Papers

We characterize the fresh-gradient oracle complexity of smooth nonconvex-strongly-concave minimax optimization, with matching upper and lower bounds up to logarithmic factors. Let $\Phi(x)=\max_y f(x,y)$, where $f$ is jointly $L$-smooth and $\mu$-strongly concave in $y$ on unconstrained Euclidean domains, and set $\kappa=L/\mu$. Each query returns a fresh unbiased joint gradient with conditional variance at most $\sigma^2$. Given initial primal gap at most $\Delta$ and dual residual $\|\nabla_y f(x_0,y_0)\|\le G$, the fixed-budget complexity of finding $\|\nabla\Phi(\widehat x)\|\le\epsilon$ with probability at least $2/3$ is $\widetilde{\Theta}(\sqrt{\kappa}L\Delta/\epsilon^2+\kappa L\Delta\sigma^2/\epsilon^4+\kappa^2\sigma^2/\epsilon^2+\sqrt{\kappa}\log_+(G/(\epsilon\sqrt{\kappa})))$, where $\log_+u=\log\max\{1,u\}$. This characterization holds when $\kappa$ and $L\Delta/\epsilon^2$ exceed universal constants, uniformly over finite dimensions, and the suppressed logarithms are independent of $G$. It establishes the necessity of linear condition-number dependence in the global stochastic cost and identifies a separate statistical refinement cost. A proximal method separates coarse primal progress from one final refinement, while the lower bounds apply to arbitrary adaptive randomized algorithms. Dual initialization contributes only an additive logarithmic cost, yet removing its control eliminates every finite dimension-free complexity bound, even with exact gradients.

We analyze a simple stochastic inertial Krasnosel'skii–Mann (iKM) method for finding a fixed point of a nonexpansive operator in a real Hilbert space. Our method is obtained simply by adding two inertial extrapolations to stochastic KM [Bravo and Cominetti, 2024], and it retains one call to a possibly biased stochastic oracle per update and achieves sharp rates in both the stochastic and deterministic regimes. Specifically, with our proposed parameter schedule, we prove the following last-iterate fixed-point residual bound: \[ {O}\!\left(\frac{1}{K} +\frac{\sigma\log K}{\sqrt K} +\frac{B_K\log K}{K}\right), \] where $K$ is the horizon, $\sigma$ is the noise level and $B_K$ is the accumulated root-mean-square bias. When $B_K=O(\sqrt K)$, this yields $\widetilde O(\epsilon^{-2})$ sample complexity that matches, up to a logarithmic factor, the stochastic-oracle lower bound given under the unbiased subclass of our model [Foster et al., 2019, Theorem 2]. It also improves the best-known $O(\epsilon^{-4})$ random-iterate guarantee for stochastic KM [Bravo and Cominetti, 2024, Corollary 5.4]. To our knowledge, this is the first single-loop method for general nonexpansive fixed-point problems to attain this near-optimal sample complexity without variance reduction or batching. When the oracle is exact, the same method attains the worst-case-optimal $O(K^{-1})$ last-iterate residual rate [Park and Ryu, 2022, Theorem 4.6], improving the $O(K^{-1/2})$ rate of classical KM [Cominetti et al., 2014; Bravo and Cominetti, 2018].

The equal-weighted portfolio is a passive, rule-based strategy that has historically been difficult to outperform, delivering higher returns than the capitalization-weighted "market" benchmark across many markets and periods. Stochastic portfolio theory (SPT) reveals that this relative performance is regime dependent, with the equal-weighted portfolio underperforming during periods of increasing market concentration and high correlations, particularly market bubbles. These observations have motivated us to formulate and solve a stochastic control problem in which an investor actively allocates between the equal-weighted and market portfolios. The investor bases their allocation decisions on forecasts made under a flexible stochastic diversity–dispersion (SDD) model. Using a quadratic surrogate for implementation frictions, we characterize the optimal allocation through a linear forward–backward SDE and obtain an explicit "aiming in front of a moving target" representation of the optimal trading rate, in the spirit of Gârleanu and Pedersen. The penalty parameters are calibrated in sample to match the cumulative wealth effect of proportional transaction costs, while out-of-sample performance is evaluated with those costs deducted directly from portfolio wealth. Using historical S&P 500 data, we show that a mean-reverting SDD specification reproduces several empirical features of market diversity and dispersion. In out-of-sample backtests from 1995 to 2024, the resulting strategies deliver higher cumulative net returns than both the equal-weighted and market portfolios, and higher information ratios than the equal-weighted portfolio after 15-basis-point proportional transaction costs.

