Papers

We study online convex optimization with one unbiased stochastic subgradient per round and an unknown finite conditional $p$th noise moment, $1<p\le2$. For every fixed interval $I$ of length $n$ and comparator path with $\Lambda_I=1+P_I/D$, one learner achieves \[ E[Regret_I(u)]\le\min(GDn, C[GD\sqrt{n(\Lambda_I+\log^2(2T))} +\sigma Dn^{1/p}(\Lambda_I+\log^2(2T))^{(p-1)/p}]). \] The learner uses none of $G,\sigma,p,I,P_I$, and the constant is universal. Interval adaptation adds to comparator complexity, preserving the distinct mean-gradient and noise exponents. The analysis controls calibration in expectation and limits the cost of observation-scale changes. Its general theorem compares to distributions over predictably available experts with relative-entropy dependence on a nonuniform prior. A common prior favors long windows and long restart lengths. With the statistics supplied, the interval cost becomes $1+\log(T/n)$, including the optimal full-horizon static rate. A change-of-measure lower bound identifies the noise power of this logarithm for learners retaining a full-horizon optimal guarantee, under explicit conditions. Static comparisons and deterministic partitions follow from the same decisions.

This paper studies a finite-horizon mixed $H_2/H_\infty$ feedback Nash game for stochastic evolution equations on a separable Hilbert space. The drift generator is unbounded, the remaining coefficients are bounded, and the one-dimensional Brownian diffusion depends on the state, control, and disturbance. The $H_2$ channel is an LQ state–control energy. For the disturbance channel, the stochastic LQ uniform-convexity characterization yields equivalence between strict induced $L^2$ attenuation and unique strongly regular mild Riccati solvability. Simultaneous bounded-generator approximation of the Lyapunov equation and the state justifies quadratic identities for strongly continuous mild operator solutions. These identities verify both Nash inequalities and full-output strict attenuation from a strongly regular coupled Riccati pair. An invertible feedback block and a contraction argument establish locally unique coupled solutions on a sufficiently short terminal interval for every positive attenuation level. For a specified stochastic heat-equation model at $\gamma=0.09$, a uniform invariant rectangle further proves full-horizon existence, bounded infinite-dimensional feedbacks, and operator-norm spectral convergence. Numerical computations reproduce the projected gains and compare selected best responses. Global coupled solvability for general coefficients remains an explicit hypothesis.

The Straight-Through Estimator (STE) is a widely used heuristic for Quantization-Aware Training (QAT), but its surrogate gradients can exhibit substantial mismatch with the underlying quantized objective, leading to noisy updates and parameter oscillations, particularly in ultra-low-bit regimes. We propose the Quantization-Aware Minimal-Norm Optimizer (Q-MINO), a temporal bundle method that combines gradient consensus, state-drift regularization, and an alignment constraint to construct stabilized, minimum-norm update directions from recent optimization states. Q-MINO solves the resulting constrained subproblem using a warm-started Frank–Wolfe procedure with a feasible fallback initialization. Theoretically, via a stochastic Lyapunov Kurdyka–Łojasiewicz (KL) framework, we show that Q-MINO achieves asymptotic neighborhood convergence. Moreover, we detail numerical experiments with Q-MINO at various quantizations.

