Papers

Density-Driven Optimal Control (D2OC) provides a principled approach to distributing multi-robot teams over non-uniform spatial distributions. Applying D2OC to nonholonomic robots, however, creates a gap between safety constraints imposed on a reference motion and the physical inputs that determine the actual robot motion. We address this issue by enforcing the safety constraint directly on the robot's physical inputs while preserving the density-driven coverage objective. The proposed framework combines D2OC with a control barrier function safety filter through a feedback-linearizing look-ahead point, allowing safety and actuator limits to be considered together during control. We further derive a safety margin that accounts for the look-ahead geometry, robot footprint, and motion during each control interval. Simulation results show that the proposed method maintains the required physical separation while achieving coverage performance comparable to a conventional reference-tracking approach, which can satisfy safety on the reference motion yet violate the corresponding physical clearance. Experiments on multiple nonholonomic robots in the Robotarium further demonstrate safe execution while driving the robots toward the desired spatial distribution. These results show that enforcing safety directly on the physical inputs can eliminate the mismatch between safety certification and physical robot motion in density-driven multi-robot coverage.

We study online convex optimization on a bounded Euclidean domain when the learner receives only one subgradient at its prediction and knows neither the horizon, a gradient bound, nor the losses' strong-convexity parameters. We present a scale-invariant algorithm that couples projected adaptive gradient descent to one cone-constrained quadratic leader through a positive wealth process. The algorithm maintains $O(d)$ state, without a learning-rate grid or restarts. For a domain of diameter at most $D$, it guarantees regret at most $\sqrt{2}D(\sum_t\|\widehat g_t\|^2)^{1/2}+2D\max_t\|\widehat g_t\|$, with no logarithmic factor. A simultaneous quadratic-distance bound yields explicit improvements for unknown strong convexity, including a refinement governed by jumps in the observed gradient scale. For varying per-round curvature, the bound depends on the total shortfall below a reference level selected in hindsight. The proof uses a clipping-compatible surrogate and a potential involving the quadratic minimum, log wealth, and two scalar curvature statistics. The resulting strongly convex bound is not uniform at the classical logarithmic rate over all curvature-to-gradient ratios; the exact parameter dependence is stated. The contribution is a single-leader construction and its analysis, within an established literature on universal and scale-free online learning.

Despite the importance for interpretability, decision trees face severe scalability challenges. Existing global optimal methods are often limited by binary feature selection and shallow tree depths, whereas traditional heuristic approaches frequently sacrifice predictive accuracy. To overcome these limitations, this paper proposes a moving-horizon approximate branch-and-reduce method to train near-optimal deep classification trees on large-scale datasets with continuous features. Built on a hierarchical root-subtree optimization framework, the method solves the root-level problem via branch-and-reduce while approximating the induced subtree problem using greedy heuristics. Although the underlying framework is capable of guaranteeing global optimality, the approximation, which functions as a lookahead rollout in a reinforcement learning context, significantly boosts efficiency for deeper structures. A low-cost moving-horizon strategy is then employed to iteratively refine model accuracy. Extensive numerical results demonstrate that our method exceeds the testing accuracy of existing heuristic baselines while offering significantly greater scalability, in terms of both dataset size and tree depth, than global optimal solvers.

This 30 years old book is posted for free download by the generosity of Oxford University Press that granted permission to allow access to future generations of young scientists. A printed copy can be purchased from the publisher. The authors received for this book "The 1999 INFORMS Computing Society Prize for Research Excellence in the Interface Between Operations Research and Computer Science" from the Institute for Operations Research and the Management Sciences (INFORMS). For a book review by Janos D. Pinter go to: Journal of Global Optimization, 16: 107-108, 2000.

We develop inference for manifold-valued population minimizers when each observation belongs to a different participant and only locally private messages reach the analyst. The method releases randomized tangent gradients and combines them through Riemannian stochastic approximation and Polyak-Ruppert averaging. Directly inserting a private data surrogate into a nonlinear loss can shift its population target, whereas conditional centring of the released gradient preserves the first-order equation. We introduce symmetric-pair regression (SPR) to estimate the asymptotic variance from the same private messages used for point estimation, without holding out participants or requesting a second release. We prove the central limit theorem and consistency of the fully transcript-based sandwich covariance and intrinsic Wald region under local differential privacy. Simulations across various statistical problems and manifolds support the predicted decrease in estimation error and near-nominal coverage under moderate privacy. An application to NHANES anthropometric data illustrates private estimation of a leading body-size direction and its uncertainty.

