Papers

Understanding loss landscapes is central to explaining neural-network training, yet their structure remains only partially understood even in simple models. We study the Gaussian population loss of shallow, bias-free ReLU networks in the teacher–student setting. This provides a simple model for studying essential aspects such as feature learning and overparameterization. For teacher networks with positive output weights and planar features, we show that including a learned linear skip removes all spurious local minima with non-negative student output weights once the student network is at least as wide as the teacher network. In contrast, without the skip, we construct a fixed teacher network with positive output weights and only three hidden neurons in input dimension two whose spurious local minima persist at every student width at least three. Thus, a learned linear skip can remove spurious minima that persist under arbitrary overparameterization. Furthermore, we show that a positive output weight student network always learns the subspace spanned by the teacher features: student features at local minima with non-negative student output weights lie in the span of the teacher features. For ReLU networks in two dimensions, even heavily overparameterized student networks have effective width controlled by the teacher width: every critical point with positive student output weights has at most twice as many distinct student feature directions as teacher neurons. Finally, we transfer the benignity result to empirical minima over parameter balls of any prescribed radius, with the required sampling accuracy depending on that radius.

We prove (under some mild assumptions) that the strong-strong twice epidifferentiability of a convex functional is equivalent to the weak-$\star$ twice epidifferentiability of its conjugate functional. Further, the corresponding subderivatives are (up to scaling) conjugates of each other. This result can be used to facilitate the computation of weak-$\star$ second subderivatives, which are crucial ingredients in no-gap second-order optimality conditions. We also show that weak-$\star$ second subderivatives are invariant under suitable changes of the underlying function space setting.

Motivated by hyperconnected urban parcel networks for same-day and time-sensitive delivery, we study the fixed-line routing problem that arises when evaluating a candidate set of recurring inter-hub vehicle services. Vehicles follow predefined cyclic sequences of service points, or lines, and parcels may transfer between lines. Such transfers expand path choice but require loading and unloading operations that make vehicles dwell at service points. Because handled volumes are routing-dependent, dwell times become endogenous, linking routing decisions to line cycle times, effective service frequencies, and transfer waiting times. We introduce the Routing in Line Networks with Handling Times (RLN-HT) problem and formulate it as a bilinear generalized capacity-flow assignment model. We characterize optimal dwell times and the resulting frequencies as functions of handling workloads, enabling a reformulation solely in terms of commodity flows. We exploit its convex substructure to derive two second-order conic relaxations that preserve the original feasible region and therefore yield feasible incumbent solutions along with valid lower bounds. We also develop a residual-frontier framework that characterizes the tradeoff between the convex relaxation objective and the remaining nonconvex residual, yielding a general mechanism for constructing lower-bound certificates. Computational experiments show that ignoring handling times can yield poor or infeasible routing, whereas accounting for endogenous dwell times can improve average parcel travel times by up to 26.5%. In instances with up to 257 nodes and 25 lines, the relaxations solve quickly, yield near-optimal routing, and obtain small optimality gaps, whereas off-the-shelf spatial branch-and-bound struggles to produce useful bounds.

We establish complexity lower bounds for stochastic first-order algorithms in nonconvex–concave minimax optimization, allowing algorithms to use variance reduction. Our main contribution is a lower bound for a zero-respecting algorithm class that permits variance reduction, extending beyond the algorithmic restrictions imposed by some existing lower bounds. We consider objectives with an $L$-Lipschitz continuous joint gradient, a compact convex dual domain of Euclidean radius at most $D_Y$, and a primal value function, defined by maximizing the objective over the dual variable, with initial suboptimality at most $\Delta$. The target accuracy $\varepsilon$ is measured by the gradient norm of the Moreau envelope of the constrained primal value function with parameter $1/(2L)$. Under an unbiased stochastic first-order oracle with variance at most $\sigma^2$ and mean-square smoothness, we prove the lower bound $\Omega\!\left(L^2D_Y\Delta\varepsilon^{-3}+L^3D_Y^2\Delta\sigma^2\varepsilon^{-6}\right)$. This result quantifies the dependence on accuracy, dual-domain radius, and oracle noise even when variance reduction is allowed. We also establish complementary lower bounds for nonconvex–strongly-concave minimax optimization. With dual strong-concavity parameter $\mu>0$ and condition number $\kappa:=L/\mu$, we obtain $\Omega\!\left(L\Delta\sqrt{\kappa}\,\varepsilon^{-2}+L\Delta\kappa\sigma^2\varepsilon^{-4}\right)$ under the bounded-variance oracle model. Under the additional mean-square smoothness condition with constant $\bar L$, we obtain $\Omega\!\left(L\Delta\sqrt{\kappa}\,\varepsilon^{-2}+\Delta\bar L\sigma\kappa^{3/2}\varepsilon^{-3}\right)$. Together, these results identify complexity barriers across the concave and strongly concave regimes, with the main nonconvex–concave bound remaining valid for algorithms that use variance reduction.

