Papers

In smooth convex optimization, the gradient norm is a directly observable measure of stationarity. Accelerating a first-order method that minimizes the gradient norm is known to be more delicate than accelerating the minimization of function values. Optimal accelerated methods such as OGM-G (Kim & Fessler, 2021) are known to exist for any prescribed finite horizon, but their coefficients depend explicitly on the length of that horizon, i.e. the number of iterations. We ask what kind of acceleration is feasible when the stopping horizon is unknown to the method, i.e. for horizon-independent methods. Diakonikolas & Wang (2022) conjectured that an $\Omega(N^{-1})$ lower bound on the squared gradient norm holds at every horizon $N$ for any nonadaptive, horizon-independent linear-span first-order method. We disprove this pointwise conjecture by exhibiting a method that achieves near-$N^{-2}$ last-iterate guarantees on a density-one set of horizons. We show instead that an $\Omega(N^{-1})$ lower bound must hold for infinitely many horizons. More precisely, if $\mathcal G_N(\mathcal A)$ denotes the bound on the squared gradient norm after $N$ iterations for a method $\mathcal A$, we prove that $\limsup_{N\to\infty}N \mathcal G_N(\mathcal A) \ge 1/2$ for any method $\mathcal A$. This bound is sharp: the constant $1/2$ is exactly attained by the horizon-independent gradient-descent schedule of Rotaru et al. (2026). In addition, we show that the above two extreme behaviors cannot be achieved by the same method: any method $\mathcal A$ with an $o(N^{-1})$ guarantee on a subsequence of iterates must satisfy $\limsup_N N \mathcal G_N(\mathcal A)=\infty$. In contrast, best-so-far output admits a uniform $O(N^{-2})$ guarantee.

Decomposing a matrix into sparse and low-rank components is central to robust principal component analysis and has broad applications in machine learning, signal processing, and computer vision. Classical formulations seek to recover the underlying sparse and low-rank structure. We instead introduce sparse-plus-low-rank matrix embedding (SLME), whose goal is to construct a computationally efficient surrogate for a large dense matrix, without requiring its components to be interpretable. Given $A\in\mathbb{R}^{m\times n}$, SLME approximates $A \approx S+HW$, where $S$ is sparse, $H\in\mathbb{R}^{m\times r}$, $W\in\mathbb{R}^{r\times n}$, and $r\ll \min\{m,n\}$. The resulting matrix-vector product can be evaluated as $Sx+H(Wx)$ in $\mathrm{nnz}(S)+r(m+n)$ operations, rather than the $mn$ operations required by $Ax$. Our primary motivation arises from optimization problems in cancer radiotherapy treatment planning, where a large dense dose-influence matrix is a major computational bottleneck. We formulate SLME as a bi-objective nonconvex optimization problem that balances approximation error against the computational cost of the downstream tasks. We then develop R3-Trust, an efficient trust-region algorithm that approximates the Pareto frontier in a single parameter-free run. Each point on the resulting frontier provides a sparse-plus-low-rank representation with a different balance between accuracy and downstream computational cost. Experiments on clinical radiotherapy matrices show that decompositions obtained from existing recovery-oriented formulations can be suboptimal for the embedding objective. Conversely, experiments on synthetic instances demonstrate that SLME and R3-Trust can also be applied to sparse-plus-low-rank recovery, where they compare favorably with state-of-the-art recovery methods in both reconstruction accuracy and computational time.

We (the author and the AI systems that did the heavy lifting) show that the quantum query complexity of minimizing a convex function over a convex subset of $\mathbb{R}^n$ with evaluation and membership queries is $\tilde\Omega(n)$, nearly matching the best known upper bound. In particular, we show this even for quadratic minimization, which is equivalent to inverting an $n \times n$ real matrix using matrix-vector queries. We also show linear or nearly linear lower bounds on the quantum query complexity of computing the trace, the sign of the determinant, and the magnitude of the determinant of a real matrix in the matrix-vector query model. We use a novel quantum lower bound technique, the determinantal witness method, based on identifying a witness whose Fourier transform vanishes on low-rank matrices and that correlates well with the function being computed.

