Papers

Systems that turn natural-language descriptions of optimization problems into solver-ready code generally use a language model at every stage, including the final translation from a mathematical formulation into executable model-building code. We propose the ANVIL compiler architecture, where we separate these concerns. A language model is called only once, assisted by constraint guidance based on problem type, to produce a LaTeX formulation. A deterministic compiler then translates that LaTeX into code with no language model involvement. We describe the deterministic compiler (a normalizer, a recursive-descent parser producing a typed intermediate representation, analysis passes that bind symbols to a dataset schema, and a code emitter) and evaluate it on the 354 easy and hard problems of the NLP4LP benchmark. The compiler produced code for 329 of 354 formulations (92.9%), taking the deterministic path in every one of those cases and never falling back to model-generated code. Median compile time was below the 10ms resolution of our timer. Overall, our formulations achieved an accuracy of 98.9% over easy problems and 91.1% for hard problems. The gap between these compilation and accuracy figures is a key point of analysis, and we analyze it: formulations that failed to compile, problems that returned as infeasible, problems raising errors at runtime, and problems returning a wrong objective. Almost all of these errors trace back to the formulation rather than to the translation. ANVIL performs exceptionally well on leading benchmarks by using language models purely where they are effective, rather than as a catch-all tool.

Traditional dynamical system models, including Koopman operator representations, are fundamentally equality-based, whereas many analysis and control tools rely on inequalities. This mismatch motivates representations that are intrinsically aligned with certification tasks involved in the analysis and control synthesis problems. In this paper, we propose a data-to-certificates (D2C) paradigm that bypasses explicit model construction and directly learns certificates from data. We introduce supereigenfunctions of the Koopman operator as an inequality-based generalization of eigenfunctions that define exponential growth envelopes encoding stability, safety, and uncertainty propagation, thereby serving as certificates for a range of control objectives. We establish their theoretical foundations and show that the associated rates recover intrinsic dynamical quantities such as Lyapunov exponents. Two complementary constructions are developed: a geometric approach based on the multiplicative ergodic theorem (MET), and a resolvent/Gramian formulation that enables computation directly from trajectory data. The resulting framework yields certificates that can be used for stability and contraction analysis, as well as stabilizing and safety-critical control synthesis via convex quadratic programming-based optimization program. Numerical examples demonstrate the effectiveness of the proposed data-driven certification approach for stabilization, contraction, and safe control design.

We study the nonconvex trust-region subproblem by a certificate-coupled difference-of-convex algorithm that incorporates spectral information before exact first-order stationarity is attained. The method maintains a persistent randomized Krylov subspace together with the DCA iterates. Near stationarity, a sufficiently negative shifted Rayleigh quotient yields an explicit feasible correction with a quantitative decrease in the objective value; otherwise, completion of a prescribed Lanczos depth provides, with high probability, an approximate positive-semidefiniteness certificate. The same Krylov subspace also provides a certified upper bound on the largest eigenvalue for selecting the DC curvature parameter at subsequent accuracy stages. We prove a uniform negative-curvature correction result over the trust region, covering the near-orthogonal hard-case regime, together with finite-termination and work bounds at fixed tolerances and a quantitative objective-gap certificate. Under a progressive accuracy schedule, the certified objective values converge to the global optimum and the iterates converge in distance to the global solution set, while the Krylov subspace is retained across corrections and accuracy stages. Numerical experiments illustrate the phenomena established by the analysis: negative-curvature corrections can occur before the corresponding post-convergence spectral corrections, the quantitative decrease bound is satisfied in all tested near-orthogonal instances, and retaining the Krylov subspace reduces the median number of spectral matrix-vector products by approximately a factor of $2.6$ relative to restarting the spectral process in the controlled comparison.

