Papers

Large language model providers are compute constrained, and a common response to congestion is to degrade service. A degraded answer fails with some probability, and a failed answer either returns as a retry or departs as churn, destroying LTV on an unaccounted ledger. We model inference allocation as a newsvendor whose stockout cost is churned lifetime value, a geometric retry multiplier in which the recycled product is dissatisfaction, and a two-regime transient fluid queue whose arrival rate is made endogenous by retries. Statically, there is a regime in which a cheaper model saves energy per initiated task while consuming strictly more capacity per initiated task, so the discount inverts when capacity binds. Dynamically, a reactive throttle fired during a surge can cross an ignition threshold beyond which it manufactures more traffic than it sheds, and a release rule set below the degraded equilibrium converts a transient surge into a permanent degraded regime. With heterogeneous customers, throttling is a transportation problem in retry-inflated load whose optimal policy rations intelligence by critical ratio, and whose dual, the shadow price of intelligence, prices a marginal query by class and by hour. Under congestion, throttling is not a cost lever but a demand lever.

In the presence of information-sharing noise, row-stochastic distributed optimization over directed and unbalanced graphs can suffer not only from noise accumulation in gradient tracking, but also from distortion of the left eigenvector-based gradient scaling used for imbalance compensation. To address these issues, this paper proposes a noise-robust distributed optimization algorithm using only row-stochastic weights, termed Robust Xi-row (R-Xi-row). The optimization update is driven by the increment of an auxiliary cumulative variable to exclude historical tracking noise, while two decaying gains and a normalized scaling gain are used to suppress the effect of noisy eigenvector estimation and asymptotically recover the required gradient scaling almost surely. Under strongly convex and smooth objectives and appropriate decaying gains, we prove almost sure convergence to the optimal solution. For polynomially decaying gains, we further establish a near-$\mathcal{O}(k^{-1/3})$ expected convergence rate. Numerical experiments validate the theoretical results.

We study the k-delete recoverable robust 0-1 problem in which a decision-maker solves a combinatorial optimization problem subject to objective uncertainty. The model follows a two-stage robust setup. The decision-maker first commits to an initial plan and may then revoke up to k components of this decision after the uncertainty is revealed. The underlying uncertainty is modeled using a budgeted uncertainty set so that the decision-maker only hedges against a limited number of deviations in the uncertain parameters. We present four reformulations of the k-delete recoverable robust problem, which can be tackled using (i) general-purpose mixed-integer linear programming solvers, (ii) branch-and-cut methods, or (iii) column-and-constraint generation algorithms. For each formulation, we identify suitable solution methods and prove their correctness. Overall, we present eight approaches to solve the k-delete recoverable robust problem, which we assess and compare in an extensive computational study on instances of the assignment problem and the single-source capacitated facility location problem.

This paper is concerned with solving the problem of finding a zero of the sum of a single-valued operator and a set-valued operator, neither of which is assumed to be monotone, in a real Hilbert space. Under the framework of pair monotonicity, we propose a generalized forward–backward algorithm and a generalized Tseng algorithm based on transformed and warped resolvents associated with an auxiliary linear operator. We establish weak, strong and linear convergence of the proposed algorithms under suitable assumptions involving pair strong monotonicity, pair cocoercivity, and pair Lipschitz continuity. Finally, numerical experiments are presented to validate the theoretical results and demonstrate the effectiveness of the proposed algorithms.

For n points in the unit square, the Heilbronn triangle problem asks for the largest possible minimum triangle area. We develop a variational stress theory for this max-min problem. At every positive-area local optimum, normalized Karush-Kuhn-Tucker multipliers assemble into a skew matrix B satisfying Bz = 2ib, where b is the outward square reaction. This equilibrium has the isotropic affine moment sum_i p_i b_i^T = Delta I_2; the identity also holds for every tight subfamily carrying weights inherited from the same multiplier, with Delta scaled by its multiplier mass. It follows that every positive stress component meets all four sides and that there are at most two such components. To handle nonunique multipliers, we introduce the intersection of the stress kernels over the whole KKT face and a canonical hybrid operator incorporating all tight determinant and boundary derivatives. A strictly convex selector removes every decomposable invisible motion, leaving a rank-one-free residual with a sharp dimension bound; the literal maximal two-dimensional residual is excluded. Two-terminal substresses satisfy an exact interface-flux law, and a five-internal-vertex rank-four block has a Pfaffian cofactor carrier. Finally, an analytic reduction followed by exact symbolic enumeration excludes every one-external completion of a specified one-silent five-cycle residual, for all orientation words. These results isolate the remaining degeneracy but do not solve the problem for arbitrary n.

