Papers

Particle Swarm Optimization (PSO) is a global optimization algorithm defined by an interacting set of particles evolving over the search space. Heuristically motivated, its theoretical analysis remains limited due to the second-order, stochastic, and highly nonlinear nature of the dynamics. In this paper, we connect classical PSO stability analysis under the stagnation assumption with more recent mean-field methods, providing new quantitative estimates for the time-discrete algorithm. We study in particular a regularized PSO model without memory, with non-degenerate noise by adding a noise floor to the original model. Studying such a surrogate model allows us to identify quantitative conditions under which the dynamics is stable and converges toward a small neighborhood of a global minimizer. We do so by first studying the Schur stability of the linearized dynamics, then analyzing the convergence properties of a nonlinear mean-field system via a Laplace principle, and finally establishing a quantitative error bound for the mean-field approximation of order $N^{-1/2}$.

\noindent We study residual-polynomial acceleration of the proximal point method (PPM) for maximal monotone inclusions, with Anderson acceleration (AA) as the prototypical adaptive scheme. We answer three questions exactly. (i)~The minimax complexity over all adaptive methods is precisely $d_0/(K+1)$ per $K$ resolvent evaluations. The upper bound is attained by the averaged-reflection estimator; the matching lower bound uses an explicit skew-adjoint instance with resolvent eigenvalues at the roots of $u^{K+1}=-1$ and $\csc^2$-distributed masses, on which every degree-$K$ polynomial method satisfies $\|r(y_K)\|\ge 1/(K+1)$. The optimal polynomial is uniquely the Fejér kernel, and the same instance certifies a per-step floor. (ii)~A sharp phase transition separates regimes: Jackson-kernel polynomials achieve $O(d_0/(K^2 s))$ when the spectral floor $s$ satisfies $sK\to\infty$, while at the critical scale $s\asymp 1/K$ the barrier is exactly $1/(K+1)$. The picture extends to normal operators and the nonlinear family $M=S+N_C$. (iii)~On linear problems AA-PPM needs no safeguarding; on nonlinear problems certification of the $O(1/k)$ envelope requires exactly two oracle evaluations per iteration, and this factor is optimal. We also correct and complete the theory for structured problems—affine, strongly monotone, piecewise-affine, and Hölderian growth—and confirm all predictions numerically.

Conventional wireless protocols such as Hybrid Automatic Repeat Request (HARQ) rely on frequent and timely feedback, which becomes impractical in low-feedback regimes including non-terrestrial networks and massive IoT. This limitation is particularly critical in heterogeneous multi-user systems with unknown and asymmetric channels, where a single weak user can dominate the overall completion latency. We propose Overhearing-driven Non-Orthogonal Multiple Access (ONOMA), a novel cross-layer transmission scheme that minimizes latency without requiring instantaneous or statistical CSI at the transmitter. ONOMA integrates Random Linear Network Coding (RLNC) with symbol-aware NOMA and explicitly exploits overhearing and acknowledgment timing. In the first phase, users overhear RLNC transmissions, and the relative timing of acknowledgments is used to implicitly infer channel strength ordering. In the second phase, symbol reconstruction enables interference-free decoding for strong users, effectively decoupling user latencies. An adaptive power allocation policy is derived from acknowledgment timing-based channel estimates. Analytical and simulation results show that ONOMA outperforms TDMA, multicast, FDMA, inter-session, and classical NOMA, reducing completion time by up to 34% in two-user and 50% in larger asymmetric networks.

In this paper, we study state-feedback fixed-time stabilization of continuous-time perturbed linear systems with nilpotent matrices in the presence of unknown large constant input/measurement delays with known bounds. By using an additional artificial delay, we design a feedback that for a precisely known linear system with a nilpotent matrix drives the state to zero in fixed time (which is larger than the bound of the unknown delay) and achieves input-to-state stability with a large exponential decay rate in the presence of additive disturbances and small system matrix perturbations and nonlinearities. For perturbed single integrators, fixed-time stabilization is achieved in the case of fast-varying unknown measurement delays with known bounds. The results are extended to discrete-time systems. Numerical simulations illustrate the efficiency of the results.

