Papers

We consider the heat equation in an interval of the real line, with null initial condition, and Dirichlet boundary controls. We study the cost of reaching an attainable state at some time $T>0$. Precisely, given a state in the interval that we know is reachable by suitable boundary controls, we estimate the size of the smallest controls that reach it, in terms of an appropriate norm of this state. Our estimate is optimal in some sense as the time horizon goes to $0$.

We study optimal input design over a finite horizon for linear dynamical systems. The goal is to minimize a weighted inverse-covariance (information) criterion subject to an energy budget. The set of covariances achievable by causal policies is convex but lacks a tractable explicit description, ruling out projection-based methods. We show that Frank–Wolfe applies naturally: each linear minimization subproblem is a budget-constrained finite-horizon linear quadratic (LQ) problem, solvable by a Riccati recursion and one-dimensional bisection over a Lagrange multiplier. Using smoothness of the objective over the feasible set, we establish an $\mathcal{O}(1/M)$ convergence rate for the objective value, while strong convexity yields an $\mathcal{O}(1/\sqrt{M})$ rate for the iterates. We further extend the framework to input design for system identification with unknown dynamics and adaptive online LQR, and illustrate the approach numerically.

In this paper we derive a Probably Approximately Correct (PAC)-Bayesian error bound for partially observed linear time-invariant (LTI) stochastic dynamical systems in state-space form with inputs and sub-Gaussian noise. Such bounds are widespread in machine learning, and they are useful for characterizing the predictive power of models learned from finitely many data points. The bound derived in this paper relates the expectation of prediction errors with the prediction error generated by the model on the data used for learning. In addition, we show that it can also be used to derive bounds for the parameter estimation error. In turn, this allows us to provide finite-sample error bounds for the prediction error and parameter estimation error for a wide class of system identification algorithms. Furthermore, as LTI systems are a sub-class of recurrent neural networks (RNNs), these error bounds could be a first step towards PAC-Bayesian bounds for RNNs.

Hydronic radiator systems are among the most widely used heating systems in buildings. In the literature, radiator heat outputs are often assumed to be independent of one another and arbitrarily adjustable in time when designing controllers. However, in practice, radiators are supplied by one or multiple common heat sources and are hydraulically coupled through the water circulation system. Maintaining occupant comfort in multi-zone buildings therefore requires not only an appropriate supply temperature but also proper hydraulic balancing to distribute the available water flow according to the heating demand of each zone. To this end, we develop a grey-box thermal model that captures the hydraulic interactions among radiators and the effects of radiator valves, circulation pumps, and heat sources. In a real building, we show that accounting for hydraulic interactions reduces the root mean square error (RMSE) between the measured and modeled zone temperatures by approximately 11% compared with a model that neglects these interactions. Additionally, we integrate the developed model into a model predictive control (MPC) framework for dynamic hydraulic balancing that jointly optimizes valve openings and the supply temperature to maintain thermal comfort in each zone while reducing energy consumption. Through both real-world experiments and numerical case studies, we demonstrate that, compared with existing MPC formulations that neglect hydraulic interactions or do not control radiator valves, the proposed MPC reduces comfort-range violations by at least 27% while requiring a similar or even lower cumulative supply temperature.

Can the classical Heavy-Ball method, with arbitrary horizon-dependent parameters chosen in advance, achieve Nesterov's $O(T^{-2})$ last-iterate rate on every smooth convex objective? We provide a negative answer. For every horizon $T\ge2$ and every predetermined schedule with nonnegative step sizes and momenta in $[0,1)$, there exists a convex $1$-smooth objective, with initialization distance at most one and zero initial velocity, for which the last iterate of the Heavy-Ball method satisfies \[ f(x_T)-f^\star=\Omega\!\left(\frac{1}{T^\alpha\log T}\right), \qquad \alpha=\frac{1+\sqrt5}{2}. \] Thus even fully nonstationary, horizon-dependent tuning cannot give the classical Heavy-Ball method a Nesterov-rate guarantee on the smooth convex class.

