Papers

We consider a left-invariant Lorentzian problem on SU(2). The following questions are addressed: complete controllability, existence of length maximizers, extremals, curvature.

Due to the nonconvex binary constraints of unit commitment (UC), no uniform linear pricing scheme supports the optimal dispatch. Convex hull pricing (CHP) and copositive duality pricing (CDP) both address this problem. CHP derives the price from the subgradient of the value function of the convex hull relaxation of UC. CDP refers to several different pricing mechanisms that can be constructed from the dual multipliers of the completely positive programming reformulation. In this work, we define a centralized convex hull price over the joint feasible set of UC and prove that, under non-degeneracy, it coincides with the marginal copositive duality price. Numerical experiments on the Scarf example validate this equivalence and quantify the pricing gap introduced by the semidefinite restriction.

We propose a new strategy for gradient-based unconstrained optimization, involving two parallel sequences of iterates that cooperate to determine their stepsizes via a Twin-Step principle. Rather than minimizing the objective function individually, the algorithm selects steplengths that minimize the Euclidean distance between the two gradient based processes occurring simultaneously at each iteration. The theoretical analysis shows that the convergence of the mutual distance is governed by the angle between the search directions. In particular the effectiveness of the overall process degrades as the directions approach parallelism. To ensure robustness against collinearity, we introduce a hybrid framework, Twin-ABB$_{\min}$, which switches to the Adaptive Barzilai–Borwein method when the geometric cooperation becomes ineffective. Extensive and very promising numerical results evidence that the Twin phase creates favorable initial conditions for subsequent BB-type iterations.

We consider constrained problems in which input data are affected by errors. In such settings, Wasserstein distributionally robust optimization provides a principled framework to mitigate model risk by optimizing against worst-case data distributions within Wasserstein ambiguity sets. However, the numerical resolution of the resulting problems remains challenging, especially in constrained settings. In this paper, we provide a general, practical way to solve Wasserstein distributionally robust formulations in the presence of constraints. Our approach only requires a linear minimization oracle for the feasible set, and combines two key ingredients: (i) an entropic regularization of the distributionally robust value function, which makes it possible to compute stochastic gradient estimators, and (ii) a stochastic Frank-Wolfe algorithm, which minimizes the regularized robust objective while naturally handling constraints. We illustrate the method, its tractability, and its interests against empirical risk minimization, on two problems: the traffic assignment and the minimum quadratic spanning tree.

In this article, we develop a robust optimization framework for obstacle-aware aerial directional sensor networks operating under positional uncertainty using a sector-based sensing model. The monitoring area is discretized into grid points and a unified geometric formulation is developed to model directional sensing while explicitly accounting for obstacle-induced visibility loss and exclusion regions. To enhance sensing performance under uncertainty, three progressive optimization strategies, namely robust aerial grid coverage (RAGC), robust aerial target coverage (RATC) and robust aerial target scheduling (RATS), are proposed. The framework employs robust orientation optimization based on the radius of robust feasibility to maximize effective grid and target coverage under sensor perturbations, followed by a target-aware sleep scheduling mechanism that minimizes the number of active sensors without compromising target monitoring. Extensive simulations under varying target distributions, obstacle configurations, uncertainty levels and sensor failure scenarios demonstrate that the proposed framework achieves robust and energy-efficient sensing while consistently outperforming representative approaches from the literature in terms of coverage quality, target monitoring and sensor utilization.

Dual-source trolleybuses alternate between overhead catenary supply and on-board battery operation, creating energy-use patterns driven by route attributes, high-frequency trajectories, and hourly weather. Existing models struggle to represent these heterogeneous inputs and rarely explain the causal drivers of consumption. This paper proposes a time-aware tabular deep learning framework for inter-stop energy management. Periodic time encoding is integrated into a parameter-efficient batch-ensemble backbone to jointly learn static and sequential features, while Bayesian optimization with tree-structured density estimation tunes hyperparameters. To move beyond prediction, a three-layer causal explanation pipeline combines feature attribution for marginal effects, a linear non-Gaussian acyclic model for causal direction discovery, and a meta-learner for net average treatment effects. Experiments on the Zurich trolleybus dataset enriched with meteorological records achieve a MAPE of 6.52% and R of 0.982, outperforming ten statistical, tree-ensemble, and deep learning baselines. Ablation results show that periodic time encoding contributes most to the accuracy gain. Causal analysis identifies regenerative braking ratio and average speed as the strongest energy-saving factors, while coasting distance is the main driver of excess consumption. The findings offer actionable thresholds for vehicle technology, driving behavior, capacity allocation, and catenary network planning.

