Papers

Rockafellar showed that every cyclically monotone operator is contained in the subdifferential of a convex function, and characterized the maximal ones. We establish the quasi-convex analogue of this theorem, with no assumption on the operator: in any separable Banach space, a setting we show to be sharp, every cyclically quasi-monotone operator is contained in the normal cone operator of a quasi-convex function. In finite dimension, we characterize maximal cyclically quasi-monotone operators and show that they are maximal quasi-monotone. Unlike in the convex case, the potential is not unique, and we build it from a preference relation obtained through successive preorders. As applications, we extend Afriat's theorem on revealed preferences in economics to infinite data sets, and obtain Kantorovich-type potentials for a supremal optimal transport problem.

Many scientific and engineering applications require the global optimization of expensive black-box functions. In high-dimensional settings, however, achieving computational scalability is a major challenge. To address this challenge, we propose a distributed spatial branch-and-bound framework that exploits partially separable structure for the global optimization of box-constrained black-box problems. The proposed framework adopts a two-layer structure in which branching is performed only on a lower-dimensional set of coupling variables in an outer layer. Within each outer node, the remaining non-overlapping (private) variables, which appear independently across blocks, have their domains partitioned and explored in parallel within an inner layer, resulting in a naturally parallelizable framework that is well suited for implementation on multi-core CPU architectures. At each node, valid upper bounds are obtained via sampling, with optional refinement using a local solver, while lower bounds are computed using data-driven convex underestimators constructed over partitions of the private variables. A residual-based correction and finite-sample validity analysis provide conditions under which the partitionwise underestimators are valid lower bounds, while the proposed branching and refinement mechanism establishes convergence to an $\epsilon$-global optimum. Numerical studies on benchmark and ODE-constrained parameter-estimation problems demonstrate improvements in both sampling efficiency and computational performance relative to monolithic approaches.

While sensitivity analysis for linear programs with parametric right- hand side vectors is well established, cases involving parametric constraint matrices remain less explored. To address this gap, we investigate linear programs where the constraint matrix depends linearly on a scalar parameter. Our goal is to compute the minimum and maximum optimal values of these parametric linear programs within a fixed parameter range. This corresponds to solving two bilevel programs. We show that both decision versions of these optimization problems are NP-complete by proving both NP-hardness and membership in NP. We additionally identify three sufficient conditions on the parametric matrix under which the bilevel programs are no longer NP-hard. We introduce a reformulation for both problems, together with a Farkas-based certificate for the lower-level feasibility condition, and compare the execution time for solving them on standard benchmark instances. The experiments highlight a practical dichotomy. Although both bilevel optimizations are NP-hard, computing the parameter value corresponding to the minimum of the linear program is fast, whereas computing the value corresponding to the maximum is not. We relate this asymmetry to the parametric equality constraints, whose free dual variables leave the relaxation of the maximum reformulation unbounded.

We develop a dual-inertial framework for convex optimization that introduces inertia in dual coordinates while retaining the Bregman proximal-type oracles. For unrestarted iterations, uniformly bounded positive directional variation gives a last-iterate $\cO((1+\log k)/k^2)$ rate, improving to $\cO(k^{-2})$ under additional weighted summability. A computable safeguarded algorithm gives a global $\cO(N^{-1})$ objective-gap bound for the best accepted point, where $N$ counts all proximal oracle calls. Under local strong-convexity and kernel regularity conditions, all three safeguarded methods restart finitely and attain a final-epoch last-iterate $\cO(N^{-2})$ rate. Numerical experiments compare objective accuracy and computational cost on entropy regression and nonnegative least squares. Comparative numerical experiments illustrate the performance advantages of the proposed methods.