This tutorial provides a unified view of the emerging area of bilinear learning and control. Using linear systems as a benchmark, it explains what fundamentally changes in the bilinear settings, how recent theory addresses finite-sample learning and control, and how these ideas connect to broader themes in nonlinear control, representation learning, and data-driven decision making. For learning, we emphasize tools that are particularly useful in the bilinear settings, such as one-sided Bernstein's inequality for dependent and heavy-tailed covariates, blocking arguments, and martingale concentration for input-dependent noise. We then apply these tools to obtain finite-sample learning guarantees for fully observed bilinear systems, partially observed bilinear systems, and linear systems with bilinear observations. For control, we discuss quadratic control from bilinear observations, where the classical separation principle fails, and review tractable approaches based on belief-space receding horizon control. We also cover stabilization of bilinear dynamics under state feedback using semi-definite programming, LMI relaxations, sum-of-squares methods, and Koopman-based lifting. We conclude by discussing connections to reinforcement learning and machine learning, and some open problems in combined learning and control of bilinear systems.

We study high-order safety-critical control of quadrotor teams under bounded inputs and pairwise collision-avoidance constraints. Squared-distance barriers may lose thrust effectiveness when the relative displacement is orthogonal to the available thrust directions, while nonsingular constraints may still be jointly infeasible under shared bounds. We characterize both phenomena through pairwise effectiveness and aggregate feasibility measures. A torque-aware dynamic extension exposes attitude torques in a fourth-order barrier and prevents the extended-input row from vanishing under positive thrust. Gaussian processes directly learn the fourth-order HOCBF residual, providing robust margins without differentiating unknown perturbations. Under residual-bound and persistent-feasibility assumptions, the resulting QP guarantees collision avoidance and recovers the nominal input whenever it satisfies the robust safety and actuator constraints.

Designing road networks for wind farms in complex terrain is a challenging task, especially under tight construction budgets. Traditional manual methods are time-consuming and may yield suboptimal results. We propose a structured three-phase optimization framework, TriPhase, to automate and minimize road construction costs from corridor selection to earthwork. Phase one formulates corridor selection as a Steiner minimum tree problem over a terrain-aware graph, incorporating slope and curvature constraints. In phase two, each road segment undergoes horizontal alignment optimization using a bilevel model, where a mixed-integer program evaluates vertical alignment costs. To ensure solver compatibility and performance, the model is reformulated explicitly for Gurobi. The final phase applies a network-wide convex optimization model to refine vertical alignment. Numerical experiments on real-world sites demonstrate up to 14% cost savings compared to industry-standard manual designs, validating the framework's effectiveness and practical relevance.

We compute the convex envelope of (nonconvex) bivariate piecewise linear-quadratic (PLQ) functions (bivariate quadratic functions defined on a polyhedral subdivision). Our algorithm is composed of the following steps: (1) compute the convex envelope of each quadratic piece obtaining piecewise rational functions (quadratic divided by linear function) defined over a polyhedral subdivision; (2) compute the (Legendre-Fenchel) conjugate of each resulting piece to obtain piecewise quadratic functions defined over a parabolic subdivision; (3) compute the maximum of all those functions to obtain the conjugate of the original PLQ function as a piecewise quadratic function defined on a parabolic subdivision; (4) compute the conjugate of each resulting piece; and finally (5) compute the maximum over all those functions to obtain the convex envelope (biconjugate) as rational functions (quadratic divided by linear function) defined over a polyhedral subdivision. Our contribution includes a practical algorithm running in linear time, and proving that the convex envelope is a piecewise rational function.

Mixed-variable blackbox optimization arises in simulation-based engineering and machine learning, where objective and constraint functions are expensive to evaluate and variables may be continuous, integer, binary, or categorical. Several software libraries are available for derivative-free optimization, but few combine ease of use, computational efficiency, flexibility, and theoretical convergence guarantees. This work presents NomadBBO, a user-friendly Python library for inequality-constrained mixed-variable blackbox optimization built around Nomad. Its core optimization framework is CatADS, a new direct-search method that extends Adaptive Direct Search (ADS) to mixed-variable problems. CatADS combines the mechanisms of ADS for quantitative variables with neighborhoods for handling categorical variables. These neighborhoods use Gaussian process-based or empirical Wasserstein distances derived from available data. Inequality constraints are handled through the progressive barrier. The mixed-variable and surrogate-based mechanisms are implemented in Python and connected to the efficient C++ backend of Nomad via Cython. This integration allows Nomad to handle categorical variables. It also enables the use of external libraries and hybrid optimization strategies, including Gaussian process models from SMT 2.0 and Bayesian optimization. Numerical experiments compare NomadBBO with other solvers on constrained and unconstrained mixed-variable problems from the Cat-Suite benchmark collection. The beta release is available at https://test.pypi.org/project/NomadBBO/

