Papers

Early discovery of at least one valid design satisfying a target requirement is a central objective in failure-prone closed-loop inverse design. A natural batch baseline ranks candidates by a product-form marginal valid-hit score, but selecting the highest-ranked candidates independently can produce redundant recommendations under predictive uncertainty and waste the experiment budget. We introduce ARC-SC(Anchored Risk-Constrained Scenario Coverage), a batch acquisition method that preserves strong marginal candidates as anchors and allocates the remaining batch positions by maximizing complementary coverage over predictive target scenarios under a risk-support constraint. In frozen-oracle closed-loop simulations on superconductivity and JARVIS materials-property benchmarks, ARC-SC yields a statistically supported improvement in first-hit discovery and remains competitive with directionally favorable first-hit performance on more challenging design space. These results establish ARC-SC as a POF-anchored, scenario-aware batch strategy for improving early valid-target discovery under structured experimental failure.

We study the edge-power cost of the labelled tree generated by the $\gamma$-strategy, a constant-gain rule for sequential Euclidean connections. Starting with $x_0=p_0$, each input point $p_i$ is attached to $x_{i-1}$, and the state is updated by $x_i=\gamma x_{i-1}+(1-\gamma)p_i$. Retaining $x_i$ subdivides the insertion segment into a spine edge and a leaf edge. The memory parameter $\gamma$ controls how long earlier points influence subsequent attachment points. We minimize the sum of the $\alpha$-powers of the edge lengths under independent uniform input and arbitrary input sequences. For uniform points in the unit ball, the stationary problem has a transition at $\alpha=1$. Its continuous extension is minimized at the boundary for $0<\alpha\leq1$, while every global minimizer is interior for $\alpha>1$. We determine the finite optimizer in the joint window $\alpha_N=1+\varepsilon_N$, $\varepsilon_N\log N\to\lambda$. Below an explicit threshold it lies on the $N^{-1/2}$ scale, at the threshold its scale is $\sqrt{\log N/(N\log\log N)}$, and above the threshold it approaches an explicit stationary root with two computable corrections. A second threshold identifies the governing correction, and differentiated estimates prove eventual uniqueness. At $\alpha=3d+8$, the linear coefficient at the stationary endpoint changes sign and a branch of strict local maxima enters the parameter interval. For arbitrary input sequences, the optimal parameter and asymptotic worst-case edge-power cost per point are explicit for $0<\alpha\leq3$. At high powers, periodic antipodal block inputs give explicit lower bounds which, with a separation argument, show that the optimized cost is asymptotic to $2\log2/\log\alpha$. Exact results for powers two and four, a rational recursion for every even power, and a high-dimensional expansion complete the analysis.

We study a PDE-constrained optimization problem for an elliptic equation with the spectral fractional Laplacian and a linear combination of Dirac measures as the forcing term; the controls are the amplitudes of these singular sources. We prove existence and uniqueness of an optimal solution and derive first-order optimality conditions. We then propose a discretization based on finite elements. Since the set of admissible controls is finite dimensional, the control variable itself does not require discretization. We conclude by deriving a priori error bounds

Efficiently and robustly analyzing shape data is critical across many scientific disciplines. While chirality is a fundamental property in numerous applications - most notably in molecular science - existing shape analysis metrics fail to distinguish between a shape and its mirror image. To address this gap, we introduce a multilinear generalization of the Gromov-Wasserstein objective. Under mild assumptions, this objective yields a distance between shapes, represented as probability distributions quotiented by a symmetry group $G$. In particular, for $G = SO(d)$, we introduce the Chiral Gromov-Wasserstein ($\mathrm{CGW}$) distance, sensitive to chirality. We establish robustness properties for the multilinear Gromov-Wasserstein distances and develop efficient algorithms to compute them, reformulating the underlying optimization problem by projecting couplings onto a low-dimensional space. We derive algorithms for both local and approximate global solutions, yielding a fully polynomial-time approximation scheme for these problems. We validate the framework through numerical experiments that demonstrate the effectiveness of $\mathrm{CGW}$ as a shape metric for chiral objects.

