Papers

We establish a stochastic control representation for $L^p$-norms on Wiener space. For every $p\ge1$ and every non-negative universally measurable functional $\varphi$, we show that $$\|\varphi(W)\|_p = \sup_a \mathbb{E}\Big[ e^{-\frac12\int_0^T\|a_t\|^2\,dt} \varphi\Big( W+\sqrt{p-1}\int_0^\cdot a_t\,dt \Big) \Big],$$ where $W$ is a $d$-dimensional Brownian motion and the supremum is taken over all progressively measurable controls $a$ satisfying $\int_0^T\|a_t\|^2\,dt<\infty$ almost surely. This identity can be viewed as a multiplicative $L^p$-analogue of the Boué–Dupuis variational formula. We also provide an extension to strong solutions of stochastic differential equations with path-dependent coefficients. Building on the variational formula, we discuss finite-$p$ estimates for small-noise limits of stochastic differential equations with path-dependent coefficients. These include quantitative Freidlin-Wentzell bounds, a functional Gaussian approximation and concentration estimates, and moderate-deviation bounds. Moreover, we discuss consequences for functionals determined up to a stopping time, obtaining in particular probability bounds that identify the sharp exponential decay rate after optimization over controls. The proof of the variational formula combines probabilistic arguments with convex duality methods, viscosity theory for Hamilton-Jacobi-Bellman equations, and a dynamic programming principle.

Proximal bundle methods (PBMs) are classical algorithms for nonsmooth convex optimization. Existing analyses of the classical PBM couple the null steps with the descent test. This coupling obscures how the bundle updates approximate the proximal subproblem. In this work, we consider composite objectives $F=f+h$ and view each null-step cycle as an inner bundle subroutine, called $\mathtt{ProxBundle}$. We analyze $\mathtt{ProxBundle}$ independently of any stopping criterion under general bundle model conditions and show that it automatically adapts to Hölder smoothness. Combining this inner-loop analysis with the descent-step analysis yields sharper complexity bounds for the classical PBM. For any fixed proximal parameter, its overall complexity is $\mathcal O\big(\epsilon^{-\frac{3-\nu}{1+\nu}}\big)$ for $\nu\in[0,1)$ and $\mathcal O(\epsilon^{-1})$ for $\nu=1$, where $\nu$ is the Hölder smoothness exponent. Choosing the proximal parameter proportional to $\epsilon$ improves the rate to $\mathcal O\big(\epsilon^{-\frac{2}{1+\nu}}\big)$ for $\nu \in [0,1)$. These are the first guarantees under Hölder smoothness with $\nu\in(0,1)$ for the classical descent test. We further introduce an absolute model-error test. The resulting PBM variant admits a clean inexact proximal-point analysis, and for any fixed proximal parameter, achieves the same complexity $\mathcal O\big(\epsilon^{-\frac{2}{1+\nu}}\big)$. Overall, our analysis separates the roles of the descent and null steps and gives a modular understanding of PBMs across different~tests.

We introduce loss-conditioned state execution, a model-agnostic method that decides whether to execute a world model's fixed feasible proposal or retain the current state. Predictive informativeness alone, however, does not establish whether an update will reduce downstream loss. Occurrence ranking can approach perfection while persistence remains the unique absolute-loss Bayes action. Two transition laws can also share occurrence information and conditional variance yet require opposite absolute-loss decisions. We formalize state movability as the existence of a loss-reducing feasible correction and distinguish it from the benefit of a particular proposal. Our method constructs a loss-specific feasible proposal from a predictive distribution and evaluates its groupwise bounded-loss gain over persistence on independent calibration units. The proposal is executed only in groups with a positive simultaneous lower confidence bound. For fixed proposals and groups with bounded unit losses, we prove that every accepted group has lower expected loss than persistence with high probability when calibration units are i.i.d. draws from the target population. Experiments on public forecasting and action-conditioned dynamics benchmarks show supported updates and a trade-off between certification and coverage. On 28,684 held-out M4 Monthly series, the method executes the proposal for 14.0% of series and achieves bounded loss 0.588, compared with 0.599 for persistence and 0.621 for always executing the proposal. The paired 95% bootstrap intervals for both comparisons lie below zero. In constrained forecasting of six unhealthy-inventory types from JD$\mbox{.}$com, a leading e-retailer in China, strong occurrence-ranking signal coexists with a loss-based preference for persistence, illustrating why event predictability and state execution must be evaluated separately.

