Papers

We show that there is a Hadamard space in which the closed convex hull of a set containing merely three points is noncompact, by providing an AI-generated construction. The seemingly innocuous problem of finding a compact or finite set with this property, or showing that none exists, was recorded by Gromov and periodically revisited by researchers in metric geometry, functional analysis, fixed-point theory, and geometric group theory. Nevertheless, it remained unanswered for over three decades. The construction itself is based on building an inductive sequence of CAT(1) metric graphs and taking the Euclidean cone over the limiting space. We discuss the history of this problem, the main ideas and details of the construction, and its relationship with existing techniques.

We study the Tsallis-to-Kullback–Leibler (KL) limit for entropy-regularized optimal transport with nonnegative bounded continuous costs. Fixing the regularization parameter $\varepsilon > 0$, we first derive an exact variational reformulation of Tsallis-regularized optimal transport in terms of the Tsallis information projection onto the set of couplings. The formula isolates an explicit correction term and thereby explains why, unlike in the KL case, the regularized transport problem and the corresponding information projection problem do not coincide exactly. We also establish existence and uniqueness for the Tsallis information projection. We then prove, with respect to the narrow topology, the $\Gamma$-convergence of the Tsallis-regularized functionals to the KL-regularized functional as $q\downarrow1$, together with narrow convergence of their unique minimizers. Finally, we obtain explicit error estimates of order $O(q-1)$ for both the regularized optimal transport values and the associated information projection values. These results quantify the passage from Tsallis regularization to the classical KL setting and clarify the relation between entropic regularization and information projection for $1 < q \leq 2$.

In this paper, we study a Tikhonov-regularized mixed-order primal–dual dynamical system with implicit Hessian damping for linearly constrained convex optimization problems in finite-dimensional Euclidean spaces, where the primal equation is second order and incorporates the viscous damping term \(\delta\sqrt{\varepsilon(t)}\,\dot x(t)\), whereas the multiplier equation remains first order. By constructing a new class of energy functions, for a general Tikhonov regularization coefficient \(\varepsilon(t)\), we prove the strong convergence of the primal trajectory and derive fast convergence rates under the same parameter assumptions, without imposing any eventual inside/outside-ball condition. More precisely, the primal trajectory converges to the minimum-norm solution, and the multiplier converges to a compatible KKT multiplier, while the convergence rates of the Lagrangian gap, feasibility violation, and objective residual are \(o(\varepsilon(t))\), and the convergence rate of the velocity norm is \(o(\sqrt{\varepsilon(t)})\). For the critical case \(\varepsilon(t)=c/t^2\), in which the damping coefficient \(\delta\sqrt{\varepsilon(t)}\) reduces to \(\delta\sqrt{c}/t\), we establish the sharper convergence rates \(o(t^{-2})\) for the Lagrangian gap, feasibility violation, and objective residual, together with \(o(t^{-1})\) for the velocity norm, which improve the corresponding \(O(t^{-2})\) and \(O(t^{-1})\) decay estimates obtained in the related literature. Most importantly, when the proposed dynamical system is specialized to the finite-dimensional unconstrained setting, our analysis answers the open question on strong convergence in this critical regime posed by Attouch and László [Math. Methods Oper. Res., 99 (2024), pp.~307–347].

Projected reflected gradient (PRG) method proposed by Malitsky is efficient for solving monotone variational inequality (MVI), while the existing upper bound of step size is not tight due to the inequality scaling in the theoretical analysis. In this paper, we construct an averaged variant of PRG method for more general MVI and present a novel Lyapunov function to establish convergent theory. This averaged PRG method provides an improvement of the golden ratio algorithm [Y, Malitsky, Math. Program., 184, 383-410, 2020], and the involved step size is compatible with that for the classical methods, such as Popov's extragradient and forward-reflected-backward methods. Moreover, a fully adaptive strategy without linesearch is presented to adjust step sizes, which generates closed-form and potentially much larger step sizes. Numerical experiments on the Nash-Cournot equilibrium, HpHard, and image reconstruction problems demonstrate that the proposed algorithm significantly outperforms existing state-of-the-art methods.

In this paper, we consider the problem of regulating and maintaining a target formation characterized by a set of bidirectional bearing constraints under disturbances. The agents in the formation are modeled by single integrators with bounded continuous disturbances of which the upper bound is unavailable for the control design. Due to the time-varying disturbances, the target formation is time-varying. We propose adaptive sliding mode control laws to uniformly globally asymptotically stabilizes the moving target formation and reject the matched disturbances. In addition, to alleviate chattering phenomena from sliding mode control, smooth adaptive control laws are then designed to guarantee uniform global boundedness of the desired formation. Finally, simulation results are given to support the analysis.