We study contraction properties of non-stationary continuous-time mean-field games (MFGs) under discounting and entropy regularization. The state of the representative agent evolves according to a controlled continuous-time Markov chain, and both the state and action spaces are finite. In contrast to the undiscounted case, we show that, under a sufficiently large discount rate, finite-horizon MFGs admit a horizon-independent contraction condition, which also coincides with the corresponding infinite-horizon non-stationary contraction condition. As a byproduct, we obtain an explicit convergence rate between finite- and infinite-horizon mean-field equilibria. For each finite horizon, we further derive a refined contraction criterion from the spectral radius of a positive operator that majorizes the propagation of policy errors, and show that its large-horizon limit agrees with the horizon-independent contraction factor. Finally, we provide an explicit error bound between discounted and undiscounted finite-horizon regularized equilibria.

Generalized disjunctive programming (GDP) expresses mixed discrete-continuous decisions through Boolean indicators and disjunctions, and can be systematically solved via a library of methods proposed in the literature. The recent InfiniteGDP abstraction lifts this modeling layer to continuous domains such as time, space, and uncertainty, but only the big-M and hull reformulations, the two endpoints of the relaxation spectrum, have been generalized to the infinite setting. This work closes this gap by generalizing four other GDP solution methods to infinite-dimensional optimization: the multiple big-M reformulation, P-split reformulation, cutting plane reformulation, and the logic-based outer approximation algorithm. It further proposes MBM-GP, a novel Gaussian-process variant of multiple big-M that learns the big-M function over the infinite domain from a small subset of the subproblem solves. Moreover, these approaches are implemented in the Julia package InfiniteDisjunctiveProgramming.jl. The methods are benchmarked on case studies arising in dynamic and stochastic optimization. The results demonstrate how the generalized solution methods can outperform big-M and hull, with MBM-GP retaining the tightness of multiple big-M at a fraction of its reformulation cost.

The Maximum Entropy on the Mean (MEM) method provides a flexible computational framework for solving inverse problems by combining data fidelity with entropy-based regularization. In practice, however, the prior distribution is typically unknown but can be estimated from data, giving rise to the empirical MEM method. We establish a parametric convergence rate of $O(n^{-1/2})$ in expectation for empirical MEM, improving upon the previously established $O(n^{-1/4})$ guarantee by King-Roskamp et al. (2026). Our proof is based on a novel stability analysis of the primal and dual optimization problems under perturbations of the underlying probability measure, relying only on foundational tools from convex analysis and probability. We further show that the MEM dual problem admits a reformulation as an expected risk minimization problem, thereby placing MEM within the modern framework of stochastic optimization and enabling scalable stochastic gradient algorithms for large-scale inverse problems. Together, these results place empirical MEM as a statistically and computationally efficient methodology for data-driven inverse problems.

Inspired by the progressive decoupling of linkages methodology for optimization and variational inequalities, we propose an algorithm for solving quasi-variational inequalities as a sequence of variational inequalities. Our method is shown to converge locally under some regularity conditions and globally when such conditions hold throughout the entire domain. Separately, under other type of assumptions, global convergence with linear rate is also established for the class of quasi-variational inequalities said to have a moving set. Advantageous computational performance is shown for large-scale Walrasian equilibrium problems, a special case of generalized Nash equilibria, as well as for some other instances encountered in related literature.

We present an inexact Augmented Lagrangian algorithm for solving nonlinear, non-convex optimization problems. Unlike most recently proposed Augmented Lagrangian methods with worst-case complexity guarantees, we utilize adaptive penalty parameter updates and full dual stepsizes. We show that the method matches the best known worst-case complexity results for Augmented Lagrangian methods (up to logarithmic factors) when both the function and constraints are deterministic, when the function is stochastic and the constraints are deterministic, and when both are stochastic. Experiments on CUTEst test problems confirm the practical advantages of the proposed approach over Augmented Lagrangian methods with non-adaptive penalty parameters and/or short dual step sizes in the deterministic setting. Numerical results on stochastic constrained optimization problems in machine learning also confirm these findings.

Motivated by a conjecture of Burq and Gérard, we investigate quantitative uniqueness principles for functions on $\mathbb T^2$ whose Fourier spectra lie in fixed-width annuli. We obtain observability estimates uniform in the radius, under a mild Sobolev regularity condition, and the generalized geometric control condition (GGCC) on the damping function. This leads to exponential decay for the damped wave equation with rough damping. We also establish an analogous uncertainty principle for spectra near dilates of convex polygonal boundaries, where no Sobolev regularity of the damping is required beyond $L^\infty$.