For the fundamental problem of prediction from expert advice, Freund asked whether one could compete with a comparator that switches $k$ times among only $m$ experts. For a discrete set of experts, this problem was resolved by Bousquet and Warmuth (2020), who gave an efficient algorithm with optimal regret matching the information-theoretic limit. We study the analogous problem in online convex optimization, where the action space is continuous. An inefficient algorithm can be derived by discretizing the action space and applying the Bousquet–Warmuth algorithm, giving regret of $\widetilde{O}(md+k)$, while the best known efficient algorithm for this problem has regret $\widetilde{O}(dk)$. We show that this computational inefficiency is inherent, establishing a separation between the information-theoretic and polynomial-time regret achievable for this problem. Specifically, under the hypergraphic planted-clique conjecture, we show that for every polynomial-time algorithm there exist instances for which it incurs at least $(k\sqrt d)^{1-o(1)}$ regret.

Momentum is a classical tool for accelerating convex optimization and a standard component of modern large language model training. Yet recent empirical work has questioned the usefulness of first-moment momentum in reinforcement learning with verifiable rewards (RLVR). We study this question through a bandit formulation of RLVR with bias-corrected exponential moving average (EMA) momentum, where the EMA coefficient $\beta$ controls how quickly the weights on past gradients decay. When policy gradients are evaluated exactly, we prove that increasing $\beta$ within a range that prevents overshoot strictly accelerates optimization. When gradients are estimated from sampled responses, a positive expected gain can be obscured by sampling fluctuations. We show that as the learning rate $\eta$ decreases, the expected gain with fixed $\beta$ vanishes faster than these fluctuations, whereas increasing $\beta$ to keep $\eta/(1-\beta)$ within the safe regime preserves a nonvanishing gain that eventually dominates them. Bandit simulations quantitatively validate these predictions. RLVR experiments on Qwen3-4B-Base using SGD with EMA and AdamW support our principle of choosing larger coefficients in practical LLM post-training. On Qwen2.5-32B with AdamW, simply increasing $\beta_1$ improves mean validation accuracy by nearly $2$ percentage points over the conventional choice, demonstrating the principle's effectiveness at a larger model scale.

Partially optimal assignments (also called persistencies) allow combinatorial optimization problems to be reduced by fixing a subset of variables to values consistent with at least one optimal solution. For the quadratic unconstrained binary optimization (QUBO) problem, such assignments can be computed efficiently via autarkies of quadratic posiforms. However, the largest autarky domain and thus the number of fixed variables can depend on the specific posiform representation. We show that every quadratic posiform with the greatest possible constant term (the roof-dual bound) yields the largest autarky domain obtainable from any quadratic posiform. By computing such a roof-dual-optimal quadratic posiform and a corresponding autarky with largest domain in polynomial time, one can therefore attain the maximum partial optimality obtainable by quadratic posiform autarkies in polynomial time. This justifies the computation of roof-dual-optimal quadratic posiforms and corresponding autarkies in popular polynomial-time algorithms and establishes that finding additional partial optimality in QUBO requires going beyond autarkies of quadratic posiforms.

We study nonconvex-strongly-concave minimax problem $\min_{\mathbf{x}\in\mathbb{R}^{d_x}}\max_{\mathbf{y}\in\mathbb{R}^{d_y}}f(\mathbf{x},\mathbf{y})$ in which the objective has $\rho$-Lipschitz Hessian and condition number $\kappa$. We propose envelope cubic regularized Newton method (ECRN) which applies a refined inexact cubic regularized Newton method to the envelope of the primal function $\Phi(\cdot) : = \max_{\mathbf{y}\in\mathbb{R}^{d_y}}f(\cdot,\mathbf{y})$. We show that ECRN finds approximate second-order stationary points of $\Phi(\cdot)$ with $\widetilde{\mathcal{O}}(\sqrt\rho\,\kappa^{3/4}\varepsilon^{-3/2})$ second-order oracles, improving the condition number dependence $\mathcal{O}(\kappa^{2})$ achieved by the best known second-order methods. Our improvement comes from the key observation that the Hessian Lipschitz parameter of the envelope is $\mathcal{O}(\rho\kappa^{3/2})$, which is sharper than the order of $\mathcal{O}(\rho\kappa^{3})$ for the function $\Phi(\cdot)$. We also conduct a lower bound on second-order methods to show that our complexity is optimal up to a logarithmic factor.