For a graph G, the Boolean quadric polytope P(G) is the convex hull of the binary solutions of $y_{ij}=x_ix_j$ for $ij\in E(G)$, and Q(G) is its standard linear relaxation. For series-parallel graphs, Q(G) together with the odd-cycle inequalities describes P(G). Lee and Skipper showed vol(Q(G)) is polynomial-time computable for bounded treewidth and gave a closed formula for vol(P(G)) when G is a cycle. We resolve their question of giving an efficient algorithm for vol(P(G)) on series-parallel graphs. With $d=|V(G)|+|E(G)|$, we compute it with $O(d^7)$ arithmetic operations, and $O(d^5)$ if G is a cactus. The algorithm is a dynamic program over the series-parallel decomposition. The same framework computes vol(Q(G)) (hence the number of linear extensions of the incidence poset of G) with $O(d^4)$ operations, and $O(d^3)$ for cacti. With every $x_v$ fixed at 1/2, the recursion reduces to convolutions of univariate polynomials and computes the cut polytope volume of every series-parallel graph with m edges in $O(m^3)$ operations. We also study how much of Q(G) the polytope P(G) occupies. Short odd cycles matter more than long ones: in every graph, the odd-cycle inequalities of a cycle of length $\ell$ cut off at most a fraction $2^{\ell-1}/\ell!$ of Q(G). Hence vol(P(G))/vol(Q(G)) $\ge 1-\sum_C 2^{|C|-1}/|C|!$ for series-parallel G, with C ranging over its cycles. The ratio does not factor over cycles sharing a vertex: in a flower of $k$ copies of $C_\ell$ it decays like $\rho_\ell^k$ for an explicit rational $\rho_\ell$ below the ratio of $C_\ell$. Nevertheless, the small fractions cut off by many long cycles compound: in the worst case, triangle inequalities, or odd-cycle inequalities up to any fixed length, close no fixed fraction of the gap between Q(G) and P(G), and the volume ratio can be exponentially small in d.

We study fixed-time input-to-state stability (FxT-ISS) for discrete-time systems subject to exogenous inputs using finite-horizon Lyapunov functions. In the proposed Lyapunov- based characterization, the effect of the initial condition is eliminated after a uniformly bounded number of iterations. This characterization encompasses existing extinction-based Lyapunov conditions while allowing more general finite-memory mechanisms. Building on this framework, we establish a finite-horizon small-gain theorem for interconnected discrete-time systems, providing conditions under which FxT-ISS is preserved under feedback interconnections. The proposed framework is illustrated through examples arising in distributed algorithms, optimization, and feedback optimization.

We study fixed-time input-to-state stability (FxT-ISS) for discrete-time systems subject to exogenous inputs. We introduce Lyapunov-based sufficient conditions for FxT- ISS based on a class of dissipation functions with a fixed-time extinction property, which encompasses commonly used two- power constructions while allowing substantially more general decay mechanisms. Building on this framework, we establish a small-gain theorem for interconnected FxT-ISS systems in the presence of disturbances. The results are illustrated through several examples, including perturbed optimization algorithms and a discrete-time feedback-optimization architecture.

This paper addresses the control of multi-input strict-feedback nonlinear systems subject to a joint capacity constraint, in which the admissible input set is a coupled subset of the individual actuator limits. Unlike existing constraint-handling methods that enforce actuator bounds channel by channel and may unnecessarily suppress admissible control directions, we develop an Anisotropic Joint-Admissibility-Preserving Input Realization (AJ-APIR) framework that explicitly exploits the geometry of the joint constraint. The proposed realization constructs a state-dependent gain matrix whose spectral decomposition separates the commanded input into normal and tangential directions relative to the constraint boundary. The normal component is attenuated as the boundary is approached, while the tangential component is preserved, which allows the admissible control effort to be redistributed without loss of tracking authority. Integrated with a backstepping controller, the AJ-APIR framework guarantees forward invariance of the joint admissible set for all time. We establish exponential convergence of the tracking error to zero together with uniform boundedness of all closed-loop signals, and characterize the resulting command-demand behavior under the joint constraint. Simulation results for a representative second-order, two-input nonlinear system subject to a power-budget constraint demonstrate the efficacy of the proposed method to enforce the joint input constraint.

As an alternative to the standard geometric analyses, we give an exact, information-theoretic analysis of stochastic gradient descent (SGD) and its variants. We show that a preconditioned SGD step is the posterior-mean update of a Gaussian Bayes model, and that its one-step regret splits into an intrinsic-time cost and a change in comparator information. The split extends to an identity for the objective itself. Convex convergence, strict-saddle-point escape, the link between flatness and generalization, the standard learning-rate schedules, adaptive optimizers, and the noisy, momentum, heavy-tailed, and gradient-free variants of SGD each correspond to a term or a special case of this identity. We measure its terms on synthetic and real training runs. On real networks it attributes the slack of classical convergence bounds to the terms their derivations drop and separates optimizers that reach the same training loss. That separation follows the number and consistency of their steps. Its relation to which of them generalizes better differs between networks. For gradient-free SGD the identity determines how a curvature preconditioner should enter the update. The sharpness-based generalization certificate it yields, with a data-independent isotropic prior, is vacuous at network scale unless the curvature spectrum is nearly flat across all parameters.

