Papers

Given a real square matrix $A$ and a nonempty closed target set of complex matrices $\mathcal E$, invariant by complex conjugation and containing real matrices, does the subset of $\mathcal{E}$ consisting of the matrices nearest to $A$ in the Frobenius distance always contain a real matrix? We give negative answers when $\mathcal{E}$ is either the set of normal matrices (solving a question posed by N. Higham) or the set of matrices whose eigenvalues lie in the closed left half-plane (solving a question posed by the first author and F. Poloni). For the latter problem, we also argue that the ratio between the real and complex distances is unbounded in every dimension $n\geq 3$, and this holds for every distance induced by a unitarily invariant norm. For both problems and $n \geq 3$, we show that a real input $A$ uniformly drawn from the unit sphere $\|A\|_F=1$ has no real minimizer over $\mathcal{E}$ with probability strictly between $0$ and $1$. The paper is complemented by some further results that are valid for more general target sets $\mathcal{E}$.

We study torsional rigidity as a function of the labeled vertices of a convex polygon. Starting from the distributed second shape derivative, we derive the Hessian with respect to vertex coordinates. At a regular polygon, dihedral symmetry makes this matrix block circulant in radial-tangential coordinates, reducing its spectrum to the eigenvalues of Hermitian matrices of order two. We also derive an exact second-variation Galerkin identity and guaranteed functional residual majorants. Finite elements approximate the PDE solutions entering the Hessian, and FLINT/Arb provides the interval arithmetic needed for certification. In the scale-invariant setting, we certify exactly four zero eigenvalues generated by similarities and $2n-4$ strictly negative eigenvalues for $5\leq n\leq25$. The regular polygons in this range are therefore strict local maximizers, modulo similarities, of torsional rigidity divided by area squared. We prove broken regularity of the first and second material derivatives at each fixed regular polygon, and deduce an $O(h^2)$ error bound for the exact Galerkin Hessian and its eigenvalues on meshes fitted to the coarse fan.

We propose a decentralized privacy-preserving learning algorithm in which each agent holds a single private sample and a shared model. Samples are learned sequentially, and each update must preserve the endpoint mappings at previously learned samples while protecting private data. This gives each agent three roles: (i) a learner that updates the model parameters, (ii) a teacher whose sample is learned at the current iteration, and (iii) a protected agent whose sample has already been learned. We build on Tuning without Forgetting (TwF) method to preserve previously learned mappings and show that TwF provides an indistinguishability guarantee for the learner whenever the set of protected agents contains another sample with the same label. For the teacher, we formulate a minimax optimal control problem that models the differential privacy noise as a worst-case disturbance to prevent performance loss while maintaining the same level of privacy for the gradient. For the protected agents, we compute the projections locally and aggregate them using a private push-sum gossip protocol. We prove geometric convergence of the decentralized gossip algorithm and of the distributed projection for TwF.

We develop a certified finite-horizon quantum optimal-control framework based on fixed-rank Hierarchical Tucker (HT) surrogates. Under a uniform rank-dependent HT truncation-accuracy condition, we establish exponentially decaying trajectory, cost, and value-function errors and near-optimality of surrogate-generated controls. We further derive a logarithmic rank–performance relation, an a posteriori cost certificate from realized truncation perturbations, and guarantees for inexact surrogate optimization and control convergence under additional regularity conditions. Numerical experiments on controlled XXZ spin chains provide finite-sample evidence consistent with the truncation condition and illustrate the trade-off between HT rank, representation size, surrogate accuracy, and control performance.

The Sinkhorn Riemannian metric characterizes the local behavior of the Sinkhorn divergence, a debiased version of entropy-regularized optimal transport. Although the Sinkhorn divergence is known to approximate the squared Wasserstein-2 distance as the regularization parameter $\varepsilon$ goes to $0$, the behavior of the Sinkhorn Riemannian metric as $\varepsilon\to 0$ remains largely open. To the best of our knowledge, the only existing argument is for measures with a density and is still a formal computation. In this work, we prove the convergence of the Sinkhorn Riemannian metric as $\varepsilon\to 0$ in the case of finitely supported measures undergoing horizontal perturbations. Under our assumptions, our techniques additionally allow us to prove the convergence of higher order derivatives of the Sinkhorn Riemannian metric, thereby ensuring the convergence of the associated Riemann curvature tensor and Christoffel symbols as $\varepsilon\to 0$. We also compare with the absolutely continuous setting and show where our approach fails for measures with a density.

