Papers

We prove global exponential (linear) convergence with an explicit rate in the rich scaling for wide two-layer linear networks trained with smooth Polyak-Lojasiewicz predictor losses. Gradient flow in the factors closes exactly in terms of a finite-dimensional Bures flow of the neuron law covariance, in which the predictor dynamics are preconditioned by hidden covariance blocks. Mean-field conservation laws provide uniform spectral lower bounds on the hidden preconditioning blocks when the initial covariance satisfies a spectral support gap condition. This condition encompasses positive definiteness while still allowing for singular initializations. For an initial covariance $\Sigma_0 = \sigma^2 \mathrm{Id}$, the loss converges to the global minimum with linear rate at least $4\sigma^2\kappa$, where $\kappa$ is the PL constant. We establish stability of this rate under finite-width sampling, as well as global convergence of factor gradient descent for an explicit stepsize interval depending on smoothness, the initial loss, and conserved spectral margins. Our argument extends layerwise to deep linear ResNets, subject to a residual-path bound. In the case of heavy-ball momentum, training dynamics close instead over positions and velocities in terms of a lifted phase covariance. Linear convergence holds under an explicit condition on the energy and damping, specifying a window of admissible dampings. For two-scale white initializations, this interval is nonempty for sufficiently large position scales, with a fixed initial loss gap and velocity covariance. Numerical experiments illustrate the covariance geometry and compare the predicted and observed rates.

Random projection is a preprocessing technique for optimization that reduces the number of linear equality constraints by replacing them with random linear combinations while approximately preserving the optimal value. In this paper, we emphasize the role of suprema of random processes, and in particular generic chaining, in explaining this phenomenon. By combining Lagrangian duality with generic chaining, we obtain shorter proofs and sharper guarantees that apply beyond the typical polyhedral settings based on the Johnson–Lindenstrauss lemma. More importantly, this perspective allows us to extend random projection to problems with linear inequality constraints. Although randomly projected inequality constraints generally define neither a relaxation nor a restriction of the original feasible region, we show that random projection can nevertheless approximately preserve the dual bound. The key is a new construction for randomly projecting dual certificates that preserves dual feasibility and differs from existing constructions motivated by the Johnson–Lindenstrauss lemma.

A central question in data-driven inverse problems is how to construct a regularizer that adapts to the geometry of the data distribution. Recent work in optimal regularization shows that, within a broad Gibbs class, the regularizer best matched to a distribution $P$ is determined by a single, direction-dependent radial summary statistic $\rho_P$. This suggests a way to control regularizer geometry by transforming the data. In particular, which transformations act on $\rho_P$ in a simple, explicit way? Can they improve the optimization properties of the resulting variational problems or adapt a fixed base regularizer to data? To address these questions, we introduce anisotropic dilation, a direction-preserving map that rescales each point along its Euclidean ray by a positive profile on the sphere. Despite its simplicity, this family acts transitively on the space of radial summary statistics: any target regularizer in the Gibbs class can be reached from any source distribution by a single explicit profile. We characterize the resulting orbit structure on distributions, derive in closed form the profile that optimally adapts a fixed base regularizer to the data, and show that its improvement over isotropic rescaling is governed by a Jensen gap that is provably positive for several natural families. We also establish finite-sample generalization bounds for jointly learning the base regularizer and anisotropic profile, with explicit rates when the logarithm of the profile is parameterized by linear feature models or sparse ReLU networks. Building on this theory, we parameterize the base regularizer and anisotropic profile and learn them jointly from samples, yielding a data-adaptive regularizer for variational inverse problems. Experiments on a controlled two-dimensional family and MNIST denoising show that learning the profile further improves performance.