Plasticity under changing environments is central to both evolutionary biology and continual learning. Motivated by recent work on genotype–phenotype maps, we study a minimal deep-learning analogue where a network is trained alternately on two Boolean label sets, and ask which biological controls of plasticity survive the translation to gradient descent. Reinterpreting four proposed biological factors as quantities of training dynamics, we find the system reduces to two dimensionless controls: the task disagreement $r$, the fraction of disagreeing labels, and the reach $\eta T$, the product of learning rate and switching period. We derive two bounds on plasticity: $r$ alone fixes an extremal geometric floor on the utopia distance, while $r$ and $\eta T$ jointly bound forgetting. Across 9,720 trajectories, an ANOVA confirms that $r$, $\eta$, and $T$ dominate, while the effect of neutral-set size (emphasized in the biological setting) is negligible. The optimal reach itself follows an approximate inverse power law $\eta T^{*}\propto r^{-1.18}$, yielding a heuristic that sets the optimal reach $\eta T^*$ from the task disagreement alone. The analogy that survives is therefore dynamical rather than geometric, and our setting enables a view of plasticity through the lens of other driven systems in physics and engineering.

Timely access to automated external defibrillators (AEDs) remains a major limitation of public-access defibrillation systems because stationary AED networks cannot adapt to spatial and temporal variation in out-of-hospital cardiac arrest (OHCA) demand. We study a platform-enabled mobile AED model in which AEDs are carried by existing urban ride-hailing fleets that can be dispatched to OHCA locations. Rather than assuming additional AED capacity, we hold the total AED supply fixed and compare alternative allocations between stationary and mobile deployment, including hybrid policies that retain both forms of coverage. Our analysis leverages real-world urban data from New York City and Toronto, integrating georeferenced public AED inventories, EMS-reported OHCA incident data, and ride-hailing mobility data. The results show that mobile AEDs can substantially improve both response times and reliability, but the benefits depend on how AED capacity is divided between stationary and mobile deployment. In New York City, average response-time gains exceed 2.5 minutes and peak when approximately 55 to 70 percent of AED capacity is mobile; beyond this range, both gains and reliability decline as stationary coverage is reduced. In Toronto, average gains approach 4 minutes and then plateau as the system moves toward a fully mobile configuration. These contrasting patterns show that mobile AEDs can materially improve access, but the best mix of stationary and mobile AEDs must be tailored to local coverage and mobility conditions.

We analyze the convergence rate of certain sequence of the duality gaps in the generalized Frank-Wolfe method, under certain regime of the weak growth condition proposed in Peña (2023). Our analysis leverages a recursion lemma that may be of independent interest.

Iteratively reweighted least squares (IRLS) methods constitute a natural approach to nuclear norm minimization, but their convergence rates and the role of the weight operator have remained poorly understood. This paper establishes sharp convergence rates for IRLS methods for constrained nuclear norm minimization in low-rank recovery. A central ingredient is a new majorization analysis for the smoothed nuclear norm: we prove that the harmonic-mean weight operator defines a valid global quadratic majorizer. Furthermore, we show that this weight operator is optimal within the family of power-mean weights, clarifying why it improves over classical one-sided reweighting schemes that use only row- or column-space information. Under a Schatten-1 null space property, we prove global linear convergence of IRLS algorithms using a variety of weight operators, including the harmonic-mean weights. For IRLS with harmonic-mean weights, we prove a dimension-independent, locally linear convergence rate. We provide a counterexample showing that this dimension-independent local rate cannot in general be obtained for IRLS algorithms using one-sided weight operators, which predominate in the literature. Numerical experiments corroborate the theoretical results and illustrate the practical advantage of harmonic-mean reweighting across square, rectangular, and adversarially initialized recovery problems.