A wide range of sparse integer programs admit a block structure in which subproblems interact through a small set of shared variables. Dualizing the linking equalities yields a decomposable Lagrangian relaxation, but generally introduces a duality gap. Recent work shows that this gap can be closed while preserving decomposability by dualizing exponentially large families of redundant nonlinear consistency constraints on the shared variables. We develop a computational framework for exploiting this idea without explicitly constructing the resulting exponentially large relaxation. Our framework combines nonlinear edge encodings of shared-variable consistency with a column-and-row generation (CRG) algorithm that generates local integer solutions by pricing and encoding constraints by separation. With complete encodings and exact separation, the framework recovers the exact relaxation while retaining independent optimization over the blocks. We introduce several encoding families and establish exponential separations among them: the Generalized family can require exponentially fewer constraints than the Vertex, Monomial, or Reflected families, yet can itself require exponentially many constraints on instances for which a single problem-specific encoding suffices. Computational experiments on decomposed stable-set and dominating-set instances show that CRG substantially outperforms a monolithic formulation across a range of tree topologies and coupling strengths. The results also show that richer encoding families need not perform better computationally, highlighting the choice of encoding as a central issue in effective decomposition.

Large language models (LLMs) are increasingly used to formulate optimization models from natural-language problem descriptions, yet realistic operations research (OR) requests are often incomplete: missing objectives, constraints, or business rules can change the resulting mathematical program. Existing evaluations largely assume a complete specification and therefore overlook whether an agent knows when clarification is needed before modeling. We introduce OR-Clarify, a benchmark for pre-formulation clarification. Each task presents a partial public problem description, withholds structured hidden slots, and evaluates agents through bounded interaction with a simulated user. The benchmark supports both openended and choice-based clarification, and measures slot recovery, stopping behavior, silent assumptions, and interaction cost. We further propose Interactive Optimization (InterOPT), a two-stage framework that identifies unresolved formulation-critical gaps and uses them to guide whether to ask the next question or to stop. In our choice-based experiments, InterOPT substantially outperforms all baselines in exact slot recovery; in the open-ended setting, it remains competitive with strong prior methods. Together, OR-Clarify and InterOPT reframe OR assistance as a selective completeness decision: clarify when needed, stop when ready, and quantify what remains missing.

We develop novel duality methods for stochastic optimal control problems under two sources of randomness and non-Markovian random coefficients adapted to just one of them. The Hamilton-Jacobi-Bellman (HJB) equation is a second-order backward stochastic partial differential equation. Our duality theory provides an alternative description of the random value function in terms of a suitable pathwise optimal control problem parameterized by the realizations of one of the Brownian motions. This allows us to regain Markovianity using the theory of rough optimal control problems, leading to rough second order HJB equations that we solve in a suitable viscosity sense.

We introduce an admissible trembling-hand refinement for stochastic mean field games formulated through relaxed controlled martingale problems. A relaxed MFG equilibrium is perfect if it is the common Wasserstein limit of two sequences of joint control-state laws. The first consists of full-support population perturbations admissible for the state laws they generate; the second consists of exact optimal responses to those perturbed state laws. This admissibility condition prevents the control and state coordinates of a perturbation from being varied independently. Under the standing continuity, growth, and coercivity assumptions, we prove existence without requiring compact controls or bounded coefficients. The proof combines a perturbed fixed-point argument in the compact-bounded case with truncation, uniform moment estimates, compactness, and stability of controlled martingale problems. A one-dimensional model with exactly two relaxed mean field game equilibria shows that the refinement is genuinely selective: only one equilibrium is trembling-hand perfect.

Real-time services, such as VoIP and large-scale neural network training, require strict transmission delay guarantees. While routing under hop constraints is tractable, real-world delays increase sharply with equipment load, typically modeled using the M/M/1 queuing function where delay is inversely proportional to available bandwidth. We investigate the resulting Delay-Constrained Maximum Concurrent Flow (DCMCF) problem, which seeks to maximize the minimum throughput across all commodities. The problem's complexity stems from the conditional and non-linear nature of the delay constraints, which are active only along the specific paths used by the flow. We prove that DCMCF is strongly NP-hard, even for single-source/single-destination instances. To address the inherent non-convexity of the problem, we introduce a new convex relaxation expressed through second-order cone constraints, obtained from the convex envelope of a function representing the conditional delay associated with a single arc of a given path. The relaxation is shown to outperform existing formulations based on disjunctive programming. Leveraging this result, we develop a polynomial-time approximation algorithm with a provable performance guarantee and present numerical experiments demonstrating the effectiveness of the proposed approach.