We consider the dynamic Schrödinger bridge problem on a finite state space and exploit its Markov structure to decompose the problem into an endpoint coupling and a collection of conditional Markov bridges. To sample from the conditional bridges, we develop a quantum Gibbs sampler based on quantum walks on the bridge path space. Under uniformly positive transition probabilities, we derive a spectral-gap bound and provide an explicit procedure for preparing the initial state. Compared with the update bounds of the corresponding classical Gibbs sampler, our quantum walk-query complexity improves the dependence on the time horizon $T$ from quadratic to linear. For the endpoint coupling, we adapt a quantum box-constrained Newton method to compute the Schrödinger potentials, improving the matrix-scaling complexity, at fixed accuracy, from quadratic to $N^{3/2}$ in the state-space size $N$. Finally, we show that exponential reweighting extends both the Schrödinger bridge formulation and the Gibbs spectral-gap estimate to models with additive running costs.

Scanning-probe x-ray tomography is useful for imaging nanoscale structures of a sample. The image resolution, which can reach down to 10 nm by the increased brightness and coherence of the x-ray optics, however, is highly susceptible to experimental error. Failure to address these errors can lead to a smeared image and, in the worst case, to misinterpretation of the imaged object's structure. In this work, we present a novel optimization-based approach to calibrate a common yet challenging source of experimental error, the drifts of the scanning positions, while simultaneously reconstructing the object. This approach uses the coupled and complementary information from different measurements to enforce consistency between the measurements and the reconstruction. We illustrate the proposed approach on both synthetic and real tomography images and show its superior performance compared with reconstruction without explicit error calibration.

Outpatient appointment scheduling must balance provider idle time, overtime, and patient waiting under uncertain service times, punctuality, and attendance. These factors may follow patient-and-time-dependent probability distributions, which existing models rarely incorporate simultaneously because of modeling and computational complexity. We propose a stochastic programming model that captures exponentially many scenarios with polynomially many variables and constraints, without sampling. It accommodates patient-dependent service times and patient-and-time-dependent no-shows and arrival times. We also discuss estimating these distributions from electronic health records. We examine how personalized reminders and changes in attendance due to incentives or negative clinic experiences affect scheduling. Our model optimally solves instances of up to 14 patients in reasonable computational time. Its schedules outperform classical Bailey-type rules. Even the best-performing variant has 97% higher expected cost on average. Incorporating patient-dependent service times reduces total costs by 34% on average. Accounting for patient-and-time-dependent unpunctuality and no-shows reduces costs by 12% and 67%, respectively. Personalized reminders can reduce costs by 23%, and sensitivity analyses show that these gains withstand moderate errors in estimated distributions. These findings show how modeling individual patient behavior and tailoring communication can reduce waiting, idle time, and overtime.