We establish a strong averaging principle for fast-slow stochastic differential equations with a time-dependent scale-separation parameter $(\varepsilon_t)_{t \geq 0}$ satisfying $\varepsilon_t \to 0$ as $t \to \infty$. In contrast to approaches based on noise-induced smoothing or elliptic regularity, our approach relies on dissipativity of the frozen fast dynamics and therefore permits degenerate diffusion coefficients. We prove a maximal $L^p$-estimate between the slow variable and the averaged ODE at late times, with the classical strong convergence rate of order $1/2$. Under an additional decay condition on $(\varepsilon_t)_{t \ge 0}$, this estimate implies that the slow variable is almost surely an asymptotic pseudo-trajectory of the averaged ODE. As a consequence, we obtain criteria for the identification of possible limit points and for convergence toward asymptotically stable equilibria for the slow variable by analyzing the dynamical behavior of the averaged equation.

In 1971, Mason conjectured that the numbers of independent sets of fixed size in a matroid constitute an ultra log-concave sequence. In 2020, this conjecture was proven by Brändén and Huh and independently by Anari, Liu, Gharan and Vinzant. Recently, this result was extended to $M^\natural$-concave functions. In this note, we make the next step by proving it for $M_2^\natural$-concave functions. This shows the same property for the intersection of any pair of matroids (which itself may not be a matroid). Furthermore, we show that this can not be further extended to the intersections of three matroids by including a counterexample of partition matroids.

Nonlinear Dynamic Inversion (NDI) has become a standard methodology in flight control law design. It offers an intuitive approach to decouple commanded variable responses, handle system nonlinearities, and adapt to operating conditions. NDI also comes with a well-structured architecture that reduces design effort by addressing various functional aspects in separate components, and that allows straightforward integration of extended capabilities, such as envelope protection, control saturation handling, and compensating for faults or damage. A development that has resulted in considerable renewed attention is the use of (angular) acceleration sensors to partially replace inverse model equations. Known as incremental NDI, or INDI, this development offers reduced sensitivity to modeling errors and lower control law complexity. Incremental NDI, however, lacks useful design degrees of freedom in the previously present inverse model equations and underlying feedback signal synthesis, and comes with pitfalls in design aspects like control allocation, disturbance rejection, and inter-disciplinary couplings. This has given rise to recently developed hybrid and mathematically restructured approaches. The aim of this article is to give an up-to-date, structured overview of the various evolved forms of NDI from conceptual, historical, architectural, and mathematical perspectives. It hereby intends to provide useful considerations for future flight control law developments by reviewing its various forms and potentials from methodological, design, and application points of view.

In this work, we study the oracle complexity of finding an $\epsilon$-stationary point for nonconvex-strongly-convex (NC-SC) bilevel optimization using only first-order oracles. Existing methods achieving the best-known complexity guarantees typically rely on double-loop, penalty-based procedures. We propose a novel single-loop algorithm based on a constrained reformulation in which lower-level stationarity is imposed as a constraint. Specifically, we construct a regularized Lagrangian by introducing a quadratic regularizer and restricting the dual variable to a bounded domain, and then apply Smoothed Gradient Descent Ascent [Zhang et al., 2020], with Hessian-vector products approximated via finite differences of gradients. We refer to the resulting deterministic and stochastic algorithms as SGHA and Stoc-SGHA, respectively. In the deterministic setting, SGHA achieves an oracle complexity of $O(\bar{\kappa}_y^{5}\epsilon^{-2})$, where $\bar{\kappa}_y$ denotes the relevant condition number. In the stochastic setting, Stoc-SGHA achieves an oracle complexity of $O\left(\bar{\kappa}_y^{17}\epsilon^{-6}\rho^{-3}\right)$ with probability at least $1-\rho$ for any $\rho\in(0,1)$, and an oracle complexity of $O\left(\bar{\kappa}_y^{17}\epsilon^{-6}\right)$ in expectation under an additional bounded-iterate assumption. Moreover, under an additional stochastic smoothness assumption imposed only on the lower-level objective, the stochastic oracle complexity of Stoc-SGHA improves to $O\left(\bar{\kappa}_y^{11}\epsilon^{-4}\rho^{-2}\right)$ with high probability and $O\left(\bar{\kappa}_y^{11}\epsilon^{-4}\right)$ in expectation, matching the $\epsilon$-dependence of the lower bounds.