A joint guidance and robust observer-based output-feedback synthesis framework is developed for discrete-time nonlinear systems subject to bounded process disturbances and measurement noise. The method co-designs a reference trajectory, time-varying Luenberger observer and observer-based linear output-feedback gains, and coupled ellipsoidal funnels that certify conditioned-invariant bounds on tracking and estimation errors. Nonlinear residuals are bounded by incremental quadratic constraints (QCs) parameterized by local Lipschitz constants, estimated via sampling approaches. The coupled stability conditions are formulated as linear matrix inequality (LMI) surrogates and solved using sequential convex programming with proximal regularization (prox-SCP) via semidefinite programming (SDP). Simulations of a unicycle model under persistent sensory noise and system disturbances demonstrate robust tracking and funnel-certified constraint satisfaction with respect to the jointly designed reference trajectory.

Semidefinite programming (SDP) certificates for feedback systems containing deep neural networks (NNs) typically scale with the total number of neurons, whereas small-gain tests are scalable but can be highly conservative. This paper develops a unified and scalable framework for incremental robust stability and performance analysis of feedback interconnections involving high-dimensional NNs and unmodeled dynamics. By combining a structured decomposition of the full-order SDP condition with scalable Lipschitz constant estimation algorithms, we derive reduced verification conditions that certify incremental convergence and incremental $\ell_2$-gain bounds. The dimensions of the resulting control-analysis linear matrix inequalities (LMIs) depend only on the widths of the last two network layers and are independent of network depth. The framework preserves the coupling between the plant and the NN, with the incremental small-gain condition recovered as a special case. To further reduce conservatism, we develop a multi-round alternating update scheme that iteratively refines the coupling variables while preserving scalability. Numerical experiments show that the proposed framework achieves state-of-the-art incremental $\ell_2$-gain bounds for large-scale NN feedback systems.

This study introduces a decision-epoch-based dynamic electric autonomous dial-a-ride problem (Dyn-EADARP), in which incoming requests are collected and processed at periodic decision epochs. A key decision is not only how to serve requests, but also when to dispatch them. At each decision epoch, the service provider decides which requests to serve and which to postpone, while jointly determining vehicle routes, schedules, and charging decisions. Unexecuted parts of existing plans can be revised as new information becomes available. To solve this problem, we develop an ML–CO policy following the combinatorial optimization augmented machine learning (COAML) framework, which combines a statistical model with a combinatorial optimization layer for decision making. The statistical model predicts prizes for available requests, and a prize-collecting E-ADARP uses these prizes to jointly determine request selection, routing, scheduling, and charging. The statistical model is trained directly to improve the decisions produced by the optimization layer. Computational experiments on 320 test instances demonstrate the efficiency of ML–CO, which solves instances with nearly 500 requests in about 6 seconds on average. It achieves 8.7%–14.3% lower objective values than benchmark policies and an average gap of 4.8% to the anticipative reference. The results provide several managerial insights. First, serving requests immediately is not always best, as selectively postponing some requests can create better ride-sharing opportunities. Second, revising existing plans preserves operational flexibility and substantially improves solution quality. Finally, more frequent decision making does not necessarily improve performance, highlighting the importance of choosing an appropriate decision frequency.

We present a candidate counterexample to a conjecture stating that the existence of a Lyapunov-Krasovskii functional with a pointwise dissipation rate is sufficient for the input-to-state stability of time-delay systems. The counterexample has been derived through interactions with large language models.

Second-order methods for smooth monotone variational inequalities reach the optimal rate $O(T^{-3/2})$, but a distributed exact Jacobian costs $d$ times more communication than an operator value. We show that similarity does part of the work for free: if the server's Jacobian differs from the global one by at most $\beta$, using it gives $O(L_1D^3T^{-3/2}+\beta D^2T^{-1})$ at first-order communication cost. A quasi-Newton approximation of the residual Jacobian $\nabla F-\nabla F_1$, built from secants already communicated, improves the model but cannot remove the $T^{-1}$ term, because any uniform bound on the Jacobian error leaves it in the rate. We therefore certify the surrogate only along the candidate step: one Jacobian-vector product tests it, and a failed test is reused as an exact correction. This attains the exact rate $O(L_1D^3T^{-3/2})$ while transmitting only vectors. Experiments on LIBSVM and synthetic instances measure accuracy against communication.