Dissipativity theory enables the stability analysis of complex, interconnected systems through the physical properties of their components. Celebrated examples of such stability results include the small gain and passivity theorems. In this paper, we prove a unifying stability theorem for dissipative linear time-invariant (LTI) systems. We use the behavioral approach to express storage functions and supply rates in terms of quadratic differential forms (QDFs). Our main result establishes conditions for the stability of an interconnection of two LTI behaviors that are both dissipative with respect to general, dynamic supply rates. As special cases, we recover existing results for static supply rates, such as those capturing finite $L_2$-gain and passivity, as well as dynamic ones capturing, for example, delta dissipativity. The key idea of the paper is to construct a Lyapunov function as a sum of storage functions and a coupling QDF, thereby allowing for a richer class of Lyapunov functions than existing works dealing with just sums of storage functions.

Traditional variance reduction methods (e.g., SPIDER, SARAH, STORM) have been extensively investigated for improving the convergence rates of stochastic optimization. These techniques typically maintain a sequence of estimators for a single function (or gradient) across iterations. However, what if we need to track multiple functions, but can only access stochastic samples of $\mathcal{O}(1)$ functions at each iteration? This scenario arises in an important emerging family of finite-sum coupled compositional optimization (FCCO) problems of the form $\frac{1}{m}\sum_{i=1}^m f_i(g_i(\mathbf{w}))$, where each $g_i$ is accessible only through a stochastic oracle. The key challenge is to track $\mathbf g(\mathbf{w})=(g_1(\mathbf{w}), \ldots, g_m(\mathbf{w}))$ over time, where $\mathbf g(\mathbf{w})$ has $m$ blocks but only $\mathcal{O}(1)$ blocks can be probed for their stochastic values at each step. To address this challenge, we propose a novel Multi-block-Single-probe Variance Reduction (MSVR) estimator to efficiently trace $\mathbf g(\mathbf{w})$ under partial block sampling. Building on the MSVR estimator, we develop several algorithms for FCCO problems, achieving improved sample complexities for non-convex, convex, strongly convex, and Polyak-Łojasiewicz (PL) objectives. We further obtain an improved dependence on $m$ when the outer function gradients $\nabla f_i$ are linear. Empirical studies on multi-task deep AUC maximization further demonstrate the superior performance of the proposed estimators.

Quadrotor navigation in cluttered 3-D environments requires global guidance while local motion remains subject to collision and motion limits. Harmonic potentials provide dense guidance from a global boundary value problem, but coupling a volumetric harmonic field to constrained physical quadrotor motion remains an open experimental problem. We couple a precomputed volumetric harmonic field with a constrained predictive planner that queries the field at predicted positions instead of extracting a global reference path. In Structured 3-D tests, harmonic guidance yields larger minimum clearance and lower RMS jerk than matched Dijkstra guidance, at the cost of longer paths; the same pattern remains when both methods use the same passage. Long maze tests span routes far beyond one prediction horizon, and Crazyflie trials validate physical execution. To the best of our knowledge, this is the first physical quadrotor demonstration of volumetric harmonic field navigation. The results show that globally constructed harmonic guidance can directly support local constrained motion generation on a physical quadrotor.