We study the computational complexity of static output feedback (SOF) stabilization for continuous-time linear systems. For rational plant data, we prove that this problem is complete for the existential theory of the reals: it is equivalent, under polynomial-time transformations, to deciding whether a system of polynomial equations and inequalities with integer coefficients has a real solution. This result holds even for controllable and observable integer plants with two inputs. We characterize the effective gain dimension as the minimum number of linear combinations of gain entries that determine the characteristic polynomial for every gain. When this dimension is fixed, both feasibility testing and the construction of a rational stabilizing gain, whenever one exists, can be performed in polynomial time. SOF stabilization also remains strongly NP-hard for a family of minimal single-input integer plants. Every feasible plant in this family admits a stabilizing gain with entries in $\{-1,1\}$. For these gains, stability is preserved under simultaneous perturbations of all entries, provided that each perturbation has magnitude at most a uniform positive constant independent of the plant dimensions.

Let $z_1,\ldots,z_5\in\mathbb C$ satisfy $\sum_{j=1}^5|z_j|^2=5$. We prove the sharp inequality \[ \prod_{1\le i<j\le5}|z_i-z_j|^2\le 5^5, \] with equality precisely for a regular pentagon on the unit circle, up to rotation and permutation. This settles the remaining five-point case of the discriminant problem considered by Mordell in 1960. The proof first identifies a maximizer with a normalized relative equilibrium of five identical planar point vortices. We then give an independent exact classification of the corresponding stationary equations. The main algebraic step proves that every collision-free physical solution of the normalized stationary equations is reflection symmetric; it is obtained from a resultant construction and exact Gröbner-basis elimination over $\mathbb Q$. The reflection-symmetric equations have exactly five physical shapes, and an exact discriminant comparison selects the regular pentagon. All symbolic eliminations, Sturm counts, and interval inequalities are reproduced by an accompanying verification script using rational arithmetic. As consequences, the same configuration gives the Gaussian weighted Fekete set and the global mode of the five-particle complex Ginibre density after the natural rescaling.

We show that input-to-state stability (ISS) and integral input-to-state stability (integral ISS) with respect to the space of essentially bounded functions are equivalent properties for linear systems. The proof combines recent results on Orlicz-admissibility and a duality result due to the authors. As a direct consequence, we show that, given an $L^\infty$-admissible control operator, the mild solutions of the linear system are continuous, resolving a question by G.\ Weiss.

Disruptions can exhaust a supply chain network's capacity, yet outside capacity is hard to use: incumbent models are private, provider profiles are unstructured, and offers stay hidden until costly engagement. We formulate open capacity pooling, making network membership a disruption-response decision. The alternating direction method of multipliers (ADMM) coordinates incumbents without sharing models, and residual capacity gaps direct search over profiles indexed by a large language model (LLM). Verification reveals offers, reduced-cost screening admits them, and warm-started ADMM re-optimizes. The procedure is optimal if every eligible offer is revealed at exact prices, heuristic otherwise. Across 200 synthetic episodes, full-information opening recovers 24.6% of capacity-scarcity cost. With 40 contacts, coordination-directed need selection and LLM ranking capture 32.0% of this value, versus 4.7% for undirected, unranked contact. LLM reading of all profiles performs within five points of a complete structured registry and error-free extraction and finds four times as many compatible providers as keyword search among 10,000 records. What profiles omit is the exact capacity each provider's single offer covers, and requests for other capacity fail; registering offers in the same index raises captured value without a contact limit from 46.8% to 98.6%, keeping net value positive at every tested contact charge.