Networked systems face a tradeoff between the scope of the information behind a decision and its freshness: a broader view of the system supports better coordination, but assembling and communicating it takes time, so it arrives older. We study this tradeoff for a team of agents that repeatedly choose actions to minimize a shared quadratic cost driven by an environment that evolves on its own, unaffected by the agents' actions. An information architecture specifies what each agent observes, from where, and with what delay; we measure an architecture by the cost it loses relative to a decision made with complete, current information. We first compare fresh local observations with a complete but delayed global view. When every component of the environment decorrelates at a common exponential rate, this comparison reduces to a closed-form threshold on the ratio of delay to coherence time. We then study intermediate architectures in which each agent acts on a time-aligned, and therefore older, snapshot of a wider neighborhood. The optimal neighborhood radius occurs where the marginal value of added scope equals the marginal cost of lost freshness. In a canonical spatial model, this radius is set by the spatial-correlation, decision-relevance, and temporal-propagation lengths. Throughout, architecture performance is governed by the predictability of the optimal decision rather than of the raw state.

We consider a Type-II variational principle driven by geometric rough path with split boundary conditions naturally suited to adjoint systems. From this rough variational principle we derive rough Hamilton's equations, establish their pathwise conservation laws and associated Hamilton–Jacobi equation. We then specialise the framework to rough adjoint systems, obtaining pathwise conservation and quasi-conservation laws that underpin adjoint sensitivity analysis with respect to initial conditions and parameters. On the discrete side, we construct a rough Galerkin discretisation of the rough Type-II variational principle and show that it generates a symplectic flow with discrete analogues of the continuous conservation laws. We establish it's equivalence to a class of Rough Symplectic Partitioned Runge–Kutta (RSPRK) methods and analyse its convergence and naturality properties. Lastly, we perform numerical experiments to validate the predicted convergence rates and demonstrate that RSPRK methods preserve the adjoint conservation laws to machine precision, yielding more accurate and stable gradients in optimisation problems than non-symplectic alternatives.

This thesis investigates Euclidean-space and $\ell_p$-norm representations of different forms of data, with a focus on efficient algorithms for computing these representations in settings where the amount of data is very large. The applications of such representations are also discussed: they may be used to summarize the data in a more compact form for downstream tasks while preserving its most salient properties; and further, they may also be used to extract insights about the data that may not be apparent in its original form. This thesis presents novel algorithms for computing Euclidean representations in the case where the data comes with linear structure, as well as in the case where the data comes only with metric, or distance structure. In addition, we give algorithms for computing $\ell_p$-norm representations in linear structured cases. The analyses of these algorithms use tools from geometry, probability, and optimization, and these tools are used to illuminate other algorithms for similar problems.

Fixed shipment costs encourage large orders, while inventory and customer backlog costs favor frequent replenishment. In a serial system, the stages must also coordinate their lot sizes and material availability. We study the classical echelon stock $(R,nQ)$ policy: the downstream stage requests a fixed lot, the upstream stage orders an integer multiple of that lot, and transfers contain complete lots. For unit Poisson demand, deterministic lead times, setup and holding costs at both stages, and full backlogging, we prove that the optimized cost in this class is at most $5/3\approx1.667$ of the optimal admissible system cost, uniformly over all nonnegative cost rates, demand rates, and lead times. The guarantee holds even when the selected policy pays a setup charge for every lot while the optimal benchmark pays only once per aggregate shipment. To obtain the guarantee, we use a common inventory calculation that allows independently chosen replenishment frequencies while respecting upstream material availability. We compare its optimized value, an upper bound on the best integer-ratio policy cost, with a novel lower bound on the optimal system cost that combines allocated single-stage costs with the minimum inventory cost when setup costs are zero. We also show that no uniform factor below $(1+\sqrt{3})/2\approx1.366$ is possible for the classical class. Existing numerical experiments place best-found classical policies within $22.7\%$ of an evaluated lower bound across three broad grids. The results quantify both the reliability and the limits of a simple operating rule and explain how shipment consolidation, replenishment timing, and flexibility affect its value.