This paper studies projection-free algorithms for stochastic constrained multi-level compositional optimization. In this context, the objective function is a nested composition of several smooth functions, and the decision set is closed and convex. Since projection onto the constraint set can be computationally expensive, we develop projection-free methods that rely on linear minimization oracles. For non-convex objectives, we propose variance-reduced projection-free algorithms and establish complexity guarantees under both the Frank-Wolfe gap and the gradient mapping criteria. We also develop momentum-based methods that achieve convergence guarantees under weaker smoothness assumptions. Additionally, by using a stage-wise design, we derive a parameter-free variant that preserves the same complexities for the Frank-Wolfe gap. Such a design can be further used to develop algorithms for convex and strongly convex functions whose rates match those of single-level projection-free counterparts. Finally, we consider finite-sum problems and derive complexities for non-convex, convex, and strongly convex objectives. Numerical experiments across multiple tasks demonstrate the effectiveness of the proposed methods.

We prove global Sobolev convergence of Howard policy iteration for finite-horizon Bellman equations with control-dependent diffusion. In one dimension, the result holds for bounded measurable coefficients under uniform ellipticity, without a large discount, a short horizon, a small diffusion perturbation, or regularity assumptions on the improving policies. We also obtain multidimensional extensions under structural conditions on the diffusion matrix. The key observation is that vanishing policy improvements force the Bellman residual to converge in measure; higher integrability then yields strong convergence and constructs the unique strong solution. We extend the argument to entropy-regularized models, including zero-temperature limits, and establish quadratic convergence for special one-dimensional models.

We study finite-horizon exit-time control of a one-dimensional affine diffusion, with an unbounded control acting on both drift and volatility. Since an admissible control can cancel the instantaneous volatility, the associated HJB equation is not uniformly parabolic. Moreover, the location of the degeneracy is not prescribed by the model coefficients but depends on the unknown value function and optimal feedback. Rather than relying on a viscosity-solution formulation, we work under suitable structural assumptions and prove that the value function is a $C^{1,2}$ classical solution throughout the interior of the state domain and is strictly convex in the state variable. Moreover, we construct a locally Lipschitz optimal feedback, explicitly identify the degenerate set as a curve, and show that the closed-loop drift on this curve equals the time derivative of its state coordinate. Finally, we prove invariance and inaccessibility before exit and derive exact squared-logarithmic asymptotics for approach probabilities.

Estimation of the derivatives of a function with bounded high-order derivative in the presence of bounded measurement noise is considered, in a deterministic setting. Theoretical fundamental limitations of causal differentiators in terms of the lowest achievable worst-case differentiation error and desired properties such as exactness—the zero-error estimation of derivatives in the absence of noise—and robustness—the small sensitivity of the estimate under small perturbations—are established for the first time for arbitrary differentiation order. Differentiators that achieve these theoretically lowest bounds on their differentiation error are formally defined, fully characterized, and their properties are studied. In particular, deterministically optimal differentiators—those featuring optimal differentiation error bounds among the class of exact differentiators—are shown to exist by means of a novel construction exhibiting, in addition to exactness from the beginning, a very strong form of robustness.

We study the joint design of a time-of-use (ToU) tariff (the partition of the day into pricing blocks and the price level attached to each block) by a profit-maximizing electricity retailer facing flexible clients who may refuse the offer in favour of an outside option. The problem is modeled as a bilevel program in which each client, at the lower level, reshapes its consumption under pointwise bounds and a fixed daily energy total, and simultaneously chooses a participation level penalized by a quadratic regularizer. We prove the client's best response is computable in loglinear time. We derive two equivalent single-level reformulations: a closed-form continuous one tackled via stochastic gradient ascent (using a continuous reparametrization of block start times and durations) and a mixed-integer nonlinear program solved via dedicated solvers. The stochastic ascent scheme provides Clarke-critical accumulation points almost surely, in a scalable way. On a case study from the French retail electricity market with $5000$ simulated households, the gradient method matches a commercial mixed-integer solver on the pointwise problem to within $0.3\%$ of the optimal profit while running two to eight times faster on $100$ representative clients. Restricting unconstrained pointwise prices to admissible ToU schedules yields an estimated profit loss of $1.3\%$- $13.8\%$ for two-period tariffs and $0.6\%$-$6.6\%$ for three-period tariffs on the instances considered.

