Papers

In this note we show that the well-known stability test proposed by O'Shea, Zames and Falb is not necessary for robust stability of Lur'e systems with slope-restricted nonlinearities and thereby disprove a conjecture by J. Carrasco. Specifically we construct a full-block multiplier which certifies stability of a constructed Lur'e interconnection that does not admit an OZF multiplier.

Variational autoencoders (VAEs) transform high-dimensional, often noisy data into a compact latent representation, making downstream optimization more tractable. Three challenges persist in VAE-based constrained optimization: (i) sampling effectively within the latent space, (ii) identifying the active decision variables that actually influence the objective and constraints, and (iii) enforcing constraints without destabilizing training. We propose a Multi-stage Constrained Optimization Framework (MCOF). First, an entropy-constrained VAE (EC-VAE) coupled with a feature selector embeds objective and constraint information into a designated subset of latent variables, so that optimization proceeds over a low-dimensional subspace while the remaining coordinates supply solution diversity. Second, a Uniform Transformation (UT) module applies a per-dimension probability integral transform, replacing the irregular aggregate posterior with a uniform distribution over a bounded box and mitigating posterior collapse and Gaussian mixture bias. Third, a constraint-priority filter method (CPFM) solves the resulting surrogate problem by alternating violation-reduction and objective-reduction steps under a filter acceptance test, returning solutions that are feasible for the learned surrogate to a specified tolerance without requiring multiplier estimation. Finally, unselected latent coordinates are resampled to generate diverse decodings of a single optimized solution. We validate MCOF on a synthetic problem, where we ablate each stage and recover the analytic optimum, and on a ZINC250k drug design task, where the generated molecules satisfy the imposed constraints and are entirely novel relative to the training set.

Determining whether multivariate polynomials of degree four or higher are nonnegative and convex is a strongly NP-hard problem. To mitigate these computational difficulties, sum-of- squares (SOS) convexity has been proposed as a tractable algebraic relaxation that yields a checkable sufficient condition for convexity and can be expressed as a semidefinite program (SDP). In this work, we introduce a structured subclass of quartic polynomials, called the Sum of Separable and Biquadratic (SPBQ) forms, and conduct a systematic analysis of the connection between convexity and SOS-convexity within this class. Specifically, we show that every convex SPBQ polynomial is necessarily SOS-convex when the associated biquadratic form has size n x 2. We then construct an explicit SPBQ example with a 3 x 3 biquadratic form that is convex but fails to be SOS-convex. Finally, we examine both unconstrained and constrained optimization problems involving SPBQ polynomials, demonstrate notable computational benefits compared to general SOS-based methods, and illustrate their use in convex polynomial regression and fluid dynamics.

In this paper we investigate discrete-time infinite horizon Markov decision processes (dynamic programming models) with recursive utilities defined by the classical CES aggregator and the certainty equivalent operator being the negative of the entropic risk measure. Under mild conditions on the primitive data we prove existence and uniqueness of a solution to the Bellman equation. Moreover, we show that for any given stationary policy, the Koopmans equation has a unique solution. These two facts imply that a measurable selector of maxima in the Bellman equation is an optimal stationary policy. The methods of our proofs are applied to study solutions to the Bellman equation for other certainty equivalents, including risk-neutral case.

Newton's method is one of the most effective second-order algorithms for smooth optimization because of its fast local convergence. However, existing globally convergent Newton-type methods typically require convexity or strong convexity of the objective function, while approaches for nonconvex optimization often rely on Hessian regularization at every iteration. In this paper, we propose a general line-search Newton framework for unconstrained optimization that avoids repeated Hessian regularization by exploiting the Newton direction only when it is well-defined and suitable. The proposed framework encompasses several existing hybrid gradient–Newton methods as special cases and naturally yields a new extragradient Newton method. We establish global convergence under mild assumptions, including the Polyak–Lojasiewicz–Kurdyka (PLK) condition, allowing both isolated and nonisolated accumulation points. We further prove local superlinear and quadratic convergence under appropriate regularity assumptions. Finally, we apply the proposed framework to strongly quasiconvex optimization and provide, to the best of our knowledge, the first Newton-type algorithm together with a comprehensive convergence analysis for this important class of nonconvex optimization problems. Numerical experiments demonstrate the effectiveness of the proposed methods.

