Papers

In dimension at least two, a class of locally Lipschitz vector fields realizes every finite correspondence between distinct inputs and distinct targets at any prescribed positive time, provided that it is linear, its members generate global flows, and it approximates every smooth vector field of bounded support uniformly on bounded sets. The input and target sets may overlap. The proof constructs a smooth autonomous reference flow and finitely many localized correction fields, then uses uniform stability and Brouwer degree to obtain exact endpoints within the approximating class. The argument uses only uniform approximation of the vector fields; it does not require control of their derivatives. Applied to shallow ReLU vector fields, the result gives exact finite interpolation without time-dependent coefficients or additional state variables, together with bounds on width and normalized coefficient strength. In contrast, no continuous autonomous semiflow can exchange and compress two disjoint balls. Its time maps therefore fail to approximate all continuous maps uniformly on compact sets. The results distinguish exact interpolation at finitely many points from uniform control of neighborhoods.

Consider Gaussian denoising with a fixed prior and varying noise variance. When the prior has a density, maximum a posteriori estimation uses its negative logarithm as a fixed regularizer; only the weight of the quadratic data term changes with the noise variance. We ask whether a fixed regularizer can also recover the posterior mean. Although a suitable regularizer exists at each variance, we prove that no single regularizer works at two distinct positive variances for any compactly supported prior other than a point mass. This holds even for nonconvex and nonsmooth regularizers. A residual computed from two denoiser evaluations gives positive lower bounds on the uniform approximation error and excess mean squared error of estimators using a common regularizer. Numerical examples illustrate these bounds and the cost of reusing a regularizer across noise levels.

This work studies the stability and convergence of augmented primal-dual dynamics when constraint values are estimated from samples. Unbiased constraint observations can produce a biased augmented multiplier signal, shifting the equilibria of the mean dynamics. For componentwise inequalities, we give a necessary and sufficient condition for preserving the Karush-Kuhn-Tucker (KKT) equilibria and construct a convex example with a locally exponentially stable equilibrium that violates complementarity. To address this bias, constraint values are estimated recursively before forming the augmented multiplier signal. For smooth convex conic problems, a joint energy analysis establishes boundedness of the primal, dual, and estimation states, vanishing estimation error, and almost sure convergence of the primal-dual iterates to a single KKT point under global regularity and bounded conditional second moments. The result allows nonunique solutions and multipliers while keeping the number of samples per iteration fixed. Numerical studies illustrate the predicted equilibrium bias and examine convergence with nonunique KKT points and nonlinear constraints.

We study the recovery of forward and reverse quadratic optimal-transport maps from unpaired samples in high dimensions. We introduce a bidirectional neural framework in which the learned maps induce forward and backward displacement interpolants, while the training objective combines bidirectional quadratic action, discriminator-restricted Jensen-Shannon endpoint objectives, and two-sided cycle consistency. The construction requires neither precomputed sample pairings nor an explicit convex-potential parameterization. For absolutely continuous probability measures supported on a compact convex set, and under the stated generator-approximation, discriminator-richness, and minimizer-attainment conditions, we prove a population recovery theorem: for every prescribed accuracy, the sum of the corresponding \(L^2\) errors between any global minimizer and the forward and reverse quadratic Brenier maps is below that accuracy, provided the discriminator level is sufficiently large and the annealing action weight becomes sufficiently small. Moreover, the cycle loss is bounded above by \(\lambda W_2^2(\mu_0,\mu_1)\). Complementary results quantify approximate invertibility and show that exact endpoint Jensen-Shannon divergence and cycle consistency control missing target mass and many-to-one map collapse, respectively. Experiments on Swiss roll, MNIST, CelebA, single-cell perturbation data, and chest X-ray images evaluate endpoint fidelity, transport cost, inverse consistency, and the geometry of the induced interpolations.

We introduce cvxgenrust, an open-source tool for generating custom Rust code that solves families of parameterized convex optimization problems modeled in CVXPY. cvxgenrust canonicalizes a problem family, extracts affine maps to Clarabel cone-program data, and generates a specialized Rust crate that updates parameters and calls Clarabel natively at runtime. The generated solver can also be exposed to Python and registered as a custom CVXPY solver. Our code generator supports a wide range of convex optimization problems up to semidefinite programs and exponential-cone problems. Numerical experiments show reduced runtime relative to direct CVXPY solves and performance comparable to CVXPYgen on shared problem classes.