Two immiscible viscous fluids in pressure-driven channel flow can be unstable through their interface when their viscosities differ, at arbitrarily small Reynolds number — the instability Yih found in 1967. The unstable state is the interface itself: a curve inside the domain, separating the two fluids, whose displacement is governed by a partial differential equation of its own and which moves the fluid domain with it. Actuation is at one wall only, so the state to be controlled is reached across a fluid. We stabilize this interface at any rate below the limit set by the streamwise mean flow, by static feedback of the two velocity components of that wall, through a Fredholm backstepping transformation acting jointly on the two layers. The target is the two-fluid Stokes problem, shifted, with the transmission of stress between the fluids kept and the coupling of the interface into them removed. Two findings about the two-fluid spectrum carry the design: joined at the interface, the eigenvalue sequences of the two layers, which collide for a dense set of layer ratios when the layers are uncoupled, avoid each other uniformly, so no degeneracy has to be assigned; and the slowest mode left to viscosity is not a bulk mode but the capillary relaxation of the interface, whose rate is linear rather than quadratic in the wavenumber, which sets the band of actuated wavenumbers. The controller is tested on the nonlinear two-fluid channel, switched on against a saturated interfacial wave.

We show that there are integers $N$ and odd $D$ such that no algorithm can decide, from the rational coefficients of a homogeneous polynomial vector field $F$ in dimension $N$ of degree $D$, whether the origin is Lyapunov stable for $\dot Y=F(Y)$. This proves a conjecture of V. I. Arnold. We provide a Lean formalization of the proof. The smallest dimension $N$ for which we were able to prove undecidability is $N = 5$. An analogous undecidability result holds for global asymptotic stability and global exponential stability for non-homogenous vector fields. For dimension $N=2$ and homogenous vector fields, we establish decidability for all commonly used stability notions, building heavily on existing results.

Spatial room impulse responses (SRIRs) capture directional distributions of acoustic sound-sources and their reflections. However, collecting SRIRs of moving sound-sources remains a challenge, requiring complex interpolations across measurements that account for multi-path spatial-temporal dynamics. This paper investigates the Wasserstein metric and displacement for evaluating interpolated SRIR echo densities in the spherical harmonic domain. We present novel sum-of-magnitude square expansions for efficiently fitting probability density functions, maximizing likelihood, inverse sampling, and computing spherical sliced Wasserstein interpolations. Experiments compare the Wasserstein displacements and metric to linear and geometric interpolations of SRIR image-source densities on a line-path, and demonstrate model-order reduction.

We introduce Multiplicatively Optimistic Regret Matching (MORM), an uncoupled learning rule for finite general-sum games. Under simultaneous full-information self-play, every player achieves external regret $O(\sqrt n\log d)$ uniformly over all horizons, using only one-step optimism. The analysis combines a potential-based regret-matching argument with multiplicative stability and Hellinger control of strategy movement. A learning-rate safeguard additionally gives $O(\sqrt{T\log d})$ regret in the face of adversarial utilities.

We study the oracle complexity of finding $\epsilon$-stationary points of smooth bilevel optimization problems with a nonconvex upper-level objective and a strongly convex lower-level problem. We consider a stochastic first-order oracle that returns unbiased stochastic gradients of both the upper- and lower-level objectives, with variance bounded by $\sigma^2$. We prove that, for any initial optimality gap $\Delta>0$ and all sufficiently small $\epsilon>0$, every adaptive randomized first-order algorithm requires $\Omega\!\left(\Delta \kappa_y^2 \epsilon^{-2}\max\{1,\sigma^2\kappa_y^6\epsilon^{-4}\}\right)$ oracle queries to find an $\epsilon$-stationary point of its hyper-objective function, where $\kappa_y$ denotes the condition number of the lower-level problem. In particular, in the noise-dominated regime, the lower bound is $\Omega\!\left(\Delta\sigma^2\kappa_y^8\epsilon^{-6}\right)$. This establishes the optimality of the $\epsilon^{-6}$ dependence achieved by the best-known first-order stochastic methods.