When can additional low-bit residual computation replace missing numerical precision for a fixed input-output map? We model a quantized residual system over a fixed horizon as a pure schedule selecting fields from a declared low-bit operation library, and use relaxed controls to characterize its infinite-depth limit. The distance from the target to the closed relaxed reachable set is the exact structural floor: no increase in depth can remove it for that library. Pure schedules approach the relaxed class at rate $O(D^{-1})$ under bounded-variation time dependence and $O(D^{-\vartheta}+D^{-1})$ under Holder dependence of exponent $\vartheta$. Execution arithmetic can reverse this conclusion: full-state write-back introduces a $D\rho_z$ penalty and can freeze residual updates, whereas increment error feedback replaces this growth by a bounded carry term and obeys an exact common-lattice conservation law. A fixed-teacher converse makes this rate sharp: for coherent depth-$L$ first-order high-precision comparators, accuracy matching requires $D=\Theta(L)$. Learned codebooks add a metadata resource, while state-dependent routing introduces hybrid event conditions. Verified primal and dual bounds yield feasible, impossible, or unresolved decisions before training. Companion software implements the workflow, and Lean 4 machine-checks the exact discrete core. Depth replaces precision only relative to a declared library, horizon, execution semantics, and routing model.

For a given real matrix $A\in\R^{n\times n}$ of order $n\geq 1$, a Pareto eigenvalue is a scalar $\lambda\in\R$ for which there exists a nonzero vector $x\in\R^n_+$ such that $Ax-\lambda x\in\R^n_+$ and $\langle x,Ax-\lambda x\rangle=0$. It is knwon that a real matrix of order $n=3$ can have at most $9$ distinct Pareto eigenvalues. We study the matrices for which this maximal number is attained and classify the way in which the $9$ Pareto eigenvalues are produced by their supports. A Pareto eigenvalue may be produced by more than one support. We therefore choose one producing support for each distinct values. The main contribution of this paper is to prove that, for every such choice, the numbers of values assigned to supports of sizes $1$, $2$, and $3$ are necessarily $$ (1,5,3),\qquad (2,4,3),\qquad\text{or}\qquad (2,5,2). $$ The proof uses bounds from the order $n=2$ problem and several restrictions on singleton, pair, and full supports. In particular, the graph formed by the pair supports producing $2$ values contains no triangle. These arguments also gives anohter proof that the maximum Pareto capacity in order $3$ is equal to $9$. For each of the $3$ profiles, we give an explicit full-capacity matrix with $9$ regular Pareto eigenvalues. We also prove that each profile occurs on a nonempty Euclidean-open set where every Pareto eigenvalue has a unique producing support.

Time-dependent routing recognizes that the same journey can take a different time depending on when it begins. Under duration minimization, even the departure time of each vehicle becomes a decision. Exact methods for this setting exist in the literature, but researchers and practitioners have lacked a ready-to-use open solver that combines rich piecewise-linear travel times, early feasible solutions and optimality claims. KAYROS fills this gap with two modes on one checker-consistent engine: an Iterated Local Search that streams improving solutions and a Branch-Price-and-Cut method that can issue computational optimality certificates under explicit arithmetic and search assumptions. It installs with one command and has no proprietary dependency. The public MAMUT-routing store currently contains 704 KAYROS certificates under a four-solve publication protocol, which has also led to the retraction and repair of invalid earlier claims. The report presents two complementary benchmark contributions to MAMUT-routing. The first integrates Blauth2024, a benchmark from the literature whose travel times derive from measured Uber speeds, for which KAYROS provides new best-known solutions on all 40 instances. The second proposes Poryos2026, a new benchmark of 1,080 paired static and time-dependent instances built from OpenStreetMap road networks and controlled synthetic traffic. Finally, the report describes the intensive human-AI collaboration behind this work and the verification practices that kept its outputs independently verifiable.