Pseudo-Boolean optimization – minimizing a polynomial over $\{\pm 1\}^n$ – includes MAX-CUT, binary quadratic optimization, and Ising models. The moment-sum-of-squares (Moment-SOS) hierarchy provides certified semidefinite relaxations that are often empirically exact at low order. We ask: once a relaxation is exact, how difficult is the resulting semidefinite program (SDP) to solve with first-order methods? Exactness does not imply regularity. At every order $d \ge 2$, a unique rank-one moment optimum is primal degenerate; moreover, if one optimal SOS certificate is strictly complementary, then the certificate set is positive-dimensional and contains singular boundary points. Thus first-order methods may converge to singular endpoints, around which pointwise analyses that assume regularity at the limit cannot establish a local linear convergence rate. We show that, for exact Boolean Moment-SOS relaxations in their canonical quotient form, this degeneracy is nevertheless benign. We develop an abstract mechanism, which we call residual-field rigidity, that propagates regularity from the relative interior of the optimal set to its singular boundary. For Boolean Moment-SOS, unique rank-one exactness and the existence of one strictly complementary certificate suffice to trigger this mechanism, yielding a uniform error bound around the entire optimal set. Consequently, exact ADMM, exact sGS-ADMM, and fixed-step PDHG converge eventually linearly from every initialization, even when their limiting certificate is singular. Finally, outside a lower-dimensional semialgebraic exceptional set of Boolean objectives, these hypotheses hold simultaneously at every uniquely rank-one exact order.

Tracking changing comparators is a central problem in online learning. For exp-concave losses, previous work established the $T^{1/3}P_T^{2/3}$ dependence on the horizon $T$ and Euclidean path length $P_T$ for fixed dimension, while the optimal cost of dimension remained unresolved. We resolve this question by establishing a sharp $d^{1/3}$ curved dimension factor, improving the $d^{2/3}$ factor of recent switching and interval reductions [1, 2]. To achieve this improvement, we develop a proper, full-information learner on compact convex subsets of $\mathbb{R}^d$ that adapts to the unknown path length and, under fixed dimension-independent diameter, gradient, and curvature parameters, guarantees $$\operatorname{Reg}_T(u_{1:T}) \le \widetilde{O}\left(d+\min\left\{d^{1/3}T^{1/3}P_T^{2/3},\sqrt{TP_T},T\right\}\right).$$ A matching lower bound on a diameter-one Euclidean ball establishes the minimax frontier up to logarithmic factors when $T\gtrsim d^2$, including the transition from the curved rate to larger path budgets. The lower bound already holds for losses that depend on only one coordinate per round, isolating an unavoidable cost of learning across directions. Our upper bound follows from square-root Gaussian KL control of comparator movement and a cancellation argument that pays for history restarts. The curved learner maintains $O(\log^2 T)$ Gaussian states, uses one gradient per round, and runs in time near-linear in $T$ under a Mahalanobis-projection oracle.

We study continuous-time q-learning for entropy-regularized diffusion control without assuming classical differentiability of the value function. Rewards and admissible policies need only be Borel measurable, which allows discontinuous rewards. The diffusion coefficient $\sigma$ is action independent and locally uniformly elliptic, and the drift is $b_0 + \sigma\psi$, with $b_0$ action independent and $\psi$ bounded. In this setting the q-function involves the value only through its gradient. For each admissible policy, we represent its value by a backward stochastic differential equation (BSDE) and prove that this value is continuous before the horizon, although the data are only measurable. Replacing the gradient with a measurable representation of the martingale integrand, we define the q-function up to Lebesgue-null sets and recover the classical definition when the value function is smooth. We then prove policy improvement, when the Gibbs update is admissible, and martingale characterizations of the value and the q-function from on-policy and off-policy observations. Here the candidate value need only be continuous before the horizon, and actions are sampled on a deterministic finite grid and held constant between sampling times. Under an integrability condition on the reward, we show that a second BSDE gives an upper bound for the optimal value, attained when its Gibbs policy is admissible, and we characterize its solution in the same way.