We introduce a nonlocal regularization of optimal transport that bridges Kantorovich and Monge formulations. The regularization penalizes oscillations through an interaction kernel singular near the diagonal. We show that, for sufficiently strong singularities, every finite-energy transport plan is induced by a map, yielding equivalence between the regularized Kantorovich and Monge problems. For fractional kernels, the regularizer reduces to a fractional Sobolev seminorm that can allow jump discontinuities, leading to existence results for broad classes of source and target measures. We derive first-order optimality conditions and analyze the vanishing-regularization limit. The regularized functionals $\Gamma$-converge to the classical Kantorovich problem, while the first-order $\Gamma$-limit selects, whenever they exist, finite-energy Monge minimizers with minimal nonlocal energy. When no such minimizer exists, we identify a different asymptotic regime and establish a sharp scaling law in a prototypical mass-splitting example. Finally, after lifting the problem to the product of transport plans, we derive a dual formulation using the cone of copositive functions and propose a fixed-point numerical scheme illustrating the regularization.

The Kolmogorov's 0-1 law is so powerful and fundamental in Probability theory that the result of the law of large numbers is its corollary. The latter is the theoretical bedrock of modern day statistical machine learning. Expected resurgence in the interest of the law's proof is challenged by its measure theoretic nature. By observing 0-1 law as False-Truth law like in logic, we show here that the proof of the law is simplified. This simplicity comes from the fact that our approach hardly needs any measure theory beyond defining an obvious probability space. A novel contribution via this article is the use of logic within the proof of a fundamental result in the theory of probability.

This paper studies the relationship between the controllability of a quasi-linear parabolic equation and that of its linear counterpart. Under suitable assumptions, we prove that for sufficiently small initial data both systems are null controllable, and that the differences between their corresponding controls and states satisfy a higher-order error estimate. Unlike standard controllability problems, the controls constructed here not only steer the system to a prescribed target at a given terminal time, but are also designed to ensure that the deviation between the nonlinear and linear dynamics obeys the aforementioned estimate. This method is applicable to general nonlinear partial differential equations, and the approximation order for controllability is optimal.

In this letter, we study finite-horizon linear-quadratic Stackelberg difference games with coupled-affine state-control inequality constraints. Under the stated assumptions, we show that generalized open-loop Stackelberg equilibria admit an exact reformulation as an optimal control problem subject to a discrete-time linear complementarity system. Eliminating the dynamic variables yields a large-scale linear complementarity quadratic program, together with an explicit recovery map for the follower strategy. This reformulation enables the numerical computation of these equilibria. We illustrate the proposed approach using a constrained network-flow game.

Minimum-weight satisfiability asks for a satisfying Boolean assignment of minimum cost, while many fuzzy decision and knowledge systems operate with graded positive and negative relations rather than crisp literals. This paper introduces a bipolar fuzzy minimum-weight satisfiability problem in which positive and negative clause–variable relations, target clause levels, and variable states are represented in the unit interval. The clause requirements are expressed by bipolar fuzzy relational equalities defined with an arbitrary continuous t-norm. Classical minimum-weight CNF satisfiability is recovered as a crisp special case, independently of the selected t-norm, and an optimal Boolean solution is shown to exist in that specialization even though the proposed formulation uses continuous variables. For the graded problem, clause satisfaction is characterized through admissible variable domains and effective clause-witness sets. This structure leads to an exact clause-witness branch-and-bound method with preprocessing, propagation, lower-point closure, and a disjoint-witness packing bound. Computational experiments verify the solver against explicit witness enumeration, mixed-integer baselines, and external SATLIB formulas; on a focused graded ablation set, the packing bound reduces the median number of explored nodes by about 45 percent relative to the single-row witness bound.