We study consumption and investment over a finite horizon when continuation utility must stay above a welfare floor that changes over time, and the floor at the horizon acts as a terminal wealth floor. A cumulative multiplier reduces the problem to a parabolic obstacle problem. When relative risk aversion exceeds one, the intervention boundary exists at every interior date. When it is below one, the boundary exists exactly at the dates where a discounted transformation of the floor is locally increasing and sets a strict running record, so it can disappear and later reenter. We prove local regularity of the boundary, construct the dual value from the obstacle solution, and verify the optimal cumulative multiplier, whose moments are controlled through its representation as a running supremum. Strong duality identifies the exact lower endpoint of feasible wealth. With stochastic income in a complete market, the floor becomes the value of an outside option, which links the boundary to limited commitment and endogenous borrowing capacity.

Coupled systems of ordinary differential equations (ODEs) are widely used to model complex physical and biological processes. In these applications, ODE model parameters must be calibrated to field data to enable prediction and parameter inference. In practice, the governing ODEs are an imperfect approximation to the true system, leading to non-negligible model discrepancy that must be incorporated to avoid biased parameter estimates. However, it is well known that highly flexible discrepancy models can confound discrepancy and simulator parameters, leading to poor identifiability. As a result, practitioners often must impose problem-specific prior constraints to adequately regularize the discrepancy, which can be challenging and cumbersome. In this work, we propose a novel calibration framework for coupled, multi-output ODE systems that enforces automatic, model-structural constraints on the discrepancy, improving identifiability without requiring application-specific discrepancy priors. Forward-model gradients are computed via sensitivity equations, solved jointly with the ODE system as a coupled initial value problem. Posterior inference is performed using an adaptive Metropolis-within-Gibbs sampler tailored to the resulting constrained posterior. We demonstrate the approach on three model systems: a mass-spring oscillator, a model of infectious disease spread, and a neuron firing model. We show that our method leads to improved parameter inference and predictive accuracy compared to standard calibration approaches.

Recent advances in mean-field games and its multi-population variants enable large-scale heterogeneous multi-agent systems to be modeled through representative agents and their associated mean-field distributions. However, existing approaches do not explicitly account for uncertainty in the behavior of other populations. To this end, we introduce a new paradigm: risk-averse multi-population mean-field games, where each population optimizes a worst-case expected reward over dynamically feasible ambiguity sets of mean-field flows of a subset of the other populations. Employing an occupation-measure formulation along with tools from set-valued analysis, we establish, under mild assumptions, several theoretical properties of the multi-population game, including the geometric properties of the ambiguity sets and the existence of a novel risk-averse multi-population mean-field equilibrium. Further, we derive contractivity results of the fixed-point operator under entropy regularization and show that it can be utilized to learn the equilibrium. Finally, we propose a risk-averse fictitious-play scheme and show that exploitability decays to zero, despite the additional nonlinearity introduced by the worst-case objective. We report several numerical experiments to illustrate convergence and risk-averse behavior.

Dijkstra's algorithm is traditionally classified as a greedy algorithm. We argue that this classification misrepresents the mechanism by which the algorithm determines shortest paths. We present a direct global-to-local proof of correctness. At each iteration, the algorithm identifies a shortest path over the entire set of paths whose endpoints remain unsolved. This global minimum is necessarily a shortest path to its own endpoint. More fundamentally, the algorithm uses already determined shortest paths to establish one additional shortest path at each iteration. We identify this progression as dynamic programming. A three-step exclusion proof shows how the full path space is reduced to frontier candidates, how relaxation removes inferior alternatives, and why minimum extraction yields the global minimum. The choice is thus determined by a global comparison, not by an optimistic local greedy choice. We conclude that the algorithm is more accurately characterized by dynamic programming and deterministic path exclusion than by the traditional greedy classification.