Communication security has traditionally been built upon one of two external resources: shared secret keys or a communication advantage over the eavesdropper. However, many practical wireless scenarios, including infrastructure-less, emergency, and highly dynamic networks, cannot guarantee either resource, motivating the need for a new communication security principle. This paper introduces a new communication security paradigm that exploits temporal dependency as a security resource. Unlike conventional secrecy techniques that prevent an eavesdropper from recovering transmitted bits, the proposed paradigm allows packet decoding but prevents correct interpretation by making original and dummy packets computationally indistinguishable. Rather than protecting individual transmissions, successive transmissions are intentionally coupled so that future communication depends on correctly interpreting previous ones. As one realization, we develop a state-chained random linear network coding (RLNC) framework in which the synchronization state required to interpret each transmission block is embedded in the previous block. Therefore, synchronization failures propagate across future transmissions, resulting in persistent eavesdropper asynchronization. We analytically characterize the probability and persistence of eavesdropper asynchronization, together with the computational complexity of resynchronization, and develop transmission strategies based on transmit power and intentional-interference optimization. Numerical results demonstrate sub-second eavesdropper asynchronization under a worst- case adversarial model with no channel advantage, no secret assumptions, complete protocol knowledge, and an arbitrarily stronger eavesdropper.

This paper proposes a stage independent multiple sampling plan (SIMSP) to improve inspection efficiency by reducing the number of samples required at each sampling stage. Unlike conventional multiple sampling plans (MSP), the proposed SIMSP eliminates the dependence of each sampling stage on the outcome of the preceding inspection. The proposed approach is developed for non-repairable products sold under a pro-rata warranty policy based on the generalized process capability index $C_{py}$. The SIMSP is designed under a Type-II hybrid censoring scheme (Type-II HCS), which provides greater flexibility in controlling test time and failure information in life testing experiments. The asymptotic distribution of the process capability index estimate is used to compute the operating characteristic (OC) function, and the exact Fisher information matrix (FIM) is obtained for further statistical analysis. A constraint optimization problem is formulated to determine the optimal design by minimizing the total cost subject to the manufacturer's and consumer's risk. Numerical investigations are conducted to examine the effects of model parameters, warranty policy characteristics, and cost factors on the optimal solution. The results demonstrate that the proposed approach provides an economically efficient and feasible method for lot acceptance while satisfying the tolerable risks requirements.

Through a "partial strong convexity" lemma, this paper shows how bounds on subproblem objective value suboptimality can be used in inexact augmented Lagrangian methods and ADMM algorithms. The ADMM result uses a small but important refinement on a long-standing criterion for approximately solving subproblems. The results enable two new approaches to computing Lagrangian bounds on the optimal values of stochastic mixed-integer programming problems, with simpler convergence analysis than the prior state of the art. In each case, the subproblems are solved by variants of the classical Frank-Wolfe algorithm. However, as compared to prior methods of the same type, there is much more freedom in the choice of Frank-Wolfe variant.

In this paper, we present a novel second-order method called Accelerated Inexact Newton Extragradient (AINE) for convex optimization using $\delta$-inexact Hessians. We show that AINE can find an $\epsilon$-solution in the inexact second-order oracle (ISO) complexity of $\mathcal{O}( (\delta/\epsilon)^{1/2} + (L_2/\epsilon)^{2/7} )$ when the Hessian is $L_2$-Lipschitz continuous, and a better complexity of $\mathcal{O}( (\delta/\epsilon)^{1/2} + (L_3/\epsilon)^{1/5} )$ when the third-order derivative is $L_3$-Lipschitz continuous. Notably, each iteration of our method can be conducted in the same running time as matrix multiplication up to logarithmic factors. In addition, we also establish matching oracle complexity lower bounds for both setups, demonstrating the optimality of our methods.

Let $\Omega$ be a bounded Lipschitz domain of $\mathbb R^N$, $N\geq 2$. In this paper, we study the asymptotic behavior of the first Robin eigenvalue of the $p$-Laplace operator as $\beta$ goes to $0$ and as $\beta$ goes to $+\infty$, deriving sharp asymptotic expansions of the eigenvalue; the expansion in the Dirichlet limit $\beta\to+\infty$ is obtained under the additional assumption that $\partial\Omega$ is of class $C^{1,1}$.