We study the long-time behavior of Wasserstein gradient flows for interaction energies \[ \mathsf E[\mu] = \frac12\iint_{M\times M}K(x,y)\,\mathrm d\mu(x)\,\mathrm d\mu(y) \] on a closed manifold $M$. For kernels diagonal in a Laplace eigenbasis with nonnegative spectral coefficients, we prove a differential inequality relating the relative entropy to the energy gap. Consequently, for any nonnegative initial density $u_0\in L^p(M)$, $p>1$, the energy gap is integrable in time and satisfies \[ \mathsf E[\mu_t]-\mathsf E_{\min}=o(t^{-1}). \] If all spectral coefficients are positive, the flow converges weakly to the constant measure. These interaction energies need not be geodesically convex in Wasserstein space, and the associated flows contain no diffusion; their global convergence therefore does not follow from standard Wasserstein gradient flow theory. The kernels covered by our results include zonal kernels on spheres, kernels arising in transformer models, regularized Riesz kernels, and inverse fractional Laplacian kernels. We also investigate the sharpness of the $o(t^{-1})$ rate. For any smooth kernel in this class with infinitely many positive spectral coefficients and any $\delta>0$, we construct a solution of the linearized flow whose energy is comparable to $t^{-1-\delta}$ along a sequence of times tending to infinity. Moreover, for any $\delta>0$, by choosing a suitable inverse fractional Laplacian kernel on the flat torus, we construct an exact solution of the nonlinear Wasserstein gradient flow whose energy is comparable to $t^{-1-\delta}$. The nonlinear construction is based on uniform-in-time estimates for the evolution of the dyadic Fourier coefficient blocks and a blockwise energy-persistence argument. These estimates also yield a uniform-in-time quantitative comparison between the nonlinear Wasserstein gradient flow and its linearization.

Let $\nu$ be a compactly supported Borel probability measure on $\mathbb R^{d}$ with $\nu(B(x,r))\leq Cr^{a}$ for some $a>0$. Refine a dyadic cube exactly when its mass is at least $t$, and let $\mathcal{L}_{\nu}(t)$ be the mean depth at which this refinement stops. We show that the lower and upper geometric-mean quantization dimensions of $\nu$ are the lower and upper limits of $\log(1/t)/\mathcal{L}_{\nu}(t)$. The dimension exists precisely when $(q-1)\sum_{Q}\nu(Q)^{q}$, summed over all dyadic cubes, converges as $q\downarrow1$, and it is then determined by this limit. The mass-threshold formula yields harmonic integral bounds in terms of the local dimensions and encloses entropy and quantization dimensions in a common spectral interval. Convergence in law of the local information rates is equivalent to convergence of the rescaled spectra in a window of width $1/k$ around $q=1$; the two dimensions are then the arithmetic and the harmonic mean of the limit law, and we quantify their difference by sharp bounds and variance identities. Without any convergence assumption, vanishing threshold variance still forces equality of the corresponding lower and upper dimensions. Bernoulli mixtures realise every local-dimension law with compact support in $(0,1]$, and a regime-switching example separates convergence in law from almost-everywhere convergence.

We study gradient descent with predetermined nonnegative stepsizes on smooth strongly convex functions. Let $p_{\mathrm{sil}}=\log_2(1+\sqrt2)$ and $\kappa$ be the condition number. We prove the iteration lower bound $\Omega\left(\kappa^{\frac{1}{p_{\mathrm{sil}}}-o(1)}\log\frac1\delta\right)$ for both relative squared distance and relative function error, uniformly over $0<\delta<1$ and sufficiently large $\kappa$. This matches the polynomial exponent of $\kappa$ for the Silver stepsize schedule established in [Altschuler and Parrilo, 2025].

We study decentralized partially observable team decision problems with low-rank latent dynamics and unknown system models. The proposed framework combines team-theoretic equivalence with low-rank model representations to address cooperative decision-making in partially observable Markov decision processes without prior knowledge of the transition model. Each team member makes decisions based on local private information and delayed common information shared across the team. Using only this available information, each member learns an approximate low-rank Markov decision process and applies least-squares value iteration to compute its policy. This yields a fully decentralized learning and planning algorithm that requires neither a centralized coordinator nor centralized training. We show that the resulting member-side solutions approximate the centralized team solution: despite partial observability, unknown dynamics, and delayed common information, each member recovers the corresponding component of an approximate team-optimal policy. We further establish finite-sample performance guarantees and derive a corresponding sample-complexity bound for the proposed algorithm.