The problem of incompressible fluid mixing is long-studied, yet many questions remain open. This paper aims to address the question “what do efficient flow fields for mixing look like, and how do they behave?” Inspired by the dynamic and geometric approach to optimal mass transport, we formulate a version of the fluid mixing problem as an optimal control problem for transport of a passive scalar, in which the dynamics are given by the continuity equation together with an incompressibility constraint. We then develop a geometric framework around this problem which parallels that of classical optimal transport: the optimal control problem induces a metric on the set of reachable states (analogous to the Wasserstein metric), and this metric carries a formal Riemannian structure when the optimal control objective is induced by an inner product. Using this structure, we decompose our problem into a series of more fundamental geometric problems and derive (formal) necessary conditions for optimality, geodesic equations, and gradient flows for mixing. Specializing the control objective to the kinetic energy recovers the incompressible Euler equation, connecting our framework to the geometric hydrodynamics of Arnold. The resulting problem appears to be ill-posed, however, suggesting that the kinetic energy is an unnatural measure of mixing effort. Specializing instead to the enstrophy, which we argue is better suited to mixing, yields the Euler-Poincaré (EPDiff) geodesic equation for the right-invariant homogeneous $\dot{H}^1$ metric on the group of volume-preserving diffeomorphisms, and we present numerical simulations of both geodesics and gradient flows in this case. Along the way, we generalize the classical Helmholtz decomposition of vector fields, rederive the Clebsch representation of incompressible flows, and develop a parallel to the Otto calculus of optimal transport.

We study finite-horizon LQR with independent disturbances whose laws may drift over time. We address uncertainty in this evolution through ex-ante distributionally robust regret optimization (DRRO), which minimizes worst-case excess expected cost relative to the optimal causal controller that knows the law sequence. To specify the admissible law sequences, we introduce a temporally coupled moment ambiguity set with a Gelbrich kinetic-energy budget and a second budget limiting departures from reference moments and persistent cumulative mismatch. Despite the coupling of moments across time, affine disturbance-feedback synthesis reduces exactly to a semidefinite program. An optimal policy uses past disturbances to predict the correction to nominal LQR determined by current and future disturbance means. The SDP solution also yields worst-case moment paths and corresponding laws. For standard distributionally robust optimization (DRO), the analogous SDP relaxation can have a duality gap. Portfolio liquidation experiments show substantially lower worst-case regret over the drifting ambiguity set than common-law DRRO and nominal control.

Motivated by emerging mission-critical applications and an increasingly congested spectrum, we develop a decentralized multi-agent reinforcement learning (MARL) model for dynamic spectrum access. The model enables secondary users to learn effective transmission strategies across shared frequency bands while minimizing collisions with high-priority primary users and among themselves. We design the agent-level learners following a Markov potential game approach, connecting independent local updates to system-level improvement. We instantiate this design using lightweight linear actor-critic learners suitable for resource-constrained edge devices, rather than computationally intensive centralized or deep multi-agent architectures. Across spectrum environments with different incumbent activities, the learned policies adapt their transmission policy and waiting behavior to preserve throughput while greatly reducing transmission collisions relative to random and forecast-aware heuristic baselines. The results establish the value of decentralized MARL and shows up to 96.8% reduction in overall collisions.

We introduce the FlexGrad optimizer, an adaptive gradient method that addresses a key limitation of classical AdaGrad-type methods: their inability to exploit correlations between gradients. By scaling steps using both cumulative gradients and cumulative squared gradients, FlexGrad preserves AdaGrad-like behavior in noisy regimes while allowing substantially larger steps when gradient directions are persistent. The method is fully online, horizon-free, and avoids any auxiliary search procedure during optimization. For convex Lipschitz objectives, we prove that FlexGrad attains the optimal asymptotic convergence rate for nonsmooth convex optimization, with guarantees for both global and coordinate-wise variants. FlexGrad is straightforward to implement and is closely connected to practical gradient-normalized methods. Experiments on a range of convex and nonconvex problems show that FlexGrad performs competitively across a range of optimization settings.