We study the effect of compression in Coordinate Descent Full Configuration Interaction (CDFCI) within an unconstrained optimization formulation of the full configuration interaction ground-state problem. Under suitable local assumptions, we prove that the compressed iteration converges linearly to the solution of an associated restricted problem. We also characterize the convergence point of the compressed algorithm. Under an additional exponential decay assumption on the target eigenvector, we show that the resulting eigenvalue error is of order $\tau^2$, where $\tau$ denotes the compression threshold. Numerical results support the analysis.

This paper studies the problem of longitudinal control, with safety and string stability guarantees, in vehicle platoons. A nonlinear adaptive cruise controller based on the Constant Time Headway (CTH) policy is designed for platoons of heterogeneous vehicles, that incorporates a speed-dependent nonlinear gain guaranteeing bounded vehicle speeds and a saturation-type nonlinearity to guarantee bounded (or non-aggressive) accelerations, while preserving collision-avoidance properties of time-headway-based spacing policies. By introducing a suitable spacing-error variable, explicit nonlinear error dynamics are obtained, allowing a direct analysis of the closed-loop system. It is shown that the collision-free region is positively invariant and that the proposed controller guarantees convergence of the spacing errors. Furthermore, nonlinear string-stability properties are established directly from the closed-loop dynamics without resorting to linearization or frequency-domain arguments. In particular, sufficient conditions are derived for $L_p$ string stability, $1\le p\le\infty$, with respect to disturbances, external inputs acting on the leading vehicle, and perturbations of the initial conditions.

Nonnegative matrix factorization represents nonnegative signals as additive combinations of latent components, but its factorization rank, and hence the model order, must usually be specified beforehand. An underestimated order discards signal structure, whereas an overestimated order produces redundant components and unstable decompositions. We propose a column $\ell_{2,0}$-regularized formulation that estimates the model order from an initial upper bound by suppressing inactive columns in both factors. A warm-started regularization path progressively removes redundant components without changing the factor dimensions, and a marginal reconstruction-loss criterion selects an order along the path. To solve the resulting nonconvex and discontinuous problem, we develop an inertial proximal alternating linearized minimization method, a scale-balanced variant, and a proximal active-set method based on P-stationarity. The balancing operation equalizes the norms of paired factor columns while preserving their rank-one products. We characterize the critical points and local minimizers of the model, provide a sufficient-condition result for rank recovery, and prove whole-sequence convergence of the proposed algorithms under explicit step-size and inertial-parameter conditions using the Kurdyka–Łojasiewicz framework. Dedicated experiments show that the warm-started $\lambda$-path is more efficient than increasing- and decreasing-order discrete $r$-paths, while scale balancing yields a more stable rank-selection path. Experiments on synthetic data and diverse signal benchmarks show that both iPALM and PASM provide reliable model-order estimates with favorable computational efficiency.

In this paper, we exhibit a randomized first-order algorithm to compute Goldstein approximate second-order stationary points of $L$-smooth functions, using tools from randomized smoothing. The algorithm has oracle complexity $\widetilde{O}({ n^2}/{\varepsilon^9}+{ n^3}/{\varepsilon^7})$, where $n=1,2,\ldots$ is the input dimension and $\varepsilon>0$ is the (common) tolerance. We also present extensions to weakly convex functions and applications to bilevel optimization.

We show that no deterministic zero-respecting algorithm (resp., (general) deterministic algorithm) can compute Goldstein approximate second-order stationary points of functions with Lipschitz continuous gradients within a finite number of (resp., no more than $n-3$ with $n$ being the input dimension) second-order oracle calls. This, among other consequences, shows that deterministic second-order weakly convex optimization is intractable.