Optimizing the sensitivity of the unbalance voltage in Wheatstone bridges with respect to changes in the bridge parameters remains a fundamental objective in circuit design and instrumentation. When finite source and detector resistances are taken into account, determining the optimal bridge configuration becomes increasingly complex, and a closed-form analytical representation of the optimal solution has not yet been established. This paper derives an analytical solution for the optimal configuration of a Wheatstone bridge with finite source and detector resistances. Furthermore, the proposed optimal solution is compared with the conventional equal-arm configuration.

Chance constraints are widely used in optimization under uncertainty. This paper aims to show the eventual convexity (EV) of chance constraints with skewed generalized hyperbolic (GH) random variables. We prove that the densities of GH distributions are $\alpha$-decreasing, and obtain exact convex reformulations for separable jointly chance constraints with GH distributions. We provide numerical results to compute the $\alpha$-decreasing threshold parameters and show the EV of the feasible set together with the computational tractability to solve the associated optimization problems.

We study robust Mckean-Vlasov Forxard Backward systems with jumps under a nondominated family of probability measures. The backward equation contains a lawdependent nonsmooth monotone source motivated by asymmetric loss aversion. We establish well-posedness and stability of the system, analyze its Yosida regularization, and develop quantitative particle approximations. The main difficulty comes from the discontinuity of the minimal selection across switching interfaces, which prevents the use of direct Lipschitz estimates. To overcome this difficulty, we combine forward propagation of chaos with occupation estimates for the limiting forward process. Importantly, no occupation estimate is required for the particle system. Under a second-moment occupation condition, we obtain an improved joint convergence rate for the particle and Yosida approximations, controlling the forward and backward states, the martingale integrands, and the accumulated source. We also verify the required occupation conditions for several concrete jump models, including affine stable-like dynamics, predictable variable jump amplitudes, and a non-affine class obtained by smooth conjugacy

We introduce the notion of a directionally fine family of cones, a local geometric property expressing the possibility of selecting arbitrarily narrow cones around every direction of a normed space. We show that this local property provides a common mechanism for finite separation and approximation. In particular, directionally compact sets can be separated from closed cones by finitely many members of the family, while locally compact cones admit finite outer approximations with quantitative Hausdorff control on their sets of directions and on bounded sections. These finite constructions lead naturally to max-type positively homogeneous separators and do not require convexity of the cones to be separated. We then study two complementary realizations of the framework. For the classical family of Bishop–Phelps cones, directional fineness is characterized by the requirement that every point of the unit sphere be a denting point of the unit ball, equivalently, by property~$(G)$; in Banach spaces this amounts to rotundity together with the Kadec property. On the other hand, we introduce transversal coercivity as a general axial construction principle. It yields directionally fine families in arbitrary normed spaces, without geometric assumptions on the norm, and includes uniform axial deviation and norm-normalized axial cones as canonical models. Finally, the explicit structure of the former yields penalization and recovery results for optimization over locally compact cones, together with quantitative approximation and convergence estimates.

We introduce a fully–discrete variational approximation of Mather measures and sets for Tonelli Lagrangians on the flat torus, together with a numerical procedure for approximating the entire Mather set. The scheme is based on a fully–discrete Lax–Oleinik operator with integer winding labels. We prove an $O(\tau+h/\tau)$ error estimate for the critical value, convergence of critical solutions, and a finite-dimensional characterization of fully–discrete Mather measures. Accumulation points of the reconstructed minimizing measures are continuous Mather measures, while the supports satisfy complementary upper and lower convergence results involving the Mañé and Mather sets. To avoid the selection of only some minimizing components by exact discrete minimizers, we introduce a mass-threshold approximation based on almost-minimizing holonomic measures. This yields a finite-dimensional constrained optimization procedure designed to recover the whole Mather set.