Power grid state estimation relies on sensor measurements that are increasingly vulnerable to adversarial corruption in cyberphysical environments, potentially leading to significant deviations in system observations. This motivates the need to identify critical measurement nodes whose compromise results in the most severe system-level impact. However, existing node criticality methods primarily rely on structural or steady-state analyses and do not explicitly account for adversarial effects on system behavior. To address this gap, this paper proposes an adversarially informed framework for identifying critical measurement nodes in linearized power systems. Within this framework, a structured attack generation mechanism is developed to construct stealthy and effective false data injection attacks (FDIAs) against an H-infinity resilient state estimator. Node criticality is then evaluated using coalition-based marginal contributions of compromised sensor subsets, estimated via permutation sampling over a prescribed set of admissible nodes, with the resulting importance scores mapped to the corresponding physical buses. Simulation results on the IEEE 14-bus system show that adversarially identified nodes induce larger deviations in frequency, voltage angle, and net power compared to randomly selected nodes, demonstrating the effectiveness of the proposed framework.

In this paper, we propose a unified framework for accelerated gradient methods, dubbed UGM, which subsumes a wide range of accelerated and conventional gradient-type methods designed for minimizing $L$-smooth and $\mu$-strongly convex functions. We demonstrate that the iteration update of the proposed framework can be intrinsically interpreted as a hybrid combination of the heavy-ball method and vanilla gradient descent. This interpretation reveals that classical accelerated gradient methods essentially integrate a conservative gradient descent step into the fast yet unstable heavy-ball dynamics, which achieves a favorable trade-off between acceleration and stability. We further establish a unified convergence analysis using Lyapunov functions. Guided by our analysis, we develop a family of enhanced accelerated gradient algorithms that leverage the inner product relationship between gradient information and iterative variables to optimize iterative updates. Extensive numerical experiments on unconstrained quadratic optimization and logistic regression validate that the proposed algorithms achieve superior performance compared with existing baseline methods under typical structural conditions.

In several articles, the authors assume that the convex roof structure of the EoF and finite-dimensionality of subsystems $A$ and $B$ guarantee the existence of the (global) supporting affine functional for the EoF at any state of the system $AB$. This means that for any state $\rho$ of $AB$ there is a Hermitian operator $\Lambda_\rho$ on $\mathcal{H}_{AB}=\mathcal{H}_A\otimes\mathcal{H}_B$ such that $E_F(\rho)=\mathrm{Tr}\Lambda_\rho\rho$ and $E_F(\sigma)\geq\mathrm{Tr}\Lambda_\rho\sigma$ for any state $\sigma$ of $AB$. We present an explicit example showing that, when $\rho$ is degenerate, this is not true even in the simplest case when $A$ and $B$ are qubit systems. The construction is based on the fact that the existence of a supporting affine functional for the EoF at a state $\rho$ is equivalent to the Lipschitz lower semicontinuity of the EoF at this state $\rho$. We use Wootters' formula and the help of Claude Fable 5 to find a state $\rho$ of the system $AB$ for which the latter property does not hold. We also describe conditions for the existence the local and global supporting affine functionals for the EoF at a given state of both finite and infinite-dimensional bipartite quantum systems. These conditions allow us to find Lipschitz lower semicontinuity bounds for the EoF at a given finite rank state $\rho$ (i.e. inequalities of the form $\,E_F(\rho)-E_F(\sigma)\leq C_\rho\|\rho-\sigma\|_1$) with and without restrictions on the support of the state $\sigma$.

The presence of bilinear terms in mathematical modeling generally yields nonconvex quadratic programs (QPs) that remain computationally challenging to solve to global optimality. While continuous piecewise linear (CPWL) approximations can reformulate these nonlinearities into mixed-integer linear programs (MILPs), the geometric construction of the domain partition heavily dictates the resulting solver efficiency. In this paper, we present efficient MILP formulations for approximating bilinear terms and rigorously evaluate their computational merits against direct QP solvers. First, we introduce CPWL approximations with arbitrary high degrees of accuracy that explicitly account for and exploit the inherent symmetries of the unit bilinear function. Second, we analytically establish the exact maximum and average approximation errors of the CPWL approximations. Third, we construct and compare three distinct MILP formulations (a “Triangle”, a “Square”, and a difference-of-convex “DC” formulation) of the CPWL functions. Fourth, we formalize a highly relevant class of optimization models, Sequentially Coupled Bilinear Programs (SCBP), where variables represent system states and state changes. Finally, through extensive numerical experiments, we demonstrate that our compact “Square” MILP formulation achieves superior computational performance on SCBP instances with long sequence horizons, allowing mature open-source MILP solvers to effectively outperform QP solvers.