We study homogenization of norm-optimal and time-optimal controls for a one-dimensional heat equation with rapidly oscillating periodic diffusion coefficients. The controls act on a fixed subinterval and are subject to pointwise constraints in space and time. We first derive final-state observability and null-controllability estimates that are uniform with respect to the oscillation scale. Combining these estimates with periodic homogenization and an abstract theory of optimal controls for parabolic systems, we prove convergence of the norm-optimal values and controls to their homogenized counterparts. We then deduce convergence of the optimal times and time-optimal controls. The controls converge weakly-star in the space of essentially bounded functions and strongly in every Lebesgue space with finite exponent, after extension to a common time interval when necessary.

We establish an exponential turnpike property for optimal boundary control of multi-dimensional first-order linear hyperbolic systems in which both the control and observation operators are unbounded and the cost functions may be nonquadratic. The main assumptions are internal and boundary dissipation conditions. These conditions ensure that the associated input-to-state and input-to-output maps are bounded, with operator norms independent of the time horizon T. This uniform boundedness is essential to the proof, which combines weighted energy estimates for the forward and adjoint equations, input-output duality, and strong convexity. Finally, we apply the result to the linearized two-dimensional Saint-Venant equations with sluice-gate control and illustrate the turnpike behavior numerically.

Constraining reaction rate coefficients is a central challenge in the development of explicit atmospheric chemical mechanisms, particularly for autoxidation systems where many reaction pathways are only indirectly observed through high-resolution mass spectrometry. In this study, we evaluate rate-coefficient optimisation methods for a toy-case autoxidation mechanism using synthetic data with known ground truth. Two complementary approaches are compared: ODE-constrained neural-network optimisation, which provides efficient point estimates of uncertain rate coefficients, and the Markov Chain Monte Carlo (MCMC) approach, which samples the posterior distribution of rate coefficients and quantifies parameter uncertainty. The methods are tested using direct concentration observations and mass-spectral observations under different noise levels. For unperturbed and low-noise synthetic observations, both methods converged towards the known rate coefficients, with the neural-network optimiser providing faster point estimates. Under high-noise conditions (with the signal-to-noise ratio approximately S / N = 1), however, MCMC was substantially more robust in recovering the rate coefficients. The posterior analysis shows that mass-spectral aggregation broadens credible intervals even at low noise, and that high-noise mass spectra can leave many individual reaction rates weakly identifiable. Posterior predictive validation nevertheless shows how broad parameter uncertainty constrained by MCMC remains consistent with accurate reproduction of the observable mass spectrum. These results demonstrate that point-estimation and Bayesian sampling methods provide complementary information: neural-network optimisation is effective for informative data, whereas MCMC is essential for diagnosing uncertainty, non-uniqueness, and identifiability in noisy or aggregated inverse problems.

This paper studies difference-of-convex (DC) composite optimization problems with conic and manifold constraints. By penalizing the conic constraint with a distance-based penalty, we propose an inexact proximal-linearized nonsmooth exact penalty (iPLNEP) algorithm. The proposed method successively finds approximate minimizers of proximal-linearized subproblems over the tangent spaces of the manifold based on computable inexactness criteria, while adaptively updating the proximal and penalty parameters. Under a boundedness assumption on the iterate and penalty parameter sequences, iPLNEP is shown to achieve an $O(\epsilon^{-2})$ worst-case iteration complexity bound for finding an $\epsilon$-stationary point. Using accelerated semi-proximal ADMM to solve the subproblems, we establish an overall oracle complexity bound of $O(\epsilon^{-4})$. Under an additional uniform error-bound condition, using semi-proximal ADMM as the subproblem solver yields an improved overall oracle complexity bound of $O(\epsilon^{-2}\log\epsilon^{-1})$. To the best of our knowledge, this provides the first proximal-linearized NEP framework with provable oracle complexity guarantees for DC composite optimization with conic and manifold constraints. If, in addition, the associated potential function satisfies the KL property, the whole sequence of iterates converges to a stationary point. Extensive numerical experiments on composite optimization problems with orthogonal-manifold, nonnegative cone, and second-order cone constraints demonstrate the effectiveness of the proposed method.