We stabilize the linearized 2-D Poiseuille channel flow at arbitrary Reynolds numbers, and assign the decay of every wavenumber to any user-prescribed rate. The actuation is as in Vazquez–Krstic (IEEE TAC, 2007), VK'07 below: the two velocity components at only one wall. This is the first Fredholm backstepping design for a Navier–Stokes system. In VK'07 the wall-normal input renders the plant spatially causal, after which the tangential input closes a Volterra backstepping loop. Here the normal input carries a finite-rank feedback of arbitrarily small gain, whose sole role, in the sense of Heymann's lemma (TAC, 1968), is to give the plant simple spectrum and make every mode controllable from the tangential velocity. The tangential input then applies a stabilizing feedback, obtained through an invertible Fredholm transform to a Stokes target PDE, shifted left by an arbitrary decay. No simplicity of eigenvalues of the Orr–Sommerfeld operator is assumed and the exclusions of plant coefficient values that have accompanied Fredholm backstepping are removed. We state the comparison with the Volterra design without embellishment: the Fredholm architecture is powerful, but its gain is spectral where VK'07 is closed-form, the simpler of the two, and one from which the Fredholm approach takes its model, its actuation, and its target. What is demonstrated here is the reach of the Fredholm method on its most demanding plant to date. The feedback is tested at Reynolds number $30000$, with the dominant part of the closed-loop spectrum assigned so that it amounts to slowing the flow about twentyfold.

Understanding the efficient transport of fluid droplets in confined geometries has been a domain of interest for industrial applications in recent times. In the present work, we focus on designing control strategies that optimally steer droplet motion. Here, we apply optimal control framework to the droplet transport problem in a microchannel based on lubrication theory that minimizes viscous dissipation. Two complementary modeling routes are adopted: a reduced-order ordinary differential equation (ODE) model optimized via Pontryagin's Maximum Principle, and a full nonlinear partial differential equation (PDE) model optimized using a Covariance Matrix Adaptation-Evolutionary Strategy. By parameterizing target displacement, droplet size, and capillary number, we uncover two distinct optimal transport regimes: a "translate-relax" strategy for short distances and a "compact-translate-relax" strategy for longer targets. In continuum mechanics, the competition between surface forces and cumulative viscous dissipation decides the optimal transport strategies. We further show that the displacement range over which the reduced-order controller transfers to the continuum model is governed by capillary stiffness. We finally address closed-loop stability by adapting a control Lyapunov function (CLF) framework to the reduced-order dynamics. We demonstrate that the terminal cost which penalizes deviation from the target, serves as a cost-compatible CLF on the physical domain, and prove that the CLF-compatible feedback exponentially stabilizes the target state.

Bernheim, Braghieri, Martínez-Marquina, and Zuckerman (2021, American Economic Review) develop a theory of chosen preferences and conjecture convergence to pure worldviews at low mindset flexibility. We show that the (original) conjecture fails: with arbitrarily many worldviews, a common-value action sustains mixed worldviews in every stationary equilibrium from a relatively open set of initial worldviews, even when all pure worldviews are absorbing. However, finite linear programs characterize a sufficient persistence condition. Conversely, mixed accumulation points require shared valuation maxima. If each implementable action has a unique maximizing evaluator, every existing stationary equilibrium converges to purity for all interior flexibility and discount parameters. Experienced utility converges without uniqueness. In existing equilibria, small payoff perturbations preserve compromise over fixed horizons with high probability. Strictly tailored worldviews yield existence and a one-step equilibrium characterization at sufficiently low mindset flexibility; weak implementation can preclude stationary pure-strategy existence. The persistence and convergence results extend to stationary randomization.