We study the optimization of convex objectives with $(L,\kappa-1)$-Hölder-continuous gradients in $\ell_q$ over $R B_p^d$, $1<\kappa\le 2$. (MG26) provides selectors with a movement bound for the problem of chasing high-dimensional convex nested sets for every $p<q$ and generally reduces Lipschitz convex optimization to bounds on the movement of selectors. We couple that movement with Hölder descent yielding a polynomial-runtime first-order method whose feasible output, in the high-dimensional regime $T\le d$ and for $p<\min\{q,2\}$, has error $$ \widetilde O_{\kappa,p,q}\!\left( \frac{LR^\kappa}{T^{\kappa(1+1/p-(1/q-1/2)_+)-1}} \right), $$ after $T$ queries to a first-order oracle, solving the COLT 2015 open problem of (Guz15), up to logarithmic factors. At $(p,q)=(1,2)$, the rate is $\widetilde{O}(LR^\kappa/T^{2\kappa-1})$, including $\widetilde{O}(LR^2/T^{3})$ cubic decay in the smooth case.

We consider distributed stochastic convex optimization with a fixed budget of $N$ independent samples split among $m$ workers. Sample average approximation reduces the problem to a regularized finite-sum problem whose local Hessians are statistically similar. This allows the Hessian of the local objective at the server to be used as an inexact Hessian of the global objective, while the workers communicate only gradients. We apply the optimal accelerated inexact Newton extragradient method of (Chen et al., 2026) and propose its distributed restarted variant for the strongly convex empirical problem. The method reaches the statistical accuracy of order $N^{-1/2}$ in $\widetilde O\left(\max\{N^{1/7},m^{1/4}\}\right)$ communication rounds. Hence, with $m=N^{4/7}$ workers, it requires $\widetilde O\left(N^{1/7}\right)$ rounds, improving the dependence on the total sample size from $\widetilde O\left(N^{1/6}\right)$ for the previous accelerated cubic Newton construction of (Agafonov et al., 2021). Each iteration uses two gradient aggregation rounds and does not require Hessian communication.

Quantization-aware training (QAT) updates a full-precision shadow weight $\mathbf{x}$ but deploys the quantized endpoint $Q(\mathbf{x})$. Existing explanations for QAT largely view its success through the lens of shadow weights: QAT can move $\mathbf{x}$ toward flatter basins, gain robustness from quantization-induced oscillations, or balance the shadow loss $f(\mathbf{x})$ against the quantization error $\|\mathbf{x}-Q(\mathbf{x})\|_2$. These perspectives do not directly explain the empirical observation that the deployed endpoint loss $f(Q(\mathbf{x}))$ improves while the shadow loss $f(\mathbf{x})$ does not, and can even increase substantially. In this paper, we offer a different explanation by treating QAT as finite-grid endpoint dynamics. Motivated by the approximate normality of rescaled pretrained weights, we propose an idealized model for the residual phase, which records where each shadow weight sits inside its quantization cell as a fraction of the cell width. This model leads to a crossing law that determines which coordinates cross quantization boundaries after a shadow update. Inspired by the idealized model and signal-imbalance phenomenon in QAT, we further propose QAR (Quantization with Amplified Routing), an algorithmic framework that directly operates on the quantization code. In contrast to QAT, QAR is both theoretically grounded and memory-efficient: it admits feasible-gradient bounds for a family of power amplifiers up to unavoidable finite-grid floors without retaining a full-precision shadow weight copy. Experiments on post-training of large language models provide evidence consistent with the endpoint view and show that QAR can be comparable to or better than QAT with smaller memory cost.

The multiobjective conditional gradient (MCondG) method is an effective descent-type algorithm for solving constrained multiobjective optimization problems. In this algorithm, the adaptive step size is derived by minimizing a one-dimensional convex quadratic surrogate function, which guarantees a sufficient decrease in all objective functions at each iteration. However, we observe that this surrogate function is relatively conservative, producing overly small step sizes and consequently slowing the convergence of the algorithm. This conservativeness stems from the fact that the surrogate function is dominated by the largest Lipschitz constant among the gradients of all objective functions. Motivated by this observation, we develop a tighter quadratic surrogate function that exploits the individual gradient Lipschitz constants. Based on this surrogate, we propose a new adaptive step size strategy obtained by evaluating this surrogate over a finite candidate set. We prove that the proposed step size preserves the descent property of the MCondG method, and has an improved worst-case complexity bound compared with the original adaptive step size strategy. Numerical experiments on benchmark and real-world problems demonstrate the effectiveness of the proposed strategy in terms of computational efficiency and solution quality.