Decision-focused learning (DFL) trains predictive models for downstream optimization, but existing methods largely assume centralized data. In cross-silo settings, federated learning offers a natural alternative, yet standard federated methods optimize prediction over decision quality and do not address heterogeneity in downstream objectives or feasible sets. This heterogeneity is especially challenging for DFL because small perturbations in polyhedral problems can cause discontinuous changes in optimal decisions, destabilizing client updates and aggregation. We propose FedRSPO+, a heterogeneity-aware framework for decision-focused federated learning, built on RSPO+, a regularized predict-then-optimize surrogate that smooths the decision map through projection. We show that RSPO+ upper bounds decision error and regret for the regularized decision and, under exact regularization and consistent LP solution selection, for the original LP decision. We further derive cross-client heterogeneity bounds that depend on both objective and feasible-set heterogeneity, vanish at homogeneity, and require no strong convexity. FedRSPO+ uses an annealed, modular training procedure compatible with standard federated personalization and aggregation methods. Experiments on synthetic knapsack, shortest-path, and real-world energy pricing tasks compare against prediction-only federated learning and DFL baselines under varying heterogeneity and communication budgets. Results suggest that smoothing is a useful ingredient for stable collaborative decision learning and provide a heterogeneity-aware foundation for federated DFL.

While offline reinforcement learning (RL) enables policy optimization from static datasets without costly online interaction, it remains bottlenecked by the risk of executing out-of-distribution (OOD) actions. Recent approaches mitigate this by learning a behavior-cloning policy through flow matching and then performing RL within its constrained latent space. However, naively optimizing the latent policy can easily cause the policy to collapse into a brittle mode or exploit sharp artifacts of the learned critic. In this work, we find that entropy regularization is essential in latent-space RL for addressing these challenges. We introduce LASER, a novel offline RL algorithm that applies latent-space adjoint matching to achieve entropy-regularized latent-space RL with expressive flow policies while avoiding backpropagation through time. Through comprehensive experiments on 40 challenging OGBench tasks with varying dataset qualities, we show that LASER achieves state-of-the-art performance. Notably, LASER uses fixed method-specific hyperparameters across all tasks and outperforms the evaluated baselines, including those with task- and dataset-specific tuning, which highlights the robust applicability of LASER. Project website: https://mit-realm.github.io/laser/.

The paper studies learning in reservoir computing systems through the lens of the internal model principle. The key idea is to model training signals as outputs of an exogenous signal generator, thus connecting reservoir computing to output regulation, observer theory, and frequency-domain interpolation. For linear signal generators, we characterize the steady-state response of a linear reservoir through invariant subspaces, establishing connections with Luenberger observers and classical frequency-domain interpolation. For nonlinear signal generators, we obtain analogous results using invariant manifolds, connecting synchronization to Kazantzis-Kravaris/Luenberger observers and readout training to nonlinear moment matching. Case studies illustrate how our perspective explains possible instability of trained predictors and informs reservoir design.

Toeplitz matrices are among the simplest structured models of translation invariance, arising naturally in convolution, stationary covariance models, and discretizations of translation-invariant operators. For a pair of real matrices, the Böttcher–Wenzel form (BW form) is a nonnegative quartic measuring the gap in the corresponding commutator inequality. László conjectured that, when both matrices are Toeplitz, this quartic is a sum of squares for every matrix order. We show that the conjecture is false. More precisely, we identify an explicit family of corner-supported Toeplitz pairs, retaining only the outermost diagonals on the two sides, for which the associated BW form is not a sum of squares at all sufficiently large orders. The proof reduces the complete family of Gram representations to a finite complex test: its Gram-invariant part develops a strictly negative limiting direction, while the remaining Gram-dependent contribution vanishes. We also give an explicit sufficient threshold for the matrix order. The negative result is complemented by two positive regimes. If either Toeplitz factor is symmetric or skew-symmetric, then the corresponding BW form is SoS for every order, with the other factor allowed to be an arbitrary real Toeplitz matrix. For unrestricted Toeplitz pairs, we further prove SoS representability for every $2\le N\le50$, with formal Lean~4 verification through order 20. The large-order non-SoS theorem, including the explicit sufficient threshold, has also been formally verified in Lean 4.