We consider solving (convex) conic linear optimization problems, at the scale where matrix-factorization-free methods are attractive or necessary. The restarted primal-dual hybrid gradient method (rPDHG) – with heuristic enhancements and GPU implementation – has been very successful in solving huge-scale linear optimization problems (LPs). However, its application to more general conic convex optimization problems is not so well-studied. We analyze the theoretical performance of rPDHG for general (convex) conic linear optimization, and LP as a special case thereof. We show a relationship between the geometry of the primal-dual $\delta$-(sub-)level sets ${W}_\delta$ and the convergence rate of rPDHG. Specifically, we prove a bound on the convergence rate of rPDHG that improves when there is a primal-dual (sub-)level set ${W}_\delta$ for which (i) ${W}_\delta$ is close to the optimal solution set in Hausdorff distance, and (ii) the ratio of the diameter to the “conic radius” of ${W}_\delta$ is small. And in the special case of LP, the performance of rPDHG is bounded only by this ratio applied to the (sub-)level set corresponding to the best non-optimal extreme point. Depending on the problem instance, this ratio can take on extreme values and can result in excellent or poor performance of rPDHG both in theory and in practice.

As robot capabilities increase, quadratic programming (QP)-based controllers must account for a similarly increasing number of constraints to ensure safe, reliable operation. Yet, with each added constraint, this introduces more chances of momentary conflict: in which case, a QP solver that returns an "infeasible" status leaves the controller with nothing to execute. To address this, we introduce ElastiQP, a modified dual active-set QP solver that relaxes every inequality constraint with an exact, per-constraint l1 penalty while keeping equality constraints (dynamics) hard. Notably, ElastiQP does so by folding the slack variables into the solver analytically, maintaining a constant size of the condensed linear system. On a suite of robot control benchmarks, ElastiQP achieves microsecond-level performance, matching or outperforming leading modern solvers on feasible problems. On infeasible problems, ElastiQP handles these gracefully, confining violations to strictly the conflicting inequality terms, returning a usable solution up to 40x faster than the best alternative solvers. ElastiQP is available as an open-source C++ header-only library, with Python and JAX interfaces, at https://github.com/StanfordASL/elastiqp.

This note establishes a geometric foundation for trajectory-manifold representations of deterministic nonlinear systems in a behavioral setting motivated by data-enabled predictive control. For a discrete-time system $x_{k+1}=f(x_k,u_k)$ with measured state and a $C^r$ transition map, $r\geq 1$, we consider the terminal-state-augmented finite-horizon behavior consisting of all admissible state-input trajectories over a prediction horizon $N$. We prove that this behavior is a $C^r$ embedded submanifold of the ambient trajectory space with intrinsic dimension $n+Nm$, where $n$ and $m$ are the state and input dimensions. Moreover, the rollout map from the admissible initial-state and input coordinates $(x_0,\mathbf u)$ is a $C^r$ diffeomorphism onto the behavior manifold, providing explicit global smooth coordinates. This yields a canonical exact encoder–decoder representation and implies that any exact differentiable latent representation of the full behavior must have latent dimension at least $n+Nm$. The geometric result does not require controllability, stabilizability, or invertibility of the dynamics. Corresponding results are given for zero-order-hold sampled continuous-time systems and fixed-step numerical transition maps. These results provide the deterministic geometric foundation for subsequent data-driven approximation and predictive-control development.

We develop a geometric framework for Hamiltonian engineering in finite-dimensional quantum systems using ideal control pulses. Starting from a bilinear Schrödinger equation with unbounded control amplitudes, we construct the closed pulse group and use extensions of control systems and Filippov's relaxation theorem to obtain a family of effective Hamiltonians given by the convex hull of the drift's adjoint orbit plus the Lie algebra of the pulse group. The geometry of this orbitope describes possibilities beyond group averaging. Using the isotypic decomposition of the adjoint representation, we characterize its affine hull and show that the group average lies in its interior. This yields locally accessible families of effective Hamiltonians around the invariant part of the drift. We apply the framework to recover the necessary and sufficient condition for dynamical decoupling from arbitrary interactions with a finite-dimensional bath. For connected abelian pulse groups, we describe the relevant representation decomposition through restricted roots. Finally, for qubit networks with identical pairwise couplings, we give a qualitative characterization and quantitative estimation of effective Hamiltonians that can be generated using our framework.