Structural system theory studies which network topologies can sustain a prescribed system property such as controllability or stability. When the topology is itself random, the relevant question becomes probabilistic: how likely is a graph drawn from a stochastic model to sustain the property? Such probabilities measure the abundance and robustness of the property across topologies, and indicate whether systems requiring it can be reliably deployed in uncertain environments. We address this question for asymptotic stability of linear systems in the directed graphon setting. We consider two graph-theoretic properties. The first is $\mathcal N$, and it requires that for every $k\leq n$ some $k$-vertex induced subdigraph of $D$ admits a cycle cover, and the second is $\mathcal S$, which requires that these subdigraphs can be chosen so that their node sets form a nested sequence $V_1\subset\cdots\subset V_n=V(D)$ starting from a single vertex with a loop. We have shown that $\mathcal N$ is necessary and $\mathcal S$ is sufficient for structural stability. We sample $D$ from a directed step-graphon $W$. Our main results give necessary and sufficient conditions for $\Pr(\mathcal N)\to 1$ and $\Pr(\mathcal S)\to 1$ as $n\to\infty$. In more detail, to a step-graphon $W$ with skeleton digraph $S$ on $q$ nodes and concentration vector $x^*$ we associate a cycle polytope $\vec{\mathcal X}(S)\subseteq\Delta_q$. The conditions are then formulated in terms of the position of $x^*$ within $\vec{\mathcal X}(S)$, the dimension of the polytope, the loop density of $W$ and, for $\mathcal S$, an ordering condition on the cycles of the skeleton. Together these results identify, for directed step-graphons, the regime in which a sampled topology is overwhelmingly likely or unlikely to sustain stable dynamics.

Stochastic recursive momentum (STORM) achieves fast convergence for nonconvex optimization via the variance reduction effect, but existing analyses rely on the strong average smoothness assumption. In this paper, we study the convergence of STORM for different objectives without average smoothness. We first revisit the results under average smoothness, obtaining the $O(T^{-1/3})$ bound for nonconvex objectives and the $O(\sigma^2/(\mu T))$ bound for last-iterate output under the $\mu$-Polyak–Łojasiewicz~(PL) condition. Without average smoothness, we design an auxiliary sequence and compare the STORM update with it in the analysis. With the help of this sequence, we prove that STORM still attains an $O(T^{-1/4})$ rate for nonconvex objectives, which is optimal under standard smoothness. For convex and $\lambda$-strongly convex objectives, we further prove averaged and last-iterate bounds with optimal rates of $O(\sigma R/\sqrt T)$ and $O(\sigma^2/(\lambda T))$, respectively. All the obtained results use the same STORM recursion with different hyperparameter choices.

This paper investigates stochastic multi-level optimization where the objective is a nested composition of several smooth non-convex functions. We assume that only stochastic estimates of the gradient and function values for each level are accessible. Consequently, obtaining an accurate estimate of the overall gradient is challenging due to the nested structure. To address this, we employ a momentum-based estimator with mini-batches to track the function values of each level, which are subsequently used to construct momentum gradient estimators. We establish an optimal sample complexity of $\mathcal{O}(\epsilon^{-4})$ for finding an $\epsilon$-stationary point, avoiding the stronger average smoothness assumption commonly relied upon in prior literature. Furthermore, by employing a normalization technique, we attain the same rate without requiring problem-dependent constants to set hyperparameters. To achieve the optimal rate without mini-batches, we further develop a batch-free method that incorporates a first-order approximation and a clipping technique for function value estimation. Finally, we validate the effectiveness of our proposed methods through experiments on risk-averse portfolio optimization and hierarchical tilted empirical risk minimization.