Solving inverse problems with differentiable physics simulators holds the potential to revolutionize scientific discovery and engineering design, as it enjoys both the strict physical correctness from rigorous numerical physics simulators, and the high efficiency and effectiveness from automatic differentiation and gradient-based optimization. However, currently, this paradigm faces performance issues in optimization. In this work, we target benchmarking the performance of different optimizers to solve various inverse problems. We construct 12 differentiable physics simulators spanning physics domains including discrete mechanics, continuous mechanics, atomistic simulations, rendering, and semi-empirical physics models. Based on these simulators, we design corresponding inverse problems that can be categorized into parameter identification, inverse design, and optimal control. Finally, we conduct extensive experiments to compare the performance of different optimizers, including regular first-order methods, approximate second-order methods, as well as global optimizers, on these inverse problems, and analyze the results to provide insights on how to choose and design optimizers for differentiable programming. We hope such benchmarks can inspire the development of more effective optimizers, and further promote the applications of differentiable programming in various scientific and engineering domains.

We study an inexact interior-point method for nonsmooth, possibly nonconvex optimization in a Hilbert space with inequality constraints ordered by a cone in a Banach lattice, with particular emphasis on infinite-dimensional state-constrained optimal control. The objective function is given by the sum of a smooth, possibly nonconvex term and a convex, possibly nonsmooth term with a computable proximal mapping. The constraints are formulated by means of an order cone in a Banach lattice. This setting covers finite-dimensional nonsmooth nonlinear problems with componentwise constraints as well as infinite-dimensional PDE-constrained optimization problems with pointwise state constraints. The method is based on barrier-regularized subproblems, which are solved inexactly by a proximal-gradient method. We consider logarithmic and power-type barriers and derive the differentiability and curvature estimates needed for the convergence and complexity analysis. Under suitable constraint qualifications and compactness assumptions, we establish approximate KKT conditions for the original problem and convergence of the inexact interior-point sequence. For logarithmic and power-type barriers, we derive complementarity estimates and outer iteration bounds; the corresponding power-barrier rates and total inner-outer complexity bounds are stated under explicit barrier-path and uniform smoothness assumptions. For convex problems, we obtain stronger convergence results. We apply the framework to state-constrained semilinear elliptic optimal control and sparse dictionary learning with nonlinear side constraints. Numerical experiments illustrate the proposed method.

The notion of s-composability was introduced to compose optimized stepsize schedules while preserving sharp guarantees. We show that the same joint potential has a second use: it can serve directly as a certificate of sharp last-iterate performance. We develop this viewpoint for constant and silver schedules. For the constant schedule, we prove the s-composability statement posed as an open question by Grimmer et al. (2025), at every finite horizon, through an explicit nonnegative smooth-convex interpolation certificate. The result identifies the unique constant stepsize minimizing the worst-case final gradient norm of smooth convex gradient descent under an initial-distance bound, thereby proving the optimal stepsize, value, and uniqueness predicted by Conjecture 3 of Taylor et al. (2017) without resolving its full worst-case curve. Through the Moreau envelope, the same constant is also the unique constant relaxation minimizing the worst-case final residual of relaxed proximal point. More generally, for every positive s-composable schedule, we determine the exact worst-case final proximal objective-gap constant and give a matching one-dimensional example. Applying this result to the already s-composable original silver schedule yields its exact last-iterate objective-gap constant, closing a question left open by Wang et al. (2025).

We establish the equivalence between the stochastic optimal control and path space formulations of the Schrödinger bridge problem (SBP) for the kinematic equation on a compact connected Lie group. Using the geometric concepts of horizontal lift and stochastic anti-development, we derive a Girsanov-type change-of-measure result, and show that the expected control energy equals the relative entropy of the controlled path law with respect to the reference Wiener measure. Thus, the SBP is equivalently a path space relative entropy minimization problem subject to prescribed endpoint marginals.Our result has three useful implications. From an analytic viewpoint, the shown equivalence helps prove the existence and uniqueness of the SB. From a probabilistic viewpoint, it helps interpret the SB as the most probable deviation of the uncontrolled stochastic dynamics consistent with the endpoint constraints. From a computational viewpoint, it allows using static Sinkhorn recursions to directly solve the relative entropy minimization problem and compute the optimal path measure. We illustrate the equivalence numerically on the torus $\mathbb{T}^2$. The code is publicly available at: https://github.com/gradslab/LargeDeviationSBP