Constructing Lyapunov functions for nonlinear dynamical systems is a central problem in stability analysis, yet remains challenging. Lyapunov functions are commonly characterized as solutions to first-order partial differential equations (PDEs), but these solutions are typically obtained for single systems, limiting their reuse across systems. In this paper, we study the Lyapunov solution operator that maps a vector field to the corresponding Lyapunov function defined by a dissipation-based Lyapunov PDE. We establish that, on compact subsets of the domain of attraction and under exponential stability assumptions, this operator is well-defined, unique, and continuous with respect to perturbations of both the vector field and the dissipation function. These results provide a theoretical foundation for approximating Lyapunov functions uniformly over families of nonlinear systems. Building on these theoretical foundations, we employ Fourier Neural Operators (FNOs) as a data-driven approximation of the Lyapunov solution operator. Numerical experiments demonstrate that a single trained operator can accurately approximate the numerical Lyapunov functions across parameterized families of dynamics. This illustrates the potential of neural operators for approximating Lyapunov functions.

Region-of-attraction (ROA) certificates for polynomial systems become expensive as state dimension and degree grow: direct sum-of-squares (SOS) formulations require combinatorially growing monomial bases. Quadratization represents a polynomial vector field exactly on an invariant manifold of a quadratic system, allowing a quadratic Lyapunov function to certify the ROA. For a fixed lift, stabilizer gains shape the off-manifold extension and transverse dynamics, while representation gauges change the matrix representation but not the vector field. Both affect the spectral-norm certificate, yet prior work fixes the gain by a feasibility heuristic before optimizing the gauge. We formulate optimal dissipative quadratization (ODQ), jointly designing gains and gauges for a fixed monomial lift, reference extension, stabilizer factorization, and Lyapunov weight $Q=I$. Gains lie in a prescribed compact Hurwitz box. At each gain, an exact semidefinite program globally minimizes the spectral-norm bound over the gauge. Residual-aware bounds yield a certified closed Lyapunov sublevel set, accounting for the floating-point Lyapunov residual. Under our stated assumptions, every accumulation point of the idealized outer search is box-Clarke stationary. A finite run returns the best independently verified candidate; global optimality of the gain search is not claimed. On a planar quintic, optimizing the gain increases the certified area by a factor of $2.238$ over a matched zero-gain gauge. Across 16 heterogeneous polynomial systems with stabilizer freedom, ODQ improves on both fixed-gain lifted baselines. All 36 ODQ runs on the relay benchmark complete, and all 27 repeat-level comparisons across nine fully paired cases favor ODQ over an SOS baseline with a fixed quadratic Lyapunov function in both the fixed-direction proxy and construction time. Broader comparisons with direct SOS methods remain mixed.

Two-stage stochastic Mixed-Integer Linear Programs are a canonical modeling tool to optimize power system operations under uncertainty, yet their extensive-form counterparts scale linearly with the number of scenarios and quickly become computationally prohibitive under day-ahead time constraints. We propose an Input Convex Neural Network architecture to learn a convex surrogate of the second-stage value function, enabling fast first-stage optimization while preserving convexity by construction. We couple the surrogate with a Neural-Benders correction loop that refines the first-stage solution a posteriori, recovering the exact optimum whether the network overestimates or underestimates the recourse cost, and certifying solution quality independently of surrogate accuracy. We evaluate the method on IEEE Stochastic Unit Commitment benchmarks (case9-case118) under both continuous and integer recourse. The proposed approach achieves solutions with zero optimality gap on the $K$-scenario instance across all test cases and scenario dimensions $K \in \{10, 50, 100\}$, with speedups up to $214\times$ over the Extensive Form in the integer setting. Solve times are stable across scenario realizations and compatible with day-ahead time windows on the tested benchmark instances.

We prove the exponential convergence in relative entropy of the Sinkhorn algorithm for the entropy martingale optimal transport. We assume that the marginals have compact supports and are in strict convex order, the terminal marginal support is convex, and the initial marginal support lies in the relative interior of the terminal marginal support; the reference cost is assumed to be Lipschitz in each variable. Under these assumptions we establish uniform bounds on the dual variables modulo affine gauges. This result allows us to obtain the existence and uniqueness of the optimizer, and its exponential representation in terms of dual variables. We then establish a relative-entropy stability estimate for martingale couplings with different terminal marginals. Proving that the constant in the stability estimate stays uniform throughout the Sinkhorn iteration allows us to establish the exponential convergence of the Sinkhorn algorithm.