We give a randomized polynomial-time $0.48$-approximation for maximizing a nonnegative non-monotone $k$-submodular function under a matroid constraint, for every $k\ge2$. The algorithm combines a top-two label rule with a nested-chain potential and maximum-weight matroid bases. Its analysis reduces the approximation guarantee to a one-dimensional time allocation: a uniform local lower bound leads to a linear differential equation, concavity closes the remaining interval, and two elementary budgets complete the proof. A strict saving in the time budget absorbs sampling and discretization errors, yielding the exact $0.48$ guarantee in the value-oracle model.

Impulsive control, in the context of continuous-time Markov decision processes (CTMDPs), refers to actions that instantaneously update the state of the process, leading to theoretical complications. Previous work has overcome these complications, but at the expense of approachability and ease of implementation: those models are mathematically complex and incompatible with pre-existing solution methods. This paper presents an alternative formulation of impulsive control for CTMDPs that is easier to employ in the modelling of real-world problems and can be solved by pre-existing solution methods. This formulation is then extended to allow for multiple objectives and decomposability. A decomposition-based heuristic method is developed that leverages Dantzig-Wolfe decomposition to find approximate Pareto fronts for large multi-objective decomposable impulsive CTMDPs. The modelling framework and solution methodology are applied to a dynamic maintenance problem to demonstrate their computational efficiency, solution space coverage, and solution quality.

This work focuses on planning daily routes for a group of employees of a condominium company. Given a set of employees and a set of customers with pre-scheduled visits, we want to build doubly-open routes to guarantee visits to all customers, ensuring that each customer is visited by an employee with the appropriate skills to carry out the required activity. The travel times between customers and the service time in each customer are known. The problem includes time windows for visiting customers, each employee has a maximum working time, and the daily working period is between 9 AM and 6:30 PM. The routes respect the lunch break period if they extend from morning to afternoon, which means that lunch break can be avoided if the route is fully completed in the morning or in the afternoon. The aim is to minimize the distance covered by all routes with a penalty for including lunch breaks. The distances and times between customers are collected via an API to Microsoft Azure; and the solutions are then displayed on Bing Maps. The results obtained show the operational advantages of an efficient and effective vehicle routing planning in a specific condominium management company. The model can be extended to other service provider companies.

Contact-rich robotic insertion is challenging because uncertainty in alignment, friction, and contact conditions can lead to unsuccessful assembly, excessive contact forces, and jamming. Designing a reliable controller therefore requires balancing insertion performance, robustness, and recovery from unexpected contact. In this paper, we propose PSO+TR, a derivative-free optimization (DFO) method that combines the exploratory search of particle swarm optimization (PSO) with a trust-region (TR) procedure for local refinement to improve the search process when progress stagnates. We apply the method to derivative-free policy optimization of an interpretable robotic insertion controller, jointly tuning its motion, feedback, and recovery parameters while preserving its predefined control structure. The proposed approach is evaluated on 93 eligible benchmark problems from the bound-constrained programs (BCP) collection and a synthetic contact-rich planar insertion task under uncertain operating conditions. Three random seeds per benchmark problem yield 279 matched problem–seed instances per method. PSO+TR reaches the prescribed target on 228 of these instances (81.72%), the highest observed target-attainment rate among the five compared methods, which also include PSO, randomized black-box optimization (VRBBO), matrix adaptation evolution strategy (MAES), and differential evolution (DE). Across eight robotic optimizer seeds, the deployed controller-selection pipeline achieves 97.78% completion under development conditions and 58.00% under withheld stress conditions, compared with 91.22% and 31.12%, respectively, for the fixed staged controller. The stress violation rate is reduced from 20.69% to 2.19%. These findings support combining global and local search for DFO and structured contact-rich controller tuning. The synthetic planar results require validation on physical robots.