LP-Newton solves a linear program through a sequence of nearest-point problems. Given an interior feasible point for an inequality-form LP, we construct the compact hull formed by the origin and its normalized constraint rows. Polar LP-Newton (P-LPN) follows the objective ray to the boundary of this row-polar hull, where it recovers a primal-dual optimum or a recession direction proving unboundedness. For rational row-polar data, we prove an outer-iteration bound quadratic in dimension and linear in binary input length, excluding inner Wolfe work. Restarting Wolfe at every outer iteration discards its terminal corral even though the hull is unchanged and the next target lies on the same ray. Corral-repair P-LPN (CR-P-LPN) instead repairs that corral and uses it to start the next projection. Verification over the full hull preserves P-LPN's projections, outer targets, and LP conclusion in exact arithmetic. Experiments in Julia and MATLAB show where repair helps. After common initialization, median P-LPN times across 180 single-LP tests are 1.62 times the corresponding CR-P-LPN times in Julia and 1.55 times in MATLAB. On three problems with 5,000 to 100,000 rows, the corresponding geometric-mean factors are 2.03 and 1.46. A separate end-to-end study includes initialization on 19 application-derived LPs. CR-P-LPN has lower median total time than the faster cold HiGHS mode, either dual simplex or the interior-point method with crossover, on 18 of the 19 workloads in each language. The geometric means of the ratios of HiGHS time to CR-P-LPN time are 4.93 and 3.06. Component comparisons identify terminal-corral reuse as the useful component, whereas candidate-first ordering alone shows no consistent gain. Repair is not always beneficial: it is slower on Klee-Minty tests, and an author-implemented simplex is faster on the retained Netlib models.

This paper develops a hyperbolic-majorization preconditioned three-term nonlinear conjugate-gradient framework for nonconvex finite minimax optimization. An analytic symmetric positive definite metric is derived from a global quadratic majorization of the hyperbolic smoothing model and is used simultaneously as a curvature absorber, a preconditioner, and the line-search energy metric. With the displacement \(s_{k-1}\) fixed and a variable curvature response \(b_k\), an enhanced three-term direction is introduced together with an adaptive parameter \(\mu_k^\star\) that is maximal under the requirement that the baseline worst-case metric-energy constant be preserved. The resulting framework yields a Dai–Liao-type conjugacy relation, enhanced sufficient descent, a smoothing-parameter-uniform Armijo lower bound, fixed-smoothing global first-order convergence and complexity, and Clarke-stationary accumulation points under continuation.

Modern real application problems involve matrix-valued parameters, yet conventional optimizers treat them as vectors, thereby motivating matrix-aware methods that exploit input-output geometry, such as Muon which orthogonalizes the momentum matrices before parameter updates. Its empirical success raises a conceptual question: can orthogonalized momentum remain effective beyond smooth optimization? This paper studies this question for locally Lipschitz functions using a generalized derivative framework compatible with backpropagation. Our first contribution is to identify a key limitation: for every fixed momentum factor $\beta\in[0,1)$, Muon can fail to approach the global optimal solution of a convex Lipschitz objective from almost every initialization, when step sizes adapt to the full gradient history. The failure can occur even along bounded iterates. Our example is inspired by the one of Parshakova et al. which only covers $\beta\in[0,\frac{1}{2})$. Then, we show that the obstruction lies in fixed momentum rather than orthogonalization. Indeed, when the momentum factor is adaptive and approaches 1 together with a vanishing step size, Muon recovers asymptotic convergence for nonconvex nonsmooth optimization under the boundedness and regularity conditions. Moreover, we propose MAGD, which combines orthogonalized momentum with gradient, weighted based on their relative progress. MAGD retains asymptotic convergence in nonconvex settings and achieves an $O(\min\{m,n\}\epsilon^{-2})$ rate in convex settings. A lower bound shows the optimal dimension dependence. Experiments on synthetic problems, image classification, and LLM pretraining show MAGD is a simple and practical alternative to Muon. Together, our results characterize when orthogonalized momentum fails without smoothness and how it can be made reliable and we hope that the analysis may be useful more broadly.