We investigate the $k$-variation bounding property of the discrete-time Hankel operator, i.e., its invariance under the set of signals with a variation (number of sign changes) of at most $k$. Building on existing sign-consistency criteria, it is shown that this property is equivalent to the external positivity of $k+1$ explicitly realized linear discrete-time systems. Thus, making the property tractable via numerical and analytic certificates. We also derive dominant-pole restrictions for the case of $k=1$. A three-node thermal example illustrates the results and distinguishes variation bounding from variation diminishing.

We analyze a stochastic algorithm with Halpern-type anchoring for constrained convex-concave problems and monotone variational inequalities. This single-loop and single-call algorithm uses one unbiased sample of the gradient operator at every iteration, to be applicable to monotone games with noisy feedback. With $t$ denoting the iteration counter, we prove an anytime last-iterate convergence rate of $O(t^{-1/4})$ for both the gradient-mapping norm and restricted gap, bypassing the $O(t^{-1/5})$ constrained-anytime bottleneck in the literature. Specializing then to multi-point oracles, we use variance reduction to achieve the $O(t^{-1/2})$ rate with an anytime single-loop algorithm using $2$ samples per iteration. Our results allow constrained problems with a potentially unbounded feasible set; as well as a structured class of stochastic oracles whose variance need not be uniformly bounded.

We study the problem of maximizing a linear function over the integer points of an origin-centered $\ell_p$ ball, which we call \BallIPp{p}. For every fixed integer $p\ge2$, we prove that the decision problem over an $\ell_p$-ball is NP-complete. We then focus on the Euclidean case and study how the difficulty of the problem depends on the numerical parameters of the instance. We give two complementary pseudo-polynomial exact algorithms. The first specializes a known radius-budget dynamic program; the same nonlinear-knapsack framework also applies to every fixed finite integer $p$. We then develop a complementary dynamic program over candidate objective values for the Euclidean case. The latter is polynomial in the encoding size of the radius and pseudo-polynomial in the magnitude of the cost coefficients. For a fixed objective value, feasibility can be formulated as a closest vector problem (CVP) instance. This connection gives an exact algorithm whose running time depends on the covering radius of that lattice. Conversely, we show that a rank-$k$ Euclidean closest-vector instance in ambient dimension $d$ reduces to the decision version of a Euclidean \BallIP{} instance in dimension $d+k\le2d$, transferring known bounds under the Exponential Time Hypothesis (ETH). Finally, we study dimension-dependent approaches based on proximity to the continuous optimizer and describe the fixed-level constructions that extend to ellipsoids.

Predict-then-optimize methods learn a mapping from features to unknown edge costs. They then solve the shortest-path problem on the predicted costs. We study a feature-free alternative for the case where the decision-maker observes many noisy cost vectors on a fixed network. The cost vectors are denoised by an eigenvalue decomposition (PCA) of their covariance matrix. Dijkstra's algorithm then runs on the result. We test this on the $5\times5$ grid of \citet{elmachtoub2022}, with $1000$ independent replications and a fresh ground truth in each. We compare against SPO+, against naive Dijkstra on noisy costs, and against two alternatives to a hand-picked rank $k$ (the Gavish-Donoho threshold and a Ledoit-Wolf filter). We also test a hybrid of SPO+ and PCA. Four findings emerge. First, the rank matters. Keeping $k=2$ components performs worse than no denoising. Second, when costs are close to linear in the features, the Gavish-Donoho threshold recovers the true rank without supervision. PCA-denoised Dijkstra outperforms SPO+ at $\deg=1$, but underperforms SPO+ at $\deg\geq 2$. The hybrid works best at $\deg=1$ and is slightly better than SPO+ at $\deg=2$. Third, when costs are strongly nonlinear ($\deg=4$), the signal is no longer low-rank, and only the Ledoit-Wolf filter clearly beats the naive baseline. Fourth, the benefit of denoising grows with network size on grids, but not on a layered non-grid network at high nonlinearity.