We study a timetabling problem faced by Finnish universities, where the challenge is to balance the preferences of three key stakeholders: the university, teachers, and students. Educational timetabling is an essential task that ensures that everything runs effortlessly and collaboratively. Its efficiency significantly impacts the smooth operation of the institution's educational programs and its financial success. As universities have evolved and competition among them has intensified, the features of the university timetabling problem have changed. This paper describes an ongoing optimization project with three Finnish universities and a leading provider of educational management systems for universities. The main goal of this cooperation is not merely to streamline the creation of timetables, but to enable an entirely new approach to manage the entire planning process. The project consists of three six-month phases: the feasibility study, the preparation phase, and the implementation. During the feasibility study, several research-related gaps and novel practical goals were identified. To address these gaps and goals, this paper presents a unique CRONOPT model for tackling three targets simultaneously: effective use of university resources, teacher workload and well-being, and smooth progression of students' studies. The optimization solution developed through the project integrates directly with the market-leading educational management system for Finnish universities.

Autocatalytic subnetworks govern sustained production and balanced growth in chemical reaction systems and can be naturally represented as directed multihypergraphs that jointly encode stoichiometric structure and reaction kinetics. Existing optimization approaches identify such subnetworks by maximizing their structural amplification, typically measured through the Maximum Amplification Factor (MAF). However, structural amplification alone does not determine growth potential, since highly amplifying subnetworks may be kinetically inefficient. Motivated by recent kinetic–stoichiometric bounds on balanced growth under scalable dynamics, we study the problem of selecting the autocatalytic subnetwork maximizing the growth bound $(\alpha^*-1)\|\mathbb{S}|_{\mathcal A'}\|_\kappa$, where $\alpha^*$ is the MAF and $\|\mathbb{S}|_{\mathcal A'}\|_\kappa$ is a kinetic consumption norm. We formulate the problem as a mixed-integer bilinear optimization model combining hypergraph selection with generalized fractional amplification constraints. Exploiting the discrete structure of the kinetic norm, we develop an exact parametric algorithm in which each subproblem reduces to MAF maximization and is solved through a Dinkelbach-type generalized fractional programming method. Finite convergence and global optimality are established. Applications to the Oregonator, the formose reaction, carbon-fixation cycles, and synthetic benchmark networks show when maximizing the kinetic–stoichiometric bound differs from maximizing the MAF alone, revealing the interplay between stoichiometric amplification, reaction kinetics, and balanced growth. The proposed framework provides an exact optimization methodology for identifying reaction subnetworks with the highest theoretical growth potential.

This paper addresses the output regulation problem for 1-D diffusion-reaction system, where both the disturbance and reference signals are generated by an unstable exosystem. We propose a constructive approach to the design of a finite-dimensional tracking error-based regulator for this class of possibly unstable systems. Based on the regulator equations, a combined plant is derived, converting the output regulation problem into a partial stabilization problem for the combined system. Using the modal decomposition method, the observability of the truncated modes is characterized under appropriate transmission zeros and observability conditions. For the controller design, unlike in the standard stabilization case, where the observer gain is dependent on the unstable modes only, the observer gain here has full order due to the coupling introduced by the exosystem. We prove that the observer gain can be designed such that its norm remains uniformly bounded with respect to the dimension of the observer. LMI-based conditions are provided for determining the observer dimension, and it is shown that the LMI is feasible for a sufficiently large dimension. Finally, the output regulation problem in the presence of unknown time-varying measurement delays is analyzed. Numerical examples are provided to validate the theoretical results.