An influential 1989 result of Naddef shows that the diameters of $0/1$-polytopes are at most their dimension. This was extended shortly after by Kleinschmidt and Onn to any lattice polytope in $[0,k]^{d}$, where they showed a bound of at most $dk$. Naddef's argument easily extends to the monotone setting motivated by the simplex method, where one requires paths to increase with respect to a linear objective function. However, the Kleinschmidt-Onn argument does not. In fact, no argument in the 30 years since has managed to fill that gap. Prior to this work, it remained open whether the monotone diameter is bounded by a polynomial in $d$ and $k$ with no lower bounds suggesting any separation between the worst-case diameter and worst-case monotone diameter. Linear upper bounds hold for $k=1$ and $k=2$. However, we exhibit a sharp threshold for this question at $k = 3$ by constructing for each $d \geq 1$ a lattice polytope in $[0,3]^{6d}$ with monotone diameter at least $2^{d}-1$. In particular, the polynomial bound does not hold. Furthermore, we show that Naddef's result does not extend to the unbounded setting by exhibiting a family of unbounded polyhedra with $0/1$-vertices and diameter exponential in their dimension.

We are interested in computing approximate dual bounds for nonconvex optimal control problems using McCormick inequalities. To handle the numerical burden concerned with the tightening of these inequalities, we locally average the bilinear term in the PDE constraint. We extend recent theory in this area to elliptic PDEs on multidimensional domains that involve bilinear terms. As an additional result, we also generalize the necessary assumptions to allow for more flexibility in the choice of the discretization for the total variation penalty term from the underlying control problem. Furthermore, we prove the existence of valid bounds on the state variable, which are crucial to enable the optimization-based bound-tightening (OBBT) procedure to tighten the McCormick inequalities. Since theoretical bounds on the state variable are rather conservative, OBBT is both crucial and the main computational bottleneck for deriving a sharp approximate dual bound. We show how the sequence of LPs arising during OBBT may sensibly be warmstarted, allowing us to successfully scale the numerical experiments to finer discretizations than what was possible before.

We study the design of feedback recommendation policies for networked multi-topic opinion dynamics. The design of recommendations is formulated as an infinite-horizon linear-quadratic optimal control problem that favors suggestions aligned with each agent's current opinion, thereby using opinion alignment as a proxy for engagement. At the same time, the performance index penalizes polarization, deviation from the uncontrolled equilibrium of the opinion dynamics, and from the current agents' opinion, recommendation effort, and favors coherence among neighbors' recommendations. An agent-wise representation exposes the interconnection structure of the social network and enables structured controller and certificate synthesis. Under the assumption that the stage cost of the performance index is strictly positive definite, the affine optimal control problem separates into a strictly convex steady-state optimization, which determines the optimal equilibrium and the affine controller offset, and an LQR problem, which yields the static state-feedback gain. The resulting centralized Riccati controller provides a performance benchmark, while structured gains are obtained from dissipativity-based and $\mathcal H_2$-based LMI surrogate formulations. Possibly overlapping certificate clusters yield scalable local sufficient conditions for the structured gain design, while a separate steady-state cluster decomposition provides an exact consensus reformulation of the steady-state optimization. Numerical results on a social network illustrate the closed-loop behavior and the trade-off between performance and controller locality.

We study the time-uniform convergence of the raw iterate of standard stochastic gradient descent (SGD) for unconstrained smooth convex objectives. We prove that, under standard noise assumptions, the time-uniform convergence rate gets arbitrarily close to $\sqrt{\log n / n}$ but never reaches it. More specifically, we prove that for every positive, eventually nondecreasing sequence $h$ satisfying $h(n) = o(\sqrt{n})$, a bound of order $h(n)/\sqrt{n}$, holding simultaneously for all $n$ with probability at least $1-\alpha$ and uniformly over the problem class, is achievable if and only if \[ \sum_{j = 1}^{\infty} \frac{1}{h(2^j)^2} < \infty. \] The constructive sufficiency result follows from a dyadic horizon-free schedule together with an additive conditional-restart inequality. The necessity counterpart applies to every deterministic nonnegative schedule and holds even for a one-dimensional analytic smooth convex objective with Gaussian noise.