This paper studies popularity dynamics in social networks, where users' engagement with competing channels is driven by intrinsic content quality, social influence, and popularity feedback. We introduce a stochastic model in which users interact through a directed influence network and channel exposure depends jointly on these three mechanisms. We first characterize the resulting dynamics in the absence of intrinsic quality, showing that popularity concentrates asymptotically on a single channel, leading to a winner-takes-all regime. When all channels have positive intrinsic quality, we establish ergodicity of the popularity process and characterize its long-run behavior. We further show that, when intrinsic quality is small, the system exhibits metastable behavior: it can remain for long periods near configurations dominated by a single channel before transitioning to another. These results highlight how weak quality differences can coexist with strong path dependence induced by popularity feedback. We then investigate platform-level interventions and propose controls that compensate for network-induced exposure biases and mitigate excessive concentration. Analytical results are illustrated through numerical simulations and an empirical analysis of YouTube data, whose observed popularity dynamics are consistent with the proposed stochastic framework. The results provide a mathematical framework for understanding the interplay between quality, social influence, and reinforcement in online popularity dynamics.

The spherical code problem asks how to place $N$ points on a unit sphere in $D$ dimensions to maximize the smallest distance between two points. It is a non-smooth and non-convex max-min problem, where few problem sizes have known optimal solutions, some have constructions, but most still only have numerically obtained solutions. We use a simple solver, with seven parameters, that anneals a LogSumExp surrogate, minimizes with L-BFGS, applies a random perturbation between anneals, and repeats with no acceptance test until a fixed limit is reached. We search the solver parameters for every instance with $N \le 1024$, $3 \le D \le 32$, and $N > 2D$, a total of 29,670 instances, using 32 parameter trials. We then run the solver for 60 seconds with the best parameters for each instance, without utilizing any symmetries or existing solutions. After enforcing monotonicity of the results and numerically polishing the solutions, we improve the best known code in 4,577 instances. When a single set of solver parameters is used for all instances, with one 60-second episode, the number of improvements is still 3,649. From our records, we find 24 codes with an exact algebraic cosine. Among our ablations, we find that 1) the repeated anneal contributes most to improvements, 2) at our $32 \times 60$-second budget only fitting one parameter per-instance is better than fitting all seven, and 3) good parameters can be predicted from the instance size. We provide the solver code, the fitted parameters, and the codes.

We present a novel class of minimum energy estimation problems for positive linear systems with linear costs subject to coupled process and measurement disturbances, where the latter is bounded by the state and the former is bounded by the noise. Through dynamic programming we show that the problem admits explicit solutions. Remarkably, the Luenberger observer structure of the optimal estimator emerges from the optimization itself, with a switching gain that inherits the sparsity of the disturbance constraints, and it is obtained from a differential equation in the finite horizon case. In the infinite horizon case, an algebraic equation characterizes the limit of the finite-horizon solution and yields a stationary estimator with a static gain. We introduce two distinct notions of detectability to characterize when the static gain stabilizes the error dynamics while preserving their positivity, and analyze under which conditions the stationary solution can be formulated as a linear program. We further establish that a disturbance perturbation of finite L1-norm yields an estimation error of finite L1-norm. The framework is illustrated on a large-scale irrigation network.