We study objective-only optimization of possibly nonconvex and nonsmooth functionals over probability measures on a separable Hilbert space, allowing the optimizer to be intrinsically non-Dirac. We introduce SCMO (Stochastic Control Measure Optimizer), a gradient-free particle method derived from entropy regularized stochastic control. After finite-particle and Galerkin approximations, a Cole–Hopf transform represents the optimal feedback as a Gibbs-weighted terminal displacement. SCMO approximates this feedback by sampling context clouds, replacing one particle at a time with candidate draws, scoring the resulting empirical measures, and applying exponential reweighting. SCMO uses separate covariances for candidate proposals and particle updates, termed matched when equal and nonmatched otherwise; our analysis covers both settings. In the matched case, we establish PDE-free qualitative convergence and a projection-first quantitative bound separating particle, Galerkin-projection, and entropy-regularization errors. For the practical multi-context implementation with nonmatched covariance, we prove finite-candidate consistency and show that the signed, curvature-dependent effect of nonmatched covariance can reduce the resulting upper bound on the approximation error. Finally, experiments on function-space, trajectory-law, and contact-rich manipulation problems show that SCMO handles nonsmooth and nonconvex objectives, escapes suboptimal local basins, and recovers prescribed multimodal, non-Dirac law structure. Code for reproducing our experimental results is available at https://github.com/HenryCHEUNG7373/SCMO.

Motivated by quantum foundations and complexity theory, Aaronson formalized a hidden-variable theory inspired by a proposal by Schrödinger. To any quantum state and unitary, this Schrödinger theory assigns a joint probability distribution via the Sinkhorn algorithm: rescale the columns and rows of the entrywise modulus of the unitary so that the marginals match the Born rule for the initial and final quantum states, respectively. He conjectured that this map is robust, i.e., inverse polynomially small perturbations in the inputs lead to inverse polynomially small perturbations of the joint distribution, which is important for complexity theoretic applications. We present a counterexample to this conjecture: We construct a pure state and two unitaries that are exponentially close, yet the joint probability distributions assigned by the Schrödinger theory differ by at least an inverse-linear term in at least one entry, and by a constant in total variation distance. We also propose a modified version of Schrödinger's theory that satisfies robustness, while retaining all of its other desirable properties. This yields a complete picture of which axioms of Aaronson can be simultaneously satisfied by hidden-variable theories.

Although flexible resource aggregators (FRAs), represented by electric vehicle aggregators (EVAs) and load aggregators (LAs), possess inherent flexibility to mitigate three-phase unbalance, the key challenge lies in how to effectively incentivize them to actively participate in unbalance mitigation. To address this, this paper proposes a generalized Nash bargaining (GNB)-based chance-constrained coordinated operation model for the distribution system operator (DSO) and FRAs. In this framework, the DSO and FRAs cooperate to mitigate three-phase unbalance, where FRAs provide flexibility to reduce the DSO's unbalance mitigation costs, and the DSO offers economic incentives to FRAs in return. Bargaining power is tailored according to each participant's contribution to unbalance mitigation to ensure fair profit allocation. Moreover, a scenario-based chance-constrained formulation is integrated to handle photovoltaic (PV) output uncertainties. To solve the model, a distributed proximal decomposition algorithm (PDA) is introduced for the independent model, while the coordinated model is decomposed into a social welfare maximization subproblem and a payment bargaining subproblem, with an improved bilinear Benders decomposition algorithm developed to solve the former. Numerical results validate the effectiveness of the proposed method in mitigating unbalance, ensuring fair profit distribution, and enhancing robustness against uncertainties.