We study several $k$-fold generalizations of the Lovász theta function associated with the maximum $k$-colorable induced subgraph problem. The first is the Narasimhan–Manber parameter $\vartheta_k$. We prove that, for graphs whose adjacency matrix belongs to a homogeneous partially coherent algebra, this parameter is recovered by the theta number of the Cartesian product with the complete graph on $k$ vertices. This class includes distance-regular and $1$-walk-regular graphs, and thus our result generalizes a theorem by Sinjorgo and Sotirov (2022) for graphs that are vertex- and edge-transitive. We introduce a new parameter $\varphi_k$ obtained from orthonormal representations of graphs and show the inequality $\varphi_k \leq \vartheta_k$. For both parameters, we study the smallest $k$ for which the parameter is equal to the number of vertices; these saturation parameters yield lower bounds on the chromatic number. We determine which vertex-weighted versions of these parameters are gauges, and discuss a natural definition for the $k$-fold theta body of a graph. We conclude with open questions comparing $\vartheta_k$, $\varphi_k$, $\vartheta(G\square K_k)$, and related convexifications.

Conditional dimension reduction (cDR) learns low-dimensional latent coordinates while accounting for observed covariates that represent known sources of variation in the data. Conditional Multidimensional Scaling (cMDS) is a cDR technique that works directly with dissimilarity data. Its standard squared-stress formulation, however, is sensitive to contaminated dissimilarity, since outliers can dominate the objective and distort the learned configuration. We proposed Robust Conditional Multidimensional Scaling (rcMDS) by replacing the squared-stress criterion with a Fair M-estimation objective. We developed a reweighted conditional SMACOF algorithm to optimize this objective. The proposed algorithm admits computationally tractable updates, and its stabilized objective values decrease monotonically and converge to a finite limit. Experiments on synthetic and real data show that the pro

Optimal transport provides a common language for allocation, equilibrium, computation, and inference. Its primal problem assigns mass or agents, while its dual variables admit economic interpretations as utilities, prices, and scarcity rents. This review explains why this combination has proved unusually effective in economics. We first present the core results, including Kantorovich duality, integrality, cyclical monotonicity, the canonical distance and quadratic costs, and entropic regularization. We then trace the field's development from planning and operations research to modern analysis, statistics, and computation. The economic literature is organized around two roles for transport: as a model of matching, trade, hedonic equilibrium, and aggregate assignment; and as a tool for coupling distributions, measuring discrepancies, constructing multivariate ranks, solving inverse problems, and certifying economic conclusions. We conclude by examining extensions beyond transferable utility, one-to-one matching, static allocation, and unconstrained transport, together with open questions involving learning and identification across multiple markets.

In the 2004 collection Unsolved Problems in Mathematical Systems and Control Theory, Erik Verriest posed the problem of characterizing Riccati stability “without invoking additional matrices.” We give such a characterization through a scalar invariant of the resolvent family. The invariant is defined by covariance balances using at most $n^2+1$ frequency–direction pairs. A finite-dimensional separation argument shows that this covariance radius equals the optimal common ellipsoidal norm of the resolvent family. Combined with the strict bounded real lemma, this yields a necessary and sufficient condition for Riccati stability involving only the Hurwitz property of the first matrix and a single scalar inequality. The invariant reduces to the ordinary spectral radius for one matrix, lies between the pointwise spectral-radius and unscaled small-gain levels of the resolvent family, and coincides with the operator-space spectral radius of Shalit and Shamovich for the natural resolvent function space. The main results have been formally verified in the Lean proof assistant.

This paper presents a regularized cyclic coordinate minimization method for solving nonconvex composite optimization problems having the objective function formed as the sum of two terms, one is twice continuously differentiable and the second term is simple and separable. We analyze the convergence behaviour of our coordinate minimization method, in particular we provide convergence rates to a first-order optimality criterion and objective residual depending on the assumptions on the problem. Then, we show that our algorithmic framework can be efficiently applied for solving quantized matrix factorization problems that arise in e.g., lossy image compression. More specifically, on KODAK and CLIC 2024 datasets, our method notably outperforms JPEG at low bit rates, achieving savings in bits per pixel without excessive degradation and remains comparable at higher bit rates. Moreover, we evaluate both the original and quantized compressed images in a classification task using 3 well-known convolutional networks AlexNet, ResNet50 and MobileNetV2 based on floating point and integer arithmetic representations. Remarkably, when comparing the two numerical representations of the convolutional networks, there is no substantially accuracy degradation on ImageNet dataset. The numerical results on image compression and classification using real data show the flexibility and efficiency of our algorithm compared to well-established methods from the literature.