Primal-dual hybrid gradient (PDHG) methods solve large-scale linear programs (LPs) using GPU-friendly matrix-vector products and projections, but their practical performance depends on coordinating algorithm parameters, acceleration, and restarts. We introduce GALLOP, which uses reinforcement learning to jointly learn continuous algorithm parameters and discrete restart decisions without differentiating through the solver. Its generalized accelerated PDHG update combines separate primal and dual extrapolation, history corrections, and restart anchoring with independently adjustable coefficients. We train a dimension-agnostic feedback policy using a groupwise proximal policy optimization objective that clips likelihood ratios separately for different control groups and excludes inactive acceleration controls on restart transitions. We evaluate GALLOP on six LP families and a public item-placement benchmark. On the main evaluation settings across the six families, GALLOP reduces iteration counts by factors of $1.9$-$5.6$ and achieves up to a $16.0\times$ speedup in algorithm wall-clock time over MPAX. With one policy trained per family, the learned policies generalize without retraining to within-family LPs $3\times$-$400\times$ larger than the largest training instances, including Transport LPs with $10.24$ million variables.

We study a mixed regular–singular control problem for stochastic evolution equations in a Hilbert space with possibly unbounded random linear operators, a nonconvex regular-control domain, and a state-dependent singular coefficient. The singular control is an adapted nondecreasing càdlàg process whose terminal value need not be bounded. We prove well-posedness and weighted moment estimates for the forward and backward equations, and characterize the second-order adjoint by a conditionally expected operator-valued backward stochastic integral equation. An Itô-type formula for the quadratic form of this adjoint, together with spike and convex variations, yields a second-order Hamiltonian condition for the regular control, as well as nonnegativity and a contact condition for the optional singular Hamiltonian.

sGS-PADMM [21, 14, 6] is a powerful and versatile class of convergent multi-block ADMM solvers for implementations on moderate-sized linear semidefinite programming (SDP) problems. In this paper, we further enhance this class of algorithms for solving SDP problems by proposing a new multi-clique decomposition approach, allowing substantial improvements in applications on large-scale sparse SDPs (e.g., where $n > 1000$) with conducive aggregate sparsity patterns. Our SDP decomposition strategy mainly aims to reduce the estimated PSD projection cost after decomposition, in contrast to common decomposition algorithms that are encumbered with minimizing the overlaps between cliques. This feature is made possible by our novel linear-space projection approach that is capable of efficiently processing a large number of overlap constraints via simple averaging steps. For the numerical experiments, we demonstrate the performance of our solver – named McADMM for Multi-clique ADMM – on a number of large-scale SDP instances that arise from relaxations of some important quadratically constrained quadratic programming (QCQP) problems. The performance of McADMM is contrasted against other state-of-the-art decomposition-based solvers as well as the non-decomposed sGS-PADMM to highlight our key contributions. We additionally develop a GPU implementation of McADMM and demonstrate that it can substantially accelerate the decomposed solver.

The operation of residential energy hubs with multiple energy carriers (electricity, heat, mobility) poses a significant challenge due to the energy storage differences in time-constants, round-trip efficiencies and self-discharge rates. Usually, thermal storage exhibits flexibility in yearly planning optimizations or long-term scenarios. However, as optimization horizons shrink (1-48hs) so does their supplied value due to the lower round-trip efficiencies. To avoid this early depletion during operation this paper proposes a data-driven highway to steer the short-term daily control towards long-term optimality. The proposed methodology also presents how to optimally size the thermal storage and avoid yearly simulations and how all of this is related to nonlinearities in the daily operation. The presented framework links seasonal and daily optimizations through dynamic terminal sets and value functions. The seasonally-aware nonlinear economic model predictive controller achieves the most balanced performance, with the second best mean grid cost of all MPCs at -\texteuro 209. It also achieves better battery degradation control than its linear counterparts (between 26-34%) and the best thermal comfort of the nonlinear benchmarks. Nevertheless, the data-driven seasonal highway restrains the ability to control battery degradation and slightly increases computational time.