We prove existence, uniqueness, and stability for fully coupled McKean-Vlasov forward-backward SDEs with jumps whose drift, diffusion, jump, and driver coefficients may depend Lipschitz-continuously, in quadratic Wasserstein distance, on the joint law of the full solution tuple $\Theta=(X,Y,Z,U)$: forward state, backward variable, Brownian integrand, and $L^2(\nu)$-valued jump integrand. The terminal function may depend Lipschitz-continuously on $X_T$ and its law. The system is driven by a Brownian motion and an independent compensated Poisson random measure with arbitrary $\sigma$-finite intensity, so infinite jump activity is admitted. Both the Lipschitz and monotonicity hypotheses are imposed only along diagonal tuple-law pairs $(\Theta,\mathrm{Law}(\Theta))$; we show that expected diagonal monotonicity is strictly weaker than pointwise monotonicity. Under a jump-extended $G$-monotonicity condition we establish an a priori continuous-dependence estimate, uniqueness, and existence on every prescribed finite horizon, by monotone continuation in the coupling strength from a small-coupling base case. A mean-field dealer-market example realises the $U$-law dependence non-perturbatively: its law interaction is monotone at every interaction strength, and its mark measure has infinite activity.

Direct policy optimization is widely used in reinforcement learning and control, but generally leads to nonconvex optimization problems. For state-feedback $H_\infty$ control, the policy objective is also nonsmooth, despite possessing a benign landscape whose hidden convexity can be revealed by the recently developed extended convex lifting framework. Motivated by recent advances in hidden convex optimization, we study zeroth-order optimization of nonsmooth, nonconvex problems admitting a convex lifting. We propose a zeroth-order proximal point algorithm: An inexact proximal-point outer loop constructs strongly convex subproblems, while an inner loop approximately solves each subproblem using only function evaluations. With probability at least $1-\delta$, our proposed algorithm returns an $\epsilon$-optimal solution using $\widetilde{O}\left(d\epsilon^{-3}\right)$ function evaluations, while all iterates remain feasible without explicit projection. Finally, we verify that the assumptions underlying our analysis hold for discrete-time state-feedback $H_\infty$ policy optimization, yielding an oracle complexity of $\widetilde{O}\left(n_u n_x\epsilon^{-3}\right)$ for attaining a prescribed objective value gap, where $n_u\times n_x$ is the dimension of the feedback gain to be optimized over.

We study a time-discrete, finite-particle Consensus-Based Optimization (CBO) algorithm in a separable Hilbert space. Our analysis provides convergence guarantees directly for the computable particle system, complementing recent continuous-time and mean-field results in infinite dimensions. Using a common-noise formulation with trace-class covariance, we first establish quantitative pairwise contraction, exponential decay of the expected swarm variance, and almost-sure convergence of all particles to a common consensus state. We then combine estimates on the exponentiated objective functional with a quantitative Laplace principle to show that, for sufficiently large inverse temperature and suitably prepared initial data, the energy of the limiting consensus state can be made arbitrarily close to the global minimum over the active subspace. A key feature of the Hilbert-space formulation is that the stochastic contribution to the convergence estimates is controlled by the trace of the covariance operator and is therefore uniform with respect to the Galerkin dimension. Numerical experiments on an elliptic energy minimization problem with mixed boundary conditions and a PDE-constrained inverse source problem validate the theoretical convergence results and demonstrate stable performance under increasing spatial resolution, in contrast with CBO based on isotropic finite-dimensional noise.

Freshness is a key attribute of perishable goods, particularly in the food sector, and strongly influences consumer purchasing behavior. Retailers often prioritize high freshness to remain competitive, yet maintaining this standard under demand uncertainty increases the likelihood of unsold inventory and waste. Retailers use markdowns to mitigate this risk, but the timing and depth of discounts must be carefully coordinated to avoid lost revenue or excess waste. This paper studies how retailers can jointly optimize order and markdown pricing decisions to balance profitability and waste reduction for perishable products within an integrated decision-making framework. The joint ordering-markdown pricing problem is formulated as a stochastic dynamic programming model that accounts for the demand cannibalization between regular price and markdown sales. The conflicting objectives of profit maximization and waste reduction are captured through a weighted objective function that reflects the retailer's trade-off preferences. To manage the complexity of the model, we propose an efficient solution approach and conduct numerical experiments to assess the performance of alternative markdown policies. Our results show that implementing multiple early price reductions with successive price reductions can simultaneously increase revenue and reduce waste. Our findings generate several key managerial insights: (i) firms can tailor their markdown policies according to their strategic priorities of profit maximization, waste reduction, or a balanced combination of both, (ii) early and well-timed markdowns based on available inventory levels allow firms to simultaneously attract quality-seeking customers and price-sensitive segments, (iii) implementing a second or multiple markdowns can further reduce waste while also increasing revenue.