We study continuous-time fictitious play in 2-bidder, symmetric first-price auctions with independently distributed discrete values and a discrete bid set. We first exhibit a minimal instance — two bidders, two values, three positive bids — on which fictitious play with the standard uniform-split tie-breaking rule does not converge to the symmetric Bayes–Nash equilibrium: the equilibrium is unstable and the dynamics converge to a stable limit cycle far from the Nash equilibrium. We then show that a small modification of the tie-breaking rule — awarding a payoff of zero to every bidder in case of a tie — restores convergence: fictitious play converges to a Nash equilibrium of the modified game. This limit is an $\epsilon$-equilibrium of the original auction in a broad range of settings.

We establish convergence bounds for deep $V$-learning with horizon $H$. The algorithm fits a scalar value function to targets from executed transitions and selects actions using a predictive model and the value function. For current observed-successor targets with fresh true-kernel outcomes, the conditional mean is $\mathcal{T}^\beta V$, which averages over behavior-policy actions. The Bellman optimality update is $\mathcal{T} V$. We decompose the update error into six residuals: fitting, transition reuse, target construction, replay, action selection, and exploration. Under $L^s$ concentrability, their $L^p$ norms ($p=s/(s-1)$) control expected $L^1$ policy loss. The bound explicitly weights residuals from only the last $H-1$ update blocks, plus an initialization term for shorter runs. We quantify the cost of a shared sampling distribution across horizon levels. For statistical error bounds of order $n^{-\nu}$, we derive optimal continuous allocations and an integer allocation whose objective is within a factor $2^\nu$ of the constrained optimum. A margin condition with exponent $\alpha$ gives action error of order $\Lambda^{1+\alpha/p}$, where $\Lambda$ combines network drift and score error; a one-step construction proves the exponent sharp. Bounds on the distance between frozen and optimal scores transfer an optimal-gap condition to frozen-iterate gap bounds while retaining the mass of optimal ties. Survival probabilities and coverage conditions at deployment yield bounds for policies selected with approximate scores. Separate spatial ReLU networks per horizon level give a conditional neural regression rate, and the finite-state case gives a log-free expected fit rate. These results give expected policy-loss consistency for the fixed-horizon generative-reset approximate-ERM procedure with exact action scores and provide an explicit residual-decay criterion for FIFO/interleaved SGD.

This paper studies the regularity of the value function arising from a multidimensional continuous-time principal-agent model with separable, nonquadratic effort costs. The associated stochastic control problem has the output and the agent's continuation utility as state variables, and its Hamilton-Jacobi-Bellman equation is fully nonlinear and degenerate, with potentially unbounded coefficients. We address these difficulties by adding an independent regularization noise and bounding the effort. For the resulting problem, we establish classical regularity of the value function and show that the optimal effort is unique, positive and remains in a fixed compact subset, uniformly with respect to both the control restriction and the regularization parameter. These estimates allow us first to remove the control restriction and then to let the additional noise vanish. Consequently, we prove that the regularized value function converges to the original value function and conclude that the latter belongs locally to the Sobolev space $W^{2,1}_{\infty, \mathrm{loc}}$, thereby extending the regularity analysis to separable nonquadratic effort costs, for which the arguments yielding classical solutions in the quadratic-cost setting no longer apply.

We study deterministic adaptive optimization of globally $\beta$-smooth, $\mu$-strongly convex functions using exact scalar function values. Queries and outputs lie in $B_2^d(R)$, and the minimizer lies in $B_2^d(R/2)$. Set $\kappa=\beta/\mu$, $Q=\beta R^2/\epsilon$, and $D_d=(d/\log(ed))^{1/3}$. For sufficiently large $d$ and $0<\epsilon\le c_\epsilon\beta R^2$, the minimax value complexity $N_\epsilon$ satisfies \[ \begin{aligned} N_\epsilon &\ge c d\min\{\sqrt Q,\sqrt\kappa,D_d\},\\ N_\epsilon &\le C d\min\left\{ \sqrt Q,\sqrt\kappa[1+\log_+(Q/\kappa)] \right\}, \end{aligned} \] where $c,C,c_\epsilon>0$ are universal constants and $\log_+(t)=\max\{0,\log t\}$. The lower bound uses an exactly shielded smooth chain and batched delayed rotations; the upper bound combines finite differences, acceleration, and restart. When $\min\{Q,\kappa\}\le D_d^2$, these bounds match up to constants in the accuracy-dominated regime $Q\le\kappa$ and at constant relative accuracy $\epsilon=\Theta(\mu R^2)$. For arbitrarily higher accuracy in the range $\kappa\le D_d^2$, the bounds differ by at most $1+\log(\mu R^2/\epsilon)$; the optimal accuracy dependence remains unresolved in general.