Developing efficient function-approximation methods for policy evaluation is a fundamental challenge in risk-aware reinforcement learning. Existing approaches either focus on restrictive classes of risk measures or rely on access to a simulator, limiting their applicability in fully online settings. In this work, we propose computationally efficient online learning algorithms for policy evaluation in Markov decision processes (MDPs) with dynamic utility-based shortfall risk (UBSR) measures under linear function approximation. Specifically, we introduce the UBSR-TD algorithm, establish conditions under which it converges almost surely, and develop several variants designed to accelerate convergence. Our formulation shows that existing policy evaluation algorithms for risk-neutral MDPs can be readily adapted to dynamic UBSR settings by incorporating a loss function into the temporal-difference error. Numerical experiments support our theoretical findings, and an application to a perishable inventory management problem with shelf-life uncertainty demonstrates the practical effectiveness of the proposed methods.

Optimization over probability measures has become an increasingly important paradigm in modern machine learning, scientific computing, and uncertainty quantification. Motivated by Nesterov's accelerated gradient method in Euclidean space, we develop Heavy-ball and Nesterov acceleration methods over the probability measure space $\mathcal{P}_2$ and establish non-asymptotic convergence guarantees that match their Euclidean counterparts. In particular, we derive convergence rates with respect to both the number of iterations and the number of particles used to represent the underlying probability distributions. Extending accelerated optimization from Euclidean space to probability measures is challenging. The natural notion of momentum requires concepts such as tangent bundles of the set of probability space and they are hard to operate numerically. To overcome these difficulties, we introduce two complementary lifting procedures. The first lifts probability measures to phase space through a Hamiltonian formulation, introducing momentum variables into the dynamics. The second lifts probability measures to a common Hilbert space, restoring the linear structure required for convergence analysis while simultaneously yielding executable particle dynamics. Together, these two complementary lifting procedures provide a systematic methodology for designing, analyzing, and implementing momentum-based accelerated optimization methods over probability measure spaces.

Natural Policy Gradient (NPG) is a well-established Reinforcement Learning algorithm that underlies widely used methods such as Trust Region Policy Optimization and Proximal Policy Optimization, both of which have demonstrated strong empirical success. In this paper, we study exact NPG in finite-horizon Markov Decision Processes with known dynamics and horizon-dependent transition kernels. We provide the first finite-time convergence guarantees for this algorithm in this setting, for which we consider both constant and increasing step size regimes. With a constant step size $\eta_t=\eta$, we prove that NPG converges sublinearly with a rate of $\mathcal{O}(H^{2}/t)$ after $t$ iterations, where $H$ is the horizon length. We also extend this constant step size analysis to linear MDPs in an exact population-projection oracle under a full support projection distribution, recovering the same sublinear rate as in the tabular setting. Furthermore, with increasing step sizes, we prove that this algorithm achieves a linear convergence rate of $\mathcal{O}\left(\left(1-\frac{1}{\vartheta_\rho}\right)^t\right)$ for a problem-dependent constant $\vartheta_\rho > 1$, and the horizon-only robust schedule of the form $\eta_t=\eta_0(H/(H-1))^t$ where $\eta_0>0$ and $H \geq 2$, attains this same geometric rate.

We resolve Problem 60 of The Open Problems Project under the natural convention that a selected vertex at the total centroid is immobile. A cyclically labeled planar $n$-vertex configuration, $n\ge3$, can be transformed exactly into a regular $n$-gon by finitely many vertex-centroid moves if and only if it is noncollinear. The affirmative direction is independent of this convention, and all moves may be chosen positive, so the selected vertex never crosses the centroid of the other vertices. More generally, any two configurations in a connected open subset of the affinely spanning configuration space can be joined by positive moves whose entire continuous execution remains in that subset. For simple labeled $n$-gons, allowing straight-angle vertices, this yields exact reachability while preserving simplicity precisely within each orientation class. In $\mathbb{R}^d$, it yields mutual reachability of all full-dimensional labeled $n$-point configurations when $n\ge d+2$, while orientation is the only obstruction when $n=d+1$. Writing configurations as coordinate matrices, each move becomes a rank-one perturbation of the identity. Explicit one-move curves and three-move conjugation words yield a finite-word endpoint map with invertible differential at every affinely spanning configuration. The inverse function theorem gives exact local reachability with all intermediate states confined to a prescribed neighborhood, and connectedness makes it global. An elementary ear-reduction argument establishes the required connectivity of simple-polygon orientation classes. The same algebra determines exactly the group generated by positive moves. The proof is existential and nonquantitative.