We prove a Filippov-type stability theorem for integro-differential sweeping processes of Volterra type with an outer multivalued perturbation, in a separable Hilbert space, under the assumptions that the moving sets are uniformly prox-regular and Lipschitz continuous with respect to the Hausdorff distance and that the perturbation is one-sided Lipschitz. Given an absolutely continuous solution of the system perturbed both in the state argument of the multivalued term and by an outer integrable term, we show the existence of a solution of the original Volterra sweeping process and estimate the distance between the two trajectories explicitly in terms of the data of the problem. The estimate becomes sharper when only the outer perturbation is present, and it recovers the Lipschitz dependence on the initial condition of Filippov's classical theorem. The proof combines a reduction of the constrained dynamics to an unconstrained differential inclusion, measurable selection arguments, and an enhanced version of Grönwall's inequality established in this work. As an application, we obtain the Lipschitz dependence of the attainable set on the initial set with respect to the Hausdorff distance, together with a one-sided estimate quantifying the effect of the perturbations. We also work out a spatially distributed fishery model with ecological memory whose harvesting rule, triggered by the aggregate biomass, is one-sided Lipschitz but is not Lipschitz continuous with respect to the Hausdorff distance, so that the classical framework does not apply. For this model, our estimates produce a stability constant governed only by the biological data.

A new robustness metric $\nu$ was recently proposed as a substitute for the structured singular value $\mu$ to better capture robustness to sparse diagonal uncertainty, but its computation has remained an open problem. In this paper, we show that computing $\nu$ is equivalent to maximizing the spectral radius of a nonnegative matrix product. This equivalence allows us to transfer existing results on spectral-radius maximization to $\nu$, including a refined upper bound and conditions under which the bounds coincide. We then provide reformulations and structural results that enable an algorithm to solve the nonconvex optimization problem for nontrivial problems using global solvers. This also enables identification of the most fragile parts of the system. Our results do not yet provide a scalable solution for computing $\nu$, but they are an important step toward computability and interpretability.

Sampling a multidomain tensor from limited measurements is fundamental in structured linear inverse problems. Kronecker-structured sampling avoids the full high-dimensional sensing matrix, but design remains difficult: sequential discrete methods can commit the cross-mode budget too early, whereas standard continuous greedy avoids such early commitment at the cost of repeated state-dependent gradient evaluations. We propose spectral-residual continuous greedy (\alg) for frame-potential (FP) tensor sampling. \alg maintains a state-dependent fractional allocation before rounding. Mode-wise Gram matrices provide safe gradient intervals for direction certification, while Shapley values prioritize unresolved gradient queries. SR-CG then applies deterministic rounding followed by path-guided exchange (PGX), which reuses the final fractional state to restrict candidate swaps and accepts only exact FP-decreasing exchanges. We establish a finite-step approximation guarantee that approaches the classical $1-1/e$ factor as direction certification becomes exact and finite-step residual error vanishes. Experiments show fewer exact gradient evaluations and better FP designs, with clearer gains on instances where the cross-mode budget allocation is difficult to determine. \alg also achieves lower average normalized mean-squared error (NMSE) than Greedy-FP at all tested noise levels, although the reconstruction gain is smaller than the FP gain.

Inspecting or servicing a non-cooperative spacecraft is a two-player game: the target can thrust to defeat the inspector's plan. Hamilton-Jacobi-Isaacs reachability answers such games exactly but its cost grows exponentially with state dimension, while learning-based controllers scale yet certify nothing. Convex formulations are the usual escape, typically cast as convex-concave saddle-point problems. That framing fails in the terminal-distance orbital game: with effort regularization, the payoff is convex in both controls, so no pure open-loop saddle point need exist. In its place we derive two exact one-sided guarantees read from the players' terminal reachable sets via support functions: an escape certificate proving the target can hold a guaranteed standoff, and a security strategy bounding the miss distance the inspector can force. Together they bracket the engagement without assuming player rationality, and over two hundred perturbed trials the escape certificate separated capture from escape without error. A projected extragradient method supplies a strategy pair in about 25 ms, certified in place by best-response gaps. The construction carries unchanged to elliptical reference orbits via Yamanaka-Ankersen dynamics, where the orbital phase moves miss distance by nearly a factor of two, reproduced by nonlinear Keplerian propagation. Under receding-horizon play the inspector captures in half of ten representative engagements. Angles-only navigation error, keep-out zones, multiple pursuers, and closed-loop play are treated as extensions.