Nonconvex optimization on Stiefel manifolds underlies many problems in machine learning and deep learning. However, existing stochastic Riemannian retraction methods often rely on stepsizes that require prior knowledge of the gradient Lipschitz constant, a prescribed iteration horizon, or restrictive batch-size choices. We develop RGR, a recursive gradient retraction method for stochastic nonconvex optimization on Stiefel manifolds. Instead of recursively updating gradient estimators in changing tangent spaces, RGR constructs them in the Euclidean space and subsequently maps them to tangent space, thereby avoiding vector transport. The method comprises three variants: RGR-SPIDER for finite-sum optimization and RGR-SRM and RGR-EMA for online stochastic optimization. RGR-SPIDER is Lipschitz-free and horizon-free, as its adaptive stepsize requires neither prior knowledge of the gradient Lipschitz constant nor a prescribed iteration horizon. RGR-SRM and RGR-EMA are mini-batch and horizon-free methods whose convergence guarantees do not rely on prescribed large batch sizes and whose parameters can be chosen without prior knowledge of the total iteration budget. We show that RGR-SPIDER reaches an $\varepsilon$-critical point using $\mathcal{O}(\varepsilon^{-2})$ iterations and $\mathcal{O}(N+\sqrt{N}\varepsilon^{-2})$ component-gradient evaluations, where $N$ denotes the number of component functions. For the online setting, RGR-SRM and RGR-EMA achieve iteration complexities of $\tilde{\mathcal{O}}(\varepsilon^{-3})$ and $\tilde{\mathcal{O}}(\varepsilon^{-4})$, respectively. Experiments on principal component analysis, low-rank matrix completion, and orthogonal convolutional neural networks demonstrate faster empirical convergence than methods that construct recursive gradient estimators directly in Riemannian tangent spaces.

We study continuous-time primal-dual dynamics for tracking solutions of time-dependent constrained optimization problems. First, for uniformly strongly convex-strongly concave saddle functions, we establish explicit finite-time tracking and additive-disturbance bounds. For convex problems, we then introduce a directional primal regularization, combined with dual Tikhonov regularization, and quantify the resulting optimization bias and constraint violation. For smooth nonconvex problems, we couple a proximal local convexification with a two-time-scale primal-dual flow and obtain a local tracking result under constraint qualification, second-order sufficiency, and temporal regularity. Numerical traffic-signal experiments at a single intersection evaluate the proposed method under vehicle-throughput and pedestrian-aware objectives, comparing it with standard signal-control schemes.

We study the optimisation landscape of the maximum mean discrepancy (MMD) over weighted atomic measures. For the one-dimensional Gaussian kernel, we show that, for any compactly supported target distribution, sufficiently many particles ensure that every local minimum has arbitrarily small MMD. Moreover, any stationary point whose MMD remains bounded away from zero is unstable under an unweighted gradient flow over the particle locations. These results hold both when the weights are constrained to the simplex, and when they are unconstrained but optimised at each step. When the target is itself finitely supported, we further strengthen this result: beyond an explicit particle threshold, there are no spurious local minima, and every spurious stationary point is unstable. To our knowledge, these are the first finite-particle overparameterisation guarantees of their kind for weighted MMD optimisation.

We study the data-driven approximation of control-affine systems, where the drift and each control vector field exhibit a low-dimensional structure. While the ANOVA (ANalysis Of VAriance) decomposition is tailored to low-dimensional structures, a straightforward implementation does not exploit control affinity, rendering the approximation computationally expensive. We propose a coupled least squares approach for given state-control samples, whose ansatz is affine linear in the control by construction and can be realized by a preprocessing of the ANOVA index sets. Based on a Marcinkiewicz–Zygmund inequality for the resulting ansatz space, we prove a probabilistic $L_2$-error bound with explicitly given constants. Separate error control on drift and input vector fields renders the approach flexible to, e.\,g., parameter changes. Finally, we numerically illustrate the identification of the coupling structure via global sensitivity indices on discretizations of a nonlinear advection-diffusion equation, the shallow water equations and the viscous Burgers equation.

Policy mirror descent (PMD) refers to a class of policy optimization methods associated with different mirror maps. For a broad class of decomposable Legendre mirror maps encompassing many standard choices, Li and Wei [2026] show that the PMD policy sequence converges to an optimal policy from any admissible initialization under any positive constant step size, even when the optimal policy is not unique. In this note, we provide a counterexample with assistance from OpenAI's GPT-5.6 model, showing that PMD with a non-decomposable Legendre mirror map need not converge in policy.