In this work, we consider the identification and control of nonlinear systems with finite action spaces. The unknown dynamics are estimated from finite samples with Koopman operator regression in a reproducing kernel Hilbert space, yielding a linear switching predictive model, the switches governed by the value of the control variable. In order to perform control in closed-loop, the learned dynamics are employed in an infinite-horizon optimal control problem with time-varying stage cost, which is solved by means of model predictive control. In a theoretical analysis, we derive learning rates for the Koopman dynamics approximation. We further quantify, under suitable assumptions, the sub-optimality of the model predictive control strategy, both in the case of exact Koopman dynamics, and in the case of learned ones. Numerical simulations on the Duffing oscillator complement our theoretical findings.

We study a continuous-time portfolio optimization problem in which an investor is evaluated relative to a non-replicable benchmark and seeks to control the persistence of benchmark-relative underperformance. We introduce a benchmark-relative drawdown-duration criterion that penalizes the expected discounted time spent in unfavorable benchmark-relative performance states. Despite the path dependence induced by benchmark-relative drawdowns, we show that the problem admits a one-dimensional Markovian representation and derive the associated Hamilton-Jacobi-Bellman equation. We obtain an explicit projection-based characterization of the optimal feedback control, establish a verification theorem, and identify geometric settings under which the associated closed-loop reflected diffusion admits a unique strong solution. Our results provide a tractable downside-risk-oriented alternative to classical benchmark-tracking formulations and reveal a novel projection-based control structure for benchmark-relative risk management.

This paper addresses the optimal control of epidemic dynamics under conflicting socio-economic objectives. We propose an economic Model Predictive Control (MPC) framework, applied to an extended SEIR-V (Susceptible-Exposed-Infected-Recovered-Vaccinated) compartmental model to govern the spread of an infectious disease while minimizing economic disruption. The control problem is formulated as a constrained nonlinear optimization problem, in which the controller dynamically adjusts social interaction levels (transmission rate beta) and vaccination efforts to minimize a composite cost function that penalizes fatalities, healthcare capacity violations, and economic losses. We conduct a rigorous sensitivity analysis of the prediction horizon N, demonstrating that the closed loop is robust to the horizon choice and that N = 35 days minimizes the realized cost. Furthermore, both the closed-loop solution and an open-loop turnpike analysis across diverse initial conditions reveal that the celebrated "Hammer and Dance" mitigation strategy emerges naturally as the mathematical optimum: the optimal trajectories anchor to a unique suppression turnpike (maximum lockdown) to drive hospitalizations toward the disease-free equilibrium before progressively reopening the economy. Through a turnpike-based argument we establish practical asymptotic stability of the optimal operating point, providing a mathematically grounded decision-support tool for pandemic policy.

We present an algorithm for solving nonlinear least-squares problems subject to a mix of nonlinear and linear constraints. The nonlinear constraints are handled by reformulating the objective as the augmented Lagrangian function while linear constraints are handled directly. Each iteration consists of approximately solving a linearly constrained problem by means of a gradient projection technique. Our approach also involves a structured approximation of the augmented Lagrangian Hessian. We show global convergence of the method and assess the performance through numerical experiments.

Audience: Tech companies such as NExT leadership; public transport operators; autonomous transit developers; charging partners; public authorities, and industrial partners considering pilots. Content: The strongest aspect of Modular Autonomous Vehicles (MAVs) is not only that vehicles can physically couple. A MAV is composed of Modular Autonomous Units (MAUs), and modularity creates operating advantages across line-level operations, intermodal services, network-level operations, stochastic real-time control, and electrified vehicle operations and charging infrastructure. Research basis and disclaimer: The brief summarizes scientific findings of four studies conducted by Dr. Xia and Dr. Sharif Azadeh on modular transit operations, including a charging infrastructure study using technical data associated with NExT MAUs.