We analyze an inexact, proximal trust-region algorithm for minimizing the sum of a smooth nonconvex function and a nonsmooth convex function. Our algorithm leverages the proximal gradient as a benchmark to guarantee global convergence. However, evaluating the proximal gradient involves solving a nonsmooth convex optimization problem that in many applications cannot be performed exactly. As such, we develop a framework that permits inexact evaluations of the proximity operator and we show the proposed inexactness conditions yield descent properties analogous to the exact proximal gradient. We further discuss practical procedures for computing approximate proximity operators for two common applications: (i) the nonsmooth term is the sum of nonsmooth convex functions where one is composed with an affine map and (ii) using alternative inner products to facilitate the evaluation of the proximity operator. We demonstrate our algorithm and confirm our analysis with two numerical experiments arising in PDE-constrained optimization.

We consider variational image denoising with spatially dependent Orlicz regularization and introduce an Orlicz–Sobolev approximation of the ROF model based on a scaled $L\log L$-type density. For a general class of uniformly superlinear integrands satisfying a $\Delta_2$-condition, we establish existence and uniqueness of minimizers and derive a Fenchel dual problem with dual attainment and pointwise optimality conditions. We then specialize the framework to a logarithmic density whose correction is activated above a prescribed local gradient scale. For this model, we obtain explicit expressions of the Fenchel dual and optimality conditions as well as a pointwise radial formula for the dual proximal map involving the Lambert $W$-function. As the logarithmic parameter tends to zero, we prove equicoercivity and $\Gamma$-convergence to the ROF functional, together with convergence of the corresponding minimizers. Numerical illustrations indicate that the logarithmic correction can reduce staircasing for diffuse transitions, while retaining behavior comparable to ROF for sharp interfaces and a natural image.

We study Polyak-type step-size selection for extragradient methods for solving deterministic and stochastic monotone root-finding problems. We show that the known projection-type correction for deterministic extragradient arises from minimizing an upper bound on the distance to a solution, paralleling the classical Polyak step-size construction. Using this viewpoint, we provide a unified deterministic analysis of the Polyak-type Extragradient Method (PolyakEG), based on a local critical condition controlling the variation of operator $F$ along the extrapolation direction. This analysis does not require global Lipschitz continuity, and covers sublinear convergence under broader conditions such as Hölder continuity or $(L_0, L_1)$-Lipschitzness and linear convergence under additional strong monotonicity, all through a single framework. We then study the stochastic extensions of this approach. We first prove convergence of a direct stochastic variant, PolyakSEG, when all stochastic component operators share a common solution. We also show that, without this condition, PolyakSEG with nonvanishing step-sizes may fail to converge to a zero of the mean operator. To address this limitation, we propose DecPolyakSEG, which combines decreasing step-sizes with Polyak-type updates, and establish a sublinear residual convergence result without requiring a common solution across the component operators. These results parallel recent developments in stochastic Polyak step-sizes from the convex minimization literature and establish an analogous research avenue in the broader root-finding regime.

We study an optimal boundary control problem for a quasilinear hyperbolic system describing isothermal gas flow in pipelines. The control acts at network nodes and models the action of compressor stations. The system admits non-constant steady states, which are strictly monotone in space. We work in the framework of H2-solutions and consider an optimal control problem that steers the system toward a prescribed steady state. We establish the existence of optimal controls and then prove the exponential turnpike property, showing that optimal trajectories remain exponentially close to the corresponding steady state over most of the time horizon. The analysis relies on a cheap control property obtained via a Lyapunov approach and on a constructed control. Numerical simulations illustrate the theoretical result.

Fast charging of all-solid-state batteries is limited by solid-state transport in the positive electrode, which can generate substantial concentration nonuniformity under aggressive charging. We formulate fast charging as a minimum-time boundary-control problem for a one-dimensional cathode diffusion model, subject to a terminal average-concentration target and a path constraint on the bulk-to-surface concentration difference. For constant diffusivity, we prove existence of a global minimizer and show that every minimum-time policy is maximal-feasible: the charging rate remains at its upper bound until the transport constraint becomes active and subsequently follows the active boundary. This structure motivates an offline-online strategy in which precomputed PDE responses are combined through low-dimensional algebraic operations, avoiding repeated PDE-constrained optimization. Numerical refinement and perturbation studies verify the predicted control structure and constraint satisfaction. We further investigate concentration-dependent diffusivity, quantify the robustness of the constant-diffusivity policy under model transfer, and introduce a blockwise nonlinear correction that restores admissibility when the transferred policy violates the transport constraint.