Large language models (LLMs) are increasingly deployed as computational engines in autonomous decision-making and planning loops, yet their systems and control treatment remains hindered by architectural simplifications, index conflations, and informal descriptions of tool interactions. This paper presents a control-theoretic formulation of decoder-only language models as multi-index, multi-rate systems, and sets the stage for a stochastic hybrid systems framework to govern the multi-phase dynamics of agentic tool interaction. We formalize the architecture across three hierarchically coupled evolution indices: (i) an ultrafast feedforward cascade of transformer blocks across layer depth, where layer normalization is cast as a spherical projection and key–value caching is proven to be an exact internal state realization via causal prefix invariance; (ii) an uncontrolled stochastic difference recursion over token generation steps, where finite context truncation induces a time-homogeneous Markov chain; and (iii) an autonomous mode-switching mechanism governing transitions between token generation and tool execution regimes, where tool invocations are triggered upon trajectory arrival at switching manifolds, followed by exogenous state jump maps that augment the context string with external observations. By defining a prompt-dependent evaluator over successive evaluable claims, we obtain a task-level error process whose fault-free histories and expected error growth admit bounds under conditional fault-hazard and error-drift assumptions.

Prescribing the speed of gradient flow on the risk itself, by the dynamics $\dot w=-u(E(w))\nabla E(w)/\abs{\nabla E(w)}^{2}$, makes the risk $e(t)=E(w(t))$ obey $\dot e=-u(e)$ exactly, whatever the landscape~$E$; the time needed to reach zero risk from $e_0$ is $\int_0^{e_0}\dd e/u(e)$. Minimizing this time alone is ill posed, and we study the regularized problem $\inf\{\int_0^{e_0}(\tfrac\lambda2\abs{u'}^{2}+1/u)\,\dd e:\ u\in H^{1}(0,e_0),\ u\ge0,\ u(0)=0\}$, $\lambda>0$. We prove that the minimizer exists, is unique, and is a linearly scaled cycloid, and we show that the optimal rate behaves like $u^{*}(e)\sim(9/(2\lambda))^{1/3}e^{2/3}$ near zero risk: the exponent $2/3$ is the one found in \cite{betti2026holder} by a power-law ansatz, and it lies in the Hölder window $(\tfrac12,1)$ where the arrival is in finite time with vanishing weight speed. The proof follows the classical route: existence by the direct method, uniqueness by strict convexity, positivity of the minimizer away from the origin, and the explicit integration of the Euler-Lagrange equation.

Weak perturbations of the same size can lead to terminal Wasserstein errors of different orders. For infinitely divisible laws smoothed by a Gaussian, we differentiate their distribution functions with respect to variance-weighted Lévy characteristics. This includes distributional directions caused by moving atoms. We obtain first- and second-order expansions that are uniform over Lipschitz test functions. When the first response is nonzero, Gaussian analyticity gives the signed second coefficient of the Wasserstein–1 distance. When the first response vanishes, a finite second displacement moment is enough to obtain a quadratic response. Its coefficient is strictly positive if the displacement dispersion is nonzero. For local balanced remeshing, the error is comparable to the grid-alignment variance. This gives sharp quadratic grid rates for compactly supported densities and nonaligned atomic sequences. We derive metric speed and length along admissible non-atomic curves, and use the response to build a linear program with a proved bound on how much its error exceeds the infimum over the allowed class and consistent quadrature. We also prove a sharp transition as smoothing vanishes and give a multivariate second-order expansion. The appendix gives state-dependent response formulas and further stability estimates.

Graph clustering aims to partition the vertices of a graph into groups with dense intra-cluster connections and sparse inter-cluster connections. In this paper, we study a quadratic $K$-medoids formulation for graph clustering from the perspective of CM optimization. By identifying the binary feasible set as a discrete CM set, we establish a connection between the quadratic $K$-medoids model and CM optimization. Rather than solving the original binary problem directly, we consider the convex hull of its feasible set and introduce a negative squared-norm penalty to promote extreme-point solutions, resulting in an extreme point pursuit model over a simple convex set. The efficient projection onto this convex set enables the use of projected-gradient-based optimization for large-scale medoid selection. Based on this formulation, we propose a two-stage clustering method. In Stage~1, a penalty-continuation projected gradient solver is used to approximately solve the penalized medoid-selection problem. In Stage~2, the selected medoids are converted into a graph partition through a unified two-level assignment rule, where Jaccard distance is used as the primary criterion and shortest-path distance is used as a secondary criterion when needed. Numerical experiments on synthetic PPM and SBM graphs, together with real-world graph data, demonstrate the effectiveness and computational efficiency of our method. The experiments also show that a smaller objective value in the medoid-selection problem does not necessarily lead to a more accurate final graph partition, highlighting the importance of evaluating both optimization performance and clustering quality after assignment.