For Frank–Wolfe (FW) methods for convex optimization over the spectrahedron, it remains open whether a block-update variant can be linearly convergent when the update rank never exceeds the (unknown) optimal rank $r^*$ at each iteration. Existing block and spectral FW methods require an update rank at least $r^*$—and typically prior knowledge of $r^*$—to obtain a linear rate. This paper develops a rank-adaptive FW method whose update ranks satisfy $\widehat{r}_t\le r_t\le r^*$ at every iteration and which converges linearly after a finite burn-in under quadratic growth and strict complementarity, the two conditions commonly used in spectral FW analyses. The method is built on two designs. The first is a nonconvex spectral oracle, motivated by the geometric connection between the simplex and the spectrahedron; it yields a thresholding rank $r_t$ of the current iterate and a closed-form low-rank solution. Computing $r_t$ exactly, however, requires a full eigendecomposition. The second introduces the efficient rank $\widehat{r}_t$ of the current iterate, a cheap surrogate that inherits the optimality properties of the spectral oracle. The algorithm switches between the thresholding rank and the efficient rank so that the actual FW update uses $\widehat{r}_t$, keeps the per-iteration cost comparable to standard FW, and eventually identifies $r^*$. These results close the gap between low-rank efficiency and fast convergence for Frank–Wolfe methods over the spectrahedron. Numerical experiments demonstrate the advantage of the proposed method.

We study obstacle-constrained variational problems whose reduced energies depend on a probability law through a lower-level optimizer. Uniform strong convexity yields a single-valued Lipschitz follower response, while convexity and a Poincare inequality give a unique upper-level policy. A type-Lipschitz reduced marginal then implies Lipschitz continuity of the policy map from the 1-Wasserstein metric to the energy space. In a one-dimensional linear-obstacle subclass, the policy derivative is the positive part of a cumulative forcing. Single crossing makes the coincidence set an interval. An algebraic crossing of order m gives a $W_1^{1/m}$ modulus for its endpoint; odd-power examples show that this exponent is sharp, while a transversal crossing recovers Lipschitz stability. For empirical laws on compact subsets of $\mathbb{R}^k$, dimension-dependent Wasserstein bounds yield finite-sample rates for policies and thresholds. At a transversal population root, the empirical threshold is asymptotically linear, with an explicit influence function and central limit theorem. For higher-dimensional regular patches, a conditional level-set argument gives local Hausdorff stability when a nondegenerate switching function is available. An explicit quadratic firm response produces a nonlinear corporate-tax schedule with an endogenous zero-tax region and a Wasserstein-stable threshold. The tax illustration is analytic and uses no empirical calibration.

In this paper we propose a new kernel-based method for the identification of the impulse responses of forward models. The resulting estimator leads to a nonlinear Tikhonov regularization problem for which we prove the existence of a solution. The latter result makes legitimate to approximate the forward model through a high-order Moving Average with eXogenous input (MAX) model, i.e. the numerical solution found using this model introduces only a negligible bias in the estimate. The optimization of the marginal likelihood to estimate the kernel hyperparameters is also taken into account. Since there does not exist a closed-form expression for the marginal likelihood, we present an evaluation method that relies on the Laplace approximation of the marginal likelihood. Finally, some numerical experiments are discussed to show the effectiveness of the proposed method.

We study the stochastic first-order oracle complexity for constrained or regularized convex-concave min-max optimization and stochastic monotone variational inequalities. We focus on the case when suboptimality is measured in terms of the gradient mapping, also known as, forward-backward or natural residual, an optimality notion that generalizes the gradient norm for unconstrained problems. In this setting, under standard unbiased oracle access with now-standard variance assumptions, the best-known complexity for making the norm of the gradient mapping less than $\varepsilon$ is $\widetilde{O}(\varepsilon^{-4})$, compared to the near-optimal $\widetilde{O}(\varepsilon^{-2})$ that is established in the unconstrained case. We bridge this gap to improve the gradient mapping complexity for constrained convex-concave min-max problems to $\widetilde{O}(\varepsilon^{-2})$. We then extend to prove the same complexity for problems without the bounded variance, by using the Blum-Gladyshev assumption.

We study zeroth-order optimization of non-convex functions with the aid of directional hints, which are cheap but potentially inaccurate approximations of the true gradient direction, given by linear subspaces at each iteration. To leverage these hints adaptively while maintaining robustness to their quality, we introduce Control-Variate Zeroth-Order Descent (CV-ZOD), a new framework that refines the classical zeroth-order gradient estimator with a control variate that can be set based on the directional hints. We first show that the oracle algorithm that optimally sets the reference vector and step size at each iteration achieves a convergence rate that interpolates between the first-order $O(1/T)$ rate and the zeroth-order $O(d/T)$ rate, depending on the quality of the hints along the trajectory. We then develop a practical variant of CV-ZOD that achieves the same oracle guarantee up to logarithmic factors, without any prior knowledge of the hint quality. We validate the method empirically on simulation-based scientific optimization tasks, demonstrating sustained progress on non-convex landscapes where zeroth-order descent is slower and existing guided methods stall as guidance deteriorates.