The marine science community increasingly relies on Marine Autonomous Vehicles (MAVs) to collect the critical environmental data required to understand global ocean systems. However, as these operations scale, manually routing and planning large autonomous fleets becomes exponentially complex and time-consuming. To address this, we propose a mixed-integer linear programming (MILP) model designed to automate and optimise MAV deployment schedules. The model accounts for strict operational constraints, including battery capacities and time windows for data collection, while aiming to maximise total data collection and minimise both the number of deployed vehicles and their energy consumption. A key novelty of this framework is integrating conventional ship itineraries, allowing MAVs to support vessels with mid-mission battery swapping or accelerated transit between waypoints. Computational experiments demonstrate that the model is highly scalable, solving routing problems for fleets of dozens of MAVs in seconds, and scaling to hundreds of vehicles in only a few minutes. Beyond operational scheduling, the framework serves as a robust simulation tool for evaluating 'what-if' scenarios and analysing the impact of varying parameters on deployment strategies. Finally, the solution generates a suite of visualisations designed to enhance explainability and support strategic decision-making for stakeholders.

In this paper, we investigate first- and second-order alignment models with non-universal interaction, time delays and possible communication failures, extending the results in [17] to interaction digraphs that are only assumed to be rooted and to a weaker Persistence Excitation Condition. In particular, we allow the amount of interaction over time intervals of fixed length to decay polynomially in time. For the first-order Hegselmann-Krause type model, we prove asymptotic convergence to consensus under a suitable condition relating the decay exponent of the communication weights to the maximal distance from the root. For the second-order Cucker-Smale model, we establish asymptotic flocking under an additional assumption on the decay of the influence function. These results show that collective behavior can still emerge under progressively weakening communication and without requiring strong connectivity of the interaction graph.

Expansion planning models usually fix storage duration before optimization, setting how much storage power to build but not for how long it can discharge. We present a generation-transmission-storage expansion framework in which candidates of many durations compete on annualized cost, making the duration mix an optimization outcome. It is a rolling-horizon, two-stage stochastic linear program, each five-year stage is a full year at hourly resolution under ten coherent inflow, wind, and solar scenarios. We apply it to the Brazilian Interconnected System over 2030-2050, where demand roughly doubles to 1,716 TWh/year and variable renewable energy (VRE) supplies most new capacity. Two cases are compared: 36 pumped-storage hydropower (PSH) candidates over four subsystems and nine durations (4-144 h) alongside 4-h battery energy storage systems (BESS), and BESS alone. With PSH available, the model builds 31.2 GW / 755 GWh of PSH and no BESS, dominated by 12-h capacity with 4.9 GW of 72-h units. Without PSH it builds 38 GW / 152 GWh of BESS and 7.8 GW more gas-fired capacity, mostly open-cycle peakers. PSH lowers 2050 annualized system cost by US$ 5.0 billion/year (6.8%), operating cost by 23.5%, thermal generation by 34 TWh/year, and long-run marginal cost by 20%; 2035 VRE curtailment falls from 8.4% to 3.2%. The magnitudes are specific to Brazil, but the underlying mechanism is general: limiting storage candidates to a single duration understates the system's optimal energy-storage requirement and overstates its residual need for thermal capacity.

In this paper, we develop a deterministic, physically constrained generative framework based on the Madelung formulation of the free-particle Schrödinger equation. A reference Born probability density and a controllable initial phase function serve as initial data for the free Madelung system, which couples the Born probability density and phase function through the Bohm quantum potential, while the phase function determines the hydrodynamic velocity field. Provided that the Born probability density remains positive and the hydrodynamic velocity field generates a unique global characteristic flow, samples drawn from the reference density and transported along the characteristic flow are distributed according to the evolving Born probability density at every time. As a consequence, randomness enters only through the initial sampling; the subsequent generation is deterministic and involves neither stochastic dynamics nor an independently parameterized time-dependent velocity field. We formulate terminal-time distribution matching as a PDE-constrained phase-identification problem and derive the underlying Hamiltonian and Fisher-information structure. For isotropic Gaussian wave packets, we obtain explicit dynamics and a necessary and sufficient condition for exact reachability of isotropic Gaussian targets by quadratic initial phase functions, together with the corresponding sampling map. For a smooth prescribed potential initial velocity field, we further establish that the characteristic flow approximates the associated first-order transport map with an O(T^2) error, both uniformly and in the 1- and 2-Wasserstein distances. A numerical Gaussian benchmark validates the fully discrete forward solver, while full-grid PDE-constrained phase identification is demonstrated for asymmetric bimodal targets.