This paper studies physical-layer authentication (PLA) in the near-field regime under impersonation attacks. Unlike the far-field case, where the steering vector depends only on the angle of arrival (AoA), in the near field it depends on both angle and distance. We analyze the attack by minimizing the mean-square error (MSE) between the signal received from a legitimate transmitter Alice and the signal generated by an active attacker Eve. For a single-antenna Eve, we derive the optimal scalar precoder and show that, for an inter-element spacing no larger than half a wavelength and under the standard second-order Fresnel approximation, perfect impersonation is possible only if Eve has the same angle and the same distance from Bob as Alice. We then extend the analysis to a multi-antenna Eve and derive the optimal precoding vector. In this case, perfect impersonation is possible only if Alice steering vector belongs to the subspace spanned by Eve steering vectors. Simulation results show that, in the near field, a distance difference is sufficient to prevent a successful impersonation attack even when Alice and Eve have the same AoA, and confirm the analytical results.

In this work, we study policy optimization under domain randomization for linear quadratic control, focusing on learning a single state-feedback controller that minimizes the average cost across systems with uncertain dynamics. We propose a policy iteration algorithm with a step-size rule that preserves stability across all sampled systems at each iteration. We show that the method yields monotonic improvement of the sample-average objective and that a stabilizing step size always exists. Under standard smoothness assumptions, the iterates converge subsequentially to stationary points, and under a gradient-dominance condition, we obtain a global linear convergence rate.

Due to resource constraints or external and internal uncertainties, clients in real-world federated learning systems are often intermittently available edge devices. In highly dynamic environments, the parameter server lacks prior real-time knowledge of clients' availability, making it challenging to adapt traditional federated learning algorithms to be resilient to uncertainties in client availability. If not carefully addressed, complex client availability can introduce significant bias, potentially harming the performance of the trained model. Most prior work either fails to account for non-stationary client availability dynamics or demands significant memory and computational overhead. This paper aims to develop efficient federated learning algorithms that are provably resilient to heterogeneous and non-stationary stochastic client availability. We propose FedSWE, which admits novel algorithmic structures to (i) compensate for missed computations, (ii) stabilize and diffuse the global updates over rounds, and (iii) evenly mix the local updates through implicit gossiping, despite being agnostic to non-stationary dynamics. Compared with the standard FedAvg, FedSWE introduces light additional memory and computation overhead. We show that FedSWE converges to a stationary point of non-convex objectives while achieving the desired linear speedup property in certain special cases. We corroborate our analysis with numerical experiments over diversified client unavailability dynamics on real-world data sets.

We study the time derivative of the matrix exponential $B(t)=\mathrm{exp}(A(t))$, where $A(t)$ is a time-parametrized curve in $\mathrm{Mat}(n)$. Starting from the Taylor series expansion, we derive a nested summation formula, which is then reformulated into a double summation. This expression is converted into an integral representation using Euler's beta function, and further expressed in terms of Lie brackets and the adjoint representation. Later, we extend this to a family of problems and verify a well-known result. Finally, we establish connections to the Gateaux and Fréchet derivatives, proving the latter's existence. These results offer a unified and explicit framework for understanding the derivative of the exponential map.

In this paper, we study games involving a continuum of heterogeneous players in the discrete-time setting with finite state spaces and continuous action spaces. We introduce a new model that incorporates joint state-action interactions within the graphon-weighted aggregate, described by a coupled forward-backward system. We establish the existence of graphon mean-field equilibria and characterize them through this forward-backward system. Additionally, we provide uniqueness results under monotonicity and contraction conditions. To illustrate the practical relevance of our framework, we solve an example of portfolio liquidation with price impact. We provide numerical results for three different graphons and two different initial distributions, illustrating the impact of the network's structure and the distribution heterogeneity on the distribution and the policy.