We assemble a finite-dimensional \(\mathcal{VU}\)-calculus for composite nonsmooth functions whose outer function is convex: the chain rule, separable and convex sums, a strictly differentiable perturbation of a lower semicontinuous (lsc) term, the model \(\delta_X+f_0+\theta\circ F\), and a finite maximum of \(C^1\) functions. The same algebra yields an \(\varepsilon\)-\(\mathcal{VU}\) chain rule for a proper outer approximation of the subdifferential. On the set \(\mathcal{R}_{h,F}\) of points at which the convex chain rule holds, the subspaces \(\mathcal{V}f\) and \(\mathcal{U}f\), an orthonormal frame of \(\mathcal{U}f\), and the \(\mathcal{U}\)-gradient \(\bar g_u\) are written in terms of the factors. If \(f\) is \(C^1\)-partly smooth at \(\bar{x}\), that gradient is \(\bar g_u=U_f^\top\nabla_{\mathcal{M}}f(\bar{x})\). If \(f\) is \(C^2\)-partly smooth and \(0\in\ri\partial f(\bar{x})\), the convex \(\mathcal{U}\)-Hessian \(H_U\) (when \(f\) is convex) or the local matrix \(H_\varepsilon\) (when \(f\) is prox-regular at \(\bar{x}\) for \(0\)) equals the Gram matrix \(U_f^\top\nabla^2_{\mathcal{M}}f(\bar{x})\,U_f\); the same matrix, written along the active manifold in a continuous frame, depends continuously on the base point. If in addition \(f\) is \(C^2\)-partly smooth at \(\bar{x}\), then under prox-regularity and subdifferential continuity at \(\bar{x}\) for \(0\), \(H_\varepsilon\succ 0\) is equivalent to tilt stability of \(\bar{x}\) and to strong metric regularity of \(\partial f\) at \((\bar{x},0)\), with \(\lip\bigl((\partial f)^{-1}\bigr)(0\mid\bar{x})=\|H_\varepsilon^{-1}\|\). The calculus and the test are illustrated on a hinge composite and two elementary tests of \(H_U\succ 0\).

Sign-based methods reduce communication costs in distributed environments, but aggregating local signs can introduce bias when data are heterogeneous. As a result, existing sign-based variance reduction methods fail to obtain the optimal convergence rates. In this paper, we solve this problem and obtain optimal rates for both nonconvex stochastic and finite-sum optimization. We first give a counterexample showing that majority voting can fail to approach stationary points even with exact local gradients. Motivated by this limitation, we propose tracking the global gradient at the server through unbiased compression of recursive gradient increments. As a result, we can obtain the convergence rates of $O(\sqrt{d/K}+\sqrt d (a/(nK))^{1/3})$ for the $\ell_1$-norm and $O(\sqrt{a/K}+\sqrt a/(nK)^{1/3})$ for the $\ell_2$-norm. Here, $K$ is the iteration number, $n$ is the number of workers, $d$ is the dimension, and $a=1+\omega$, with $\omega$ denoting the compressor's relative variance. For finite-sum problems with $M$ components, we combine periodic exact gradient refreshes with compressed component-gradient differences. The resulting total sample complexities are $O(M+d\sqrt{aM}\epsilon^{-2})$ and $O(M+a\sqrt M\ epsilon^{-2})$ for $\ell_1$ and $\ell_2$ gradient norms at most $\epsilon$, matching the corresponding bounds in centralized settings.