GPU-accelerated first-order methods are increasingly effective for solving large-scale linear programs (LPs), yet LP preprocessing remains predominantly CPU-based and can become costly. A key challenge is that dependencies among presolve reductions and frequent modifications to sparse LP data make efficient GPU parallelization difficult. To address this challenge, we develop GPU-Preprocessor, a GPU-native LP preprocessing framework that integrates folding and presolve. To efficiently implement folding on GPUs, we adapt a randomized partition refinement algorithm to LP folding, using sparse matrix-vector products and parallel sorting to construct stable row and column partitions. For presolve, we introduce a new piecewise-linear reformulation rule that exploits underlying convex piecewise-linear structure to obtain sparser LP representations. We also develop a framework that combines phase scheduling with a Scan-Plan-Apply design to coordinate presolve rules and reduce sparse-matrix reconstruction overhead through batched updates. Numerical experiments on three LP benchmark sets show that GPU-Preprocessor achieves greater average problem-size reductions than PSLP and Gurobi Presolve, with SGM10 preprocessing-time speedups of up to $43.9\times$ and $677.7\times$, respectively. Here, SGM10 denotes the shifted geometric mean with a 10-second shift. On the largest tested quadratic assignment problem LP relaxation, these speedups reach $126.3\times$ and $2{,}207.9\times$. When integrated with the GPU-accelerated LP solver HPR-LP, GPU-Preprocessor achieves SGM10 speedups of $2.5\times$ to $5.0\times$ in combined preprocessing and solving time over HPR-LP without preprocessing. The code is available at https://github.com/PolyU-IOR/GPU-Preprocessor.

This study develops and validates an adaptive parametric method for sizing safety stock when forecast errors are autocorrelated, heteroscedastic, asymmetric, and heavy-tailed. Using four years of weekly forecasts from a fast-moving consumer goods company, the analysis identifies substantial positive autocorrelation, departures from normality, systematic forecast optimism, and heterogeneous relationships between forecast volume and error variance. The proposed method integrates several corrections into a modular, sequential safety-stock architecture. It constructs cumulative forecast errors over the operational action period, selects an SKU-specific parametric distribution, and determines a baseline protection point using an expected-shortage condition consistent with the target beta fill rate. A forecast-scaled module adjusts dispersion when error variance depends on forecast volume, while a Cycle-End Exposure safeguard limits within-cycle shortage risk when large replenishment lots increase exposure. Performance is evaluated through discrete-event simulation against the company's current method. The proposed method reduces average inventory by 9.9% and maximum inventory by 19.5%, with no statistically significant deterioration in beta fill rate or average stockout duration. The proportion of weeks without stockouts declines slightly, without evidence of more persistent or operationally severe shortages. The heteroscedasticity module provides selective benefits, with greater service improvements as variance dependence intensifies. The resulting framework is interpretable, adapts to SKU-specific statistical profiles, links distributional features to safety-stock outcomes, and can be integrated into existing ERP environments.

Repeated high-fidelity state and gradient evaluations make large-scale PDE-constrained optimization computationally expensive. Reduced-order modeling provides a natural way to reduce the cost of these repeated high-dimensional computations. However, existing reduced-order modeling approaches often rely on converged high-fidelity states for model construction or updating, while their reduced spaces may lose accuracy as the design moves beyond the region represented by the available data. To address these limitations, we present Reduced-order Trajectory-based Optimization (RETO), an online framework that constructs local reduced-order models (ROMs) directly from unconverged primal solver trajectories. Each local ROM is built from snapshots collected during a finite number of high-fidelity primal iterations, without requiring full-order convergence, and supports multiple low-cost state evaluations, adjoint-based gradient evaluations, and design updates. New solver trajectories are generated at updated design points to reconstruct the local ROM and continue the optimization. Across three distinct PDE-constrained optimization problems, RETO achieves solution quality comparable to that of full-order optimization, with speedups of approximately 8 to 22 times in cases without discontinuities in the state fields. Even in the transonic inverse-design case involving moving shocks, RETO retains an approximately threefold speedup. The transonic results further show that the online local ROM progressively captures evolving shock-related flow features as new primal solver trajectories are incorporated. These results demonstrate the value of unconverged primal solver trajectories as an online information source for reduced-order optimization, allowing the reduced basis to evolve with the design rather than requiring a basis that covers the entire design space in advance.