This paper studies the isolated calmness property of solution mappings arising from convex regularized linear inverse problems. We mainly establish a tangent-cone characterization of this property. For convex piecewise linear-quadratic regularizers, the isolated calmness coincides with the solution uniqueness, leading to simple verification procedures. In particular, for analysis sparsity regularization, our condition can be verified by linear programming.

We compare closed-loop and open-loop Nash equilibria in a finite-horizon stochastic linear-quadratic $N$-player game with decoupled state dynamics and interaction through the state costs. We introduce a block-diagonal reference game and a nearby perturbed game, and study each under both information structures. The closed-loop and open-loop problems lead to different Riccati systems, but for the reference game their induced equilibrium state and control processes coincide exactly. We then prove solvability and stability under weak perturbations of the state costs, yielding quantitative bounds between the corresponding equilibria. In particular, when the perturbation decreases sufficiently fast with the population size, the closed-loop and open-loop equilibria of the perturbed game become asymptotically equivalent. The comparison is carried out directly at the finite-player level, without requiring exchangeability or a mean-field limit.

We study policy optimization for discrete-time robust $\mathcal{H}_\infty$ control with static output-feedback, and present the first feasibility-preserving algorithm with a deterministic, non-asymptotic complexity guarantee. This problem naturally leads to a nonsmooth and nonconvex optimization over the set of stabilizing feedback gains. We first establish several structural properties of the $\mathcal{H}_\infty$ cost. In particular, we show that the cost is weakly convex on every convex subset of a sublevel set. For the state-feedback case, we further establish a weak Polyak–Łojasiewicz inequality, which ensures that every stationary point is globally optimal. Building on these properties, we develop a proximal bundle method for $\mathcal{H}_\infty$ policy optimization. The proposed method can be viewed as an implementable approximation of the proximal point method and uses only function value and subgradient information. We show that all iterates remain stabilizing and establish a deterministic non-asymptotic complexity bound of $\mathcal{O}(\max\{\eta^{-4},\epsilon^{-2}\})$ for finding an $(\eta,\epsilon)$-stationary point. Numerical experiments illustrate our theoretical results.

We study continuous-time full-order dynamic output-feedback $H_\infty$ policy search, a nonconvex and nonsmooth problem. Direct policy search is a central paradigm in reinforcement learning and continuous control, but rigorous guarantees remain scarce in robust output-feedback settings. The $H_\infty$ problem is a canonical benchmark because it captures disturbance attenuation and robustness while exposing the hard nonsmooth geometry of policy-space optimization. We prove that on the exact identity-gauge slice of the extended convex lift, $\varepsilon$-stationarity yields $O(\varepsilon)$-suboptimality on compact exact slices, which in turn yields convergence-rate guarantees for nonsmooth policy-search methods. This result addresses the finite-time optimality-gap question raised by Guo and Hu [2022] in the more general dynamic output-feedback $H_\infty$ policy-search setting. We further use the established value equivalence supplied by extended convex lifting to formulate a nonstrict-feasibility bisection method with one final strict-feasibility recovery step, yielding an explicit $\varepsilon$-optimal stabilizing controller. These results provide a quantitative and algorithmic strengthening of prior qualitative optimality theory for nonsmooth $H_\infty$ policy search.

This paper addresses a fundamental question in non-convex optimization: How should a stochastic optimizer allocate computation between global exploration and local exploitation? We study this question in continuous time, using Langevin dynamics for global exploration and deterministic gradient flow for local exploitation. The simplest algorithm in this class is the best-state Langevin–gradient method: several Langevin trajectories explore the objective landscape, the best state encountered is retained, and a single gradient trajectory then refines this state to high terminal accuracy. Our main objective is to characterize the computational work required to achieve a prescribed accuracy with prescribed confidence. We develop low-temperature approximations for this probably approximately correct (PAC) work–accuracy tradeoff, both for the best-state method and for related Langevin schemes. Global exploration is governed by energy barriers, through spectral-gap and Eyring–Kramers asymptotics. The resulting approximations expose the competing effects of temperature, computational budget, and the number of exploratory trajectories, and quantify their dependence on the confidence parameter. In contrast, terminal accuracy is largely separated from global exploration: once a suitable region of attraction has been reached, gradient flow requires only $O(\log(1/\varepsilon))$ additional work to reach accuracy at least $\varepsilon$. Numerical experiments on challenging objectives serve to compare algorithms and to test the quantitative predictions of the asymptotic theory. They identify regimes in which the low-temperature approximations provide useful guidance, as well as regimes exhibiting substantial temperature sensitivity and potentially significant high-dimensional limitations.