Classical Bayesian games assume that the joint distribution of players' types is common knowledge, an assumption that rarely holds in practical applications. We address this issue by studying a group of myopic players who repeatedly interact under the Bayesian Nash conjecture and learn a parametric joint type distribution from type profiles observed over time within a Bayesian learning framework. Players may hold heterogeneous priors and may be risk averse toward both epistemic uncertainty about the distributional parameter and aleatoric uncertainty in their rivals' types. We propose two models: a Bayesian Nash equilibrium among risk-averse Bayesian learners (BNE-RABL), in which epistemic and aleatoric risks are evaluated separately, and a BNE based on Bayesian predictive distributions (BNE-BPD), in which the two sources of uncertainty are integrated into a Bayesian predictive distribution and evaluated through a single risk measure. Under suitable conditions, we establish the existence and uniqueness of both equilibria, and derive non-asymptotic convergence rates of the equilibrium sequences toward the corresponding oracle BNE under the true type distribution as the game is repeatedly played and more type profiles are observed. These results further show that the discrepancy between the BNE-RABL and BNE-BPD strategies vanishes as the sample size grows, which illustrates that the two models can be regarded as effective approximations to each other. We apply the proposed models to a price competition problem and numerically illustrate the theoretical results.

Solving fixed-point problems by reducing residual bounds is one of the central approaches in fixed-point iteration, and, in particular, the Halpern iteration is known to effectively reduce the residual. Recently, a new type of duality theorem has been discovered in the convex optimization and fixed-point problem literature and establish a certain symmetry in the convergence behavior of algorithms that are dual to each other. However, these duality theorems rely on a Lyapunov-style proof structure and mostly focus on Hilbert spaces. In this work, we first establish an M-duality theorem for Mann iterations with Lipschitz operators in general normed spaces, assuming only mild conditions on the coefficients of the iterations. Moreover, leveraging this M-duality, we derive new algorithms that attain a strictly smaller worst-case residual bound than that of the Halpern iteration, which was previously known as the state-of-the-art method in general normed spaces.

This paper introduces rlaopt, a PyTorch-based package for large-scale optimization and scientific computing using randomized numerical linear algebra (RandNLA). Despite substantial progress in RandNLA-based algorithms, few implementations combine GPU acceleration with a simple interface for specifying optimization problems. rlaopt addresses this gap by providing GPU-enabled solvers for positive-definite linear systems and convex empirical risk minimization with constraints and regularizers. These solvers use RandNLA to accelerate conjugate gradient (NystromPCG), operator splitting (NysADMM), and stochastic gradient methods (SAPPHIRE). Moreover, rlaopt includes a modeling language that lets users specify problems using natural mathematical syntax. rlaopt automatically checks compatibility with the selected solver and performs the required problem decomposition. The solvers also support differentiation through their iterations, enabling applications such as hyperparameter tuning. Experiments on ridge regression, bounded multinomial logistic regression, and bounded elastic net identify when randomized preconditioning improves performance and demonstrate substantial speedups from GPU execution. The package is open-source under an Apache license, with source code at https://github.com/udellgroup/rlaopt and version 0.1.0 available on PyPI.

In nonconvex optimization problems arising in geometric machine learning, data augmentation is commonly used to promote invariance by averaging empirical losses over transformations of the data. Computing the fully augmented objective, however, requires access to every element of the transformation group $G$, which may be prohibitively expensive when $G$ is large or accessible only through sampling. We study whether full augmentation can instead be approximated using a small, fixed sample of transformations acquired before optimization and reused thereafter. Under suitable regularity conditions, we show that, with probability at least $1-\delta$, gradient descent (GD) on the resulting sparsely augmented objective returns an $\varepsilon$-stationary point of the fully augmented objective using $\mathcal{O}\bigl((\log |G|+\log(1/\delta))/\varepsilon^2\bigr)$ group-transformation-oracle queries. By comparison, standard group stochastic gradient descent (group-SGD), which samples a fresh transformation at every iteration, uses $\mathcal{O}(1/\varepsilon^4)$ transformation queries. Therefore, gradient descent with fixed sparse augmentation requires fewer transformation queries than both GD applied to the fully augmented objective and group-SGD. Our proof techniques, which may be of independent interest, establish a uniform approximation of the full group-averaged gradient field by a random group average using spectral properties of group-induced operators and tools from representation theory.