Transaction costs can make or break a trading strategy, particularly in relative-value trading of commodity and macro markets, where edges are a few basis points. Price impact is a central component of transaction cost. Price impact models usually include self-impact (a trade in a contract moves that contract's price) but omit two well-documented effects: cross-impact (a trade in one contract also moves the prices of related contracts) and transient impact (price impact decays over time, so an unwind recovers part of the entry cost). A model without these effects overprices the impact of relative-value trades, whose correlated legs are built and unwound over days, and so forgoes potentially profitable trades. This paper models both effects with a convex quadratic cost. In each period, a positive semidefinite matrix built from volatility, volume, and correlation forecasts couples trades across contracts. A power-law decay kernel then couples trades across periods. The resulting cost admits no price manipulation even when liquidity varies over the planning horizon. The model is demonstrated empirically on calendar spread trading of crude oil futures around the commodity index roll.

Forsythe's conjecture, published in 1968, asserts that for each restart length $s$, every exact-arithmetic restarted conjugate-gradient iteration on a real symmetric positive definite problem either terminates or has normalised residuals that converge separately along the even and odd restart subsequences. Apart from the classical steepest-descent case, this asymptotic question remained unresolved in full generality for nearly six decades. We give a complete classification by restart length in the original finite-dimensional setting and identify a sharp threshold. For $s=2$ and $s=3$, every problem either terminates or has convergent even and odd residual directions. For every $s\ge4$, there is a diagonal positive definite counterexample of dimension $s+4$ which never terminates and whose even residual directions do not converge. Together with Akaike's theorem for $s=1$, this shows that the conjectured universal conclusion is true precisely for $s\in\{1,2,3\}$ and false for every $s\ge4$. The positive results follow from a degree-independent double-orthogonality identity and an analysis of the low-degree fixed-point sets. At restart length four, rational interval arithmetic and Sturm sequences certify a transverse Hopf point of the leading vector field of the rescaled squared-weight map. Analytic periodic-orbit and shadowing arguments yield the counterexample at restart length four, and degree elevation extends the construction to every larger restart length. The classification for all $s\ge2$ is also formally verified in Lean.

Blind random search repeatedly draws a candidate point and replaces the current estimate whenever the candidate has a lower loss. In the absence of noise, the true loss is observed directly. It decreases strictly at every accepted update and is monotone nonincreasing over all iterations. Measurement noise can make a worse candidate appear better and thereby break this monotonicity. To recover almost sure convergence under noise, we incorporate averaging and thresholding into the original decision criterion. These two classical tools are coupled. As the sample sizes grow, the positive threshold shrinks at a matched rate. These modifications allow blind random search to recover eventual monotonicity of the true loss under noisy measurements and to converge almost surely.

We study a class of McKean-Vlasov control problems with discrete path dependence. The coefficients and cost functional may depend on finitely many past values of the controlled state, observed at fixed deterministic times, and on their joint law. We establish well-posedness of the controlled state equation and derive necessary and sufficient optimality conditions through a stochastic Pontryagin maximum principle. The adjoint process is characterized by a backward stochastic differential equation with jumps at fixed observation times. Each jump represents the conditional sensitivity of future costs with respect to the corresponding observed state and its distribution. In the linear-quadratic case, we prove global solvability of the resulting forward-backward system by combining a continuation argument with a mean-field Riccati reduction. We finally discuss an application to time-series generation, where the terminal cost is given by a kernel-based discrepancy between the law of the sampled path and a target path distribution.

We study stochastic gradient descent with random reshuffling for finite sums \[ F(x)=\frac1n\sum_{i=1}^n f_i(x). \] For fresh reshuffling with a constant component stepsize, if each $f_i$ has an $L$-Lipschitz gradient and the average $F$ is $\mu$-strongly convex with a Lipschitz-continuous Hessian, we prove the last-epoch rate \[ \mathbb E[F(y_K)-F(x_\star)] =\widetilde O\!\left(T^{-2}+n^2T^{-3}\right), \qquad T=nK, \] matching the known quadratic lower bound in its $(n,K)$-dependence. The components may be nonconvex, and no componentwise Hessian continuity or separate bounded-iterate assumption is required. More generally, a $\nu$-Hölder-continuous average Hessian adds only $\widetilde O(n^{1+\nu}T^{-2-2\nu})$, so every $\nu\ge 1/2$ preserves the quadratic rate. Under convex components, a decreasing-stepsize result removes the large-epoch requirement and recovers the same two-term scale once $nK$ exceeds the condition-number scale. We also analyze epoch-wise ProxRR for $\mathcal P=F+\psi$. Writing $x^\dagger$ for the composite minimizer and $\beta_\star=\|\nabla F(x^\dagger)\|$, we prove \[ \mathbb E\|y_K-x^\dagger\|^2 =\widetilde O\!\left( \frac{\beta_\star^2}{K^2} +T^{-2}+n^2T^{-3} +n^{1+\nu}T^{-2-2\nu} \right). \] For $\nu\ge 1/2$, we show that the $\beta_\star^2/K^2$ splitting term is unavoidable and obtain a matching lower bound up to logarithms in the stated constant-stepsize regime.