Supporting Uncrewed Aerial Vehicles (UAVs) with mobile charging stations enables persistent UAV autonomy in infrastructure-sparse environments. In this setting, trajectory optimization for UAVs is challenging because it couples task scheduling with when and where to recharge, as well as terrain-access constraints on where charging is available. We propose a smooth nonlinear trajectory optimization model for UAV with mobile charging support. Compared with existing results, the proposed model allows nonlinear charging dynamics mode via a unified battery dynamics model with disjunctive constraints on the time allocated to each mode. Furthermore, it provides smooth approximations of the disjunctive constraints with bounded approximation errors. By avoiding integer variables, these approximations enable efficient solution using smooth nonlinear optimization algorithms. We evaluate the proposed model on UAV missions with multiple spatially distributed tasks, nonlinear constant-current–constant-voltage charging dynamics, and terrain-access constraints on mobile charging support. Compared with mixed-integer nonlinear programs, the proposed model provides high-quality approximate solutions while reducing the computation time by orders of magnitude.

The Wind Farm Layout Optimization problem consists of placing a given number of turbines within a given area so as to maximize energy production. One of its main challenges is the wake effect between turbines, which can strongly affect production and is a cumbersome function to evaluate and to embed in an optimization framework. In this paper we introduce a new endogenous wake model (EndoWake), based on the idea of treating the wind speed at every grid point as a decision variable, linked to its upwind neighbors through linear propagation constraints. This makes it possible to incorporate the wake equations directly into a Mixed-Integer Linear Programming (MILP) model. We then develop such a MILP model and, in particular, introduce a new class of valid "band" inequalities, showing that they substantially strengthen the root bound. Building on this, we design a branch-and-check scheme: the master problem retains only the site variables together with one hypograph variable per cell and scenario, integer solutions are evaluated by an exact linear-time oracle, and the optimality cuts are problem-specific "upwind cuts". Computational results are reported on a ladder of square grids under a four-sector axial wind rose, demonstrating the effectiveness of the proposed improvements.

We establish exponential iteration lower bounds in the number of states for Howard's policy iteration on deterministic discounted and average-reward Markov decision processes (MDPs), with a constant number of actions per state. This rules out strong polynomiality of Howard's policy iteration in both settings. We further prove that a Dantzig counterpart of Howard's policy iteration runs in strongly polynomial time on deterministic average-reward MDPs. Together with the known strongly polynomial bound for Dantzig's rule on deterministic discounted MDPs, these results establish an exponential separation between the two algorithms in both settings. This separation between Howard's decentralized and simultaneous selfish improvements and Dantzig's coordinated selection of a single action with the largest improvement across all states reveals a “price” of algorithmic anarchy.

Motivated by the computational challenges of large-scale Cox regression, we study stochastic minimization of LogSumExp objectives over large sets. Mini-batch normalizer estimates generally yield biased gradients. We instead use a softplus surrogate that introduces one auxiliary scalar per normalizer and admits unbiased single-sample gradients. For smooth convex LogSumExp objectives, we prove an $O(T^{-1/2})$ averaged objective bound, improving the previous $T^{-1/4}$ analysis. With a strongly convex regularizer on the original variable, we also obtain a last-iterate squared-error rate of $\widetilde{O}(T^{-1})$ without strong convexity in the auxiliary variables. For Cox regression, the normalizers are defined over nested risk sets. We exploit this structure by grouping neighboring failures and sharing one auxiliary variable per group. The resulting compressed objective admits uniform score and curvature bounds that control the errors from grouping and softplus approximation. Together with the general optimization result, these bounds give a mean-square rate of $T^{-4/5}$, up to logarithmic factors, relative to the full Cox solution. The compressed estimator also matches the full estimator's asymptotic distribution. Experiments on synthetic and real survival datasets with slowly decreasing risk sets show a favorable performance relative to stochastic baselines.

We introduce quantum tilted walks, a quantum algorithmic framework for solving exact combinatorial optimization problems. The framework applies an average of powers of a tilted Hamiltonian that biases the discriminant matrix of a base Markov chain (mixer) with the objective function. Our starting point is quantum short-path algorithms, which prepare the ground state of such a Hamiltonian and obtain super-quadratic speedups over exhaustive search for certain combinatorial optimization problems. Recently, Le Gall and Tamaki~(arXiv:2604.12131) developed a classical conditioning-and-search algorithm for weighted MAX-E$k$-LIN2 and weighted MAX-$k$-CSP. Under the same assumptions, their algorithm is only sub-quadratically slower than quantum short-path algorithms. Consequently, existing short-path algorithms do not establish a super-quadratic speedup over this stronger classical baseline. For maximization problems, we give conditions under which tilted walks increase the amplitude on the target state with high objective value when initialized from a starting state with lower objective value. This framework captures conditioning-and-search and yields super-quadratic speedups over it for the same problems. While our framework recovers quantum short-path algorithms as a special case, it neither requires ground-state preparation nor initialization in the ground state of the base mixer. We demonstrate these advantages on a synthetic optimization problem for which tilted walks achieve a super-quadratic speedup whereas the short-path algorithms do not.