Low-rank tensor modeling has become an effective tool for hyperspectral anomaly detection. However, existing methods still suffer from high computational cost and limited flexibility in characterizing spatially structured anomalies. To address these issues, this paper proposes a hyperspectral anomaly detection method based on group sparse low-rank tensor factorization with automatic anomaly grouping (GSAA). Specifically, the low tubal rank background is characterized by imposing group sparsity on tensor factors, which provides an efficient alternative to direct tensor rank regularization. For anomaly modeling, a latent grouping map is introduced to build an automatic anomaly grouping penalty, allowing anomaly groups to be adaptively inferred from the data rather than predefined at the pixel level. To further exploit complementary spectral and spatial information, GSAA is applied in both domains, and the resulting detection maps are fused to form a spectral–spatial version of GSAA, termed GSAA-SS. An efficient linearized alternating direction method of multipliers algorithm with convergence guarantee is developed to solve the resulting model. Experimental results on five real hyperspectral datasets demonstrate that the proposed method achieves superior detection performance and competitive computational efficiency compared with several state-of-the-art methods.

This paper studies lower-level-constrained bilevel optimization in a response-oracle setting, where lower-level model information is unavailable and the induced response mapping is locally Lipschitz but potentially nonsmooth. In this setting, the classical response Jacobian and reduced hypergradient may fail to exist. We propose Bilevel Zeroth-Order Learning (Bi-ZOL), a structure-guided zeroth-order method for finding stationary points of the nonsmooth reduced problem. Instead of estimating the gradient of a fully smoothed reduced hyperobjective, Bi-ZOL separates the bilevel chain-rule structure: it keeps the exact upper-level partial gradients at the queried response and uses zeroth-order sampling only to estimate the response Jacobian. This construction yields an approximate hypergradient that is more directly aligned with the Clarke chain-rule subdifferential. We show that the Bi-ZOL direction admits a partial-smoothing interpretation, quantify its pointwise structural bias, and prove finite-time convergence to a $(\delta,\epsilon)$-Bi-ZOL Frank–Wolfe stationary point. The bias is $O(\delta)$ for piecewise $C^{1,1}$ responses under local regularity and vanishes for piecewise affine responses on active-cell neighborhoods. Experiments on incentive-based tracking problems show that Bi-ZOL achieves smaller stationarity gaps and lower hyperobjective values than vanilla zeroth-order smoothing under comparable response-oracle budgets.

Extending the classical $K$-means and $K$-medians models, this paper introduces an $\ell_{p,q}$ mixed-norm clustering model where the centroid updates and cluster assignments are under the $\ell_p$ and $\ell_q$ norms, respectively. The model is formulated as a mixed-integer program (MIP) with Heaviside composite constraints that describe the nearest-center assignments. The framework recovers $K$-means and $K$-medians when $p=q=2$ and $p=q=1$, respectively, and yields new models when $p\ne q$. To address the computational challenges, we develop a progressive integer programming (PIP) method that adaptively fixes confident assignments and solves restricted mixed-integer subproblems. For $q=1$, we develop a convex inner approximation of the difference-of-convex constraints in the restricted subproblems, for which a global solution can be computed. Importantly, we establish the connection between the local minimizer and the strong center-local minimizer of the mixed-norm clustering problem and the global optimal solution of the restricted subproblems under certain assumptions. This connection provides a practical certificate of a local minimizer of the nonconvex mixed-norm clustering model. We further develop techniques for constructing adaptive fixing sets and working sets for $q=1$. Extensive numerical experiments demonstrate the superior performance of the mixed-norm clustering model and the efficiency of PIP for solving the MIP model, which may be intractable otherwise. In particular, the $\ell_{2,1}$ mixed-norm clustering model is effective under coordinate-sparse, mean-balanced contamination, whereas the $\ell_{1,2}$ model is preferred under dense coordinatewise Cauchy contamination. The numerical results also show that PIP can escape poor alternating solutions and obtain substantially better feasible clustering, while preserving strong warm starts when no improvement is found.