In this paper, we investigate the weak observability of second-order infinite-dimensional evolution systems generated by skew-adjoint operators of the form $iA_0$, where $A_0$ is a self-adjoint elliptic operator. We first establish a spectral characterization of weak observability by introducing the notion of spectral coercivity for the observation operator and proving its equivalence to a suitable resolvent estimate. Our main result reveals a direct link between resolvent estimates for the elliptic operator $A_0$ and the weak observability of the associated evolution generator $A$. More precisely, we prove that a resolvent inequality for $A_0$ implies a Hautus-type spectral observability estimate for $A$, which guarantees the weak observability of the system. This provides a unified spectral framework for weak observability based on the coercivity properties of the observation operator. As an application, we establish explicit weak observability estimates for the wave equation on a rectangular domain under several geometric configurations of the observation region. The analysis combines frequency-domain methods, resolvent estimates, and Fourier analysis, yielding new insights into the interplay between resolvent inequalities, spectral coercivity, and weak observability in infinite-dimensional systems.

We determine the rate of convergence of the value functions of the $N$-particle stochastic optimal control problem to the value function of the corresponding mean field control problem, for mean field costs that are merely Lipschitz continuous in the 1-Wasserstein distance, a class that covers problems whose mean field optimizers are neither unique nor stable. For $d\geq2$, the optimal rate is the empirical-measure rate ($N^{-1/d}$ for $d\geq3$, $N^{-1/2}\sqrt{\log N}$ for $d=2$): this proves the rate conjectured by Daudin, Delarue and Jackson, and removes the semiconcavity hypothesis made there. In dimension one, we discover that the empirical-measure benchmark is not optimal: cooperating particles beat it, and the optimal polynomial exponent is $4/7$, strictly between the accuracy of independent samples and that of quantization by freely placed points. The proofs are control-theoretic: from each realization of an $N$-particle control we build a pathwise Fokker-Planck flow (a "shadow flow") which, repeatedly recoupled to the particles by optimal transport, shadows the empirical measure at the optimal rate. The one-dimensional rate requires additional constructions: we correct the shadow flow with a filter built on the future of the discarded noise, draw the cooperating particles from a Gibbs law, and prove the optimality of the exponent by a Schrodinger ground-state estimate. The empirical-measure rate also holds under additive common noise, uniformly in its intensity.

In this paper, we study last-iterate convergence of stochastic first-order methods for constrained smooth convex–concave minimax optimization under the standard bounded-variance stochastic oracle. A fundamental challenge is that the last iterates of vanilla stochastic extragradient (S-EG) and stochastic optimistic gradient descent–ascent (S-OGDA) may fail to converge in the presence of stochastic gradient noise, even for simple bilinear problems. To overcome this difficulty, we introduce a simple perturbation framework that regularizes the original convex–concave problem into a strongly convex–strongly concave one. Applying S-EG and S-OGDA to the perturbed problem yields two simple single-loop methods, referred to as perturbed S-EG (PS-EG) and perturbed S-OGDA (PS-OGDA). We establish last-iterate convergence by first deriving convergence in terms of the squared distance to the saddle point of the perturbed problem and then translating this estimate into guarantees for the restricted primal–dual gap. Based on this framework, we establish two types of convergence guarantees. When the optimization horizon is known a priori, both PS-EG and PS-OGDA achieve an $\mathcal{O}(T^{-1/4})$ last-iterate convergence rate for the restricted primal–dual gap, which coincides with the standard primal–dual gap on compact feasible domains. When the optimization horizon is unknown, we develop an anytime variant based on diminishing perturbations and diminishing stepsizes. For general closed convex feasible sets, both PS-EG and PS-OGDA achieve an $\mathcal{O}(T^{-1/5})$ last-iterate convergence rate for the restricted primal–dual gap. Furthermore, in the unconstrained setting, PS-EG admits a sharper $\mathcal{O}(T^{-1/4})$ anytime convergence rate in terms of the gradient norm.