We study free-material optimization for nonlinear bond-based nonlocal diffusion and its localization as the interaction horizon $\delta$ vanishes. Bond conductivities are subject to pointwise bounds and a global material budget constraint. After normalization, the asymptotic behavior of these problems is governed by the fraction $\theta_\delta$ of potentially active bonds that can be assigned the maximal conductivity. We identify three distinct limiting regimes. If $\theta_\delta\to0$, the normalized optimal values converge to those of the classical vanishing-material optimal design problem, equivalently formulated as a Lipschitz-constrained state problem or a measure-valued flux problem. For $\theta_\delta\to\theta\in(0,1)$, we obtain a directional local free-material problem governed by the upper tail of the directional $p$-strain over position-direction space. Thus the state variable localizes, while the limiting constitutive law retains an internal directional variable inherited from the bonds. If $\theta_\delta\to\theta\geq1$, the limit is the isotropic local $p$-Dirichlet problem. A convex minimax argument eliminates the normalized conductivity variable exactly, after which the analysis combines convex duality, nonlocal compactness, and $\Gamma$-convergence for convex integrands. We prove a unified $\Gamma$-convergence result for the reduced state energies whenever $\theta_\delta\to\theta\in[0,+\infty)$, yielding convergence of normalized optimal values and compactness and identification of optimal states throughout the full parameter range. At the vanishing-material endpoint, we additionally recover optimal measure-valued local fluxes from optimal nonlocal two-point fluxes.

Completely positive programs are known to provide an exact representation to several NP-hard optimization problems. In this work, we look at the maximum of random quadratic functions and compute the maximum expected value over ambiguity sets defined with moment and support information. We propose exact completely positive programs and semidefinite programs when the random variables are Bernoulli, nonnegative and unrestricted in support. Our experimental study demonstrates the utility of these formulations.

This work analyzes a $C^1$-conforming virtual element method for the optimal control of generalized Oseen equations, formulated using the stream-function on simply connected polygonal domains. The method is based on a conforming discretization in $H^2$ of arbitrary order $k \ge 2$. The discrete scheme considers an approximation of the state and co-state equations which involves a specifically designed operator for the convective term and a variational discretization of the control variable. We derive quasi-optimal a priori error estimates in the $H^1$ and $H^2$-norms for the stream-function, and in the $L^2$-norm for the control, under standard assumptions on the family of polygonal meshes. The proposed method recovers the fluid velocity, vorticity, and pressure using suitable projections. Finally, we report several numerical experiments to support the theoretical findings using different polygonal meshes including the cavity problem. We highlight that this analysis also covers OCPs governed by Brinkman and Stokes equation.

This paper studies infinite-horizon equilibrium dividend distribution for a firm whose surplus follows a diffusion under observable Markov regime switching and is absorbed at bankruptcy. Dividend payments are modeled as singular controls, and non-exponential discounting leads to time inconsistency. We formulate equilibrium control laws and establish a verification theorem for a general finite-state model with surplus-dependent coefficients. For a two-state model with constant positive drifts and volatilities and bi-exponential discounting, we prove the existence of a regime-dependent equilibrium barrier policy for arbitrary positive switching intensities. The policy retains surplus below the prevailing regime's threshold and distributes any excess immediately. Numerical experiments examine how shareholder patience and economic conditions affect dividend decisions. Over the parameter ranges considered, greater discount-rate dispersion at a fixed initial average rate raises both barriers, while switching speed can generate nonmonotone responses and reverse their ordering.