We study the control of Markov decision processes in which the quality of a policy is evaluated by a dynamic, time-consistent Markov risk measure rather than by an expected discounted cost. The main obstacle to combining such measures with reinforcement learning is that a transition risk mapping depends on the transition kernel in a nonlinear way, and therefore cannot be estimated from a single observed transition. We remove this obstacle by employing mini-batch transition risk mappings: the mapping is applied to the empirical measure of $N$ independent next-state samples, and the result is averaged. The resulting mapping is again coherent. However, as an expected value of a function of $N$ next-state values, it admits an unbiased one-sample estimator. We embed this mapping into a double deep Q-network, analyze the two sources of estimation bias that arise, and obtain a risk-averse Q-learning method applicable to state spaces far beyond the reach of tabular schemes. The method is applied to an underwater robot navigation problem, in which a vehicle must visit collection points, gather stochastic information payloads, and deliver them at transmission points, while exposed at each step to the risk of destruction. A hierarchical decomposition delegates path execution to an exact graph search and confines learning to the high-level “collect or transmit” decision. A low-dimensional feature map, invariant under the symmetries of the problem, replaces the raw state–configuration encoding. In experiments on $300$ held-out environments, the resulting policies transfer to instance sizes never seen in training, and already $N=2$ reduces the upper semideviation of the outcome distribution while simultaneously improving its mean whenever the simulator is misspecified—an empirical counterpart of the duality between coherent risk measures and distributional robustness.

Fourier-Motzkin elimination computes an inequality description of the projection of a polyhedron onto a subset of its coordinates by eliminating one variable at a time. It is used in several areas of optimisation and computer science, and it is a standard way of obtaining the entropic constraints of a causal structure, where the marginalisation over the latent variables produces such a projection. Its limitation is the growth of the intermediate systems of inequalities, which can be doubly exponential in the number of eliminated variables even though the projection itself grows only as a single exponential. In practice the computational overload of the method therefore depends on two choices: how the redundant inequalities are removed after each step, and the order in which the variables are eliminated. We consider both. We first show, by an explicit example, that Imbert's redundancy test cannot be interleaved with redundancy removal by linear programming. We show that the two methods, however, can be combined soundly if the derivation records used by Imbert's test are re-initialised after every step at which linear programming is used. We then propose a rule for choosing the elimination order of the variables that gives a significant computational advantage, however, at the cost of increased resource usage. We demonstrate this advantage on some random polytopes, where the rule reduces the running time by factors of between 6 and 25 compared with the same elimination under a fixed order. For entropic descriptions of causal structures, with more than 250 inequalities and more than 100 variables to eliminate, our rule keeps the number of inequalities handled at each step one to two orders of magnitude lower than a fixed order.

For more than four decades, the 60/40 stock/bond portfolio has served as a benchmark for delivering reasonable returns without excessive risk. More recently, a 50/30/20 stock/bond/alternative portfolio has been suggested. We use gold as the alternative and as an inflation hedge. In this paper we ask: how much improvement over these benchmark fixed-weight portfolios can be obtained using widely available public data and standard methods from quantitative finance? We restrict ourselves to long-only dynamic portfolios of stocks, bonds, and gold, plus cash, rebalancing monthly, using only publicly available data. We evaluate portfolios on the conventional metrics: return, volatility, Sharpe ratio (computed in excess of the federal funds rate), drawdown, and turnover, in addition to consistency of performance over time, judged by the consistency of the realized annual volatility. Over the 20–year period 2006–2026, using a conservative estimate of trading costs, we show that all risk-adjusted and drawdown metrics are improved using simple volatility control, where we dynamically mix the fixed-weight portfolios with cash so as to target a fixed volatility. This method relies on a simple estimate of portfolio volatility derived from past returns. We also demonstrate that more sophisticated portfolios based on convex optimization—similar to those used in quantitative hedge funds—yield further substantial improvement in return and risk-adjusted return. We consider two such portfolios, one that uses a simple estimate of future returns based on past returns, and one that forecasts future returns based on past returns and just a handful of widely available public economic data. These portfolios also outperform a suite of standard risk-based allocation methods, such as risk parity and minimum variance, evaluated on the same assets and data.

We consider Pólya's problem of finding, among convex sets of prescribed area, the one with the longest shortest fence, in the polygonal setting, namely when the class of competitors is restricted to polygons with a prescribed number of sides. While it is straightforward to show that, among triangles, the optimal shape is the equilateral one, we prove that symmetry breaking occurs in the case of quadrilaterals: the optimal quadrilateral is not the square. More precisely, we identify it as a specific isosceles trapezium, which is uniquely determined, up to homotheties and rigid motions, by an elementary equation for its base angle. The proof combines analytical arguments and rigorous interval-arithmetic computations.