Reliability theory gives a mature language for layered systems, but its formal tools are not yet standard in frontier AI control. We apply them to Google DeepMind's defenses against rogue deployment. The same control stack can have cubic, quadratic, or linear rare-failure suppression depending on its failure domains. Birnbaum importance identifies which component improvements buy the most nominal reliability, while prevention changes the population on which recovery is demanded. These results give concrete guidance about what to separate, improve, measure, and test.

We study the infinite-horizon optimal investment and consumption problem in a general class of continuous financial markets, where uncertainty is driven by a continuous non-decreasing stochastic clock representing accumulated variance. This framework encompasses classical Markovian and non-Markovian stochastic volatility models as well as singular realized-variance models in which no spot volatility process exists. We characterize the value process and optimal investment and consumption strategies in terms of a non-linear infinite-horizon backward stochastic differential equation driven jointly by calendar time and the stochastic clock. We develop a general well-posedness theory for this new class of IVC-BSDEs based on the method of sub- and supersolutions, establishing existence, uniqueness, and stability under natural conditions that might be of independent interest beyond the financial application at hand. We are moreover able to identify the sign of the $Z$-component of the solution using Malliavin calculus. We then apply our results to Volterra Heston models with locally integrable kernels, covering both rough and hyper-rough regimes. Exploiting the affine structure of the model, we verify the optimality of the candidate strategies in incomplete markets and obtain an explicit representation of the solution in the complete market case. Owing to the generality of the framework and the weak assumptions imposed on the stochastic clock, our results unify and extend several existing results for optimal investment and consumption, including classical Markovian stochastic volatility models.

We study prophet inequalities for a random walk reward stopping problem with sample-based information. The goal is to stop as close as possible to the maximum of a random walk with i.i.d. increments, measuring performance by the ratio between the expected reward when stopping and the expected true maximum. We consider a sample-based model in which the increment distribution is unknown and the decision maker has access to $K$ independent sample paths of the reward process. For the infinite-horizon setting, we establish a sharp prophet inequality with constant $(K/(K+1))^{K+1}$. The guarantee is attained by a randomised stopping rule based on the ladder height decomposition of random walks. As $K\to\infty$, this recovers the classical $1/e$ prophet inequality from the full-information setting. For the finite-horizon setting, where the process terminates after $n$ steps, we first prove a tight no-information prophet inequality with constant $1/H_n$, where $H_n$ is the $n$-th harmonic number. For $K\ge1$ samples, we show that a prophet constant of $1/4$ is attainable. Finally, we prove that, with $K$ samples, the prophet constant is at most $(K/(K+1))^{K+1}+(6+6H_K)/H_n$ for $n\ge 2K^2$, implying convergence to the infinite-horizon constant as $n\to\infty$. Our approach combines random walk theory, including ladder heights and Spitzer's identity, with linear programming duality. Our results contribute to random walk stopping theory and to sample-based prophet inequalities for correlated rewards, an area that remains largely unexplored. To the best of our knowledge, our tight sample-based prophet inequalities are the first whose performance is parameterised exactly, rather than only up to constants, by the number of available samples.