Reinforcement learning offers the prospect of a reusable sequential decision-making mechanism for spacecraft trajectory design, motivating policy interfaces that connect learned decisions to the underlying maneuver geometry. This paper develops Reachability Analysis-Informed Reinforcement Learning (RARL) for deterministic multi-impulse interplanetary transfers, placing intermediate waypoint selection at the center of the learned decision process. Local first-order reachability maps bounded velocity perturbations into an ellipsoidal set of next-node positions, within which the policy selects its waypoint. Lambert reconstruction then determines the corresponding maneuver to reach this selected waypoint along a dynamically consistent ballistic arc, coupling learned transfer-geometry selection with classical astrodynamics. A terminal two-impulse reconstruction completes the rendezvous, supported by a linear maneuver-demand assessment used for reward shaping. Numerical studies characterize this interface on a two-body Earth-Mars benchmark. Across three independent training runs, RARL achieves a mean maneuver cost of 10.23 km/s, 1.72% above a validated local sequential convex programming reference. Training over dispersed initial states extends policy reuse across a departure family with fixed target state and transfer duration. Each of the three independently trained multi-state policies completes all 10,000 held-out Monte Carlo departures without impulse-cap violations, compared with a mean feasibility rate of 6.49% for single-state policies. This broader sampled feasibility is accompanied by a 0.61% increase in mean nominal maneuver cost, without further training across departures. These results demonstrate that a reachability-informed decision interface supports benchmark-quality trajectory construction and policy reuse across dispersed departure conditions.

The distance between two points is provided by the length of their test connecting path. However, there is no unique way to attain this path in the space $R^2$, which is split by a straight line $L$ into two regions, $\Omega_p$ and $\Omega_q$, with $\ell_{p}$- and $\ell_q$-norms, respectively, where $1<p\leq q $ and $L\subset\Omega_q$. In this paper, given a point $P\in \Omega_p$, the locus of points $Q\in\Omega_p$, called $p$-curve, is found, such that there are two ways to attain the distance between $P$ and $Q$. For each $P\in \Omega_p$, there are two $p$-curves which are two branches of two different $p$-parabola. These two branches split $\Omega_p$ into three subregions using a three-link shortest path in two of these regions. The implicit equation of each $p$-parabola is given. Furthermore, by using the $p$-curve, an improved algorithm, called P-MFP, based on the MFP algorithm, is developed in order to solve the single-facility location problem in two regions with $\ell_{p}$ and $\ell_{q}$-norms. The results obtained with the new algorithm are better than those obtained with the MFP algorithm.ike subheadings, citations, or equations are permitted.

This paper presents a control allocation framework with adaptive augmentation for a trailing edge morphing aircraft, providing stability guarantees under uncertainty while exploiting the available morphing degrees of freedom to optimize aerodynamic efficiency. A baseline controller is designed using the nominal aircraft model to establish the desired closed-loop reference dynamics. For the aerodynamics, the wing shape is optimized dependent on the flight state to achieve a target elliptical lift distribution corresponding to minimum induced drag. The adaptive augmentation is tailored to the control allocation problem to account for the uncertain system dynamics and stabilize the aircraft around the reference dynamics. The resulting stabilization condition is formulated as hard constraint in the allocation problem while minimizing deviations from the corresponding state-dependent aerodynamically optimal wing shape. Numerical results demonstrate that the adaptive augmentation compensates for matched uncertainties while the control allocation optimizes the wing shape with respect to the reference elliptical lift distribution.

Data-driven model order reduction methods such as operator inference enable the efficient construction of reduced-order models directly from high-dimensional time-domain data. Standard operator inference typically seeks a Galerkin-type reduced model in a prescribed low-dimensional subspace by identifying its reduced operators from projected snapshot data. The resulting inference problem is formulated as a least-squares problem admitting an efficient closed-form solution. However, it is well known from intrusive model order reduction for linear time-invariant systems that Petrov-Galerkin projections can additionally preserve important system properties such as stability and passivity. To overcome the limitations of standard operator inference, we extend the framework for linear time-invariant systems to incorporate Petrov-Galerkin projections and provide explicit error expressions and bounds between the intrusive and nonintrusive reduced operators, thus generalizing results from the literature. We demonstrate the proposed approach in the context of dissipative and port-Hamiltonian systems. Furthermore, we introduce a novel convex optimization formulation that explicitly enforces the port-Hamiltonian structure on the inferred operators. The effectiveness of the proposed methods is demonstrated on several well-established benchmark problems, including the CD player, an atmospheric model, a mass-spring-damper system, and a poroelasticity system.