We present shape calculus techniques and a scalable automatic shape differentiation framework for multi-phase topology optimisation on unfitted discretisations defined by several level-set functions. First we establish general, exact shape calculus expressions in the discrete case for multi-phase systems by leveraging concepts from convex geometry. To complement this theoretical foundation, we introduce an open-source multi-phase automatic shape differentiation framework based on polytopal cutting. This computational framework is validated against both finite differences and our established exact expressions, matching the latter to near machine precision. Furthermore, the proposed automatic shape differentiation is scalable across distributed computing environments, demonstrating near ideal weak scaling up to 1.65 billion finite elements across 13,824 computer cores. We demonstrate our implementation by solving unfitted multi-phase topology optimisation problems for anisotropic diffusion, linear elasticity, and fluid–structure interaction. Together, these theoretical and computational contributions provide a robust and accessible foundation for advancing multi-phase topology optimisation using unfitted finite element methods. In particular, the methods enable the solution of topology optimisation problems involving multi-phase and multi-physics systems with non-trivial boundary conditions. The open-source software is available at https://github.com/zjwegert/GridapTopOpt.jl.

The Matrix Spencer conjecture asserts that for all symmetric matrices $A_1,\ldots,A_n\in\mathbb{R}^{n\times n}$ with $\|A_i\|\le1$ there are signs $\varepsilon_1,\ldots,\varepsilon_n\in\{-1,1\}$ with $\|\sum_{i=1}^n\varepsilon_iA_i\|=O(\sqrt n)$. We prove it: a signing of discrepancy below $8\sqrt n$ always exists. We also give a randomized algorithm that finds a signing of discrepancy below $12\sqrt n$ with failure probability at most $p$. The algorithm uses $n^{3+o(1)}\operatorname{polylog}(1/p)$ arithmetic operations in the real-arithmetic model. This matches the size $n^3$ of the dense input up to subpolynomial factors. In the other direction, we prove that for every $n$ there are collections of symmetric matrices such that every signing has discrepancy at least $(2-o(1))\sqrt{n}$. We present three different proofs of the matrix Spencer conjecture. The key to every proof is a hereditary small-ball estimate. This is a lower bound on the Gaussian measure of the spectral body $\{x\in \mathbb{R}^n:\|\sum_ix_iA_i\|\le R\}$ that holds for every subfamily of the matrices. The other ingredient turns that Gaussian measure into a partial signing. We give three approaches to obtain such a signing. The first one covers the cube by partially signed faces through Gaussian concentration with a constant $7\cdot10^9$. The second proof replaces the covering by a projection lemma with explicit parameters for a constant $156000$. The third proof turns Gaussian measure into signs by a lossless coding, with no union bound. It proves the estimate at the right radius with smooth spectral barriers and certified coefficients. It gives a constant below $7.88$. For algorithms, the main idea is to project Gaussian points onto a smoothed spectral body. The $n^{3+o(1)}$ time algorithm tracks the Gibbs matrix of that body across coordinate-descent steps with sketched increments and random refreshes.

We study continuous-time and possibly high-dimensional stochastic control problems where drift coefficients and running reward functions are unknown. Due to these missing model primitives, we take the exploratory, reinforcement learning (RL) framework of Wang, Zariphopoulou, and Zhou(2020) with relaxed controls and entropy regularization. The objective is to develop theoretically grounded, efficient and scalable RL algorithms to learn both the optimal value functions (which also solve the exploratory HJB equation) and optimal exploratory feedback control policies. When the diffusion coefficients do not contain control, we employ probabilistic representations of both the optimal value function and its gradient based on an auxiliary state process depending only on the diffusion part of the original dynamics. With a delicate analysis on some properly defined mappings and their fixed points, this leads to the introduction of our policy iteration algorithms and their convergence. We demonstrate the performance of our algorithms through various numerical examples. Finally, we study a special control-dependent diffusion case where probability representation of the Hessian is called for.