We study decentralized stochastic optimization over a network of $N$ agents under compressed communication. We propose CED-EF, an exact diffusion-based method with error feedback that directly accommodates biased $\delta$-contractive compressors while communicating one compressed model-sized vector per node per iteration. For smooth nonconvex objectives with unbiased stochastic gradients whose variance is bounded by $\sigma^2$, where $\sigma\geq0$, we establish a convergence rate whose leading stochastic term is $\mathcal O(\sigma/\sqrt{NK})$. For $\sigma>0$, the dominant dependence of the corresponding transient time on the number of agents, compression level, and spectral gap $\Delta_\lambda$ is $\mathcal O(N^3/(\delta^4\Delta_\lambda^4))$, with fixed problem-dependent factors suppressed. Under the Polyak–Łojasiewicz condition, CED-EF attains a leading stochastic term $\widetilde{\mathcal O}(\sigma^2/(NK))$ with transient time on the order of $\widetilde{\mathcal O}(N/(\delta^2\Delta_\lambda^2))$. These dependencies improve the compression and/or network dependence of existing results. Numerical experiments on least-squares and logistic-regression problems illustrate the performance advantages of CED-EF.

Two-stage sample robust optimization with linear decision rules is a standard data-driven approach to two-stage stochastic linear programs with unknown distributions. \cite{bertsimas2022two} established that this approach is asymptotically optimal under several conditions. Most are mild, but one is technical, and whether it follows from the others was left as an open question. We resolve this open question through a single geometric feature of the support set: simpleness, the property that every vertex is incident to exactly as many edges as the ambient dimension. When the support set is simple, the technical condition is automatic, and asymptotic optimality holds under the remaining standard assumptions. When the support set is not simple, there exists a problem instance on which the technical condition fails, and asymptotic optimality breaks down. This settles the open question in the negative and yields a sharp dichotomy. The dichotomy has direct practical consequences, since widely used support sets may not be simple. For such supports, we provide a polynomial-time algorithm that certifies the technical condition at any given vertex using only its incident edge directions. Our case study illustrates the effectiveness of our algorithm.

We analyze a variant of stochastic gradient descent with initial regularization (SGDIR) and derive dimension-free upper bounds on its expected excess risk for the squared loss. In the noiseless case, we obtain new bounds for both averaged and non-averaged SGDIR under moment, source, and capacity assumptions. For a particular value of the source parameter, these bounds are of order $m^{-2}\log^{2}m$, where the number of training samples is of order $m$. For another value of the source parameter, we obtain, for any $\epsilon>0$, bounds of order $m^{-3+\epsilon}$, provided that the capacity parameter exceeds $\epsilon^{-1}$. We also establish a lower bound that matches our upper bounds in certain regimes up to a polylogarithmic factor. In the noisy case, we provide an instance-based comparison between SGDIR and ridge regression. Under general assumptions and a mild lower bound on the regularization parameter, we show that the expected excess risk of SGDIR is no larger than that of ridge regression, up to a polylogarithmic factor. Numerical experiments on synthetic and real data are consistent with our theoretical findings.

Federated learning has traditionally been formulated as a single-objective optimization problem, primarily focused on maximizing model utility. In real-world applications, however, machine learning models often need to optimize multiple and potentially conflicting objectives simultaneously. This motivates federated multiobjective optimization (FMOO), which provides a natural framework for jointly handling multiple task-specific objectives in federated learning. In this paper, we propose a momentum-based variance-reduced algorithm for federated multiobjective optimization. The method incorporates a momentum-driven gradient estimator into the local updates to reduce the variance of stochastic updates, leading to an improved convergence rate. We establish theoretical guarantees showing that the expected Pareto stationarity measure of a randomly selected output iterate decays at a rate of $\mathcal{O}(T^{-2/3})$, improving upon the $\mathcal{O}(T^{-1/2})$ rates established for existing methods such as FSMGDA and FedCMOO. Numerical experiments on federated multiobjective optimization benchmarks demonstrate the effectiveness and competitive performance of the proposed algorithm.