In a series of three papers, we propose a distributed intraday scenario planner for energy management in 16.7-Hz single-phase railway power systems, which is realized and tested as open-source Python package bahnstrom\_ems v0.9.1~\cite{noroozi2026bahnstromems}. The first part of our work develops the mathematical foundation of a two-layer energy management system for a 16.7-Hz single-phase railway power network, where the network is an interconnection of several control areas. Given a day-ahead plan, last measurement of stored energy in wayside batteries, and forecasts of the tractive, available regenerative and renewable energy, we provide a mathematical modeling of the 16.7-Hz single-phase railway power network in intraday time-scale. Then the intraday energy planning is formulated as a risk-neutral convex stochastic optimization problem. The overall intraday planning is shown to be a sparse convex quadratic program. Hence, this problem is solved recursively by adopting a distributed scenario model predictive control (MPC) setting. The corresponding area problems, coordination updates, convergence conditions, and closed-loop key-performance-index accumulation are derived. In practice, detailed train-motion and microscopic motoring and regenerative-power profiles are usually needed for accurate simulations and numerical validation purposes. Due to data privacy and security concerns, detailed train-motion and microscopic motoring and regenerative-power profiles in a real power network may not be publicly available. Therefore, we also propose a mathematically-solid approach to estimate such microscopic information by converting causally timetable and train-motion information into quarter-hour motoring and regenerative-power profiles while exactly preserving each hourly energy target.

This paper presents the executable Python implementation~\cite{noroozi2026bahnstromems} of the two-layer energy-management system developed mathematically in Part~I~\cite{norooziP1I}. If needed, the software first converts hourly railway operation data into a 15-minute timetable and calculates the amount of gross motoring and available regenerative powers in a 15-minute resolution using the timetable handling layer. Given the energy prices are available either cached or by sending request to ENTSO-E Transparency Platform through the price handling layer, the software constructs a deterministic hourly day-ahead plan with either in the so-called must-run converter commitment mode or in a sub-hourly feasibility certificate mode, where the latter demands more computational complexity. Given the inputs from the timetable, day-ahead planning layers and initial measurements of the intraday-resolutions network's physical quantities, the intraday handling layer executes a risk-neutral intraday scenario model predictive controller by area-wise quadratic programming and consensus ADMM. The retained S1 forecast layer combines a one-week seasonal-naive point forecast with causal residual-path resampling, while the current measured stage is identical in all scenarios. We specify the day-ahead and intraday data contracts, absolute-UTC coupling, local-QP assembly, warm starts, residual tests, solver-status interpretation, physically validated recovery paths, state and key-performance-index updates, and the railway-owned generation and pumping increment. The Python package version, dependency lock, frozen-plan manifests, deterministic fixtures, run logs, and hash manifests make the software configuration executable and auditable. The paper describes software behavior and numerical contracts; comparative performance and stress-test claims are reserved for Part~III~\cite{norooziP1III}.

We study the minimisation of the first solenoidal Maxwell eigenvalue $\lambda_1^{\mathrm{Max}}(\Omega)$ in perfectly conducting cavities through the scale-invariant functional $J_{\mathrm{Per}}(\Omega):=\mathrm{Per}(\Omega)\lambda_1^{\mathrm{Max}}(\Omega)$ in $\mathbb R^3$. While the corresponding unconstrained problem is known to be ill-posed, we focus on bounded convex cavities and prove that $J_{\mathrm{Per}}$ has a positive infimum in this class. The proof relies on the study of the three main possible behaviours for sequences of convex domains with bounded perimeter: convergence to a bounded convex domain with non-empty interior, collapse to a planar domain or collapse to a one-dimensional domain. The case of a planar collapse is handled through projection arguments, yielding a lower bound, while that of a one-dimensional collapse is controlled by a different argument, showing that in that case $J_{\mathrm{Per}}$ must diverge to $+\infty$. Although the question of existence of an optimal shape (that is, ruling out planar collapse) remains open, we derive several negative results and obstructions to existence; typically, no minimiser with a $\mathscr C^{3,+}$ boundary exists. This relies on a fine analysis of first-order optimality conditions.