As in Bouchard et al. (2010) and Bouchard and Nutz (2014), we study a utility maximization problem with expectation constraint. We first consider a uniformly elliptic case in which the endogenous state boundary associated with the constraint in expectation is proved to be smooth. This allows one to derive a proper Dirichlet condition for the value function of the optimal control problem on this boundary. We then propose a new truncation argument in the martingale representation of the expectation constraint. This leads to an approximating sequence of auxiliary systems of PDEs for which comparison holds. Convergence to the initial optimal control problem is proved. In the degenerate case, we propose another approximation which consists in adding a small noise term to recover uniformly ellipticity. Convergence is also proved. To the best of our knowledge, it is the first time that a full analysis is performed for such control problems, so as to open the doors to the use of numerical schemes. Numerical resolution in a toy example is performed using neural networks. It is complemented by an estimation of the numerical error, also performed by using a neural network approach.

We study a continuous-time rational addiction model where addiction capital follows a piecewise deterministic Markov process with state-dependent jumps capturing relapse and recovery. The instantaneous utility combines consumption and addiction capital via a power specification, leading to Hamilton-Jacobi-Bellman (HJB) equations with nonlocal jump terms. In the capped, bounded-control case we obtain a unique bounded viscosity solution and prove that optimal policies are bang-bang, switching between minimal and maximal consumption, while in the uncapped case we derive explicit linear feedback controls and closed-form value functions in several parameter regimes.

High-order methods can improve worst-case evaluation complexity, but for orders $p\geq3$ their Taylor subproblems are nonconvex polynomial optimization problems and are generally difficult to solve. We develop a radius-controlled boundary approach based on homogeneous tensor representations. By augmenting the step with a constant coordinate, any $p$th-order Taylor polynomial can be represented exactly as an order-$p$ homogeneous tensor form; at a prescribed radius, the boundary model is a spherical polynomial optimization problem. The representation applies to arbitrary $p$, while the algorithmic development focuses on the cubic case $p=3$. For an inhomogeneous cubic on the sphere, we introduce a quadratic shift and prove, under an explicit shift bound, equivalence with a three-block multilinear formulation at global optimality. This motivates a proximal alternating minimization (PAM) method with closed-form block updates; its objective values decrease and every accumulation point is stationary. We embed the boundary-step mechanism in an Adaptive Homogeneous Tensor Method (Ada–HTM). Under explicit smoothness, safeguarded-decrease, weak-curvature nondegeneracy, and local-refinement conditions, Ada–HTM attains the adaptive-regularization-type (AR$p$-type) evaluation complexity $\mathcal{O}(\epsilon^{-(p+1)/p})$ for first-order stationarity. Numerically, PAM matches order-$2$ moment–sum-of-squares (SOS) certificates on the structured cubic instances for which certification is tractable, scales particularly well for low-rank tensors, and makes Ada–HTM competitive with trust-region and cubic-regularization methods, with its largest gains on ill-conditioned and badly-scaled problems.

We revisit contextual optimization from the perspective of policy class design. A desirable policy class should be expressive enough to learn rich context-decision relationships, should enforce hard feasibility constraints rather than soft penalty terms, and should remain smooth enough for gradient-based training on downstream decision losses. Existing approaches usually emphasize only part of these requirements. We propose Legendre-regularized policies, which parameterize decisions as solutions of regularized optimization problems over the original feasible region. This construction yields policies that are feasible by construction and differentiable with respect to learned latent parameters. We prove that the associated optimizer map is single-valued, maps onto the relative interior of the feasible set, admits an explicit Jacobian, is Lipschitz continuous, and can be made arbitrarily smooth. We also establish a universal approximation result showing that the proposed class can approximate any continuous feasible policy on compact context sets. The framework unifies explicitly regularized optimizers and implicit perturbation-based smooth optimizers. Experiments on contextual newsvendor and resource allocation problems show that our approach improves prescriptive performance relative to the benchmark methods.

This paper investigates a class of stochastic linear-quadratic (SLQ) control problems over an infinite horizon for Markov regime-switching jump-diffusion systems. Unlike classical diffusion models modulated by a Markov chain, we assume that the state process undergoes abrupt jumps that are synchronous with the regime switches of the Markov chain. In contrast to conventional Poisson jump-diffusion models, the jumps in the state process are entirely induced by the state transitions of the Markov chain, which can be interpreted as losses or gains of state process incurred during regime changes. Under this formulation, we thoroughly discuss the closed-loop solvability of the SLQ control problem and provide a feedback representation of the optimal control via the stabilizing solution of a system of coupled algebraic Riccati equations (CAREs). Finally, we further apply our results to a lifetime wealth tracking problem and derive the corresponding optimal investment strategy.

