a result of the stability analysis, we show that there exist lower bounds on the controller gains that guarantee global stability of the system. Under the scenario of unknown parameters, we augment the original system with additional dynamic equations of the feedback gains to adaptively determine the feasible gains that stabilize the system. The analysis also sheds insight on the role of thermal coupling in the stability of the control system, which allows us to extend the results to negative $z$-state coupling coefficient as well as a class of nonlinear $z$-state coupling functions. Numerical simulations further demonstrate the effectiveness of the proposed controller design

Reliable power grid control requires sequential decisions under transmission constraints, time-varying demand, renewable variability, and imperfect load information. We investigate how load information quality affects control in a customized transmission network simulator inspired by Grid2Op and based on a DC power flow model. The environment includes stochastic spatial loads, temporally correlated renewable-like generation, energy storage, line protection, generator dispatch, and line reconnection. An Ensemble Score Filter (EnSF) corrects noisy load information before it is passed to the controller. A heuristic study first isolates the effect of this correction under a fixed feedback rule. Proximal policy optimization (PPO) agents are then trained for sequential generator and reconnection control, frozen, and evaluated with Forward, EnSF-corrected, and Truth load information on paired stochastic scenarios. EnSF consistently lowers load estimation error. Under heuristic control, the correction extends survival and reduces warning and overload exposure. With frozen PPO policies, it increases cumulative return and keeps performance closer to the Truth benchmark, with a larger advantage when load uncertainty is amplified beyond the nominal training distribution. In both experiments, EnSF reduces load-estimation error and improves the resulting control performance in the simplified transmission network.

Sparse communication in decentralized zeroth-order learning requires compatible peer-state coordinates. We characterize this one-hop condition and develop ZO-COSMO, coupling two-query estimation with average-preserving masked consensus using $q$ values per active link. Global supports serve all-neighbor mixing; matching updates require agreement only within each pair. We derive a sharp contraction-per-scalar bound within the matching class and convergence guarantees for the core and sparse-momentum updates. At fixed matching, exact moment identities characterize how shared directions preserve gradient-heterogeneity cancellation and redistribute estimation error and disagreement. Mechanism experiments cover unequal curvatures, noise, and sparse momentum. Further tests span $64$ synthetic agents and eight logical Qwen LoRA workers. At matched payload budgets, Qwen2-7B QNLI gains $3.65$ accuracy points over explicit-index Rand-$k$; edge-local updates gain $3.42$ and $2.53$ points over all-neighbor mixing on eight-worker complete and ring graphs. A matched-first-step ablation gives a $3.92$-point momentum benefit. Seed-aware and same-matching controls distinguish encoding, scheduling, and query correlation.