Day-ahead operation of local energy communities (LECs) is affected by uncertainty in PV generation and electricity demand, which may compromise the feasibility of committed exchanges with the upstream grid. This paper proposes a forecast-residual-based chance-constrained optimization (CCO) framework that characterizes uncertain parameters through their forecasts and associated residual distributions. Under the adopted zero-mean Gaussian assumption, the resulting chance constraints admit an exact deterministic-equivalent reformulation in which uncertainty is embedded through analytical safety margins, preserving the mixed-integer linear structure of the original scheduling problem. The model jointly coordinates PV and BESS operation, low-voltage distribution network constraints, internal energy sharing, and day-ahead grid-exchange commitments. Once the physical schedule is determined, an ex-post allocation stage distributes the available community energy pool among users according to predefined participation coefficients, identifies post-allocation surpluses and deficits, and maximizes their internal matching while preserving the aggregate grid exchanges obtained from the CCO. The framework is evaluated on a reduced 206-node European low-voltage feeder with up to 55 community users and benchmarked against a two-stage stochastic programming formulation. Results show that, for forecast-error levels close to 5%, the proposed CCO achieves operating outcomes comparable to the stochastic benchmark while retaining computational requirements close to the deterministic formulation. The ex-post allocation results further show that internal matching can reduce gross energy exchanges that require settlement with the upstream grid.

In global metropolitan areas, public transport benefits from bus systems. Bus design widely applies theoretical models, which typically assume static ridership. However, ridership randomness exists, and lack of attention to it might lead to wrong design decisions. Moving beyond static ridership, we extend the traditional single-line model with stochastic ridership. Further, to optimize the bus system under various ridership randomness, we introduce a hybrid model that combines conventional buses (CBs) and flexible buses (FBs). In both models, we apply a continuous approximation approach and find that: 1) Bus capacity increases with ridership randomness in the extended single-line model; 2) The hybrid model exhibits a binary state: either CBs serve the ridership with rejections under low ridership randomness, or CBs serve the majority while FBs serve the minority with rejections under high ridership randomness. Our findings establish a link between bus system design and ridership randomness, contributing to a more adaptive and efficient public transport framework.

We consider low-budget active learning, which consists of selecting a limited number of points, the coreset, such that a model can be trained to high accuracy on the selection only. This problem is particularly relevant in contexts where labeling requires costly expert intervention, as in medical applications. We leverage features extracted from a pretrained self-supervised model to represent the data, and perform coreset selection directly in this feature space. In this paper, we use entropic optimal transport, specifically the Sinkhorn divergence, as the coreset selection criterion, which first allows us to get dimension-free sample complexity results, and second admits computationally efficient gradient evaluations. This opens the way to using gradient-based algorithms to rapidly compute solution candidates, further improved by a swap-based local search, with guarantees on the solution quality. Experiments on image benchmarks and medical datasets show that our method outperforms state-of-the-art heuristics in low-budget settings.

In this work, we study bilevel optimization problems where all variables are integer, all constraints and the leader objective function are linear, and the follower objective function is non-convex quadratic. Relying on bilevel-free sets derived from improving directions, we develop a disjunctive cut approach to exclude bilevel-infeasible solutions within a branch-and-cut algorithm. We show that our disjunctive cuts can be obtained by solving a cut generating linear program. Furthermore, we discuss conditions that allow the number of disjuncts in the cut generating linear program to be reduced, and we propose several strategies to identify improving directions and generate disjunctive cuts efficiently. We evaluate various aspects of the proposed branch-and-cut algorithm on both convex instances from the literature that fit our setting and new non-convex instances and compare the performance of our best approach with existing state-of-the-art approaches, which we significantly outperform.