Economy-wide electrification is driving steep load growth, while renewables and batteries are increasing pressure at the distribution grid edge often requiring costly upgrades. Local Electricity Markets (LEMs) can coordinate distributed energy resources (DERs) more efficiently than conventional designs. This paper extends the LEM framework by modeling photovoltaic (PV) inflow as an Ornstein-Uhlenbeck (OU) process to capture uncertainty and temporal correlation. Using rolling-horizon alternating optimal power flow on the IEEE 123-node feeder, we compare operator, customer, and system costs plus storage value under three market designs: flat tariff (FT), real-time pricing (RTP), and LEM. Results show that the LEM achieves the lowest customer and system costs, under uncertainty, while RTP performs worst. It outperforms the other markets across most storage and solar penetration levels. However, it may worsen customer outcomes at very low PV penetration, indicating that LEM adoption should accompany PV expansion and precede storage expansion. We examine two approaches to estimating battery storage value. Storage value depends on storage penetration, imbalance between storage and PV penetrations, location, and whether batteries are colocated with renewables and/or load. Poorly placed units can worsen congestion, whereas well-sited units benefit owners and the system. Both the marginal value of new storage and the stored energy value of existing capacity decline as PV penetration rises. However, they respond oppositely to battery penetration: marginal value falls because of diminishing returns as the fleet grows, while stored energy value rises as more batteries compete for the same local PV generation. These metrics capture different dimensions of storage value under a LEM: one measures the benefit of expanding the fleet, while the other measures the scarcity premium on energy already stored.

Generative AI relies on finding reverse time models for a discrete-time forward diffusion with non-Gaussian initial state, but uses indirect approaches as there is no theory for direct reversal in discrete time. This paper develops a theory for directly finding reverse diffusions for discrete time nonlinear processes with non-Gaussian states and process noise. We also give a necessary and sufficient condition for the reverse model to be input-affine when the forward process is linear and the process noise Gaussian, and show that for a wide variety of state densities an input-affine reverse diffusion does not exist. This is among several differences between the reversal of stochastic difference equations and their continuous time counterparts.

A battery-swapping station must provide every arriving vehicle with a charged battery while minimizing the time-of-use cost of recharging returned units. Coordinating heterogeneous compatibility, vehicle-specific return times, and finite charger capacity requires service-aware recharge decisions across the planning horizon. We formulate a per-battery mixed-integer linear program that captures these operational features under a hard no-stockout constraint and derive a provably equivalent reduced form with fewer explicit binary variables. In the synthetic scaling study, a price-guided battery-path heuristic returned a full-service schedule for every instance; regime-level median solve times ranged from 0.24 to 8.0 seconds. Its median cost premiums were 7-8% over certified reference costs for small- and medium-scale instances, and its certified ex post optimality-gap upper bounds were 9-12% for large- and extra-large-scale instances. For each operational baseline, the certified reference schedules reduced charging-energy cost by 50-60% on instances that the baseline fully served and for which a certified reference was available. In a 30-day replay of 1,002 swaps recorded at a commercial station, the reduced-model and heuristic rolling controllers served every swap and reduced charging-energy cost by approximately 50% relative to immediate charging.

The Discrete Fourier Transform (DFT), the Discrete Cosine Transform (DCT), and their block-wise variants underpin most deployed image and video codecs. Their effectiveness rests on three properties: their runtime is near-linear (up to a polylogarithmic factor) in the image size, they are exactly invertible, and they carry few to no parameters. We generalize these transforms to dataset-adaptive isometric multilinear bases that keep all three properties at the cost of a polylogarithmic number of parameters, by parameterizing each basis as an isometric tensor network, inspired by quantum many-body theory, and training one shared basis per image dataset with Riemannian optimization. Because the family contains the DFT and the DCT-IV (a variant of the DCT) as exact points, training can start from these transforms, and it improves on its starting point at the compression level it is trained for. We evaluate the bases on natural photographs and line drawings. On photographs, which the cosine basis already fits well, the best trained basis leads the DCT only under aggressive truncation. On Quick Draw line drawings, under a common codec at matched quality, the best trained basis stores each image in about 20% fewer bytes than the 8x8 block cosine transform of the Joint Photographic Experts Group (JPEG) format, not counting the one-off storage of the basis.