This work addresses decentralized online Riemannian optimization on Hadamard manifolds. Prior work under geodesic convexity (g-convexity) may require curvature information in the optimization analysis, typically through a finite lower bound on the sectional curvature. Curvature may also enter the step size or contraction factor of tangent-space Riemannian consensus schemes. In this work, we relax the curvature dependence for a narrower class of horospherical convex (h-convex) functions. We study Distributed Riemannian Online Gradient Descent (D-ROGD), which combines local Riemannian h-subgradient updates with an implicit Fréchet-mean consensus. For h-convex and strongly h-convex local objectives, we establish $O(\sqrt{T})$ and $O(\log T)$ static regret, respectively, matching the corresponding Euclidean rates with respect to $T$, with network dependence governed solely by the spectral gap. To our knowledge, these are the first curvature-independent regret guarantees for decentralized online optimization on Hadamard manifolds. Experiments on hyperbolic embeddings corroborate the predicted rates, with no observable degradation due to curvature.

We give an explicit extremal function for one step of gradient descent when the gradient error is bounded relative to the true gradient. The function is the sum of a quadratic in one coordinate and a Huber function in the other. It attains, for every admissible relative error level and every stepsize in the intermediate regime, the proposed worst-case constants for both the final squared gradient divided by the initial objective gap and the final squared gradient divided by the objective decrease. This proves Conjecture 2.6 of Vernimmen and Glineur, first stated in June 2025. A change of variable reduces the parameter description to a scalar cubic with a rigorously specified root. We prove all parameter inequalities, derive a nonnegative-square certificate for the upper bound, and verify the matching construction directly. We also characterize the exact one-dimensional worst-case value and prove that two dimensions are necessary in the open intermediate regime.

Products of positive semidefinite matrices arise when successive steps of a randomized optimization method use different data points. Their average can have a negative eigenvalue even though every individual matrix is positive semidefinite. Lai and Lim proved at ICML 2020 that $\sum_{i\ne j}A_iA_j\succeq-\frac{1}{4}n(n-1)I$ whenever $A_i\succeq0$ and $\sum_i A_i\preceq nI$, and conjectured that the constant is sharp for every $n\ge2$. We give an explicit family of $n$ rank-one matrices in dimension $n$ that attains this bound for every $n$, proving that sharpness statement. The construction compresses one direction of an orthonormal basis by a factor $1/\sqrt{2}$. We also give a short, self-contained proof of the known inequality and classify all equality cases: every extremizer contains a common invariant $n$-dimensional block equivalent to the construction. Consequently, dimension $n$ is necessary for equality, and positive definite matrices approach but never attain the lower endpoint. Finally, strongly convex quadratic objectives realize the extremal negative eigenvalue as a two-step expected gradient-descent update.

Modern design optimization frameworks aim first and foremost for models with the most accurate predictions without balancing computational overhead. It remains a reason why scaled architecture and multidisciplinary design optimization problems are difficult to address, even with sample-efficient Bayesian optimizers. In this paper, a metric quantifying the computational energy footprint is introduced within a Bayesian optimization framework to guide the parameter setting of a model towards configurations that balance both performance and frugality. The computer experiments highlighted existing tradeoffs between optimum convergence and the underlying energy footprint, and sometimes resulted in both a better-found optimum and lower energy consumption.

Control allocation for multirotors with bidirectional propellers is commonly formulated in signed-thrust variables, where the wrench map is linear. The signed-quadratic map from physical rotor speed to thrust, however, is not a local diffeomorphism at zero speed. This work derives two distinct consequences. Under nonredundant full actuation, a single-propeller reversal removes one instantaneous task direction; hence, no global continuously differentiable exact static allocator exists over the complete task space. Under redundant actuation, the physical task map may remain regular, yet a transverse pseudoinverse zero crossing requires an unbounded rotor-speed derivative. We define differential realizability as regularity of the physical lift of an actuator-output section, derive exact and first-order validity conditions, and distinguish structural rank loss from an allocator- induced rate singularity. A local nullspace deformation repairs isolated pseudoinverse reversals, while a global fixed-orthant construction establishes existence of regular sections at the cost of persistent task-preserving internal actuation.