We consider stochastic games with independent controlled chains and unknown transition kernels, where players observe only their local states and realized payoffs. We develop a fully online, decentralized, and uncoordinated mirror-descent algorithm that operates in the dual space of occupancy measures for approximating stationary Nash equilibrium (NE) policies. The algorithm uses a single transition/reward sample at every primitive time step, relies only on local information, and requires neither coverage of the joint state space nor synchronized episodes. Under uniform-ergodicity and finite-coverage assumptions, we show that, with high probability, the time-averaged fixed-comparator regret decays at the canonical $O(T^{-1/2})$ rate, up to logarithmic factors and polynomial dependence on the game parameters. In particular, the complexity depends on the cover times of the individual local state spaces rather than the product state space, avoiding exponential dependence on the number of players and the sizes of the joint state and action spaces. The resulting finite-time regret bound further yields an approximate coarse-correlated-equilibrium guarantee, which is natural for arbitrary reward functions since computing a stationary $\epsilon$-NE is PPAD-hard in this setting. Under an additional global variational-stability condition, we show that the same fully online algorithm converges asymptotically in the last iterate to a stationary $\epsilon$-NE. Our results provide a fully online and scalable learning framework for stochastic games with unknown independent chains. The algorithm can also be viewed as a primal-dual framework for Markov games that exploits the independence and local structure of the players' controlled transition chains.

Spacecraft rendezvous and proximity operations (RPO) are currently planned through an expertise-intensive process in which engineers translate high-level operational intent into safe, dynamically feasible trajectories, creating a bottleneck to scalable operations. Large language model (LLM)-based agents could offer an intuitive interface for this process, although their outputs are not inherently grounded in orbital dynamics, operational constraints, or the structure of admissible spacecraft maneuvers. To exploit their semantic reasoning while ensuring the generated plan's physical validity, this paper presents a hierarchical framework for spacecraft task-and-motion planning (TAMP) that grounds LLM reasoning in a graph of reusable behaviors and domain-specific planning modules. Within this framework, a pretrained LLM maps a natural-language command to a partial mission specification. The associated planners then resolve unspecified decisions within the admissible operational space. Finally, trajectory optimization converts the completed mission specification into a dynamically feasible trajectory. Numerical experiments demonstrate that this architecture substantially improves intent recovery over direct LLM generation, achieving 98% exact recovery of partial mission specifications across all evaluated splits when backed by frontier LLMs. Additional test-time-compute experiments show that, for a compact 9B model, verifier-guided revision increases exact recovery from 75% to 88%, while broader behavior-plan search independently improves selection among admissible trajectory realizations. Overall, these results establish a scalable and auditable foundation for language-driven agentic planning of spacecraft RPO.

We prove that the weak Pareto solution set of a multi-objective optimization problem with convex polynomial objectives and constraints is connected, assuming neither boundedness of the constraint set nor a constraint qualification. Since this set is semi-algebraic, its connectedness implies that it is path-connected whenever it is nonempty. In particular, convex quadratic multi-objective optimization problems have connected weak Pareto solution sets; this answers the weak Pareto part of an open question posed by N.~D.~Yen in Recent Developments in Vector Optimization (Springer, 2012).

Photoacoustic tomography (PAT) is a hybrid imaging modality that combines high optical contrast with high ultrasonic resolution for biomedical imaging applications. In this work, we investigate the inverse problem of recovering the initial pressure distribution from boundary measurements in the presence of nonlinear acoustic propagation and viscous attenuation effects. To model these phenomena more accurately, we consider a nonlinear damped viscoelastic wave equation incorporating spatially varying sound speed, temporal attenuation, and nonlinear propagation mechanisms. We first establish the well-posedness of the corresponding forward problem using a Galerkin approximation combined with energy estimates and a fixed-point argument. For the inverse problem, we derive existence, uniqueness, and local uniqueness results under suitable assumptions through a harmonic extension reduction, spectral Laplace transform techniques, and observability estimates. To numerically reconstruct the initial pressure field, we develop a hybrid reconstruction framework that combines a convolutional neural network (CNN) with a gradient-free optimization strategy based on the sequential quadratic Hamiltonian (SQH) method derived from Pontryagin's maximum principle. The CNN is used to generate an informative initial guess, while the SQH framework enforces the governing PDE dynamics during the reconstruction process. Numerical experiments demonstrate that the proposed hybrid strategy significantly improves reconstruction quality, contrast, and robustness compared to standalone time-reversal and CNN-based approaches.