This paper develops a model-free reinforcement learning framework for continuous–time extended mean field control problems, where both the dynamics and reward may depend on the joint distribution of states and controls. We adopt deterministic feedback policies, under which the state–action distribution is induced directly as a push–forward of the state law. This avoids optimization over stochastic kernels and bypasses key limitations of existing approaches in extended mean field settings. We first establish a model–free sensitivity formula for parameterized McKean–Vlasov dynamics and use it to derive a deterministic policy gradient formula expressed through an advantage–rate function on the Wasserstein space. We then refine this formula by introducing local value and advantage–rate representations that depend on the state, action, and joint state–action distribution, yielding a policy gradient that includes both action derivatives and measure–derivative terms with respect to the control distribution. These characterizations lead to a martingale–based learning principle and motivate a continuous–time deep deterministic policy gradient algorithm combining particle approximations, measure–dependent neural networks, temporal–difference learning, and exploration in either action or parameter space. Numerical experiments on stochastic Cucker–Smale consensus control and optimal liquidation with trade crowding demonstrate the efficiency, stability, and robustness of the proposed method, including problems with explicit dependence on the control distribution.

Autonomous-vehicle motion planners must resolve conflicts among safety, regulation, comfort, and efficiency in real time while exposing those decisions for audit. We present W-SQP, a weighted tiered-slack nonlinear model predictive controller (NMPC) that compiles nine driving-rule families into a four-tier shared-slack nonlinear program solved online with CasADi and IPOPT; the name denotes the weighted quadratic slack penalty, not a sequential-quadratic-programming solver. Strongly separated tier penalties bias residual violations toward lower-priority rules while leaving actuation bounds hard. The controller replans from its executed state at $10$\,Hz and records per-rule residuals on every cycle. A $90$\,ms solver-time limit returns an anytime iterate that is projected through the vehicle dynamics before execution; median and maximum observed wall-clock solve times were $28$ and $104$\,ms. We evaluate W-SQP in closed loop on 150 Waymo Open Motion Dataset scenarios in Waymax against reactive and proposal-and-select baselines, and introduce a log-independent protocol that separates safety and regulatory compliance from resemblance to the recorded human trajectory. Under this protocol, W-SQP shows no systematic group-level deficit relative to expert replay on the log-independent safety and regulatory rules, with several localized regressions in the hardest, highest-divergence scenarios. The results characterize W-SQP as an auditable, priority-biased, anytime-capable NMPC prototype rather than a hard-real-time or formally safe controller.

Standard learning rate schedules such as cosine annealing are tied to a fixed training horizon, limiting their ability to accommodate post hoc horizon extension. Warmup-stable-decay (WSD) partially addresses this issue by maintaining a long constant-rate phase before a short linear cooldown, allowing training to resume from a pre-decay checkpoint. However, its peak learning rate is still tuned based on the original training horizon and can become suboptimal when training is extended. Motivated by stochastic convex optimization, we propose WSqD (Warmup with Square-root base and linear Decay), a learning rate schedule that replaces WSD's constant stable phase with a shifted inverse-square-root base while retaining the final linear cooldown. In the stochastic convex setting, WSqD provably attains the minimax-optimal $O(1/\sqrt{T})$ last-iterate convergence rate. Importantly, its base learning rate schedule is horizon-independent, and the training horizon is needed only to determine when to begin the final cooldown. Empirically, on language-model pretraining using the SlimPajama corpus, WSqD matches or outperforms carefully tuned WSD and other baselines across multiple training horizons while reusing a single peak learning rate.

We study stochastic simple bilevel optimization with smooth, possibly nonconvex upper- and lower-level objectives accessed only through stochastic gradient oracles. A key challenge is that the dual multiplier induced by the lower-level constraint may become unbounded near lower-level stationary points, invalidating bounded-dual analyses and destabilizing stochastic gradient estimates. To address this, we propose Stochastic Dynamic Barrier Perturbed Gradient (SDBPG), a single-loop method that adaptively perturbs the dual formulation to regularize this degeneracy. The perturbation stabilizes the multiplier and yields controlled bias and variance even near the lower-level stationarity region. Under a rare-visit assumption governed by a parameter $\delta \in (0, \tfrac{1}{2}]$, SDBPG finds an $(\epsilon, \epsilon)$-stationary point in $\mathcal{O}(\epsilon^{-1/\delta})$ iterations, with sample gradient complexities $\mathcal{O}(\epsilon^{-2/\delta})$ and $\mathcal{O}(\epsilon^{-3/\delta})$ for the upper- and lower-level objectives, where larger $\delta$ corresponds to rarer visits to the bad region describing the negative alignment between the two objectives when the lower-level gradient is small. We further develop PR-SDBPG, a penalty-regularized variant that eliminates the rare-visit assumption, and VR-PR-SDBPG, which improves the resulting sample complexities entirely through variance reduction. To our knowledge, these are the first explicit $(\epsilon_f,\epsilon_g)$-stationarity guarantees for stochastic nonconvex-nonconvex simple bilevel optimization.