Uniform exponential tracking is addressed for nonstationary trajectories of the nonlinear Kuramoto–Sivashinsky equation subject to a constant input delay. The reference belongs to a family of complete trajectories contained in the global attractor and need not be stationary, periodic, slowly varying, or generated by a finite-dimensional exosystem. The delayed input is represented by a first-order transport equation coupled with the fourth-order tracking-error dynamics. A predictor–backstepping transformation compensates for the temporal mismatch between command generation and actuation by mapping the augmented closed-loop system into the nominal delay-free error dynamics driven by the outgoing trace of a homogeneous transport subsystem. This subsystem vanishes after one delay interval. Uniform attractor bounds permit the feedback parameters and stability constants to be selected independently of the initial time and the reference trajectory. Global well-posedness and uniform exponential stability are established in the augmented state space. Numerical results show that, for the considered configuration, uncompensated delayed feedback amplifies the tracking error, whereas predictor compensation restores sustained decay. Predictor-consistency, finite-time-extinction, and discretization-refinement diagnostics support the numerical implementation.

The Integrated Healthcare Timetabling Competition 2024 (IHTC-2024) defined a benchmark that integrates surgical case planning, patient admission scheduling and nurse-to-room assignment. Its instance set remains live: the organizers keep a public table of best-known upper bounds that is open to new submissions. We report five improved upper bounds on that table, for instances i02 (1263, previously 1264), i05 (12744, previously 12760), m01 (3380, previously 3384), m03 (6692, previously 6697) and m04 (3315, previously 3318). All five solutions were checked with the official IHTP validator and carry no hard-constraint violations. The solutions are not built from scratch: each is derived from the corresponding solution published on the benchmark page, holding its admission days, room assignments and operating-theater assignments fixed, and reoptimizing only the nurse-to-room-and-shift layer to proven optimality. The improvements are small in magnitude, between 0.075% and 0.125%, but they carry a structural observation: on the instances where the nurse layer can be closed exactly, it closes within seconds on all but one of them, and several of the published state-of-the-art solutions fall short of optimality in that layer.

Time-dependent density functional theory, though exact in theory, is in practice applied in an adiabatic approximation using exchange-correlation functionals with only local temporal dependence. Simultaneously, the exact correlation potential formally depends on, among other quantities, the time-history of electron densities. Here we develop a framework to learn neural network models of correlation functionals that feature explicit memory-dependence. Our framework includes two different approaches, both linked through their use of adjoint-based optimization. One approach decouples the learning of the functional from inversion to find ground truth values of the correlation potential. The other approach learns the functional directly without requiring inversion. We apply these methods to modeling the electron dynamics of two-electron systems in two spatial dimensions. In both cases, our methods yield correlation functionals with low test set propagation error, outperforming standard local density and generalized gradient approximation functionals by one to two orders of magnitude. Overall, our framework points at one strategy to move beyond the adiabatic approximation and develop memory-dependent correlation functionals that yield accurate propagation for excited state and/or non-equilibrium dynamics.