This paper considers the local convergence for solving general nonlinear equations. We establish randomized quasi-Gauss–Newton methods based on the approximation of the Gram matrix, addressing both the underdetermined and the overdetermined settings. For the underdetermined case, we show that our methods achieve local superlinear convergence to the optimal solution. For the overdetermined case, we show that our methods achieve local condition-number-free convergence to the stationary point of the nonlinear least-square formulation of the problem. In contrast, existing quasi-Newton methods mostly focus on square systems and additionally require the initial Jacobian estimate to be sufficiently close to the exact one.

We classify the facial exposedness of completely positive cones over symmetric cones in terms of the rank of the associated Euclidean Jordan algebras. The completely positive cones are facially exposed when the rank is at most $2$, but are not facially exposed when the rank is at least $5$. Facial exposedness is not completely determined by the rank when the rank is $3$ or $4$, but we provide a characterization in each case. The resulting classification of facial exposedness in fact agrees with the corresponding classification of spectrahedrality for completely positive cones.

We study exact low-dimensional reformulations for a regularized spectral approximation problem and its weighted variant. In the unweighted setting, weak-majorization monotonicity of the fidelity term enables an exact reformulation over singular values. In the weighted setting, we establish such a reformulation under compatibility and simultaneous diagonalization conditions. We further show, via a counterexample, that this exact reformulation can fail in the absence of simultaneous diagonalization.

We study the maximization of the minimum eigenvalue under combinatorial and integrality constraints. We propose a new approach based on branch-and-bound combines entropic smoothing of the minimum eigenvalue function and Frank-Wolfe methods over concave relaxations of the constraints, thereby exploiting combinatorial structure through linear optimization oracles. We establish approximation and convergence guarantees, including for truncated gradients computed from partial eigendecompositions, and introduce rank- and eigenvalue-based pruning and duality-based variable fixing. We evaluate the method on E-optimal experimental design and maximum algebraic connectivity problems and compare it with SCIP-SDP. The results show that our approach is particularly effective for large-dimensional instances and problems with additional combinatorial structure, whereas SCIP-SDP performs better on moderately sized instances with simpler constraints.

We study long-horizon Wasserstein-penalized minimax control for discrete-time stochastic linear systems with empirical disturbance data, in which adversarial disturbance distributions induce time-varying mean and covariance dynamics, making standard turnpike arguments not directly applicable. For possibly uncentered data, we characterize the generally nonzero mean reference through a reduced convex-concave Hamiltonian saddle problem. We prove horizon-uniform, two-sided exponential turnpike estimates for the mean state, adjoint, control, worst-case disturbance mean, and closed-loop covariance, showing that they spend the majority of time near static references when the horizon is long. We further construct a hybrid policy combining time-independent affine feedback with finite-horizon steering over a terminal layer, proving that its worst-case cost gap decays exponentially with the terminal-layer length uniformly in the horizon, which helps reducing the computation cost for long-horizon robust controls. Numerical examples illustrate the estimates and their dependence on the Wasserstein penalty.

Graphs are invariant under node permutations, motivating permutation-equivariant architectures in generative models. In flow matching, however, symmetry also affects the source-target coupling: one must choose both which graphs to pair and which node representatives to align. We study these two forms of alignment, termed outer and inner alignment, and their effect on equivariant graph flow matching. We connect inner alignment to transport on the graph quotient space, whose Euclidean quotient metric coincides with Gromov-Monge distance, and show that quotient couplings admit aligned representative lifts while symmetrization yields equivariant flow-matching minimizers, including for categorical endpoints. In practice, we compare random augmentation, permutation-blind minibatch optimal transport, approximate Gromov-Wasserstein alignment, and combinations of inner and outer alignment. Across continuous graph and molecular generation, alignment can substantially simplify trajectories and improve few-step generation, while its benefits depend on the alignment and computational budget. Our results highlight that effective graph flow matching benefits from alignment both inside and across graph pairs.