We provide generalized linearization theorems for skew-product flows on abstract Banach bundles over locally compact metric spaces, allowing for a unified treatment of uniform and nonuniform, exponential and strong exponential dichotomies. This is accomplished by constructing an operator whose fixed points are exactly the conjugacies that are bounded perturbations of the identity, and by giving criteria under which this operator is a contraction. Under stronger assumptions we refine the argument to obtain Hölder regularity of the conjugacy. Finally, we apply our results to recover known results for non-autonomous differential equations and to obtain new linearization results for control-affine systems on manifolds.

The classical linear quadratic regulator (LQR) minimizes the expected cumulative return but fails to account for performance variability, rendering it inadequate for risk-aware applications. To address this, we introduce the variance of the cumulative return as a risk measure in LQR. We derive the first exact closed-form expression for the variance of the discounted in?finite horizon return within the discrete-time Distributional LQR framework, for i.i.d. disturbances with symmetric probability densities. For Gaussian disturbances, this expression elegantly simplifies to a form dependent only on the disturbance covariance. Leveraging these theoretical foundations, we formulate a mean-variance optimal control problem that explicitly manages the trade-off between expected return and performance variability. To address the resulting non-convex optimization problem, we propose a novel adjoint gradient descent algorithm for a penalized formulation of the original problem, and establish that all iterates remain stabilizing and converge to a stationary point of the penalized objective. The effectiveness of this framework and the inherent risk-performance trade-off? are demonstrated through numerical experiments.

The input-to-state stability of a class of infinite-dimensional second-order port-Hamiltonian systems on a one-dimensional spatial domain is analyzed under nonlinear dynamic boundary feedback and boundary disturbances. Using energy methods in the port-Hamiltonian framework, we obtain sufficient conditions that guarantee uniform input-to-state stability of the closed-loop system.

We study when concave weighted power means of positive semidefinite quadratic forms admit sharp quadratic upper bounds. The starting point is the scalar fact that each such mean is the lower envelope of its supporting affine majorants. After the scalar variables are replaced by quadratic forms, the resulting bounds may cease to be extremal among quadratic majorants. We prove that sharpness is preserved under a lower-envelope convexity condition on the joint numerical range. In that case, a nonlinear inequality involving power means of quadratic forms is certified, with no gap, by a single linear matrix inequality. The condition is weaker than convexity of the full augmented range and holds automatically for two quadratic forms. This yields a sharp Peter-Paul-type majorant for the whole concave power-mean scale, including Yuan's lemma and the weighted geometric mean as special cases. We also derive variational formulas for the extrema of sums of power means of quadratic forms on the unit sphere; under the same lower-envelope condition, the maximization becomes an exact convex eigenvalue minimization. A finite-horizon control application illustrates how the certificate can be used constructively.

How much unknown process noise can a bounded experiment tolerate while still establishing that the plant can be stabilized? We answer this question for scalar discrete-time systems with bounded inputs, exact state measurements, and a disturbance-energy budget proportional to the experiment length. We distinguish individual stabilizability of every consistent model, stabilization by one common gain, and a common quadratic certificate. We determine their exact noise ceilings for every drift. The last two requirements coincide; individual stabilizability generally tolerates more noise. Bounded periodic inputs approach the ceilings, while adversarial disturbances prevent their attainment. We also determine the long-horizon limits. For unstable plants, a short experiment can establish individual stabilizability at noise levels where every sufficiently long experiment fails. The same obstruction yields upper bounds for multidimensional systems with real eigenvalues. Input design may use plant knowledge, but certification uses only the recorded data and noise bound. The results therefore provide fundamental robustness benchmarks for unknown-plant designs.