We investigate a joint optimization problem of dividend payments and capital injections for a surplus process driven by a general diffusion. Dividend payments are assumed to be absolutely continuous in time, with the dividend rate bounded by a nonnegative concave function of the current surplus; while capital injections are modelled by a general nondecreasing process. We first analyze an auxiliary bail-out problem in which capital injections are required to keep the surplus nonnegative at all times. Under a concavity assumption on the drift, we prove that the associated value function is concave and is a classical solution of the corresponding Hamilton-Jacobi-Bellman (HJB) equation. We further characterize an optimal policy as a refraction-reflection strategy: the surplus is reflected at zero by capital injections, while dividends are paid at the maximal admissible rate whenever the surplus exceeds an optimal threshold. Our main contribution establishes that the general optimization problem exhibits a Lokka-Zervos type dichotomy. More precisely, an optimal policy is either a dividend refraction strategy without injections, in which ruin occurs, or a refraction-reflection dividend-injection strategy. This optimal injection decision is characterized through a simple comparison of the auxiliary value functions at the origin.

We study online bipartite matching with reusable server capacity and non-stationary rewards. Jobs arrive sequentially, reveal compatible servers, reward rates, and processing durations, and must be accepted or rejected irrevocably. An accepted job occupies one unit of server capacity only during its processing interval, so an assignment may displace an unknown sequence of future jobs. Existing guarantees are typically calibrated by a global reward range, which can become arbitrarily large when rewards drift over a long horizon. We instead impose a locally bounded reward condition: reward rates of jobs that can compete for the same server within a relevant time window differ by at most a factor $\delta$. Under this condition, we develop two BALANCE-type algorithms with time-aware opportunity-cost losses. TS-BAL maximizes cumulative blocking losses over feasible reuse schedules and achieves a competitive ratio of $2\ln(\delta D)+\mathcal O(\ln\ln(\delta\vee D))$. GR-BAL uses a greedy relaxation of this loss and achieves $\ln(\delta D)+\mathcal O(\ln\ln(\delta\vee D))$, matching a lower bound of $\ln(\delta D)$ in the leading term. Numerical experiments demonstrate robust performance under substantial global reward drift and favorable finite-capacity performance.

We study simulation-optimization of systems of optimizers (SOSO): agent-based simulations in which every agent solves a structured optimization - a linear program (LP), mixed-integer program, or dynamic program - at every decision epoch. Such systems arise in supply chains, electricity markets, and logistics, yet standard simulation optimization treats the simulation as a black box, discarding the inner optimization's geometry. We formalize SOSO and develop a framework that converts this geometry into computational advantage. We prove that the inner LP's optimal basis and dual variables propagate through the dynamics to yield an exact, unbiased, single-replication infinitesimal perturbation analysis (IPA) gradient of the outer objective, with measure-zero basis changes as the only obstructions. We derive a common-random-numbers covariance bound governed by a computable basis-disagreement count from the forward simulation. We further prove that IPA variance grows exponentially with feedback depth, formalizing the bullwhip effect, and introduce surrogate-as-decision with an error budget unifying LP-horizon and reinforcement-learning surrogates under one bound. We compose these into PRIME, a stochastic-approximation solver integrating the IPA gradient, adaptive step sizing, and multi-start spatial diversification. On six testbeds spanning the SOSO taxonomy, PRIME achieves the best or tied-best optimality gap at equal budget with near-zero oscillation and narrow seed-to-seed variance; IPA yields a ~2000x per-replication variance reduction over independent finite differences, and common random numbers cut paired-difference variance by over 10,000x in a 1,000-SKU, six-distribution-center supply chain. These results establish exploiting embedded-optimization geometry as a practically significant direction for simulation optimization.