This paper proposes a novel data-driven algorithm to approximate the dominant eigenfunctions (aka.~modes) of the Koopman operator of nonlinear dynamical systems using neural networks. The relevance of learning the dominant Koopman modes is to approximate nonlinear dynamics by linear ones in a lifted space, thereby enabling simplified control and analysis. To fight the curse of dimensionality arising from using expressive templates (here neural networks) for the mode approximation, the proposed method leverages a power-iteration scheme that directly learns the dominant Koopman modes without explicitly constructing the projection of the Koopman operator on the template of functions. Our approach connects to other approaches in the literature that avoid the curse of dimensionality by learning small dictionaries of functions, but differs from them in that we do not require “anti-collapse mechanisms” to ensure that the learned dictionary is expressive enough to approximate the Koopman operator since our power-iteration scheme is designed to converge toward the dominant modes of the projected Koopman operator. The approach is fully data-driven, requiring only sampled state transitions. Theoretical guarantees are provided, showing convergence under increasing sample size and network width (in connection with the neural tangent kernel theorem). Numerical experiments demonstrate that the method achieves accurate and smooth approximations of dominant modes while avoiding the limitations of traditional techniques such as extended dynamic mode decomposition.

We revisit a degree-only arc Hamiltonian for fixed-fleet, homogeneous, uncapacitated vehicle routing. Its local penalties define only a cycle cover, so ground states may contain customer cycles disconnected from the depot. We construct a polynomial-size quadratic unconstrained binary optimization (QUBO) repair using capped single-commodity flow and prove connected, cost-optimal ground-state routings under explicit penalty assumptions. For $N-1$ customers and $K$ nonempty routes, the unreduced encoding uses exactly $|E|(1+\lceil\log_2(N-K+1)\rceil)$ logical problem qubits. A reversible compute-phase-uncompute implementation evaluates the flow penalties in $O(N^2\log N+N\log^2N)$ logical gates on a complete graph, using $O(\log N)$ reusable workspace and no product register. On complete loopless graphs, a position encoding that arranges all routes in one sequence separated by depot occurrences uses fewer problem qubits and written terms when the number of bits per flow register grows. Under the same reversible accounting model, the flow formulation admits a smaller logical-gate upper bound. Exact Hamiltonian and circuit audits and a benchmark of 1,200 cost matrices verify the formulation and quantify the connectivity gap. A 32,000-shot Amazon Braket task on IQM Emerald characterizes depth-one termwise Ising circuits for a diagnostic $N=4$, $K=1$ counterexample. In the degree-only circuit, 78.05% of selected $p=1$ shots realize the invalid disconnected ground state; the reduced 14-qubit flow-augmented circuit yields no fully feasible sample. These device results characterize mapped Hamiltonians and compilation, not an asymptotic advantage in solving routing problems.

Connected and automated vehicle (CAV) platooning offers a promising approach to improving road safety and traffic capacity. However, platoon control in real-world traffic is challenging due to uncertainty and heterogeneous driving behaviors. Reinforcement learning (RL) has strong potential for addressing such control problems, but its practical deployment raises challenges related to safety and learning efficiency. This paper proposes a generic modeling and simulation framework for investigating CAV platoon joining maneuvers and comparing deep reinforcement learning (DRL)-based control algorithms. The problem is particularly challenging in mixed-traffic environments, where CAVs coexist with human-driven vehicles exhibiting heterogeneous longitudinal and lateral behaviors. The objective is to achieve safe and efficient joining maneuvers by either incorporating penalties for risky behaviors into the learning process or using an external safety controller to constrain the learned policy. An agent-based modeling framework coupled with the Simulation of Urban MObility (SUMO) simulator is used to evaluate Deep Q-Network (DQN), Double Deep Q-Network (DDQN), and Proximal Policy Optimization (PPO). Results show that PPO outperforms DQN and DDQN, achieving a joining success rate of approximately 98 % and a collision rate below 1 %, largely due to risk-related penalties incorporated into the reward function. However, this improved performance requires more decision steps to complete the maneuver, revealing a trade-off between safety, joining effectiveness, and decision efficiency. An external safety controller effectively prevents collisions, although its interventions may reduce joining efficiency. The results highlight the importance of jointly considering safety and efficiency when designing RL-based controllers for CAV platoon joining in mixed traffic.