We study decentralized learning of Nash equilibria (NE) in infinite-horizon discounted Markov games under bandit feedback, focusing on Markov $\alpha$-potential games. We develop KL-projected natural policy gradient (NPG) algorithms in two settings: an episodic setting with frozen policies during sampling and a fully online setting in which players receive a single realized cost sample per time step and update their policies asynchronously along a continuing trajectory. We establish finite-time high-probability NE regret bounds of order $\widetilde O(T^{-1/4})$ and $\widetilde O(T^{-2/15})$ for the episodic and fully online settings, respectively, up to fixed approximation terms. Crucially, our bounds eliminate the distribution-mismatch coefficient, which can scale prohibitively with the size of the state space, while accommodating potential approximation, estimation-oracle bias, and transition sensitivity. We further identify a state-wise potential structure that yields sharper guarantees with additive dependence on the potential approximation error $\alpha$. We specialize the framework to independent-resource Markov congestion games (IMCGs), establish their approximate-potential and transition-sensitivity properties, and construct decentralized estimation oracles from realized costs. As an application, we introduce strategic online job scheduling on stochastic machines and obtain a scalable decentralized algorithm for learning stable dispatching policies. Overall, our results provide the first finite-time high-probability NE regret guarantees for fully online asynchronous decentralized learning in Markov $\alpha$-potential games, remove distribution-mismatch coefficients from the regret bounds, accommodate fixed estimation-oracle bias, and provide scalable decentralized learning with finite-time guarantees for IMCGs.

This paper aims to develop new and efficient distributed algorithms for solving a class of monotone inclusions, $0 \in \sum_{i=1}^n (G_ix + T_ix)$, over a connected network of $n$ agents, where the single-valued operator $G_i$ and the possibly multivalued operator $T_i$ remain private to agent $i$. Existing distributed algorithms for this problem class are primarily non-accelerated, and their exact convergence rates in the original primal space are largely unexplored. To bridge this gap, we propose two Decentralized Fast Fixed-Point-based algorithms, ND-DFFP and NI-DFFP, which integrate Nesterov-type acceleration with primal-dual techniques under two prominent settings: (i) \textit{Lipschitz continuity of $G_i$ and maximal monotonicity of $G_i+T_i$}; and (ii) \textit{co-coercivity of $G_i$ and maximal monotonicity of $T_i$}. While ND-DFFP utilizes a homogeneous network-dependent stepsize, NI-DFFP reformulates the problem into a three-operator inclusion to decouple the network topology, enabling heterogeneous network-independent stepsizes. Under appropriate assumptions, we establish an $\mathcal{O}(1/k)$ convergence rate for the consensus error and an $\mathcal{O}(1/k)$ rate for both the restricted gap function and the squared forward-backward splitting residual, with the latter two metrics evaluated at the network-average iterate or its projection onto the effective domain. Finally, numerical experiments on distributed bilinear matrix games and a virtual power plant problem demonstrate the competitive performance and computational efficiency of our methods over recent decentralized baselines in the literature.

We consider boundary stabilization of a heavy chain hanging from a moving trolley with no tip load. Because the tension vanishes at the free end, the wave speed vanishes there; in Riemann coordinates the model becomes a degenerate $2\times2$ hyperbolic system in which the coupling is singular and the free-end reflection is generated in the domain rather than by a boundary condition. We construct a Volterra backstepping transformation that maps this system to the same chain with uniform damping of an arbitrarily prescribed rate and an elastic restraint at the trolley, yielding exponential convergence of displacement, velocity, and strain to zero. The singular kernel equations are solved by selecting their bounded Frobenius branch at the free end, which replaces the missing boundary datum, and the four kernels are generated by a globally convergent power series. The transformation is boundedly invertible on the energy space, with inverse obtained by reversing the prescribed decay rate. The result extends the radial backstepping structure developed for parabolic equations on disks and balls to a degenerate hyperbolic system.