Certifying the safety of a control action in real time, from streaming partial observations of a nonlinear, interconnected system under non-stationary disturbances, is a problem no existing data-driven framework solves. Batch methods such as data-enabled predictive control require a pre-collected dataset and offer no stability certificate for nonlinear dynamics; informativity-based approaches characterize data sufficiency offline and non-recursively; neither exploits the known graph topology of networked systems as a structural prior. This paper addresses both limitations. First, we develop a streaming contraction certificate beta_cert(t) = beta_hat(t) - rho(t), where beta_hat(t) is estimated recursively via integral regression on a sliding window of partial input-output observations, and rho(t) is a data-dependent uncertainty radius mapping estimation error to a conservative bound on the true closed-loop contraction rate. The certificate issues a provably safe deployment signal the moment beta_cert(t) crosses and sustains above zero. Second, we introduce a topology-aware estimator enforcing known graph adjacency as exact zero constraints on the Jacobian, reducing the effective parameter count per row from O(N) to O(d_max) for maximum node degree d_max. On a five-node nonlinear benchmark under heavy-tailed Laplace disturbances with two observed nodes, the streaming certificate achieves certified deployment at t*=2.6s from 130 samples, 17 seconds earlier than an offline batch baseline, with 16x lower accumulated error during the unprotected window. The topology-aware estimator cuts certification time by 59% (1.62s vs 3.98s) and accumulated disturbance cost by 58%, with the advantage persisting across all window sizes below 40 samples. The framework is domain-agnostic and applies to any large-scale nonlinear networked system under streaming data and partial observations.

Growing interest in space interferometry for detecting bio-signatures of exoplanets has led to the development of several low-cost mission concepts involving Earth-orbiting formation flying techniques to perform nulling interferometry of incoming light from exoplanetary systems. Pursuing these developments, this work proposes a multi-objective dynamic programming scheme to optimize the balance between fuel expenditure and scientific outcomes of low-cost formation flying space interferometry missions by leveraging several insights from mission design. This scheme accounts for nominal concept of operations constraints, fuel expenditure, and observability conditions to produce a globally optimal Pareto front policy for exoplanetary system observation, and represents one of the first applications of multi-objective optimization to distributed space telescope astronomy.

We study the population gradient flow of an infinitely wide two-layer neural network learning a misspecified single-index model in high dimension. The two layers are optimized jointly, with a perturbative parameter tuning the relative training speed between the first and second layer. This setting was considered by Berthier, Montanari and Zhou in \cite{berthier2024learning}, who conjectured a hierarchical learning scenario with explicit timescales as the second layer is trained faster than the first. In this paper, we prove that the constant and linear components of the hidden link function are indeed recovered within the predicted timescales, at sharp explicit thresholds. We then analyze the onset of learning of the quadratic component and show that the components learned at earlier stages continue to influence the dynamics in an essential way. Our proof is based on quantitative approximation results for singularly perturbed flows evolving near a manifold defined by integral constraints. At a phenomenological level, we also show that the empirical measure of the weights displays singular behaviour when reaching the quadratic component of the hidden link, with a small fraction of neurons growing significantly while the remaining ones rearrange to preserve the components already learned.