We study a new class of mean-field quadratic backward stochastic differential equations (qBSDEs) arising from mean-field portfolio games with exponential utility. Typical examples of such games include a mean-field portfolio game with price impact, and a finite-contract pricing model with market clearing conditions. Generators of these mean-field qBSDEs contain quadratic terms $\mathbb{E}[Z]^{\top} Z$ and $|\mathbb{E}[Z]|^2$, instead of the classical pathwise $Z^\top Z$ term. We prove local well-posedness under $L^q$-integrability assumptions on terminals and their Malliavin derivatives, and global well-posedness under an extra exponential integrability condition on the Malliavin derivatives. Then we show the existence and uniqueness of global equilibria of the above two mean-field games via our qBSDE theory.

For $(Q_t,P_t)$ governed by suitably damped underdamped Langevin dynamics, we quantify the convergence in KL divergence of the positional marginal to a $\sigma$-strongly log-concave target $\pi(dq) = \frac{1}{Z}e^{-V(q)}dq$ as \begin{align*} \operatorname{KL}(\operatorname{Law}(Q_t) \| \pi) \leq e^{-\sqrt{\sigma} t}\operatorname{KL}(\operatorname{Law}(Q_{0}, P_{0})\, \|\, \Pi_0), \end{align*} where $\Pi_0$ denotes an appropriately selected reference measure. When $\pi$ is merely log-concave, the estimate \begin{align*} \operatorname{KL}(\operatorname{Law}(Q_t) \| \pi) \leq \frac{\tau^2}{t^2}\operatorname{KL}(\operatorname{Law}(Q_{\tau}, P_{\tau})\, \|\, \Pi_{\tau}) \end{align*} is derived, where $\Pi_\tau$ denotes another correspondingly chosen reference measure at time $\tau > 0$. Both rates match precisely the canonical rates of the corresponding accelerated gradient flows in $\mathbb{R}^d$. They are achieved by the novel interpretation of the underdamped Langevin dynamics as a combination of strategies in an online sampling game and by estimating the KL divergence using a cost function informed by fictitious competitors.

This paper develops a convex tube model predictive control formulation for constrained nonlinear systems. We consider dynamics described by a discrete-time state-space model with parametric and additive uncertainty that admits a difference-of-convex decomposition. Convex directional bounds of the nonlinear dynamics are combined with configuration-constrained polytopic tubes, whose predefined combinatorial structure yields an affine parameterization of their vertices. The resulting finite-dimensional sufficient conditions certify robust one-step tube propagation under parametric uncertainty and additive disturbances, while allowing the tube geometry and an associated vertex control law to be optimized jointly in a single convex program. An implicit terminal condition guarantees recursive feasibility and convergence of the predicted tube to a target robust control invariant set. Numerical results illustrate the closed-loop properties and the trade-off between geometric flexibility and computational complexity.

We prove boundary null controllability in any positive time for a finite system of Euler–Bernoulli beams with possibly non-diagonalizable coupling. For $0<\rho<2$, a Fourier–Jordan reduction yields a vector moment problem with polynomial–exponential modes. Under the Fattorini–Hautus condition, global spectral non-collision, and nonvanishing modal transfer factors, a suitably estimated block-biorthogonal family produces an $H^2$ boundary control. Geometric multiplicities of the coupling matrix determine the minimal control dimension, and Jordan-block lengths determine the degrees of the generalized moments. At $\rho=2$, an exact heat-system reduction gives the same result for $A=A^*\geq0$, despite the time-differential linkage of the reduced boundary inputs.

Greedy algorithms efficiently approximate combinatorial optimization problems, but their guarantees weaken when feasibility couples combinatorial structure with global physical constraints. We study monotone nondecreasing supermodular minimization over the bases of a graphic greedoid under physics-induced constraints. We model physics-informed selection using a look-ahead oracle that identifies candidates extendable to a feasible basis, yielding the Conditioned Sequential Greedy Algorithm. We derive a closed-form approximation bound, which we call the price of feasibility, based on the variability of oracle-restricted candidate sets and a probabilistic correction for unobserved elements. As a case study, we show that FORWARD, an algorithm for multi-source radial network reconfiguration, instantiates this framework. Numerical results demonstrate the tightness of the bound and quantify the feasibility-optimality trade-off.