In model predictive control (MPC) with time-varying objectives, predicted signals need to be often incorporated in the cost function, such as prices in energy system operation. These are, however, often difficult to predict from the historical trajectory of these signals alone, as they may depend on other contextual events. We propose LiFT-MPC, an MPC framework that integrates a LiFT (Language-in-the-Loop Feedback Tuning) correction scheme to refine such predictions within the MPC loop. The prediction mechanism is updated online via a control-performance loss function, and we establish a performance guarantee for the resulting closed loop system. Numerical experiments using a realistic example of energy-storage management with real prices and news context to improve predictions, demonstrate an improved economic performance

The discovery of scaling laws has motivated training neural networks on ever increasing quantities of data. This is typically done with a constant decoupled weight decay which causes the network weights to shrink steadily over the course of training. Taking inspiration from the Robbins–Monro conditions, we propose to scale weight decay by the fraction of the peak learning rate $\eta/\eta_{\max}$. We prove that this scaled weight decay preserves the asymptotic stationarity guarantees of the corresponding unregularized methods for both stochastic gradient descent and the non-Euclidean spectral optimizer Muon, thereby avoiding the additional asymptotic bias introduced by constant decoupled weight decay. This retains the stability benefits of weight decay without changing the asymptotic optimization target. Using a steady-state analysis, we explain why under standard weight decay the weight norm shrinks steadily as training proceeds, whereas under scaled weight decay it settles to a roughly constant value. When applied to the training of mixture-of-experts models, Muon with scaled weight decay (Muon-SW) consistently outpaces Muon with identical hyperparameters, reaching the same validation loss $\mathbf{30\%}$ faster at our largest scale across models from $72 - 930$ million parameters trained at $\sim 600$ tokens per active parameter. If this trend continues to hold, the method promises to substantially accelerate the pre-training of frontier models while requiring only a few lines of code to implement.

We study continuous data assimilation for the two-dimensional Navier–Stokes equations on a smooth, bounded, connected domain with Navier-slip boundary conditions, using no interior observations. The available data consist only of finite-dimensional measurements of the tangential velocity on a non-empty relatively open subset $\Gamma\subset\partial\Omega$. We prove that sufficiently strong boundary feedback, constructed from sufficiently fine observations, generates a coercive spectral gap for the assimilation error. The limiting gap is identified with that of a mixed-boundary problem obtained by imposing a homogeneous Dirichlet condition on $\Gamma$, and is shown to be of order $\nu$. Combining this feedback-induced coercivity with an estimate of the non-linear error production in terms of the long-time averaged symmetric-gradient energy of the reference solution, we obtain a sufficient criterion for exponential synchronisation. We verify this criterion in the unforced case, for sufficiently small forcing when the unnudged Navier-slip form has an $\mathrm{L}^2$ spectral gap, for sufficiently large viscosity on domains without tangential rigid motions, and for sufficiently large viscosity in the presence of positive boundary friction. We also treat perfect slip on domains admitting tangential rigid motions, where synchronisation follows under a smallness condition on the non-rigid solenoidal component of the forcing.