We study the problem of constructing the nearest graph Laplacian matrix to a given Laplacian while enforcing a prescribed connected-component structure. Let the vertex set be partitioned into nonempty disjoint blocks $C_1,\ldots,C_k$, and let $U=[u_1,\ldots,u_k]$ be the matrix of the corresponding block-indicator vectors. The constraint $MU=0$ ensures that these prescribed indicators lie in the nullspace of the optimized Laplacian $M^\star$, and hence the associated graph has at least $k$ connected components. To guarantee exactly the prescribed components, we impose additional block-connectivity constraints on the principal blocks $M_j=M[C_j,C_j]$. These constraints ensure that each prescribed block induces a connected weighted subgraph. The resulting problem is a convex semidefinite optimization problem with a strictly convex Frobenius-norm objective. We prove existence and uniqueness of the minimizer and show that the optimized Laplacian has exactly the prescribed connected components, with nullspace $\operatorname{span}\{u_1,\ldots,u_k\}$. The framework proposed in this work provides a principled tool for quantifying the minimum structural intervention required to transform a graph-based network into one having a prescribed group-separated structure. Numerical examples, including the Sampson monastery positive-affection network, illustrate the nearest faction-consistent weighted reconstruction and the minimum Laplacian perturbation required to realize the prescribed faction structure.

CAR T-cell circuit design enables targeting tumour cells while keeping healthy cells unaffected to a cancer treatment. Designing CAR T-cell circuits that maximize the number of tumour cells eliminated (called efficacy) while meeting the safety constraint of keeping a large number of healthy cells unaffected is a challenging task. We provide a new multistart local search approach for this problem and show that it outperforms the current SOTA approach on a single-cell cancer atlas. Furthermore, we introduce a multi-objective local search approach which provides insights into the trade-offs of efficacy and safety of the treatment. Our experiments also demonstrate that the multi-objective approach yields improved solutions even for the constrained single-objective optimisation problem.

We study adaptive gradient descent for continuously differentiable, possibly nonconvex objectives under one-sided Hölder regularity. Unlike classical Hölder- or Lipschitz-gradient assumptions, which control the full gradient variation, our condition bounds only the directional term appearing in the descent inequality. This can allow less conservative step sizes when large gradient changes are orthogonal to, or favorable along, the update direction. We propose an adaptive scalar-step method based on an estimate of positive one-sided Hölder curvature, combined with a simple sufficient-decrease safeguard. For nonconvex objectives on a convex region containing the accepted update segments, we prove an explicit best-iterate stationarity bound with a rate determined by the Hölder exponent. Unlike predetermined diminishing step-size schemes, the method adapts to the local descent geometry. We evaluate the approach on two full-batch benchmarks designed to separate directional curvature from full gradient variation. On a binary classification problem, the method achieves the lowest final cross-entropy, objective value, and gradient norm, together with the largest classification margin among the compared scalar gradient methods. On a nonconvex Hölder regression problem, it attains the lowest final objective gap and gradient norm. These results indicate that one-sided Hölder curvature is an effective adaptive step-size signal when full-gradient variation is inflated by directions that do not hinder descent.