We study the problem of identifying nontrivial, i.e., nonzero, periodic fixed-points in discrete-time Lur'e feedback systems. Using the circulant matrix constructed from the transfer function of the linear subsystem, whether stable or unstable, we introduce an algebraic framework that allows us to determine when such fixed-points exist. This framework yields a sector bound defined by two vectors, whose slopes correspond to the maximum and minimum positive singular values of the circulant matrix. Assuming that the nonlinear feedback function is memoryless, we show that a necessary condition for the existence of nontrivial $P$-periodic fixed-points is that the intersection of the continuous completion of the nonlinear feedback function with that sector bound contains at least one point other than the origin. Our characterization provides a unified condition valid for all periods $P$, and further enables us to derive upper bounds on the amplitudes of admissible periodic fixed-points with bounded feedback functions. In particular, for relay feedback systems with passive feedback functions, we derive both upper and lower bounds for the amplitudes of such periodic fixed-points.

Low-rank tensor factorization provides a flexible framework for completing multidimensional data from incomplete and corrupted observations. However, unweighted spectral regularizers impose a common shrinkage profile across singular components, which may excessively attenuate dominant low-rank components, and factorized variants either lack component-specific weighting or require costly singular value decompositions (SVDs). This paper proposes two weighted Schatten-$p$ tensor factorization models, termed \WSpTFI{} and \WSpTFII{}, under the tensor-tensor product (t-product) framework to address these limitations. \WSpTFI{} is motivated by a factorized weighted tensor Schatten-$p$ norm identity and permits flexible, possibly asymmetric factor exponents. \WSpTFII{} constructs a regularizer from transform-domain column-pair energies, yielding SVD-free main factor updates and a column-pruning mechanism for reducing redundant rank components. This paper further develops an iteratively reweighted alternating direction method of multipliers (ADMM)-type scheme for \WSpTFI{} and an iteratively reweighted least squares (IRLS)–block successive upper-bound minimization (BSUM) scheme for \WSpTFII{}. Theoretical analysis establishes the weighted factorization relation and provides a conditional limiting Karush–Kuhn–Tucker (KKT) characterization for \WSpTFI{}. For \WSpTFII{}, the actual damped quadratic block updates yield a quantitative sufficient-decrease mechanism for the fixed-$\delta$ smoothed factor objective. This implies asymptotic regularity, and every accumulation point of the fixed-dimensional tail is stationary. Experiments on synthetic tensor completion, color-image restoration, hyperspectral inpainting, and printed-circuit-board defect detection demonstrate competitive reconstruction quality and robustness under various degradation conditions.

Complex engineering optimization problems are often characterized by multimodality, high dimensionality, and nonlinear constraints, posing significant challenges for efficient and reliable computation. To address these challenges, this paper develops a hybrid crossover-based optimization framework that enhances population interaction and improves search efficiency. The proposed framework integrates two complementary mechanisms, namely a Levy long jump crossover strategy for global exploration and a horizontal-vertical crossover strategy for effective information exchange and local refinement, thereby improving convergence behavior and robustness. To support scalable computation, the method is implemented within a unified multi-backend computational framework based on FEALPy, enabling consistent and efficient execution across heterogeneous platforms, including NumPy and PyTorch on both CPU and GPU. This design enhances portability, reproducibility, and computational efficiency in large-scale optimization tasks. Extensive experiments on the IEEE CEC2022 benchmark suite demonstrate that the proposed framework achieves competitive or superior performance compared with several representative metaheuristic algorithms, as validated by Wilcoxon rank-sum and Friedman statistical tests. In addition, the method shows strong performance on constrained engineering design problems. Finally, the proposed framework is applied to UAV path planning, formulated as a constrained optimization problem, demonstrating its effectiveness and scalability in complex engineering scenarios.