We develop single-loop stochastic projected damped extragradient methods for nonconvex–(strongly) concave minimax optimization, with complexity guarantees for both game stationarity (GS) and optimization stationarity (OS). Our approach combines a stochastic projected damped extragradient (SPDE) method with a recursive variance-reduced variant, VR-SPDE, both of which retain a single-loop structure. Under an unbiased stochastic gradient oracle with uniformly bounded variance, SPDE finds an $\varepsilon$-game-stationary point with stochastic first-order oracle (SFO) complexities of $O(\kappa\varepsilon^{-4})$ and $O(\varepsilon^{-5})$ in the nonconvex–strongly concave and nonconvex–concave settings, respectively, where $\kappa=L/\mu$. Under an additional mean-square Lipschitz condition on the stochastic gradients, VR-SPDE improves these GS complexities to $O(\kappa^{3/2}\varepsilon^{-3})$ and $O(\varepsilon^{-9/2})$, respectively. For an $\varepsilon$-optimization-stationary point, SPDE achieves SFO complexities of $O(\kappa\varepsilon^{-4})$ and $O(\varepsilon^{-6})$, while VR-SPDE achieves $O(\kappa^{3/2}\varepsilon^{-3})$ and $O(\varepsilon^{-6})$, in the two settings, respectively. These OS guarantees match the best-known bounds achieved by multi-loop methods while preserving a single-loop implementation. To the best of our knowledge, our results provide the best-known SFO complexity guarantees among single-loop stochastic first-order methods for the respective stationarity criteria and problem classes.

Fully-actuated multirotor aerial vehicles must not only track nominal wrenches but retain the "readiness" to modulate them rapidly under disturbances. Classical effort-minimizing allocators ignore this dynamic limit, whereas maximizing readiness leads to topologically disconnected optimal sheets demanding physically impossible actuator rates. Enforcing a readiness safety floor on fixed-geometry symmetric platforms further encounters a zero-sum degeneracy: motor-speed redistribution cannot improve authority without conceding wrench tracking. This paper uses active morphology to break the degeneracy, treating servo tilt as a geometric resource supplying authority-recovery directions unavailable to static rotors. We construct a configuration-dependent, motor-only readiness certificate - the log-volume of the reachable wrench-rate set - that explicitly excludes servo capacity, preventing a "ghost capacity fallacy" in which the certificate would falsely credit slow mechanical kinematic limits instead of collapsing accurately at motor saturation. The certificate is enforced as a Control Barrier Function (CBF) within a Unified Physical-Command Quadratic Program acting on motor torques and servo setpoints. Closed-loop simulations of an articulated octorotor under severe gust disturbances show classical allocators diverging and uncertified articulated allocators violating the safety floor, while the proposed CBF filter bounds the system state and preserves vehicle authority.

Residential heating accounts for a large share of building energy use, and predictive control strategies can reduce it by anticipating rather than reacting to the room temperature alone. The usual manner of implementing predictive control, model predictive control (MPC), requires an accurate model of each individual residential unit for heating control. Obtaining this model is time-consuming to obtain manually or even impossible, and such models do not transfer across a heterogeneous building stock. Data-enabled predictive control (DeePC) removes this modeling step by designing the controller directly using measured trajectories of the system. The foundations of DeePC, however, are built on the class of deterministic linear time-invariant systems, which is an unrealistic assumption for residential heating. This work applies three recently proposed nonlinear extensions of DeePC, collectively referred to as data-driven predictive control (DPC): Select-DPC, gain-scheduling DPC, and linear parameter-varying DPC. These are compared against standard DeePC and the hysteresis controller, the industry standard in residential heating. The comparison runs over a full heating season on a calibrated digital twin of an occupied research unit, the NEST research building in Switzerland, assessed by heating energy and comfort-band violation. Moreover, gain-scheduling DPC was deployed on the real residential apartment and validated the conclusions from the simulations. Each of the DPC controllers consumes significantly less energy than the hysteresis controller, amounting to roughly $11\%$ over the season, validating the use of DPC. The nonlinear methods further outperform linear DeePC in all assessed scenarios, and we analyze the relative costs and benefits of each nonlinear method.

We study the problem of tiling an $N\times N$ square with axis-aligned squares whose side lengths are integers strictly less than $N$, with the objective of minimizing the number of tiles. The standard placement-based exact-cover MILP contains $\Theta(N^5)$ placement-to-cell nonzero coefficients and is also affected by the dihedral symmetry of the square domain. We introduce a Corner-Difference MILP (CD-MILP) that represents each selected square by at most four signed corner coefficients. Two-dimensional prefix reconstruction shows that the resulting constraints are equivalent to the original cell-cover equations, while reducing the coverage-related nonzero count to $\Theta(N^3)$. We also introduce lightweight corner-ordering constraints that retain at least one representative from every $D_4$ orbit, although ties may leave residual symmetry. Computational experiments on nine prime-size instances compare the baseline, CD-MILP, the baseline with corner ordering, and their combination using Gurobi and COPT. Under the stated experimental settings, the combined formulation solves eight instances with Gurobi and seven with COPT, compared with four and five, respectively, for the baseline. Its average presolved matrixs contain approximately $10^5$ nonzero coefficients on average, compared with approximately $10^7$ for the baseline.