In this paper, we establish the theoretical foundations for the generic optimization problem in a Hilbert space whose feasible set is an affine-transversal Hilbert submanifold given by the intersection of a nonlinear manifold and an affine subspace. We develop the geometric and analytical toolkit, such as the tangent-space characterization, projection operators, and implicit retraction operators, to essentially ensure the feasibility of iterates for algorithmic design. We also derive weak-form expressions for the derivatives of lifted objective functionals. In particular, we introduce a projection-induced Riemannian metric whose induced norm is uniformly equivalent to the ambient norm, under which the projection-induced gradient becomes an exact Riemannian gradient with an explicit formula. This construction replaces the implicit tangent-space Riesz representation underlying classical Riemannian optimization with directly computable operator evaluations, yielding a practical variable-metric framework for algorithmic design while preserving the geometric structure required for convergence analysis. With these theoretical foundations, it becomes possible to apply standard techniques in Euclidean spaces to design algorithms with strictly feasible iterates for the optimization problem on an affine-transversal Hilbert submanifold. We also propose the theoretical frameworks for algorithmic design on affine-transversal Hilbert submanifolds by showcasing the Riemannian line-search and trust-region algorithms with rigorous convergence analysis.

This paper provides two results that are useful in the study of the existence and the stability properties of almost periodic solutions for a given dynamical system. The obtained results are generalizations of recent results for periodic systems and are applied to the global entrainment problem in nonlinear time-invariant control systems. It is shown that local exponential stability for the unforced system and input-to-state stability with respect to small inputs can guarantee global entrainment to small almost periodic inputs. In this way, global entrainment is shown in Lotka-Volterra systems with a Volterra-Lyapunov stable interaction matrix. All results can be extended to the uniformly recurrent case.

Variance reduction is indispensable in Byzantine-resilient decentralized stochastic optimization over multi-agent systems (MASs), as it effectively mitigates gradient noise to facilitate the resilient aggregation process. However, most existing Byzantine-resilient decentralized variance-reduced (VR) stochastic gradient algorithms rely on random data sampling, which proves inefficient in data-scarce yet high-dimensional tasks, for instance, image deblurring. This paper aims to pursue an alternative approach that achieves variance reduction via gradient sketching. We first formulate a class of structural risk minimization (SRM) problems over time-varying networks, where the local objectives are not necessarily decomposable and their gradients may be inaccessible. To solve the SRM problems in a decentralized manner, we incorporate a localized gradient-sketching technique into a preconditioned stochastic proximal gradient framework, proposing a novel gossip-based decentralized algorithm with variance reduction, dubbed Gossip-SEGA. To further enhance its resilience, we substitute the weighted-average aggregation in Gossip-SEGA with a norm-penalized approximation, thereby developing a Byzantine-resilient decentralized algorithm, namely RED-SEGA. Theoretically, we derive sufficient conditions for both consensus among reliable agents and (sub)linear convergence rate of RED-SEGA over time-varying networks in the presence of Byzantine agents with unknown identities. This is challenging, as the convergence and consensus errors must be bounded within a transformed variable-metric geometry, where the inherent complexities are further compounded by adversarial Byzantine agents and time-varying networks. The effectiveness and resilience of RED-SEGA are validated by solving a constrained least-squares problem and an image deblurring task under various Byzantine attacks.

Probing and dual fixing are two powerful presolve techniques in mixed integer programming (MIP) solvers. Probing tentatively sets some binary variables to 0 or 1, applies linear constraint based domain propagation techniques to derive better variable bounds, and extracts useful information such as stronger variable implications and better global variable bounds. Dual fixing attempts to fix variables to lower or upper bounds while ensuring that at least one optimal solution is retained, as long as the problem was feasible. In this paper, we investigate how to combine the two approaches in MIP solvers to achieve a better performance. In particular, we first embed dual fixing into the probing framework, deriving more useful variables' implications for enhancing the capability of probing. Then, we develop an improved dual fixing technique where more variable fixings can be applied, and use the probing framework to detect the reductions. Computational results on the MIPLIB 2017 benchmark instances demonstrate the potential of the two proposed techniques in combining probing and dual fixing on the open-source MIP solver HiGHS.

We analyze a variant of the cubic-regularized Newton method for nonconvex optimization. This variant is parameter-free in that it requires no prior knowledge of problem-dependent parameters. Under the generalized smoothness condition $\|\nabla^3 f(x)\| \leq L_0 + L_1 \|\nabla f(x)\|$, we derive an oracle complexity bound for finding an $(\varepsilon, \delta)$-second-order stationary point. This assumption is weaker than the generalized smoothness conditions used in existing analyses of second-order methods, while the complexity bound improves upon existing guarantees for parameter-free second-order methods. In particular, when $L_1 = 0$, the bound matches the optimal dependence on $L_0$ as well as on $\varepsilon$, $\delta$, and the initial function value gap, up to additive logarithmic terms. To establish this bound, we derive Taylor-type inequalities and prove their equivalence to the generalized smoothness condition.