Lee, Skipper, and Speakman (2018) asked for formulae or asymptotic expressions for the volumes of the strong and weak relaxations $\mathcal{P}_{m,n}\cap \mathcal{D}_{m,n}$ and $\mathcal{Q}_{m,n}\cap \mathcal{D}_{m,n}$ of the uncapacitated facility-location polytope with $m$ facilities and $n$ customers (their Problem~3, “Texas Hot”). We give exact formulae for all $m$ and $n$, and we give asymptotics that hold uniformly as $m+n\to\infty$, whatever the relative growth of $m$ and $n$. The volume ratio satisfies $Vol(\mathcal{Q}_{m,n}\cap \mathcal{D}_{m,n})/Vol(\mathcal{P}_{m,n}\cap \mathcal{D}_{m,n})\sim \prod_{j=2}^{n}\mleft(1+\tfrac{1}{(m-1)j}\mright)^{m}$. From this, we read off the behavior in every growth regime of $m$, and we identify a sharp transition at $m\asymp\ln n$. Below it, in a uniformly random point of the strong relaxation, each facility variable is close to $1$ with high probability, and the ratio grows superlinearly in $n$. Above it, in a uniformly random point of either relaxation, each facility variable is asymptotically uniform on $[0,1]$, and the ratio grows linearly in $n$.

We propose a bounded degree SOS+SONC hierarchy for constrained polynomial optimization, termed B-SOS+SONC. Starting from Lasserre's bounded-degree SOS framework, we enlarge the certificate cone from SOS to the recently introduced SOS+SONC cone, thereby combining the algebraic strength of semidefinite relaxations with the sparse structure captured by circuit polynomials. We show that, for each fixed certificate degree, the resulting hierarchy is complete, that is, its optimal values are monotone and converge to the global optimum. Moreover, we derive an explicit SDP-REP reformulation, so that each relaxation can be solved within a tractable convex optimization framework over semidefinite and relative entropy cones. Beyond the optimization hierarchy itself, we investigate structural properties of the SONC cone and introduce the notions of first-order and second-order SONC-convexity. This leads to a new sufficient condition for first-level exactness of the B-SOS+SONC hierarchy. Numerical experiments illustrate that the proposed hierarchy often yields tighter lower bounds than the B-SOS relaxation while remaining tractable.

We study a dissipativity-based model predictive control (MPC) scheme for energy-optimal constrained output stabilization of a discrete-time nonlinear SISO port-Hamiltonian system described by difference and differential representation. For the optimal control problem to be solved in each MPC step, we establish a measure turnpike behavior. The turnpike set in our work is obtained from the underlying port-Hamiltonian structure without assuming existence of steady-states or periodic orbits. In particular, unlike existing results which typically establish the turnpike property w.r.t. a controlled forward invariant set such as optimal steady-states or optimal periodic orbits, we do not require the turnpike set to be forward control invariant. For MPC, we establish recursive feasibility and show practical stability to the turnpike set w.r.t. the MPC closed loop leveraging both dissipativity and the port-Hamiltonian structure. Finally, we demonstrate our results using two numerical examples.

We study the cyclic steepest descent method (CSD) for strictly convex quadratic minimization. CSD repeats, for j consecutive iterations, the exact steepest-descent step size computed at the beginning of each cycle. The only rigorous result for the real algorithm has so far been restricted to two dimensions, where the cycle-starting gradient sequence is known to converge doubly exponentially. For the simplified ("simple") model obtained by discarding bounded logarithmic terms, Dai and Fletcher predicted that CSD is superlinear whenever the number n of distinct eigenvalues is below twice the cycle length m, and linear otherwise. Let A be symmetric positive definite with q distinct eigenvalues, and suppose 2j > q. We prove the superlinear side of this threshold for the real algorithm in arbitrary dimensions, and in fact obtain a faster, doubly exponential, decay. Except for a Lebesgue-null set of initial points, for every kappa below an explicit positive threshold kappa_0, the cycle-starting gradient satisfies ||g_k|| <= exp(-C exp(kappa k)), and the full gradient and iterate errors satisfy analogous bounds with exponent kappa/j after m iterations. This is the first rigorous proof for the real CSD in arbitrary dimensions, including repeated eigenvalues, and confirms the threshold 2j > q (the simple-model prediction n < 2m); the doubly exponential rate is faster than the superlinear one predicted by the simple model. The proof combines arbitrary-reference ratio coordinates, inverse-image volume contraction, thin-band estimates, and a per-component Borel-Cantelli argument.