We consider \[ \min_x f(x)+g(x), \] where $f$ is smooth but its value and gradient are expensive, while $g$ is nonsmooth and has a cheap proximal map. Douglas–Rachford splitting (DRS) then requires a sequence of smooth proximal solves. These solves have different centers but share the same nonlinear curvature. We introduce curvature-recycling DRS (CR-DRS). The method transports an old exact residual to the new proximal center, reuses a quasi-Newton model, and moves the old proximal state before the first new gradient is evaluated. We give paired direct and inverse BFGS and safeguarded symmetric-rank-one realizations. A certified variant accepts a transported quasi-Newton step only when it passes descent and residual tests; otherwise it uses a fixed number of safe gradient steps. Under strong convexity and smoothness, a small-gain condition gives global linear convergence with a constant number of new gradients per outer iteration. A transported Dennis–Moré condition gives local superlinear reduction of the new proximal error. We show that this condition is not automatic, derive full-model and active-subspace sufficient conditions, and relate the active subspace to the local DRS dynamics induced by the nonsmooth proximal map. Experiments on $\ell_1$- and total-variation-regularized problems show large reductions in expensive-gradient calls relative to restarted and curvature-only quasi-Newton proximal solves. For directly proximal $\ell_1$ models, we also compare with FISTA.

We investigate finite-horizon, zero-sum controller-stopper games in which the stopper observes the state process and implements a feedback stopping rule. Building on a nonlinear Snell envelope representation for a related game established in a companion paper, we prove that our game admits a value by extending the associated first-contact principle. Our approach is purely probabilistic and yields an optimal feedback stopping rule for path-dependent systems while allowing for degeneracy in the underlying stochastic differential equation (SDE). As an application, we consider nonzero-sum controller-stopper games and show that the optimal feedback stopping rule for a zero-sum game appears as a component of a $\varepsilon$-Nash equilibrium for every $\varepsilon>0$.

This paper considers the general problem of the design of boundary controllers for distributed parameter systems. The control objectives are the asymptotic stability of the closed-loop system, as well as the disturbance attenuation of exogenous disturbances affecting the measurements and boundary conditions. More specifically, this work studies input-to-state stability (ISS) and stabilization for systems governed by coupled linear ordinary differential equations (ODEs) and homogeneous linear hyperbolic partial differential equations (PDEs), where actuation, sensing, and disturbance inputs are located at the system boundaries. First, we extend a previously established Lyapunov-based ISS condition for a similar class of systems to the H$_1$ setting under mild assumptions on the disturbance inputs. This extension not only ensures that the H$_1$-norm of the system states driven by non-vanishing disturbances and compatible initial conditions are bounded, but also provides an upper bound on the system H$_1$-norm. Subsequently, these theoretical results are applied to derive stability analysis and control design conditions expressed in terms of linear matrix inequalities. Numerical examples are provided to illustrate the effectiveness and potential of the proposed approach.

Finding Nash equilibria in the pure coordination game is trivial when every player interacts with every other player evenly. However, when players have a relational structure, the question becomes harder to answer. Here, we show that Nash equilibria to the structured coordination game represent a local notion of the Min-cut partition by way of the potential function. Using the standard relationship between partitions and subgraphs in the planar dual, we define a quasimetric space for which Nash equilibria are local minimizers of a global objective. When the game is restricted to only two strategies, these results allow us to construct a dynamic subgraph process which recapitulates the Myopic Best Response dynamics of the coordination game.

Nonlinear optimization problems with complicated, nonconvex, yet geometrically structured constraints can be tackled by projected-gradient methods: under weak regularity assumptions, these approaches were recently proved to possess convergence properties to the strongest stationarity conditions. In this work, we show how momentum terms, commonly used in nonlinear optimization to speed up the convergence process, can be integrated within this algorithmic framework without harming convergence guarantees. Preliminarily, we highlight an intrinsic issue induced by the direct replacement of the negative gradient with a general descent direction within the projected approach. Then, we present suitable backtracking mechanisms for the pre-projection step, allowing us to integrate momentum terms in the direction. By this technique, we can specifically ensure, without any smoothness assumptions, convergence to Mordukhovich stationarity as long as the base directions asymptotically revert to the negative gradient for small stepsizes; moreover, if the base search direction reverts exactly to the negative gradient for the smallest steps, the algorithm is proved to converge to Bouligand and Proximally stationary points, with and without (local) smoothness assumptions respectively. Finally, the proposed procedure is numerically tested on some classes of problems, namely, sparsity and bounded-rank constrained problems; the results indicate that the proposed method is computationally effective, taking advantage of the additional information provided by the momentum term.