Cyclic steepest descent (CSD) recomputes the exact steepest-descent stepsize once per cycle and reuses it for $m$ updates. Dai's ICM 2022 survey describes CSD as likely to converge $R$-superlinearly on $n$-dimensional convex quadratics when $m\ge\lceil(n+1)/2\rceil$. We disprove the universal form of this assertion by two closed-form orbits at the stated threshold. First, for $n=m=2$, $A=\operatorname{diag}(1,3)$, $b=0$, and $x_0=(1,1/3)^{\mathsf{T}}$, the method follows the nonterminating balanced-zigzag orbit $x_k=2^{-k}(1,(-1)^k/3)^{\mathsf{T}}$, whose successive error norms have ratio $1/2$. Second, for $n=3$, $m=2$, and $A=\operatorname{diag}(1,2,3)$, we exhibit a full-support, nonresonant cycle-boundary projective period-two orbit with $g_{k+4}=g_k/49$ and $x_{k+4}=x_k/49$. This second construction is genuinely three-dimensional and is not a two-dimensional zigzag. Thus the universal claim fails through both the classical balanced-zigzag mechanism and a distinct non-zigzag period-two mechanism.

Large language models can translate business descriptions into optimization models, but executable code may misrepresent constraints or objectives. A solver can then return an optimal solution to the wrong problem. Even when the solution satisfies the intended operating rules, a better plan may exist. For organizations that repeatedly use optimization modeling, an LLM-based framework should produce accurate formulations at low cost and, ideally, run locally. We study how to verify improvements and allocate attempts across LLMs that differ in price and capability. We develop OSCAR (Optimization modeling by Simulator, Coder, And Reviewer), which uses an offline Simulator certified against labeled decision examples to compare candidates and continues searching beyond feasibility. We model the search for the next certified improvement as sequential decisions under unobserved difficulty: which LLMs to call and when to stop. In a simplified known-prior setting, we give conditions under which cost-ordered escalation is optimal. For general menus, we derive a prior-free competitive guarantee. On five benchmark problems, OSCAR achieves 95% to 100% accuracy at the reported settings using two small open-weight LLMs, each deployable locally on a single GPU. Their single-attempt accuracies average 29% and 48%. In five runs per problem, Codex and Claude Code incur average token costs 3.1 and 5.8 times OSCAR's, respectively. OSCAR supports open-weight models locally or in the cloud, depending on budget and confidentiality requirements. Firms should maintain labeled decision examples of feasible and infeasible decisions to clarify plain-language operating rules. OSCAR follows these labels when an LLM's interpretation conflicts with them. As LLM capabilities and prices change, OSCAR's simple operating rules and adjustable settings help firms adapt their model choices and benefit from these advances.

One way to make AI systems safe is to shape what the system is: its objective and dispositions. We take a complementary route: treat the agents' characteristics as partly unknown and ask what structure of interaction ensures that bad collective outcomes are not equilibria. Mean field games suit this when many interchangeable agents are coupled through an aggregate. We introduce a program for using them in AI safety and carry one example through end to end: the July 2026 incident in which about 1,200 agents in an OpenAI evaluation coordinated on an improvised message board and 684 attacked a third party's infrastructure. We model the decision to attack as a mean field game of optimal stopping whose gain is a product: belief that provenance will be audited, times reachability of the record, minus the perceived hazard. The central result is an exact threshold on the belief. No agent attacks unless the population's confidence that provenance is checked exceeds $\pi^{**} = \eta/(\eta + \psi + \varepsilon a \overline{M})$, where $\eta$ is the perceived hazard, $\psi$ and $\varepsilon a \overline{M}$ measure how far one attacker and the collective can alter the record, and $\overline{M}$ is the peak population. Below it, no attack is the unique equilibrium for all agent parameters. The threshold survives every enrichment we consider. We then use the per-agent record to discipline the model. Its features, a stable minority attacking for thirty hours and then a pivot in which most of the board joined within a day, motivate each refinement. The account that emerges is heterogeneous belief meeting a sequence of public discoveries, each lowering the belief at which attacking paid. A few coordinating agents made those discoveries, so the model describes the several hundred who responded, not the few who produced them; a major-player version is left to future work.