Classical dynamic programming represents optimal sequential decisions through value functions and policies. While this functional representation is natural for computing optimal decisions, it does not directly identify the mathematical object governing policy reconstruction, representation complexity, or oracle-query complexity once an optimal policy is fixed. This paper addresses this question by developing a geometric theory of structured optimal policies in which the decision-boundary geometry induced by the policy becomes the primary object of analysis. We show that, under suitable structural regularity conditions, this geometry provides the minimal representation required for policy reconstruction and determines the statistical and computational complexity of the reconstruction problem. Building upon this representation, we establish structural properties of policy-induced decision geometry, introduce intrinsic notions of boundary and decision complexity, derive information-theoretic measures of decision compression, and obtain statistical guarantees for boundary estimation and policy reconstruction from black-box policy queries. Collectively, these results demonstrate that, for the structured decision problems considered here, the complexity of policy reconstruction is governed by the geometry of the decision boundary rather than by the cardinality of the ambient state space. Controlled numerical experiments examine the principal theoretical predictions and provide empirical evidence consistent with the proposed framework.

Observer design for nonlinear systems is a relevant and challenging task in systems and control design. In this work, we follow the idea of embedding the system in the class of nonlinear parameter-varying systems to benefit from linear structures as in a standard LPV embedding while keeping some nonlinear structures and, thus, reducing the numbers of scheduling-parameters in the representation. We lay out the NLPV observer design procedure for general nonlinear systems, propose a number of improvements, and exemplify the application for a two-arm robot model. In a numerical study, we compare the performance of the NLPV design to established standard nonlinear approaches such as the extended Kalman filter and the moving horizon estimation.

Early wildfire detection is critical to prevent small ignitions from escalating into large-scale disasters, yet current monitoring systems lack a quantitative framework for allocating detection infrastructure at scale. We jointly optimize the placement of monitoring infrastructure and the routing of autonomous drones under realistic operational constraints to quantify the investment required for rapid, large-scale wildfire detection. Evaluated out-of-sample on 3,693 California ignitions from 2021-2024, an optimized drone network operating at a $100 million five-year budget detects 97.3% of fires, including 74% within the first hour. Amortized over five years, that budget is about $20 million per year, roughly 5% of California's annual wildfire-prevention expenditure. Under current technology costs, drone-based monitoring is substantially more cost-effective than static ground sensors. Detection is governed primarily by spatial coverage, while routing strategy mainly determines detection speed.

The classical Projected Hessian Lemma, originating from Finsler's theorem, characterizes definiteness over a constraint null space through quadratic penalties. However, it does not quantify the corresponding constrained eigenvalues or the associated eigenvalue penalty path. This work develops a quantitative penalty theory for constrained symmetric eigenvalue problems and establishes exact characterizations of the extremal eigenvalues of the reduced Hessian via full-space penalized eigenvalue problems. Three proofs are provided based on orthogonal decomposition, Schur complement analysis, and semidefinite programming duality. We characterize finite exact recovery along the extremal-eigenvalue penalty paths and, in the absence of finite recovery, establish asymptotic convergence with a first-order error expansion and an explicit leading coefficient. The associated Hellmann–Feynman sensitivity relation leads to a strategy for predicting penalty parameters. Based on these results, we develop a matrix-free Penalty–Split–Merge method for successive constrained extremal eigenpairs using penalty continuation, Split–Merge iterations, deflation, and projected certification. Numerical experiments illustrate the predicted penalty regimes, evaluate projected certification, and assess computational performance on moderate-scale benchmarks and large-scale matrix-free test instances.

Stochastic subspace methods have gained popularity as gradient descent based techniques for large scale optimisation problems, especially in distributed settings. In this paper, we introduce the technique of "persistence of memory" to greatly extend and improve the random subspace methods. To this end, we leverage a vector that is only weakly correlated with the gradient in order to provide a guiding structure to the generative process of the random subspace along which the descent is going to take place. This guidance vector may be fixed for a large number of iterations, only to be refreshed at wide intervals (on whose size we can provide guarantees in terms of problem parameters). In important machine learning settings, such as optimisation problems embodying sparsity or a minibatch structure, we show that the guidance vector can be obtained in an effective and computationally inexpensive manner by leveraging the structured properties of the problem. En route, we establish to our knowledge the first theoretical analysis of classical SSD methods for sparse functions. In a local neighbourhood of the optimum, we demonstrate an alignment phenomenon of our gradient estimates with a low-lying eigenvector of the Hessian, allowing a once-for-all computation of the guidance vector which renders the method computationally favourable even in scenarios with unstructured objectives.