In this paper we propose first-order methods for a class of nonsmooth composite strongly convex–strongly concave and nonconvex–concave minimax optimization. We first develop an inexact proximal method and an accumulative regularizated method for strongly convex–strongly concave problems. The latter achieves the optimal dependence on the curvature parameters and places the smaller curvature inside the accuracy logarithm. Using this method as a subsolver, we propose a proximal point method for nonconvex–concave problems. Under suitable assumptions, it finds an $\epsilon$-stationary point with an operation complexity of $O(\epsilon^{-5/2})$, which improves the best-known $O(\epsilon^{-5/2}\log(1/\epsilon))$ bounds by removing the logarithmic factor. We further develop parameter-free variants for both problem classes, and achieve the same complexity without knowledge of any problem constants. All proposed methods are equipped with verifiable termination criteria.

We study a geometric singular perturbation theory (GSPT) approach to optimal control problems involving entry-exit phenomena in planar fast-slow dynamical systems. In contrast to previous work, we assume that the control acts only on the slow dynamics and we consider minimum-time optimal control problems (i.e., we aim at minimizing the time to reach a certain target). We apply this method to a simplified fast-slow toy model, characterized by a repetition of entry-exit processes, enabling a transparent analytical exploration of the control dynamics and allowing us to derive the key results explicitly. First, we analyze each non-autonomous entry-exit phenomenon separately, and we derive the optimal way to control them. Second, by artificially generating a stable limit cycle, we determine under which conditions it is possible to reach the target. Finally, under such conditions, we combine all entry-exit processes together and we demonstrate the existence of an optimal control. Our approach mainly exploits techniques of GSPT and dynamic programming. Indeed, we show that our problem can be formulated in a way resembling the Bellman equation, even though not in its standard setting. Numerical simulations, built on such Bellman-like formulation, illustrate our analytical results. In particular, the optimal control is derived with an algorithm that converges in a finite number of steps. Lastly, we extend this novel approach to more complex planar fast-slow dynamical systems.

We study the problem of designing optimal damping for a system of $n$ coupled linear oscillators. Using the Weyl–Horn theorem, we construct an explicit damping matrix that realizes the optimal asymptotic decay rate of the system. We prove that this decay rate is sharp and is determined by the geometric mean of the natural frequencies. The resulting damping strategy is shown to outperform the classical Rayleigh (proportional) damping approach. We conclude by characterizing the parameter regimes in which the optimal damping matrix necessarily ceases to be positive definite and discuss the implications of this phenomenon.

Many statistical procedures compute an entire solution path rather than a single estimator. For one class the path is piecewise linear, traced corner to corner by an active-set homotopy. Two of its members coincide exactly: under $\Sigma=X^\top X$ and $\mu=X^\top y$ the gross-exposure-constrained mean–variance program and the constrained LASSO trace the same piecewise-linear curve, and they keep doing so under arbitrary linear equality and inequality constraints. Which parameter is swept, a leverage budget or a return tilt, is not a choice the curve notices: under a homogeneous mandate the two sweeps differ by a scalar. A mandate that holds the portfolio invested costs a radial rescaling instead, and the efficient frontier is that curve rescaled; its corners pass unnoticed because the frontier is continuously differentiable where they fall. We give the map between the two parametrisations and the single way it degenerates. That mean–variance selection and the LASSO instantiate one parametric quadratic program is prior art (Gärtner, Jaggi and Maria, 2012). The Critical Line Algorithm itself lies off that route, and the correspondence established here is new. The identity is one of curves, not of statistical experiments. The map between the parametrisations is computed from the data, so the geometry of the path transfers while quantities averaged over the response do not. We cross that boundary deliberately and give the degrees of freedom of the constrained fit under arbitrary linear constraints. In the long-only, fully invested case it is the expected number of holdings away from a bound, less one.