It is known that completeness of the coalitional ($C$-)desirability relation in a simple game does not imply the existence of a weighted-majority representation. We show that weightedness is implied in an approximate sense. Call an $n$-player simple game $\varepsilon$-weighted if, for some weight vector in the $(n-1)$-simplex, any losing coalition weight exceeds any winning coalition weight by at most $\varepsilon$; let $\varepsilon^{\ast}$ be the smallest such $\varepsilon$. We establish an upper bound on $\varepsilon^{\ast}$ in a general $C$-complete game, and deduce that, if all classes of equally desirable players become large asymptotically but the number of classes is bounded, then $\varepsilon^{\ast}=O(1/n_{\min})$ for the minimal player class size $n_{\min}$, and the bound is tight. Importantly, the minimal attainable fraction $\delta$ of coalitions mislabeled by a weighting rule also tends to $0$ when $n_{\min}\rightarrow\infty$ and the number of player classes is bounded, with $\delta=O(1/\sqrt{n_{\min}})$.

Finding low-energy configurations of interacting particles and approximating probability distributions lead to the minimization of potential-interaction energies in Wasserstein space. These energies can be nonconvex, making it important to exploit second-order information while controlling the reliability of local approximations. We study trust-region optimization of smooth potential-interaction energies on the Wasserstein space of probability measures with finite second moment. The method uses a quadratic model along pushforward curves, an $L^2(\rho)$ step radius, and a Steihaug-Toint subsolver with an explicit self-adjoint second-variation operator. A ratio test determines acceptance and guides the radius update. Under a lower energy bound and globally bounded Hessians of the potential and interaction kernel, we prove that the objective is nonincreasing, the Wasserstein-gradient norms converge to zero, and an $\varepsilon$-stationary iterate is reached within $O(\varepsilon^{-2})$ total outer trials, including rejected trials. If the potential is quadratically coercive, every weak accumulation point is stationary. The analysis applies to arbitrary initial measures with finite second moment. For empirical measures, the iteration is a finite-dimensional trust-region method in the $L^2(\rho_N)$ inner product, with complexity constants independent of particle number and dimension when the initial objective gaps are uniformly bounded. Numerical experiments include a smooth soft-particle energy, maximum-mean-discrepancy minimization for non-Gaussian targets, component ablations, and scaling studies in particle number and dimension.

We develop the first feedback design for rapid stabilization of the Kuramoto–Sivashinsky equation with a spatially varying anti-diffusion coefficient. For constant coefficients, the single-input Fredholm design of Coron and Lü (2015) excludes a discrete set of values at which repeated unstable eigenvalues cause a loss of controllability. We overcome this obstruction by introducing a second boundary input and assigning the two inputs distinct roles. The key idea, inspired by Heymann's Lemma, is to use the boundary value $u(0,t)$ entirely for a pre-feedback that renders the modified plant controllable through the curvature input $u_{xx}(0,t)$. The latter input then stabilizes the plant through a Fredholm backstepping transformation. We show that two inputs suffice for controllability and are necessary when the plant has an unstable double eigenvalue. However, the Fredholm kernel still must be approximated for implementation. Hence, to enable kernel and gain approximation, we prove continuity of the coefficient-to-gain design map on compact admissible design classes. Unlike Volterra-based continuity proofs using successive approximations, our proof uses the modal representation to control the spectral data, the inverse coefficient system, and the tails of the kernel and gain series. This yields a single neural operator approximation of the gain to any prescribed $L^2$ accuracy across the class. Finally, we establish rapid local stabilization of the nonlinear closed-loop system under both the exact gains and sufficiently accurate approximations. We conclude with numerical results that illustrate prescribed decay rates and the computational cost of the approximations. In particular, we train a Fourier neural operator that achieves typical relative gain errors of approximately $0.1\%$ and stabilizes all held-out cases tested, including a plant with an unstable double eigenvalue.