We study the deterministic first-order oracle complexity of finding stationary points of the value function in smooth nonconvex-Polyak-Łojasiewicz (NC-PŁ) minimax optimization. We assume that the objective is jointly $\ell$-smooth and satisfies the $\mu$-PŁ condition in the dual variable, and that its value function $\Phi(x):=\max_y f(x;y)$ satisfies $\Phi(0)-\inf_x\Phi(x)\leq\Delta$. When $\kappa:=\ell/\mu\gtrsim 1$ and $0<\epsilon^2\lesssim\ell\Delta$, we prove that every deterministic first-order method requires $\Omega(\ell\Delta\kappa/\epsilon^2)$ oracle queries in the worst case to find $x$ satisfying $\|\nabla\Phi(x)\|\leq\epsilon$. This rate matches the known upper bound in its dependence on $(\ell,\Delta,\kappa,\epsilon)$ [Yang et al., 2022] and shows that the linear dependence on $\kappa$ is unavoidable for deterministic first-order methods.

We determine the exact worst-case value, at every horizon $N\geq7$, of the smallest queried gradient norm generated by Nesterov's fast gradient method on smooth convex functions. Let $t_0=1$ and $t_{k+1}=(1+\sqrt{1+4t_k^2})/2$, and let $x_0,\ldots,x_N$ denote the points at which the method evaluates gradients. For every such $N$ and every dimension $d\geq N-4$, we prove \[ \sup_{\substack{f\in\F_{0,L}(\R^d),\ x_\star\in\arg\min f \norm{x_0-x_\star}\leq R}} \min_{0\leq k\leq N}\norm{\nabla f(x_k)}^2 =\frac{L^2R^2}{\sum_{k=0}^N t_k^2}. \] The relaxed-PEP upper bound is due to Kim and Fessler, who also reported tight numerical solutions of the exact-interpolation PEP at selected horizons. What remained missing was an analytic matching family valid uniformly over the horizon. For every $N\geq7$, we construct such a family using an FGM-specific spherical polytope $K_N$ and the standard projection-envelope function \[ f_N(x)=\max_{g\in K_N}\left\{\ip{x}{g}-\frac12\norm{g}^2\right\}, \qquad \nabla f_N(x)=\Proj_{K_N}(x). \] Every queried gradient has the same norm, and the vertices of $K_N$ are generated from a three-dimensional seed by a one-dimensional spherical cone lift. The lift preserves all projection inequalities and raises the adversary dimension by one at each horizon. The projection/Moreau-envelope template itself is classical; the new ingredients are the FGM-specific algebraic seed, the proof that it attains the relaxed bound, and the common-latitude lift that propagates this exactness to every $N\geq7$. We state precise hypotheses for that propagation and do not claim that every rank-one relaxed PEP admits such a seed.

We study the first-order oracle complexity of finding a queried point with small gradient in $\ell_p$ geometry, with particular attention to the information needed to adapt the unknown smoothness and distance scales. In the strict counted local value–gradient model, no finite complexity bound can depend only on $LR/\eps$ without a nondegenerate local scale observation: a one-dimensional construction keeps $LR/\eps=4$ while defeating every prescribed finite query budget. We resolve Diakonikolas's general-$\ell_p$ parameter-free extension question for every fixed $1<p<\infty$. Under a nondegenerate secant initialization, the method knows neither the smoothness constant $L$, the initial solution distance $R$, nor $f^*$, and returns a queried point $\widehat x$ with $\|\nabla f(\widehat x)\|_q\le\eps$. For fixed finite $p>2$, we first establish the dimension-free deterministic known-parameter upper exponent $p/(p+2)$ in $K=LR/\eps$, matching the published lower polynomial exponent under its horizon and dimension qualifications. The finite local routine fits the same observable scale–radius procedure, so this exponent is preserved without knowing $L$ or $R$. Writing $\Kbar=\max\{1,LR/\eps\}$, the post-initialization pair-oracle complexity is $O_p(\Kbar^{1/2})$ for $1<p<2$, $O(\Kbar^{1/2})$ for $p=2$, and $O_p(\Kbar^{p/(p+2)})$ for $p>2$, together with the additive calibration cost $O_p(\log(e+L/M_0))$ in every regime.