A scalar hyperbolic PDE, actuated at one boundary, is coupled with a fast PDE in two spatial dimensions: transport in the axial coordinate and in an internal coordinate, and possibly diffusion in the internal coordinate. Backstepping designs exist for coupled hyperbolic systems, for their ensembles, and for their continua, but in all of these the second variable of the fast subsystem carries no transport or diffusion; when it does, no design is available, and from a scalar input no exact control of the two-dimensional state is to be expected. The pair is stabilized here by time-scale separation: in the quasi-steady limit the two-dimensional subsystem collapses into a spatial Volterra operator inside a one-dimensional reduced plant, backstepping applies there, and the two-dimensional subsystem is left to be a stable boundary layer. One theorem establishes exponential stability of the pair for every time-scale ratio below an explicit threshold, uniformly in the internal diffusion coefficient down to the pure-transport case.

For constant-stepsize stochastic approximation (SA), the iterates converge in distribution to a stationary law that depends on the stepsize $\alpha.$ Steady-state convergence (SSC) concerns the limit of the scaled stationary distribution as $\alpha \downarrow 0.$ Existing SSC theory requires i.i.d. or additive noise and global differentiability of the mean operator, and yields suboptimal rates. We develop a unified SSC theory for constant-stepsize contractive SA driven by Markovian, multiplicative noise, covering both locally differentiable and locally nondifferentiable mean operators. A key methodological contribution is a multi-step universality framework that progressively reduces the original stochastic recursion to tractable auxiliary dynamics while preserving its steady-state limit. Under local quadratic linearization at the fixed point, we obtain a Gaussian approximation of the scaled steady state at the optimal rate $O(\sqrt{\alpha})$ in Wasserstein-2 distance, which further gives finite-time Gaussian approximations for the raw iterates. In the locally nondifferentiable regime, we establish a general SSC result and show that the leading-order asymptotic bias can be of order $\sqrt{\alpha}$, in contrast to the $\alpha$-order bias in the smooth regime. We apply the theory to Markovian linear SA and asynchronous Q-learning, neither of which is covered by prior results. We further propose a bias-reduction scheme for Q-learning that requires no knowledge of the local smoothness regime, validated by numerical experiments.

We introduce a verification framework to numerically analyze inexact model predictive controllers (MPCs) in the constrained non-linear discrete-time setting. Rather than modifying the controller so that guarantees hold by construction, we treat the controller as given. In particular, we focus on two types of inexact controllers: (a) one whose input is extracted from a primal-dual point satisfying the Karush-Kuhn-Tucker (KKT) conditions of the non-convex MPC problem, and (b) one whose input is obtained by linearizing the dynamics and solving a convex quadratic program. The main idea of our verification framework is to formulate an optimization problem that searches over the worst-case initial state within a given set and control inputs consistent with the inexact controller to maximize a carefully-chosen performance metric. Using this framework, we show how to certify (i) the worst-case suboptimality gap of a single MPC problem, (ii) the worst-case closed-loop suboptimality gap over a given number of dynamical system iterations, (iii) closed-loop stability, and (iv) feasibility of the closed-loop system. Through numerical examples, we showcase the ability of our framework to precisely quantify both types of suboptimality, and to test the stability and feasibility of the inexact controllers.

In this paper, we derive an explicit solution of the stationary prediction with expert advice PDE for five experts. The formula is given in three regions. In the first two regions, it is the four-expert solution plus a single integral with an elementary positive density. In the third region, it is a finite sum of hyperbolic products whose coefficients are determined by one scalar quadrature. Our formula establishes that the direction $(1,0,1,0,0)$ is optimal throughout the ordered sector, and that the COMB strategy $(1,0,1,0,1)$ is optimal only on a lower dimensional subset of the sector (where $x_1=x_2$ and $x_3=x_4$). This disproves the COMB optimality conjecture of Gravin, Peres and Sivan. The verification of the Hamiltonian inequalities is a tedious task, part of which is completed with a computer assisted proof. The verification reduces to 21 scalar inequalities, which we prove using 147 exact rational Bernstein polynomial certificates. The exact certificates and their independent arithmetic checks are included in a supplement to this paper, and a Lean 4 formalization machine-checks the verification and both main theorems, apart from the viscosity characterization.