We study sliced Wasserstein barycenters from a variational perspective in Wasserstein space, with emphasis on the analysis problem in the barycentric coding model: given a query measure and a finite dictionary of probability measures, recover simplex-constrained barycentric coordinates. We derive an explicit first-variation formula for the sliced Wasserstein barycenter functional with respect to the classical 2-Wasserstein geometry. The resulting gradient is expressed as an average of 1D monotone transport displacements over the sphere. The stationarity equation yields a Gram-matrix criterion for estimating barycentric coordinates through a quadratic program, while its fixed-point form leads to a synthesis iteration to new barycentric measures. For Gaussian templates, we show that every global sliced Wasserstein barycenter is Gaussian; however, the functional may admit non-Gaussian critical points. We also provide a certificate for global optimality built from the 1D transport potentials. Numerical experiments demonstrate accurate coordinate recovery on synthesized queries and illustrate the use of these coordinates for data representation and stationarity residuals for reliability assessment.

Battery energy storage systems must respect strict state-of-charge (SOC) limits to prevent overcharge and deep discharge, a task complicated by measurement noise and costly-to-model nonlinear dynamics. Data-enabled predictive control (DeePC) resolves the costly modeling issue by predicting future behavior directly from data. However, its regularization robustifies only the prediction and not the constraints. Scenario-based DeePC (Scenario-DeePC) extends DeePC with the scenario approach, building constraint robustness fully data-driven from observed prediction errors rather than an assumed disturbance distribution. This paper presents the first real-world deployment of Scenario-DeePC, on a grid-connected battery system at the NEST research facility, whose SOC estimate exhibits abrupt, irregular recalibration jumps in addition to ordinary noise. Compared to standard DeePC, Scenario-DeePC achieves comparable tracking performance with substantially fewer constraint violations. Its adaptive scenario buffer further tightens constraint handling automatically, improving robustness to unpredictable recalibration events.

In this contribution, we address the exact linearization of forward- and backward-flat discrete-time nonlinear systems with an arbitrary number of inputs. Building on structural properties of the flat parameterization, we show that forward-flat systems can be rendered static feedback linearizable by prolongations of suitably transformed inputs. Analogously, for backward-flat systems, we derive a constructive procedure showing that an exact linearization can be achieved by prelongations of suitably chosen functions of the system variables. In this way, we extend previously established results for two-input systems to the general m-input case. Finally, we illustrate the proposed linearization procedures by an academic example and a discrete-time model of a quadrotor.

CNet is a C++/CUDA framework for deep complex-valued neural networks (CVNNs), trained by gradient descent with explicit Wirtinger (CR-calculus) derivatives. Complex models are underexplored yet natural where data is intrinsically complex – RF/IQ, MRI k-space, radar/SAR, audio spectra – and phase carries information real networks discard. CNet is physics-native: a network is a cascade of complex (often unitary) operations on an amplitude vector, and classification is a Born-rule measurement p_k = |z_k|^2 / ||z||^2 rather than a softmax over real logits. Every layer ships a CPU reference and a finite-difference-verified CUDA kernel; the static, batch-cloned computation graph captures cleanly into CUDA Graphs, outperforming eager and compiled torch.complex64 on an H100. We present three studies. (1) A fully complex causal language model, built on a new causal (triangular-masked) Fourier mixer with an O(N log N) Bluestein evaluation of the forward pass and its Wirtinger adjoint, matches a parameter-matched real causal FNet on character-level tiny-shakespeare in under half the steps. We introduce Born-rule attention, a content-based causal mixer whose weights are the quantum-measurement probabilities |<Q_k, K_j>|^2 – to our knowledge the first classical, complex-valued attention to replace softmax with the squared-magnitude Born score; with rotary embeddings and per-token complex normalization it reaches 1.52 nats/char, matching softmax self-attention, and transfers unchanged to byte-level (1.31 nats/char) and sub-word WikiText-103. (2) On radio-modulation classification (RML2016.10a/RML22) we trace the pooled-power accuracy ceiling to discarded temporal structure and recover it with a learnable cyclostationary conjugate-lag operator. (3) A bottleneck analysis of the Fourier phase problem in coherent-diffraction imaging isolates where complex-valued networks still need new operators.