Space situational awareness (SSA) requires purpose-matched models across spatial, temporal, and fidelity scales. Building on our previously reported three-dimensional (3D) cell formulation and implementation, this study establishes a reproducible, resolution-aware, catalog-conditioned framework for macroscopic assessment of the low Earth orbit (LEO) collision environment. The framework maps supplied catalog or scenario populations to time-averaged spatial density and target-specific impact metrics while retaining individual-object information. Using a 2025 Space-Track snapshot, we evaluate radial, declination, and right-ascension resolution sensitivity and computational performance for six targets, including two Korean space assets. Normalized expected impact counts range from 0.615 to 1.599 and vary nonmonotonically; for a synthetic 500-km circular target, the result at a 0.25-km radial width is 38.5% below the 10-km reference. Runtime and memory show direction-dependent trade-offs. Ten annual snapshots show catalog growth from 15,723 objects in 2016 to 28,540 in 2025 and a 7.86-fold increase in the 500-km target metric, driven primarily by Starlink, other payloads, and unknown/TBA records. In a conditional stress test of the proposed 998,240-satellite SpaceX Orbital Data Center population, exact annual probabilities of at least one impact reach $3.75\times10^{-3}$ and $1.48\times10^{-3}$ for the 700-km and 1,000-km targets. The framework provides a reproducible, resolution-aware basis for catalog-conditioned environment monitoring, comparative scenario assessment, and prioritization of cases for higher-fidelity follow-up analysis.

This work derives convergence guarantees for mirror descent and proximal mirror descent algorithms when a logarithmic barrier is used as a distance-generating function. Standard approaches cannot be applied when the solution lies on the boundary, where the Bregman divergence blows up. We show that, in a specific setting, both methods enjoy an $O(\log k / k)$ rate, which is also tight. In addition, our contributions include: (i) a new technique for handling the blow-up; (ii) a resolution of a gap in the theory of relative smoothness; and (iii) a comparison of the proposed approach with interior-point methods.

Stone (Ph.D.\ thesis, Department of Operations Research, Stanford University, 1981) proved that every matrix in $U \cap Q_0$ is a $P_0$-matrix and conjectured that the same conclusion holds for the larger class $E_0^f \cap Q_0$ of fully semimonotone $Q_0$-matrices. Murthy and Parthasarathy [SIAM J.\ Matrix Anal.\ Appl.\ 16 (1995), 1268–1286] verified the conjecture for matrices of order up to $4 \times 4$, for $5 \times 5$ and $6 \times 6$ matrices under additional hypotheses, and for several special subclasses of arbitrary order, but the conjecture remains open in general. In this paper we prove that every $E_0^f$-matrix with positive determinant is a $P_0$-matrix, for matrices of arbitrary order $n$; our proof proceeds by induction on $n$, via an algebraic analysis of principal minors under principal pivotal transforms. We further exhibit a matrix $A \in E_0^f$ with $\det A > 0$ that fails to belong to $Q_0$, showing that the hypothesis $\det A > 0$ used in our theorem cannot, by itself, be deduced from membership in $Q_0$, and hence does not on its own yield a proof of Stone's conjecture. Stone's conjecture itself remains open.

We consider the finite horizon continuous-time Principal–Agent problem under deterministic discount factors. Following the Sannikov reduction to a stochastic control problem, we provide a further characterization of the Principal's value function in terms of a backward SDE inducing the corresponding optimal contract. In particular, this allows to bypass the fully nonlinear HJB equation satisfied by the Principal value function in the Markovian setting. This new approach allows to handle a new class of Principal-Agent problems which was not accessible with the existing method, namely the setting where the Agent faces a regime-switching control problem.

We study mixed-integer quadratic optimization (MIQO) problems with indicator variables. We propose a unified framework, based on decision diagrams, that serves both to solve the associated optimization problems and to construct ideal conic quadratic extended formulations of the closure of the convex hull of the underlying mixed-integer set. The construction applies to arbitrary quadratics and to any combinatorial constraints admitting a tractable dynamic programming representation. The resulting diagrams and the ensuing convex hull descriptions are of polynomial size when the quadratic is low-rank, or when the support graph of the Hessian or of its inverse is a tree, recovering and generalizing several results from the literature. For structured sparse and inverse-sparse quadratics, we show that approximate decision diagrams have size linear in the dimension while yielding solutions with arbitrarily low optimality gap. Computational experiments demonstrate the effectiveness of the proposed approach.