We study the deterministic first-order oracle complexity of smooth nonconvex-concave minimax optimization over a bounded convex dual domain. Let $\ell$ denote the joint smoothness constant, $D_{\mathcal{Y}}$ the diameter of the dual domain, and $\Delta$ the initial gap. We prove that every deterministic first-order algorithm requires $\Omega(\ell^2D_{\mathcal{Y}}\Delta/\epsilon^3)$ oracle queries in the worst case to find an $\epsilon$-optimization-stationary point whenever $\epsilon\lesssim\min\{\ell D_{\mathcal{Y}},\sqrt{\ell\Delta}\}$. We then develop Tracked-FOAM, a first-order method that attains a matching upper bound, removing the logarithmic factor from previous upper bounds. Together, these results establish the optimal dependence on all problem parameters in the stated regime.

We study lower bounds on the first-order oracle complexity of smooth nonconvex-concave minimax optimization. We consider objectives $f$ that are jointly $L$-smooth in the primal and dual variables $(x,y)$, concave in $y$, and whose primal value function $\Phi(x) := \max_{y\in\mathcal Y} f(x,y)$ satisfies the initial-gap condition $\Phi(0)-\inf_{x\in\mathcal X}\Phi(x)\le \Delta_\Phi$, with a bounded dual domain satisfying $\operatorname{diam}(\mathcal Y)\le D_{\mathcal Y}$. We measure stationarity by the norm of the gradient of the Moreau envelope of $\Phi+\iota_{\mathcal X}$ with parameter $1/(2L)$. We prove that any deterministic zero-respecting first-order algorithm requires $\Omega\left(L^2D_{\mathcal Y}\Delta_\Phi\epsilon^{-3}\right)$ oracle evaluations to find an $\epsilon$-stationary point. Under an unbiased stochastic first-order oracle with bounded variance, any stochastic zero-respecting algorithm requires $\Omega\left(L^3D_{\mathcal Y}^2\Delta_\Phi\epsilon^{-6}\right)$ oracle evaluations. The same lower bounds hold when $\Delta_\Phi$ is replaced by the initial primal-dual gap $\mathcal G_0$. These deterministic and stochastic lower bounds match the corresponding upper bounds of [14] and [29], respectively, up to a logarithmic factor in the deterministic setting.

Global inversion of nonlinear multivariable systems raises the question of whether every target value of the selected output derivatives can be realized by a plant state. We prove that, under globally regular modified inversion with fixed row selection and finite termination at full input rank, every fixed homogeneous linear output assignment yields a surjective output-derivative map with a smooth global section. The result requires a state bound on each finite interval in terms of the initial state and finitely many output derivatives, uniformly over inputs. This condition is strictly weaker than weak uniform zero-state detectability, making the separate surjectivity requirement in a global inversion approach to stabilization a consequence of detectability under the stated inversion hypotheses. The section selects a realizing state smoothly for every derivative target, including in the presence of internal dynamics, and is constructed through complete lifting of compatible smooth references. For known plants, the same lifting method yields a smooth dynamic controller using full state feedback and current reference derivatives, with initialization from the plant state. From any initial state, it achieves exact tracking of any smooth desired output after any prescribed positive time.