The Monge-Kantorovich problem with the Euclidean distance cost is degenerate, typically admitting infinitely many optimal plans. A unique optimal plan is selected by perturbing the distance to a strictly convex or increasing strictly concave cost and passing to the limit: the convex side gives the plan induced by the map which is monotone on each transport ray, while the concave side was understood only on the real line, where Juillet proved that power-cost optimizers converge to the so-called excursion coupling, which is induced by a map when the source is atomless. This paper extends the concave selection to $\mathbb{R}^n$ for mutually singular finite positive Borel measures $\mu$ and $\nu$ with equal total mass and finite first moments, assuming that $\mu$ is absolutely continuous with respect to the Lebesgue measure: for a broad class of increasing strictly concave perturbations of the distance with a well-defined first-order profile, the corresponding optimal maps converge in $\mu$-measure to the same intrinsic limit $t_\#$, independent of the perturbation family and its profile. The map $t_\#$ is obtained by disintegrating the transport along its maximal rays and applying the one-dimensional excursion coupling on each ray. The same limit is established for the power costs $\|x-y\|^{1-\varepsilon}$ under a finite $\|x\|\ln(1+\|x\|)$ moment condition, establishing a conjecture of Juillet.

The growth of e-commerce has intensified the need for last-mile delivery systems that can jointly manage customer choice and operational efficiency. We study the Dynamic Offering and Pricing of Mixed Delivery Options problem, in which a logistics service provider dynamically selects and prices attended home-delivery time slots and out-of-home pickup options for sequentially arriving customers. Each decision affects immediate revenue, customer acceptance, and the route-dependent fulfillment cost realized at the end of the booking horizon. We formulate DOPMDO as a finite-horizon Markov decision process and propose State-Value Anchored Pricing via Approximate Dynamic Programming. The method learns a continuation-value approximation on an aggregate state representation of the mixed-delivery system and uses accepted-versus-rejected value differences to estimate option-level opportunity costs. These opportunity costs are embedded in an anchored trust-region pricing problem around a calibrated fine-static benchmark. Computational experiments on the real-world Seattle instance show that SVAP–ADP increases mean episode profit by 7.0% relative to current practice (95% CI: 6.7–7.2%), primarily by reducing terminal fulfillment cost while maintaining a stable home–locker–opt-out mix. Assortment-control experiments show that dynamic pricing is most effective when the menu exposes operationally valuable locker alternatives, with richer candidate pools delivering substantially larger gains than restricted nearest-locker menus. These results indicate that anticipatory pricing and assortment control are complementary: pricing steers customers toward lower-cost options, but the menu determines whether high-value consolidation opportunities are available in the first place.

We develop a spectral three-term modification of the classic Hestenes–Stiefel conjugate gradient algorithm, preserving its anti-jamming characteristic and, simultaneously, taking care of the sufficient descent property. We discuss how a modified secant equation can be extracted from our modification scheme, yielding a memoryless BFGS updating formula. Then, the spectral parameter of our method is obtained by steering its direction toward the given BFGS direction within a least-squares context. Using our technical improvements, we outline the general framework of our algorithm and discuss its theoretical features, including the descent and convergence properties, without the convexity assumption. We put our algorithm to the test in comparison with the three other conjugate gradient algorithms on a set of CUTEr unconstrained optimization test models, comparing the outputs using the Dolan–More measure. Next, we provide a concise evaluation of the results, highlighting the practical advantages of our algorithm. As a real-world case study, we introduce a reduced quadratic surface SVM with the rescaled loss for robust binary classification and apply the proposed algorithm to assess its accuracy and training time against several other SVMs.

We study the relation between positive semidefiniteness and entrywise nonnegativity for solutions of continuous-time Lyapunov equations. For a real Hurwitz matrix $A$, we show that the solution of $AP+PA^\top=-Q$ is positive semidefinite for every symmetric entrywise nonnegative $Q$ if and only if the solution of $A^\top X+XA=-R$ is entrywise nonnegative for every positive semidefinite $R$. This equivalence follows from the adjointness of the two solution operators and extends to all real unmixed matrices. We then prove that both properties hold for every real Hurwitz companion matrix, settling a conjecture previously established under the additional assumption of a real spectrum. The proof combines a controllability-Gramian normalization with a pairwise positivity theorem for the coefficients of the adjugate polynomial of a real accretive matrix. The latter is obtained from a coefficient-sign property of bivariate polynomials, an auxiliary determinant that does not vanish on the product of two open right half-planes, and a rank-one perturbation argument. Covariance and energy interpretations connect these results with dissipative realizations, damped second-order systems, and comparisons between input Gramians.