Policy mirror descent (PMD) enjoys fast convergence in regularized Markov decision processes (MDPs), but existing guarantees often rely on exact or increasingly accurate policy evaluation. We analyze PMD coupled with a persistent critic advanced by one temporal-difference (TD) update. For finite discounted MDPs, we establish global linear convergence in value for exact coordinate-wise Bellman updates, with any positive constant actor stepsize and arbitrary finite critic initialization. The proof combines a resolvent-based auxiliary distribution with a decaying Bellman-violation correction and a potential weighted by inverse coordinate weights. We then study stochastic TD-PMD with general strongly convex mirror maps under a single off-policy Markov trajectory. With suitably chosen constant stepsizes and a finite-batch TD update, the method achieves an expected value gap of $\epsilon$ after $\widetilde{O}(1/((1-\gamma)^5 \widetilde{\sigma}_b \epsilon))$ transitions. The stochastic analysis relies on the trajectory-wise Lipschitz continuity of the regularizer, derived from uniform bounds on vertex Bregman divergences, together with a visitation-weighted resolvent estimate for signed critic-error propagation that yields an inverse-linear dependence on behavior coverage $\widetilde{\sigma}_b$. In contrast to many prior guarantees for regularized policy optimization, our sample-complexity guarantee holds without trajectory resets, generative-model access, or nested policy-evaluation loops. Numerical results are consistent with the theoretical convergence analysis.

Fall Armyworm (FAW) poses a serious threat to maize production in many regions due to its aggressive feeding habits and rapid development cycle. In this study, we developed and analyzed a stage-structured mathematical model to evaluate the impact of different FAW larval instars on maize dynamics during the vegetative and reproductive stages. Analytical results indicate that the two models have unique and positively bounded solutions for all time $t\geq 0$ and admit four equilibrium points: the trivial, non-trivial, maize extinction and coexistence equilibria. The behavior of the model was studied using stability analysis to find conditions under which FAW dies out or continues to spread. Furthermore, sensitivity analysis and numerical simulations were conducted to examine how key parameters affect FAW population dynamics and maize. Numerical simulations of the model in both stages indicate that there is high destruction of maize plants in both vegetative and reproductive stages of maize production due to increased egg production and larval population density. The extensive damage caused by large populations of eggs, larvae and adult moths motivated an extension of the two models to include intervention measures such as traditional methods like handpicking and chemical pesticides. Numerical results indicate that these control strategies significantly suppressed the FAW population with a resultant increase in the maize plant population towards its maximum capacity during the vegetative and reproductive stages, respectively.

We study deterministic first-order bilevel optimization under weak lower-level convexity, allowing nonconvex lower-level objectives and without assuming strong convexity, the Polyak-Łojasiewicz condition, or an error-bound property. We consider a $\delta$-relaxed Moreau-gap constraint, with $\delta>0$, for the lower-level stationarity condition and propose an inexact variable-smoothing penalty method (IVSP) for computing its approximate Karush-Kuhn-Tucker (KKT) points. For any fixed relaxation level $\delta$, under a standard extended no-nonzero-abnormal-multiplier constraint qualification (ENNAMCQ), we prove finite stabilization of the adaptive penalty parameter and an overall $\widetilde O(\varepsilon^{-3})$ first-order complexity for computing an $\varepsilon$-KKT point. Notably, we give verifiable sufficient conditions for ENNAMCQ covering convex lower-level objectives without nonconstant affine segments (including the strictly convex case), and a class of nonconvex sample-reweighting models. The positive relaxation avoids the intrinsic constraint-qualification degeneracy of the exact Moreau-gap constraint while achieving an $\mathcal O(\sqrt{\delta})$ lower-level near-stationarity guarantee. Numerical experiments on synthetic and real-world bilevel learning problems illustrate the practical performance of IVSP.