Single-machine asymmetric weighted earliness–tardiness (AWET) scheduling asks how to sequence jobs around a common synchronization date when early and late completion incur unrelated job-dependent penalties. At the boundary nonrestrictive date $d=\sum_jp_j$, a compact V-shaped schedule reduces the continuous-time problem to a quadratic choice of a nonempty early set. We establish four complementary results for this model. First, the positive-integer problem is strongly NP-complete by a unary-polynomial reduction from Restricted Exact Cover by 3-Sets. Second, unrestricted AWET admits a polynomial-time $(3+2\sqrt2+\varepsilon)$-approximation based on an anchored semidefinite relaxation and deterministic marginal thresholding. Third, when the earliness and tardiness ratio orders are strict reversals, the problem is weakly NP-complete but has an exact two-resource pseudopolynomial dynamic program. Fourth, for fixed total refinements whose ratio permutation is separable, an exact separating-tree recurrence and coordinated geometric trimming yield an FPTAS. The proofs use different manifestations of the same canonical objective: scale-separated prefix penalties, positive-semidefinite minimum-kernel covariance, a dominant completed load square, and a bounded four-coordinate decomposition interface. Together, the results show that the decisive issue is not merely whether the two ratio orders agree, but whether their interaction can be controlled by a global certificate or compressed into a bounded constructive interface.

Extremum seeking control (ESC) relies on deliberately injected excitation to extract optimization information from measured outputs, making its networked implementation particularly vulnerable to denial-of-service (DoS) attacks. This paper develops a resilient discrete-time ESC architecture for cyber-physical systems subject to DoS attacks. By holding the most recently successfully transmitted signals during DoS intervals, the proposed mechanism yields averaged error dynamics with an exponentially contracting mode under successful communication and a neutral mode under attack. For deterministic DoS attacks, practical exponential convergence to the extremum is established under an average bound on the attack duration, with an explicit convergence rate depending on the fraction of time under attack. For probabilistic DoS attacks, almost-sure and in-probability convergence properties are established via stochastic averaging, with the attack success probability explicitly entering the convergence rate. A key finding is that a seemingly natural zero-input strategy can fundamentally compromise ESC: attacks synchronized with the dither generate an $\mathcal{O}(1/a)$ bias in the averaged dynamics, where $a$ is the dither amplitude, and can displace the equilibrium from the true optimizer. Thus, under the proposed hold-based architecture, increasingly severe DoS attacks primarily slow the optimization process rather than destroy its stability, provided that communication is not permanently blocked. Numerical simulations illustrate the theoretical guarantees and the failure mechanism induced by dither-synchronized attacks.

The homogeneous form associated with a symmetric tensor, restricted to the sphere, is the objective of the best rank-one approximation problem and, with random coefficients, the energy of a mean-field spin glass. Its critical points have been studied extensively, but the global organization of the landscape they form is far less understood. We study this global structure for positive orthogonally decomposable tensors. We determine the persistent homology of the sublevel and superlevel filtrations in closed form, for every homological dimension $q$, every ambient dimension $D$ and every tensor order $k$, via a recurrence in the ambient dimension. Taking the coefficients to be random, we further prove laws of large numbers for the resulting persistence diagrams at dimension $0$ and $D-2$, giving an exact description of the typical global topology of a spin-glass-like energy landscape. We illustrate our results with numerical computations and simulations.

We study a class of distributionally robust optimization (DRO) problems for the statistical risk problem, formulated as minimax problems over the product of a Euclidean space and a Riemannian manifold. Because the resulting minimax landscape is nonconvex nonconcave in general, no globally convergent first order method is known to be available. We instead introduce the notion of a basin saddle point, a Nash equilibrium defined locally on the Cartesian product of a $\delta$ basin around a connected component of the local minima critical set and a geodesic ball on the measure manifold. We develop an abstract convergence framework for a Riemannian gradient ascent multistep descent iteration to a basin saddle point under a local Łojasiewicz type growth condition, with exponent $\beta \in (1,2]$, in the $\delta$ basin around connected components of the local minima critical sets. Under Lipschitz regularity of critical sets we establish linear convergence for $\beta = 2$ and polynomial convergence for $\beta \in (1,2)$ to a basin saddle point, with explicit dependence on the sectional curvature of the manifold. We then instantiate this framework for the statistical risk DRO problem over Gaussian measures, where the ambiguity set is naturally modeled as the product of Euclidean space and the Bures Wasserstein manifold of covariance matrices, which we relax to a penalized DRO formulation. We derive nonasymptotic Hessian estimates for the resulting Lagrangian, establish existence and local uniqueness of its maximizer, and prove that an alternating Riemannian gradient scheme converges to a basin saddle point of the penalized DRO problem, recovering the linear and polynomial rates of the abstract theory with all constants explicit in terms of data dimension, loss moments, and the reference covariance.