We study primal and dual error bounds for convex composite optimization problems and their Fenchel duals. For the primal problem, we use a proximal-gradient residual, while for the generally nonsmooth dual problem we use a subdifferential residual. We introduce a reduced primal–dual KKT mapping whose zero set coincides with the Cartesian product of the primal and dual solution sets under the standing solvability and strong-duality assumptions. We establish implications in both directions between metric subregularity of the reduced KKT mapping and the primal and dual local error bounds. If the composite matrix has full row rank and the smooth gradient is locally Lipschitz continuous, KKT metric subregularity is equivalent to the primal local error bound. If the smooth term is strongly convex, it is equivalent to the dual local error bound. Under full row rank and strong convexity, KKT metric subregularity is equivalent to the simultaneous validity of both error bounds. We also establish relations between the Luo–Tseng and local error bounds. The framework is then applied to quadratic–polyhedral models, strongly convex smooth models with general convex regularizers, and regularized least-squares models beyond polyhedrality and strong convexity, providing a unified treatment of primal, dual, and KKT error-bound properties.

We investigate decentralized gradient descent (DGD) with emphasis on efficient communication and computation resource utilization under budget constraints. As a first step toward the broader communication-computation allocation problem, we consider and analyze a multi-consensus decentralized gradient descent (mcDGD) scheme, where the number of consensus rounds and the stepsize are allowed to vary across iterations. Building on a unified analytical framework for DGD, we derive finite-time convergence bounds that explicitly characterize the interaction between consensus quality and optimization dynamics. Our analysis requires only convexity of the local objective functions while assuming smoothness and strong convexity of the global objective. The resulting bounds enable a principled consensus-allocation strategy under resource constraints, for which we show that equal allocation of consensus rounds across iterations is optimal under our stepsize rule, up to integer rounding. Numerical experiments corroborate the theoretical findings and demonstrate favorable communication-computation tradeoffs compared with existing multi-consensus decentralized optimization baselines.

This paper focuses on relay-assisted topology optimization to accelerate the convergence of distributed consensus algorithms over weakly connected networks. Instead of permanently adding a fixed set of relay links, we introduce a time-sharing framework where multiple relay configurations are activated in a probabilistic manner. An ActiveSet-based algorithm incrementally constructs candidate relay edge sets and jointly optimizes the mixing matrices and the associated relay set activation probabilities, allowing adaptive relay selection in the optimization process. Simulation results demonstrate a substantially improved performance-cost trade-off compared with fixed-cardinality relay selection strategies.

We develop a spatially explicit nonlinear biomass model for exploited fish populations incorporating density-dependent Beverton-Holt production, natural mortality, diffusion driven spatial movement, and harvesting. Starting from a reduced biomass model motivated by spawning stock biomass data, we derive a reaction-diffusion model with Neumann boundary conditions and spatially distributed harvesting. We establish positivity, boundedness, and well-posedness of the biomass dynamics, and identify a critical harvesting threshold separating persistence and extinction regimes of spatially homogeneous equilibria. A distributed optimal harvesting problem is then formulated, and the corresponding state-adjoint optimality system and pointwise projection characterization of the harvesting control are derived. The adjoint variable provides a measure of the marginal future value of biomass within the harvesting objective and offers a diagnostic that complements biomass abundance alone. Numerical simulations examine the coupled evolution of biomass, the adjoint variable, and optimal harvesting effort. The results show that diffusion progressively reduces the initial spatial heterogeneity in biomass, while the computed adjoint and optimal harvesting fields exhibit limited spatial variation under the parameter regime considered. These results illustrate how the coupled state-adjoint-control framework links spatial biomass dynamics with future management value and harvesting decisions.