Based on second-order inertial dynamics, we develop two accelerated Lagrange multiplier (LM)-based optimization methods: Linear LM-based second-order flow method and linear LM-based variable and operator splitting method for unconstrained non-convex optimization problems. Relaxed and adaptive variants of both methods are proposed to prevent degeneration of the Lagrange multiplier and to enable automatic adjustment of the time step size. Within the novel second-order LM-based optimization framework, the associated pseudo-energy functional is shown to dissipate monotonically along the iterative trajectory, and convergence of the generated iterates to stationary points is rigorously established. Extensive numerical experiments on function optimization and partial differential equation benchmark problems, including applications as optimizers for physics-informed neural networks (PINNs) and deep operator networks (DeepONets), demonstrate that the proposed methods effectively escape local minima, achieve high accuracy, and exhibit strong robustness with respect to the choice of the initial learning rate during neural network training.

This article develops a systematic characterization and exact computational framework for the maximal robust positively invariant set of linear difference inclusions subject to hard state constraints. The set is characterized as the greatest fixed point of the robust predecessor operator, leading to three equivalent decreasing set iterations—the standard, self-restricted, and incremental iterations—and exact stopping tests. Under uniform exponential stability of the matrix family and bounded disturbances, the limiting disturbance-reachable set is characterized and used to derive conditions for nonemptiness and finite determination. When the state constraint set is polyhedral, the three iterations admit exact half-space implementations for bounded disturbances and finite or polytopic matrix families. Although these implementations generate the same set sequence, they distribute computational effort differently. The complexity analysis and numerical study make these differences explicit and reveal how the underlying problem structure informs the choice of implementation.

Feature-mapping methods represent structural designs by explicit geometric primitives on fixed analysis grids, an interpretable and parametric alternative to the density fields of density-based topology optimization (SIMP). In contrast to SIMP, however, feature-mapping optimization is sensitive to the initial design. We present a gradient-based approach to align a feature-mapping configuration to a given (pseudo) density field, e.g., to initialize a subsequent feature-mapping optimization. The staged approach is based on a least-squares tracking formulation, preceded by a variant that only rewards alignment and does not penalize mismatch. Without an underlying finite element simulation, iterations are cheap, but isolated features receive little sensitivity information. As a remedy, we propose an asymmetric transition function based on automatically parametrized Bézier curves. We demonstrate the approach for a 2D cantilever and the five-bar design; for the latter, we show optional feature minimization based on the feature scaling variable of the geometry projection method. The approach works well with first-order optimizers. The absence of a state problem and the small number of variables, however, also make a second-order formulation attractive. To the best of the authors' knowledge, we present the first Hessian formulation for feature mapping and report the behavior of first- and second-order optimizers on our benchmark problems. For completeness, we also give the exact Hessian of the state-based compliance, which requires one additional solution of the FEM system per feature variable.

In this work, we introduce a new clustering method, namely T-ARC (Topology-Aware Randomized Clustering), that corrects the geometric bias of K-means by embedding topological information directly into the optimization objective. Building on the assumption that the data admits an underlying hidden structure modeled via a latent graph, the idea is to uncover this information through the interplay between the standard K-means data-fidelity term and a graph-cut penalty, which discourages cluster assignments inconsistent with the connectivity structure of the data. To render this coupling tractable, the latent graph is modeled as a random realization from a Stochastic Block Model (SBM), whose scalar parameter is optimized within a Distributionally Robust Optimization (DRO) framework, yielding a closed-form proximal update. Both SBM and DRO are informed by a persistence-based similarity matrix derived from zero-dimensional persistent homology ($H_0$), which translates the multiscale connectivity structure of the data into a pairwise topological prior. The overall optimization proceeds via Block Coordinate Descent; convergence is established through a global Lyapunov functional: the deterministic blocks satisfy monotonic descent, while the stochastic graph update satisfies descent in expectation, so that the expected energy converges. Experiments on synthetic datasets with non-convex geometries and on random subsets of Fashion-MNIST show that T-ARC recovers latent topological structures where K-means fails, achieving the highest accuracy on curved and interleaved clusters while remaining competitive, and markedly more stable than K-means, on real data.

We show how to compute the passivity radius of a given generalized state-space model of a rational transfer function that represents a passive system, both in the continuous-time and discrete-time cases. We then use this to construct robust descriptor realizations. For the continuous-time case, these realizations have a port-Hamiltonian descriptor form. For the discrete-time case a similar result holds for the so-called normalized realizations. The computation of such robust realizations is linked in both these cases to a particular solution of a linear matrix inequality that characterizes the passivity of the transfer function.