For computing the ground state of the defocusing Gross–Pitaevskii energy, we propose and analyze two efficient schemes based on the Davis–Yin three-operator splitting, which treats the potential and interaction terms explicitly, and the kinetic energy by a resolvent. One iteration costs one inversion of $I-\gamma\Delta$ for the first scheme, and of $I-\gamma\Delta+\gamma V_1$ for the second, with $V_1$ denoting the separable part of the potential, and on structured meshes both operators can be inverted by simple fast GPU solvers. For monotone discrete Laplacians, including the second-order finite difference scheme and the lumped linear finite element method on simplicial meshes with suitable angle conditions, we prove global convergence to the unique positive discrete ground state, for every positive normalized initial vector, for any constant step size below an explicit threshold. In contrast, the methods previously proven to converge globally to the ground state all invert a more difficult elliptic operator that depends on the iteration variable. In three-dimensional tests with up to $999^3$ unknowns on one GPU, a simple variable step size rule makes the proposed splitting schemes efficient in practice, comparable in wall-clock time to Riemannian conjugate gradient methods that also invert only a shifted Laplacian operator, and much more robust with respect to the choice of the initial guess.

Recent developments in the US knowledge economy have created a significant growth in datacenter loads, with two major consequences for power generation. First, to compensate for growth in load, datacenters are encouraged to bring their own generating units. Second, in search of the remaining pools of dispatchable generation, utilities are increasingly turning to subtransmission and distribution level generating assets. Coupled with increasing loads and the resulting tighter grid conditions, both trends are likely to create a need to commit a large number of localized generating units under grid constraints. We propose an extension of Relax-and- Round Unit Commitment (RRUC) that is capable of committing generating units for larger problems faster than conventional methods, while staying within intertemporal and spatial MVA constraints. We demonstrate the performance of RRUC using synthetic congestible test systems ranging from 100 to 20,000 buses. RRUC consistently finds low cost solutions, independent of the problem size, and its run time increases sub-quadratically in the number of buses. RRUC can solve the 100 bus system in less than a second and the 20,000 bus system 7 minutes. In contrast, a leading state of the art solver cannot find a feasible solution to the 100 bus system in 15 minutes.

Understanding how humans respond to incentives, both at the individual and collective levels, is crucial to the design of effective policies. Within the continuous-time stochastic framework for large interacting populations, mean field games (MFGs) model populations of non-cooperative agents, whereas mean field control (MFC) describes the fully cooperative benchmark, interpreted in our setting as fully altruistic behavior. Mixed-individual MFGs interpolate between these two extremes through a parameter governing the degree of altruism. A central challenge for regulators and policymakers, however, is that intrinsic altruism levels and other private structural parameters, such as individual labor costs, are typically unobservable. To address this challenge, we develop an inverse learning framework for mixed-individual MFGs. Our approach enables the recovery of (latent) altruism and labor cost levels from noisy observations, with experiments demonstrating the feasibility and accuracy of our method. These findings underscore the promise of inverse MFG methodologies for uncovering latent preference structures in large populations, with important implications for incentive design, empirical behavioral modeling, and data-driven policy analysis.

We establish near-optimal higher-order oracle bounds for smooth monotone variational inequalities. For fixed $p\ge2$, let $F$ be monotone on a known compact convex set $X$ of diameter at most $D$, with $\operatorname{Lip}(D^{p-1}F)\le L_p$. Each feasible query returns the complete jet $(F,DF,\ldots,D^{p-1}F)$, and the goal is to find $x$ with tangent residual $\operatorname{dist}(0,F(x)+N_X(x))\le\varepsilon$. Writing $Q=L_pD^p/\varepsilon$, we improve the $\widetilde O_p(Q^{1/p})$ upper bound of Chen et al. to $\widetilde O_p(Q^{2/(3p-1)})$ via a dimension-independent deterministic algorithm that returns an explicit tangent-residual certificate. We prove a matching $\Omega_p(Q^{2/(3p-1)})$ lower bound for arbitrary adaptive deterministic algorithms and randomized algorithms with per-instance success probability at least $2/3$, without span or tensor-update restrictions. Hence the high-dimensional worst-case oracle complexity is $\widetilde\Theta_p((L_pD^p/\varepsilon)^{2/(3p-1)})$. The same method applies to smooth convex–concave minimax problems, improving the fixed-geometry accuracy exponent from $4/(3p+1)$ to $2/(3p-1)$ and matching the known lower-bound exponent.