Structured feedback controllers provide rigorous stability guarantees, but often require manual parameter tuning to achieve good closed-loop performance. Policy-gradient methods offer a systematic approach to parameter optimization; however, conventional gradient evaluation requires sequential forward state rollout and backward costate propagation. This letter develops a time-parallel policy-gradient framework for discrete-time nonlinear control-affine systems. We derive the policy-gradient expression where the state and costate rollouts required for policy-gradient evaluation are formulated as residual-minimization problems and solved using Gauss-Newton (GN) iterations with parallel associative scans. For closed-loop systems that are globally asymptotically stable and locally exponentially stable, we show that the residual-minimization problems satisfy a local Polyak-Lojasiewicz (PL) inequality and that the GN iterates converge locally at a quadratic rate. Moreover, the PL constant, the size of the convergence neighborhood, and the quadratic convergence bound are independent of the rollout horizon T. We also prove that, for any finite horizon T, the state solver recovers the exact trajectory from any initialization in at most T iterations. Finally, an inertia-wheel pendulum example with interconnection and damping assignment passivity-based control (IDA-PBC) demonstrates improved closed-loop performance and the computational benefits of the proposed parallel policy-gradient framework.

Cutting stock problems are of large relevance to a variety of industry branches. Here, we present a linear programming formulation for a restricted version of the multistage 2D cutting stock problem with Guillotine cuts and test it on published benchmark instances. After this, we derive a non-exact reformulation in quadratic unconstrained binary optimization (QUBO) form via unbalanced penalization, which we show to have possible benefits in modeling the problem with usable leftovers. This work mainly considers the restricted formulation, in which the dimensions of each cut are determined by those of a single required piece. We further discuss how the approach can be extended to the unrestricted problem, which allows more general cutting patterns involving cuts with dimensions of multiple required pieces and naturally gives rise to quadratic terms in the inequalities of the problem formulation. We showcase solutions via a Simulated Annealing solver for the restricted problem class, where we mainly study the performance of the iterative augmented Lagrangian method. We also discuss possible applications of Quantum Annealing in this setting, as a strong motivation to research QUBO formulations.

We consider the problem of objective inference in the context of receding-horizon linear-quadratic regulator (LQR). In this setting, we are given sequential state-action observations, where each observed action is the first control of a newly solved finite-horizon LQR problem. We characterize when the objective of that problem is uniquely identifiable from these observations and when additional observations provide no new information about the objective. We then analyze action prediction at unseen states and show that all objectives reproducing the observed actions yield identical actions throughout the affine hull of the observed states; outside this hull, we derive an upper bound on the prediction error. % and deriving a prediction-error bound outside this hull. Additionally, we show that, when only the linear objective terms are unknown, exact prediction holds at every state. Finally, numerical results show that, even under stochastic observation noise, re-optimizing an inferred objective enables accurate action prediction at unseen states across different planning horizons.

Existing operation formulations for AC/DC distribution grid often oversimplify the interplay of AC/DC interconnections and VSC characteristics. Moreover, current distributed algorithms depend on gradient-based methods, which achieve only linear convergence rates and struggle to scale efficiently for large systems. This paper proposes a convex ALADIN variant that exploits second-order derivatives of convex subproblems, achieving locally quadratic convergence for distributed nonlinear but convex optimization problems. Then, this convex ALADIN is applied to a separable AC/DC OPF model for AC/DC hybrid distribution grid, explicitly incorporating complicated AC/DC interconnections and multiple droop-controlled VSCs while retaining computational traceability. To bridge theory with practice, we implement the distributed convex second-order optimization framework using the PROcess Orchestration Framework, a co-simulation platform that automates workflow design, model integration, and distributed execution via a user-friendly graphical interface, eliminating manual coding. Numerical results illustrate effectiveness of the proposed model.