Singular value soft-thresholding can be computed via a reduction to the matrix polar decomposition, which allows one to exploit GPU-friendly algorithms for computing the polar decomposition. Empirically, there is a significant speed-up on GPUs compared to the standard approach using the SVD. We leave the investigation of robustness to future work, but note that due to the discontinuous nature of the sign function, the reduction to the polar decomposition is likely only suitable for low-accuracy applications.

A Rotating Detonation Combustor (RDC) is a promising technology for increasing efficiency in propulsion and power generation applications. The dynamics of the RDC are governed by continuously propagating detonation waves within an annular combustion chamber. Multiple operating modes can be observed, including nonlinear interactions between counter-rotating waves and the emergence of standing wave patterns. Koopman operator theory provides a framework to globally linearize nonlinear dynamical systems by representing their evolution in the space of observables rather than states. In this work, finite-dimensional approximations of the Koopman operator are constructed using variants of Dynamic Mode Decomposition (DMD) applied to high-speed video data capturing the natural flame luminosity of the detonation waves in the RDC at the Technical University (TU) Berlin. By introducing time-delay embeddings as a dictionary of observables, this approach overcomes the limitations of standard DMD methods, particularly for accurate reconstruction of standing wave patterns and for capturing nonlinear interactions. In addition, a technique is presented to mitigate the influence of sensor noise in the luminosity measurements. Finally, it is shown that the DMD-based models provide insight into the dynamics of different operating modes by decomposing the reconstructed signal into its characteristic features.

We study small-noise asymptotics for controlled functional differential equations driven by additive Gaussian noise. The Gaussian noise is modeled on an abstract Wiener space, covering both classical Brownian perturbations and non-Markovian perturbations such as fractional Brownian motion. The drift coefficient is assumed to be non-anticipative, Lipschitz continuous in the state path, and of linear growth. For bounded uniformly continuous cost functionals, we prove game-theoretic lower and upper bounds for the small-noise logarithmic value functions and identify their limit whenever the associated deterministic zero-sum game has a value. In the limiting game, one player chooses the drift control, while the other selects a Cameron–Martin shift of the Gaussian noise, penalized by the corresponding quadratic energy cost. We further provide sufficient Fan-type convexity and concavity conditions under which the game has a value, thereby obtaining a full small-noise Laplace principle. The proof combines the Boué–Dupuis variational representation on abstract Wiener spaces with pathwise stability of the controlled solution map and adapted finite-dimensional approximations of Cameron–Martin shifts.

We study Langevin-based methods for non-convex optimization under smoothness and dissipativity assumptions, focusing on non-asymptotic expected excess-risk bounds rather than sampling guarantees for the full target distribution. A main methodological message is that relative-entropy sampling guarantees can be converted directly into expected objective-value guarantees, without passing through Wasserstein distance. This direct KL-to-objective route yields sharper dependence on the Log-Sobolev constant, which may scale exponentially with inverse temperature and dimension in non-convex problems. We first analyze the Unadjusted Langevin Algorithm with exact gradients and derive explicit bounds in terms of inverse temperature, dimension, stepsize, smoothness and dissipativity parameters, and the Log-Sobolev constant. We then extend the result to inexact ULA, allowing biased and stochastic gradient surrogates whose mean-square error grows at most quadratically in the state. This framework covers stochastic gradients and zeroth-order estimators based on function evaluations. We show that Gaussian and spherical finite-difference estimators fit the theory and obtain explicit function-evaluation complexity bounds. To the best of our knowledge, these are the first non-asymptotic global non-convex optimization complexity bounds for zeroth-order ULA. We also provide numerical experiments illustrating the spherical zeroth-order scheme.