This work presents an interpretable data-to-control framework for actuator-constrained flutter suppression by combining higher-order dynamic mode decomposition (HODMD), dynamic mode decomposition with control (DMDc), and model predictive control (MPC). The primary case is a pitch-plunge typical section with a trailing-edge flap and rational unsteady aerodynamics. HODMD identifies the dominant frequency and growth rate in subcritical, near-critical, and post-critical regimes from four measured structural channels. With 2% RMS sensor noise, the near-critical growth-rate error is 1.8 x 10^-3 s^-1, substantially lower than with standard DMD without delay embedding. A five-state real reduced-order model is formed from the HODMD subspace, and its flap-input matrix is estimated from a small-amplitude PRBS record using DMDc. For an independent chirp input, the normalized error over all channels is 2.4 x 10^-3. The constrained MPC stabilizes the true post-critical plant for an initial pitch perturbation of 8 degrees with a flap limit of 1 degree, whereas saturated LQR designed from the same model reaches the prescribed validity limit. Three supporting studies assess transferability. SU2 simulations show HODMD frequency recovery consistent with the available spectral resolution; OpenFOAM dynamic-mesh calculations show nearly linear articulated-flap moment authority from 2 to 8 degrees with frequency-dependent gain; and a two-way SU2 Python-FSI case demonstrates bounded disturbance rejection using a proportional-derivative baseline. These studies support modal identification, actuator authority, and closed-loop feasibility, but are not presented as CFD-level MPC validation. The framework provides a reproducible route from early-time measurements to constrained aeroelastic control.

We develop a geometric framework for distance-based formation control that separates the evolution of inter-agent distances from its realization by compatible node motions, reducing the stability problem to the edge space. We show that local exponential convergence of the edge dynamics implies local exponential convergence of the formation, and that stability is certified by spectral properties of a linear edge operator. We introduce a hierarchy of generic spectral properties — weak admissibility, admissibility, and strong admissibility — that provide necessary conditions for local exponential stability. Specializing to directed sensing, we obtain a necessary and sufficient spectral condition for local stability at an arbitrary target, together with a quadratic sufficient certificate. These conditions reveal that stability depends jointly on the graph orientation and target geometry, and show that persistence is neither necessary nor sufficient for local convergence. We show that every generically rigid graph admits an admissible orientation and, for acyclic orientations, we give an exact combinatorial characterization of admissibility. Finally, the quadratic certificate leads to a semidefinite program for synthesizing stabilizing edge gains.

We introduce an adaptive multi-parameter variant of ADMM and prove that it exhibits local superlinear convergence once the set of active constraints has been identified. In simulations, the proposed algorithm consistently outperforms OSQP, a standard ADMM solver, in terms of iteration count. We then implement an MPC solver based on our method and compare its runtime against a broader selection of state-of-the-art algorithms. Evaluations on challenging benchmark problems reveal that our approach delivers competitive performance both in terms of average and worst-case solve times, without being limited to coarse tolerances, as is typically the case for standard ADMM implementations and first-order methods.

We prove that the solution map of the classical bilateral parabolic obstacle problem is directionally differentiable when interpreted as a function between suitable Lebesgue and Sobolev spaces. To the best of our knowledge, this paper is the first to establish this kind of differentiability for a parabolic evolution variational inequality with two-sided fully distributed pointwise inequality constraints. Our analysis relies on known differentiability results for unilateral parabolic obstacle problems and a new theorem on the directional differentiability of solution operators of non-expansive parameterized fixed-point equations. The latter may also be of interest for other applications. Our results can be used, for example, to derive optimality conditions for optimal control problems governed by bilateral parabolic obstacle problems.

This paper studies the problem of optimal control of Caputo linear fractional-order systems of order $\alpha \in (0, 1]$ with continuous-time state constraints and input constraints. The resulting problems are semi-infinite, since the state constraints are imposed over the sampling intervals and not merely at the sampling instants. To address this, we derive a sampled-data representation of the inter-sample trajectory and use Bernstein polynomials to construct polytopic approximations of the continuous-time state constraints. Explicit bounds are obtained for the conservatism introduced by this approximation. By subdividing each sampling interval and applying the Bernstein-based construction on every subinterval, the approximation is made arbitrarily tight. Numerical examples illustrate the tightness of the proposed enclosures, the computational tractability of the proposed method, and the trade-off between approximation accuracy and computational complexity.