We develop a new framework for the analysis of co-radiant sets based on a new notion of regularity. This framework extends the classical theory beyond co-radiant sets admitting a norm-base and provides a unified setting for separation and optimization. We introduce two new tools: inner approximating sequences and the radial profile function. The former provides a sequential approach to regularity and is the key ingredient in the proof of a general separation theorem for regular co-radiant sets, extending the existing separation theory. The latter is defined for arbitrary co-radiant sets and yields new characterizations of the existence of norm-bases, together with sequence-based characterizations of regularity under a solidness assumption on the associated cone. It also gives rise to a scalar function leading to scalarization results for approximate efficiency without additional structural assumptions on the underlying co-radiant set. Finally, we provide a variational interpretation of the separation residual by establishing radial-depth and metric error bounds. These yield sufficient conditions for approximate radial feasibility in equilibrium problems and variational inequalities, and an exact penalization principle on the zero level set of the residual.

In this paper, we propose a mathematical model for class scheduling using binary linear programming with the aims to automate the tedious task and to minimize the leftover seats distributed over all the classrooms. A case study at the Department of Mathematics, King Mongkut's University of Technology Thonburi (KMUTT), Thailand, is also presented. The result from our model is shown to have improved significantly over the current manual scheduling method.

Training data constrains optimizer geometry through the covectors visible to a declared information channel. We study how such partial information determines a full positive cometric relative to a reference and which degrees of freedom remain unidentified. Our central result resolves full-column-rank positive-definite compression under affine-invariant Riemannian geometry. The compression map is a split-Hadamard metric submetry and admits an explicit unique completion that is the affine-invariant nearest full geometry realizing a visible target and yields exact full-to-visible variational reduction. When the channel moves, the completions form a gauge-invariant rank stratification of the positive-definite cone. Its closed-form pullback pair metric separates visible-metric motion from subspace rotation through a reference-mismatch weight, yields an explicit positive-semidefinite multi-direction Gram matrix, and exposes the precise singularity of reference-valued modes. The mechanism is explained by a metric theorem equating ball submetry, attained fiber distance, and lossless reduction of every monotone radial visible decision problem. A smooth split-Hadamard theorem supplies coherent information sheets, proximal commutation, and solution-wise gradient-flow lifting. The positive-definite realization also gives closed-form prior-data shrinkage. Diagonal and block optimizer families reduce to relative-interior conic image tests with valid facial certificates, while deterministic and finite-sample bounds quantify recovery of the visible geometry and its subspace. Together these results characterize exact reduction, reference-dependent completion, and structured expressivity for the stated finite-dimensional affine-invariant model.

This geometric study evaluates a compact seven-petal, 6:5-resonant tulip-shaped orbit constellation for lunar south-pole positioning, navigation, and timing (PNT) and direct-to-Earth relay services. The tulip-shaped orbits are compared against elliptical lunar frozen orbit (ELFO) constellations over the NASA LunaNet Service Volume 2 (SV2), covering lunar latitudes south of -75 degrees. We compare a six-satellite tulip baseline with a minimum-cost five-satellite variant; both use the same shared three-body orbit and differ only in satellite count and along-track phasing. Performance is scored against three Initial Operating Capability C (IOC-C) metrics: line-of-sight (LOS) link availability, Lunar Augmented Navigation System (LANS) geometric dilution of precision (GDOP) below 6, and daily extravehicular activity (EVA) usable-PNT windows. Both tulip constellations satisfy all three IOC-C metrics across SV2. The six-satellite configuration meets requirements with wide margin: 75% worst-point daily GDOP availability and 18 hours of daily EVA support. The five-satellite variant also passes, but with thinner margin: 44% availability and 10 hours of EVA support. Unlike the ELFO configurations, each spacecraft in the tulip-shaped constellation maintains continuous Earth line of sight, providing persistent geometric opportunity for direct single-hop Earth relay. Because all spacecraft share a single three-body orbit, initial phasing and post-failure reconstitution reduce to along-track drift maneuvers between neighboring orbits, with screening-level estimates indicating low maneuver cost.