Clinical decision support frequently involves heterogeneous measurements, expert assessments, and guideline-based knowledge whose relationships to clinical decisions are gradual rather than purely binary. This paper develops a clinical decision-support framework based on bipolar fuzzy relational optimization, in which favorable and unfavorable clinical relationships are represented jointly through positive and negative fuzzy relational matrices together with a fuzzy requirement vector. The simultaneous participation of each decision variable and its complement in the relational constraints enables bipolar clinical information to be incorporated within a unified optimization model. The minimum \(t\)-norm is adopted to provide a noncompensatory, bottleneck-type interpretation of the interaction between fuzzy relational grades and decision levels. We characterize the feasible recommendation set by deriving clinical admissibility intervals and effective evidence activation sets, which provide the basis for feasibility analysis and systematic reduction of the relational system. The complete feasible set is then represented as a finite union of clinical recommendation regions associated with admissible Clinical Evidence-Assignment Functions. A region-wise optimal candidate is constructed for each admissible evidence assignment, and comparison of the resulting finite collection of candidates yields a globally optimal recommendation. An algorithm integrating feasibility analysis, system reduction, region-wise optimization, and global selection is developed and illustrated through a numerical CDSS example. The results show that bipolar fuzzy relational optimization provides a structured and mathematically rigorous mechanism for transforming fuzzy clinical relational information into feasible and optimized decision-support recommendations.

Distributed systems often operate under uncertainty, where minimizing expected loss may overlook rare but severe events. This paper studies a distributed risk-averse convex optimization problem in which agents cooperatively minimize the average of local conditional value-at-risk (CVaR) objectives over a time-varying network. Each agent has access only to noisy evaluations of its local loss function, rather than to its CVaR objective or gradient. We therefore develop a zeroth-order algorithm that uses sampled losses to construct empirical CVaR estimates and their gradient estimates. At each iteration, agents combine neighboring decisions and perform a local update. Under convexity and Lipschitz continuity assumptions, we prove that the agents reach exact asymptotic consensus. We also establish a finite-time expected suboptimality bound for the weighted ergodic iterate. With diminishing step sizes and fixed sample sizes, the local last iterates converge almost surely to a common optimum, and their limiting expected CVaR gap is bounded in terms of the smoothing and finite-sample errors. This distributed bound matches the parameter dependence of the centralized benchmark provided in this paper. Finally, simulations on a distributed sensor network estimation problem illustrate the efficacy of the method.

This paper introduces and studies a Principal-Agent competition model in which two risk-neutral Principals compete to hire a single Agent. The Agent chooses to work exclusively for one Principal, making the Agent's reservation utility endogenous, as it is determined by the competing offers. We characterize the Nash equilibria of the resulting game by relating them to the individual Principal-Agent contracting problems and analyzing the Agent's selection bias. We further introduce two numerical algorithms to approximate Nash equilibria by locating sign changes of the value functions of the Principals' individual problems. The first is a Newton-type method that exploits differentiability of the value functions, while the second is a bisection method that does not require differentiability. The proposed procedures are illustrated through several examples involving exponential, logarithmic, and constant relative risk aversion (CRRA) utility functions within the polynomial framework of Renner and Schmedders (2015). We also provide technical results regarding the regularity of the value function of a classical Principal-Agent problem, including differentiability under suitable conditions.

Classical extremum seeking (ES) is commonly interpreted as approximating gradient descent, but this interpretation is less clear for nonsmooth objectives in the continuous-time multivariable setting. We propose a minimal modification of the classical multivariable perturbation–demodulation architecture: rationally independent perturbation frequencies and matched demodulation signals. For any locally Lipschitz static objective, the Kronecker–Weyl theorem shows that, at every fixed perturbation amplitude, the long-time averaged dynamics are exactly the negative gradient of a kernel-smoothed objective. Because rationally independent frequencies render the perturbation and demodulation signals nonperiodic, we employ general averaging theory rather than periodic averaging theory. If the gradient flow of the smoothed objective is globally uniformly asymptotically stable, then the ES dynamics are practically globally uniformly asymptotically stable. We also derive a general matching condition relating the perturbation occupation density, demodulation signal, and smoothing kernel, yielding a family of alternative designs. Numerical examples include a nonsmooth objective function, which may be interpreted as the penalty function of a nonlinear program, and the Rastrigin function, for which smoothing eliminates all undesired local minima.