In this paper, we propose an inexact Riemannian gradient descent algorithm on the Stiefel manifold (IRGS-StieONS) using an adaptive step size, where the “inexact” refers to the inexactness of retraction. It is proven that one single Newton-Schulz iteration for the retraction is sufficient for global convergence and local linear convergence. Compared to the landing and augmented Lagrangian-based algorithms, the proposed algorithm is the first infeasible algorithm that permits adaptive step sizes with a practical initial step size and guarantees global convergence and local linear convergence under mild assumptions. Moreover, we show that the local convergence rate depends on the condition number of the Riemannian Hessian, which matches the Riemannian steepest descent algorithm. This result implies that the infeasibility in the proposed algorithm does not influence the local convergence rate. Furthermore, a stochastic gradient version of IRGD-StieONS is proposed and is shown to achieve the same convergence rate as Riemannian stochastic gradient descent with decreasing step size. Numerical experiments demonstrate that both IRGD-StieONS and its stochastic counterpart exhibit superior performance and robustness.

Coupled-dynamics environments expose the one-step outcomes that would follow from several possible counterfactual actions under a common realization of exogenous randomness. The ordinary Markov decision process formalism allows one to reason about the marginal law of each action but discards dependence across these counterfactual outcomes. The Joint Markov decision process (JMDP) formalism preserves that dependence. Prior work established the formalism and solved the fixed-policy joint moment evaluation problem in JMDPs. This paper develops optimal-control methods. We define a nonparametric distributional Bellman optimality operator for JMDPs, and prove that when the induced marginal MDP has a unique optimal policy, its iterates converge in Wasserstein distance to the optimal joint return law. For the first two moments, we establish convergence under a weaker condition that permits several mean-optimal actions as long as their tie resolutions share a second-moment fixed point. We also derive sampled targets for neural approximation.

This paper studies the Sinkhorn distributionally robust hypothesis testing (SDRHT) problem, seeking a robust detector against least-favorable distributions in Sinkhorn discrepancy-based ambiguity sets centered at the empirical distributions. Existing approaches solve this problem by solving large-scale conic programs, which are not scalable. To overcome this, we propose a generative framework that learns least-favorable distributions and supports efficient training and end-to-end sampling. For the Sinkhorn discrepancy-based ambiguity sets, we first derive an equivalent conditional-KL-divergence representation with respect to kernel-smoothed reference distributions. This property allows us to prove strong duality for both constrained and unconstrained minimax SDRHT formulations. Based on the closed-form optimal detector and Brenier's theorem, we reformulate the max-min dual formulation as a maximization problem over convex potentials whose gradients characterize invertible transport maps between kernel-smoothed distributions and their least-favorable counterparts. We efficiently approximate these potentials using Hyper Input Convex Neural Networks (HyCNNs) equipped with stochastic gradient estimators and prove the representation power of HyCNNs and the distributional universality of their induced transport maps. Numerical results show that the proposed method achieves superior accuracy and robustness across different sample sizes and dimensions, while avoiding the scalability limitations of classical SDRHT methods.

We study the cone $\mathcal{M}_{n,2d}$ of nonnegative mean polynomials—real $n$-variate forms of degree $2d$ that can be expressed as weighted power means $M_{q,p}(Y,w)$ with $q>p$. This cone simultaneously generalises the cone of sums of squares $\Sigma_{n,2d}$ and the cone of sums of nonnegative circuit polynomials $\mathcal{C}_{n,2d}$. We prove that every square of an arbitrary polynomial belongs to the mean polynomial preprime $T_{\mathrm{mean}}$, that $T_{\mathrm{mean}}$ is strongly generating, and consequently that every polynomial strictly positive on a compact semialgebraic set admits a representation with mean polynomial certificates. We exhibit the Robinson form $\hat{R}$ as a separating example that lies in $\mathcal{M}_{4,4}$ but outside $\mathrm{SOSONC}_{4,4}$. Finally, we outline a convergent hierarchy of lower bounds for polynomial optimization based on the mean polynomial cone and discuss tractable depth-truncated approximations via signomial programming.