We study convex optimization over a compact convex set when the objective is smooth and convex only on an open domain. Under a boundary-blow-up condition, every feasible initialization yields a compact invariant sublevel set separated from the complement of the objective domain, and an optimal solution exists. For a domain-aware Armijo projected-gradient method, a safe-neighborhood analysis establishes well-defined objective evaluations, finite backtracking, sufficient decrease, and a run-specific but iteration-independent positive lower bound on the accepted step sizes. These properties yield explicit sublinear objective and stationarity guarantees, together with convergence of the full iterate sequence. Positive curvature restricted to feasible displacement directions further guarantees uniqueness and linear convergence. We apply the framework to controllability scoring with prescribed input directions and compact convex allocation constraints. Feasibility is characterized exactly by controllability of the input directions eligible for positive allocation, while restricted injectivity of the Gramian map guarantees uniqueness of the optimal allocation and provides explicit strong-convexity bounds. A directed-network example illustrates how candidate exclusion can preserve or destroy feasibility and alter the optimal allocation in a criterion- and horizon-dependent manner.

Discrete optimization algorithms are often analyzed through continuous-time limiting ODEs, but a convergence certificate for the ODE is not automatically one for the discrete algorithm. We develop contact Hamiltonian systems as a setting where the transfer can be made precise. A contact Hamiltonian $H$ on $J^1(\mathbb{R}^n)$ obeys the intrinsic decay identity $\dot H = -H\,\partial_s H$, so an augmented energy $\mathcal{E}$ built from $H$, together with the conformal rate $\partial_s H$, is a continuous-time rate certificate whenever $\mathcal{E}$ controls the objective gap. Our main theorem states, under three named and independently checkable hypotheses, that an order-$r$ contact splitting with step $h$ transfers this certificate over the finite horizon set by backward error analysis. The discrete decay envelope is governed by the modified conformal factor up to $O(h^r)$ perturbations plus a backward-error shadowing defect, and the mechanism is inherited exactly because the modified Hamiltonian is itself a contact Hamiltonian. Quadratic heavy ball is a fully solvable example: its projected dissipative-leapfrog spectrum agrees with established conformal-symplectic optimization theory, while the augmented contact Hamiltonian yields a sharp objective-to-certificate comparison that verifies the transfer hypotheses. For strongly convex objectives with state-dependent damping, an explicit Bregman-type Lyapunov certificate instead transfers by an auxiliary-shadowing corollary. The decomposition $H=K+V+D$ into kinetic, objective-encoding potential, and dissipation terms serves as a design template, with a catalogue of closed-form sub-flows including contact-specific damping families. Numerical experiments confirm the predicted conformal-factor tracking orders and show competitive performance on ill-conditioned benchmarks and deep-learning tasks.

Designing effective reward functions for model-free reinforcement learning under non-holonomic constraints remains a persistent challenge, often resulting in severe local minima such as policy paralysis or over-conservative hazard avoidance. In this work, we present a parameterized reward shaping framework featuring coverage-gated alignment feedback, drive-direction switch regularization, and an aligned episode termination mechanism evaluated on an autonomous parallel parking task. Crucially, we show that environmental reward parameters and algorithmic hyperparameters are deeply co-dependent, requiring joint meta-optimization to achieve stable convergence. By employing surrogate-based Bayesian optimization, our co-optimized Deep Q-Network (DQN) agent resolves characteristic control failure modes, significantly outperforming uncalibrated baselines across both success rate and trajectory smoothness.

Different from most existing literature in the analysis of distributed optimization algorithms that reports sufficient convergence conditions leading to a conservative convergence rate, this work provides the exact worst-case convergence rates for two typical gradient tracking algorithms, DIGing and AugDGM. By eigen-decomposition, we show that two algorithms share the same average-state dynamics, while they differ from each other in the gradient tracking subsystems, which entirely govern algorithm convergence. Exploiting the diagonal structure of this decomposition, we reduce the stability analysis of MIMO systems to that of a set of parameter-varying SISO systems, from which explicit formulas for the exact worst-case convergence rate can be derived. These formulas clearly show that the optimal worst-case convergence rate of DIGing is larger than that of AugDGM under the same assumptions on objective functions and communication networks. Furthermore, we find that there is an inflection point in the graph connectivity, which is $\sigma =1/3$. For graphs with connectivity better than this inflection point, the optimal convergence rate of centralized gradient descent can be achieved by AugDGM provided the condition number of objective functions is worse enough. On the other hand, for graphs with connectivity worse than this inflection point, the centralized optimal rate can never be achieved. Numerical experiments validate the theoretical results.