In this paper, we study a stochastic optimal control problem for forward-backward stochastic control systems with quadratic convex generator and unbounded terminal condition, where the control domain is not necessarily convex. Since the existing bounded mean oscillation (BMO) martingale approach used in the setting of quadratic BSDEs with bounded terminal conditions is not directly applicable to the present setting, we introduce a new probability measure, under which all subsequent analysis is then carried out. Finally, a global stochastic maximum principle is obtained.

We consider moving-average iteration for fixed complex matrices and place the theory in the general context of Krasnoselskii and Mann iteration. Necessary and sufficient spectral conditions for the asymptotic stability of moving-average iteration are presented in dependence on the fixed window width. For the corresponding region of stability in the complex plane explicit formulas are derived and it is shown that the region is open, convex and semi-algebraic. The regions are shown to be strictly increasing in the window length and the limiting stability region is determined as the window size tends to infinity.

We study the optimization landscape for infinite-horizon discounted Markov decision processes (MDPs) with general state and action spaces under structured stationary policy classes. A general weighted policy-iteration approach to establishing global convergence guarantees for policy gradient methods requires closure under weighted policy improvement at every policy, a property that may fail even when the policy class contains an optimal policy. To address this issue, we propose weaker conditions that guarantee the absence of suboptimal stationary points and establish the Polyak–Lojasiewicz–Kurdyka (PLK) condition for the policy gradient objective with a finite concentrability coefficient. We also establish the PLK condition from a policy-improvement bound that holds at every state, without a concentrability assumption. Our general results encompass settings covered by the earlier framework when the common standing assumptions hold for the same policy class and parameter domain. We further verify our proposed conditions for two operations models: inventory systems with Markov-modulated demand and stochastic cash-balance problems. For both models, the Bellman equation yields approximate convexity of the Q-value functions in the action variable, with deviations controlled by the first-order stationarity measure. These estimates establish exponent-one and, under additional curvature assumptions, exponent-two PLK conditions, which, together with Lipschitz continuity of the policy gradient, imply an $\mathcal{O}(1/\epsilon)$ iteration complexity and linear convergence, respectively, for projected gradient descent using exact policy gradients. To the best of our knowledge, we provide the first non-asymptotic convergence rates for solving infinite-horizon discounted inventory systems with Markov-modulated demand and stochastic cash-balance problems using policy gradient methods.

Quantile-based reinforcement learning provides an interpretable approach to risk-sensitive decision-making by optimizing a prescribed quantile of the cumulative-return distribution. Despite this appeal, learning under a point quantile objective is challenging: quantiles can change abruptly under small perturbations of the return distribution, and exact quantile-sensitive planning requires computationally demanding distributional optimization. Lower-buffered quantiles alleviate the former difficulty by averaging neighboring quantiles immediately below the target level, providing a smoother surrogate while preserving the underlying point-quantile objective. Existing methods based on this principle, however, remain model-based and rely on explicit return-law planning, limiting their applicability beyond small tabular problems. We develop Deep-BQRL, a model-free distributional reinforcement-learning framework that extends buffered-quantile learning to neural function approximation. The method learns conditional return quantiles directly from sampled transitions, constructs buffered action scores from the relevant region of the learned quantile function, and uses ensemble disagreement to guide exploration. An augmented input representation allows the learned policy to respond to trajectory information without explicitly reproducing the quantile-state recursion required by exact planning. Experiments on an asset-selling optimal-stopping problem and slippery FrozenLake compare Deep-BQRL with model-based UCB-BQRL and tabular PPO and TRPO implementations. In asset selling, Deep-BQRL attains smaller mean cumulative point-quantile policy gaps than PPO and TRPO at the reported target levels, while UCB-BQRL retains the smallest gaps. The learned stopping decisions also vary with the target quantile, providing an interpretable illustration of the method's risk-sensitive behavior.