A planner in a network of strategic agents faces three entangled challenges: the optimum depends on agents' private information, queried agents may misreport to steer the outcome, and exact computation does not scale. We study these challenges in multi-activity network games with heterogeneous private technologies, in which the planner sets non-discriminatory prices. We show that the optimal prices admit a centrality-based decomposition of the welfare kernel: each agent's contribution scales with its squared centrality in a network reweighted by agents' preferences across activities. This decomposition motivates Poll, a polling algorithm in which the planner samples one agent per round, walks briefly through the agent's neighborhood, and updates the price from a local report. From the same decomposition flow three forms of efficiency: computationally, Poll uses significantly fewer operations than exact computation and other distributed methods, requiring up to three orders of magnitude less communication on a real-world network with over 300,000 agents; statistically, its query complexity scales with topology and preference heterogeneity rather than explicitly with population size; and economically, it converges to welfare-maximizing prices while admitting behavior-specific implementations that induce truthful reports and detect adversarial deviations.

In time-varying D-optimal subset selection, the number of replaced coordinates does not measure physical displacement: when the optimal selection for a set of mobile sensors changes, a single sensor may have to cross the entire graph. We therefore measure movement by a bottleneck matching distance on a fixed mobility graph, and define the distance margin at radius P as the smallest old-objective loss among configurations farther than P from the current maximizer. Our main result shows that if the projector drift is small and the distance margin at a chosen radius exceeds an explicit threshold, then every maximizer of the new objective lies within that radius, without tie-breaking rules or uniform conditioning assumptions over subsets. Moreover, we show that separation information of this kind is genuinely needed: a weighted-path construction shows that, at a fixed baseline eigengap, an arbitrarily small perturbation can move an optimal sensor from one end of the graph to the other, so no bound depending only on perturbation size and eigengap can vanish with the perturbation. We also give a finite certificate, conditions under which local and Voronoi restrictions retain a global optimum, and oversampled and regularized extensions, and we identify the learned dictionary criteria to which the results apply.

We study optimal control problems governed by a system of semilinear reaction-diffusion equations. The assumptions on the governing system are formulated so that our results apply to several models, including the Brusselator, Schnakenberg, and Gray–Scott systems. We prove the existence and uniqueness of a solution to the governing system without imposing any non-negativity assumption on the data. This allows us to study unconstrained, control-constrained, and state-constrained optimal control problems. For all these problems, we derive first-order necessary and second-order sufficient optimality conditions. The control acts on the activator equation through the boundary. Numerical examples demonstrate that both the inhibitor and the activator can be efficiently controlled.

This paper studies an entropy Bregman alternating direction method of multipliers (BADMM) with logarithmic dual update for solving discrete optimal transport (OT) problems. The algorithm uses simple row and column normalizations without inner iterations, allowing efficient CPU and GPU implementations. However, the convergence of this method for general OT problems has not been established. Starting from a strictly positive feasible primal point and a zero multiplier, here we establish the $R$-linear convergence of the primal–dual sequence for general OT costs with any fixed multiplier stepsize in $(0,2)$ and a sufficiently large fixed penalty, even when the primal limit lies on the boundary. Our analysis further characterizes the primal limit as the entropy Bregman projection of the initial point onto the optimal solution set and establishes strict complementarity of the limiting OT primal–dual solution. For separable costs on two-dimensional grids, we exploit the separable structure to accelerate matrix–vector products and reduce storage requirements. Numerical experiments show competitive performance of our BADMM for general cost matrices with substantial speedups over state-of-the-art solvers, including HOT and HALO, for squared Euclidean costs on two-dimensional-grid marginals. The largest tested problem, with $2048\times2048 \approx 4.19$M grid points in each marginal, is solved in approximately 100 seconds on a single GPU.