The paper studies open optimization programs, that is, parametric optimization programs viewed as open systems, and their interconnection. Each open optimization program is assigned a behavior and interconnection is defined through variable sharing. For a class of open quadratic programs (QPs), we characterize closure and well-posedness under interconnection and, for modules interconnected by adding their objectives, analyze the convergence of a sequential module-by-module solution method, deriving error bounds for the resulting approximation. We further illustrate that the variables through which modules are interconnected indicate an appropriate behavioral description, and each description requires dedicated interconnection rules. This perspective suggests a route toward compositional analysis and design of optimization architectures.

Helmke and Shayman showed that, for generic targets, least-squares distance to the manifold of rank-r matrices is a nondegenerate Morse function whose indices are governed by Bruhat-cell codimensions. We develop the analogous theory for Monarch matrices, products of a fixed permutation matrix with two block-diagonal factors. The parametrization's gauge dimension, as well as the codimensions and closure relations of the associated Bruhat-type cells, is determined by a common transfer-matrix invariant, although these quantities are generally distinct. In the equal-block case, we prove the Monarch-Schur theorem, characterizing the permutations that minimize gauge freedom and maximize manifold dimension, and derive a general lower bound for arbitrary partitions. We further establish generic second-order nondegeneracy, a Morse-index formula, and an explicit focal set of measure zero. Finally, we show that the unregularized gradient flow lacks coercivity because of unbounded flat directions. A quartic regularizer restores global convergence, selects a canonical minimum-norm exact solution when one exists, and otherwise yields a well-defined best-approximation image. The regularizer preserves the Monarch architecture and provides a geometric interpretation of weight decay in structured factorized layers.

We present the first method for native, continuous gradient descent for machine learning models with $p$-adic parameters. Existing native optimizers are discrete, mostly combinatorial searches, as the $p$-adic numbers $\mathbb{Q}_p$ are totally disconnected, with standard losses that are flat away from their minima. To enable continuous optimization, we propose working with $\mathbb{Q}_p$ via its Berkovich affine line: a canonical, path-connected expansion of $\mathbb{Q}_p$ that preserves its isometries and uniquely extends its analytic maps. This hull is a metric tree with interpretable points and local derivatives, which we show enables effective optimizers and backpropagation. We formulate gradient descent and show that its approximations efficiently learn linear models with coefficients in $\mathbb{Q}_p$ to do modular arithmetic, an XOR-like task not expressible by linear models in $\mathbb{R}$. We also demonstrate momentum and Adam variants, linear regression, and classification on binary-encoded hierarchies (Quillian semantic networks), addressing open problems posed by Martins (2025). Library at https://github.com/google-deepmind/padic-ml

Autonomous spacecraft guidance and control requires a fast solution to non-convex trajectory optimization, which can be accelerated by providing a near-optimal initial guess to an optimization protocol, i.e., warm-starting. A robust warm starting method is especially useful for rendezvous, proximity operations, and docking (RPOD) in cislunar space, where the underlying dynamics become severely nonlinear and chaotic compared to those in Earth orbit, especially at perilune. This paper extends the Autonomous Rendezvous Transformer (ART), a transformer-based warm-start trajectory generation method, to cislunar RPOD scenarios for the first time. To accurately and reliably solve the nonconvex optimal control problems (OCPs) posed by these scenarios, a new and enhanced version of ART, ART-TWIN (Two-Way INference), is introduced. Inspired by forward-backward shooting methods used in other trajectory design applications, ART-TWIN autoregressively generates two arcs, one from the initial state and one from the desired terminal state, that are patched together at the midpoint of the timeseries. When evaluated on a set of simulated rendezvous scenarios that are initialized at perilune, ART-TWIN is demonstrated to substantially accelerate convergence and increase feasibility guarantees when used as a warm-start to sequential convex programming (SCP), compared to convex relaxations and the original ART. These results illustrate the necessity of ART-TWIN's dual-arc generation to enable the viability of and gain benefits from using transformer-based warm-start methods in the most challenging areas of the cislunar dynamical regime.