In this paper, we develop and analyze techniques for recovering a linear image $Bx$ of an unknown signal $x$ from indirect noisy observation $\omega=Ax+\xi$. It is a priori known that $x\in \cX$, a given convex compact set, and that $x$ is $s$-sparse—has at most $s$ nonvanishing entries. The proposed estimates belong to a large family of recovery routines by $\ell_1$-minimization. However, unlike the classical result describing performance of such estimates, we do not make any special (and hard to check) assumptions about the sensing matrix $A$ such as nullspace or Restricted Isometry condition and the like. As a consequence, parameters of the estimates and the upper bounds on their risks are not available in a closed analytic form, but are delivered instead by efficient computation as solutions to explicit convex optimization problems.

We study the problem of low-rank matrix recovery from linear measurements via the global nonconvex landscape of a low-rank factored formulation of the matrix LASSO (nuclear-norm–regularized least-squares). If the landscape is benign, that is, has no bad local optima, then practical and scalable algorithms can compute good statistical estimates. Previous state-of-the-art landscape guarantees have typically assumed that the linear measurement operator has the restricted isometry property, that is, the operator is approximately an isometry over all low-rank matrices. This is an unrealistic assumption for many applications; in particular, when the individual measurement matrices are themselves low-rank, we typically have poor upper isometry constants. To overcome this, we establish new guarantees of a benign landscape under a weaker isometry condition: rather than requiring upper isometry over all low-rank matrices, we only require it over the linear low-rank tangent space to the low-rank ground truth matrix. To illustrate the utility of this result, we apply it to the problem of matrix recovery from random rank-one linear measurements; via high-probability concentration bounds on the random measurement operator, we prove a novel landscape guarantee with statistically near-optimal sample complexity and recovery error.

This review develops a control-theoretic perspective on measurement-based feedback for continuously monitored open quantum systems, with the main analysis focused on finite-dimensional systems governed by diffusive stochastic master equations. We introduce the relevant state-space, invariant-subspace, and quantum non-demolition structures, interpret quantum filtering as nonlinear observer dynamics, and review open-loop asymptotics, filter stability, state-feedback stabilization, robustness, and reduced-order observer-based control. Particular emphasis is placed on a recurrence–contraction framework, in which Hamiltonian feedback removes non-target invariant obstructions while measurement-induced dynamics provide local exponential contraction. Although the detailed analysis is developed for finite-dimensional diffusive models, the underlying measurement–estimation–feedback architecture is relevant across a broad range of quantum platforms. We further discuss implementation challenges and open problems involving scalable estimation, sampling and delay, adaptation, hybrid and non-Markovian dynamics, practical stability, optimal control, and learning-based design. By organizing these developments around invariance, estimation, recurrence, and contraction, the review provides a tutorial bridge between measurement-based quantum feedback and nonlinear stochastic control.

This paper studies a dynamical system approach for solving indefinite quadratic programming problems subject to linear constraints. We investigate the convergence of the trajectory generated by the system to a Karush-Kuhn-Tucker point of the quadratic programs. In addition, we derive an estimate for the distance between the trajectory and a solution of the problem. We further prove that the objective value is decreasing along the trajectory. An illustrative example and a numerical test are presented to demonstrate the behavior and performance of the proposed method.

Stochastic min-max optimization has attracted increasing attention due to its applications in modern machine learning, while existing theoretical studies mainly rely on the bounded variance assumption for stochastic gradients. Under heavy-tailed noise, where stochastic gradients only possess a finite $p$-th moment for $p\in(1,2]$, gradient clipping or normalization is commonly believed to be necessary to guarantee convergence. In this work, we revisit stochastic min-max optimization under heavy-tailed noise and provide a comprehensive theoretical study of stochastic gradient descent ascent (SGDA). We first show that vanilla SGDA, without any modification to its update rule, can converge under heavy-tailed noise in both nonconvex-strongly-concave (NC-SC) and nonconvex-concave (NC-C) settings, establishing the first convergence guarantees for SGDA in these regimes. Beyond unregularized problems, we further investigate regularized stochastic min-max optimization, where directly incorporating gradient normalization into proximal updates is nontrivial due to the incompatibility between normalization and proximal structures. We overcome this difficulty by developing new clipping-free algorithms, i.e., Stoc-TRGDAM and Stoc-TRGDmax, and they both can achieve the optimal dependence on the target accuracy without using gradient clipping.

We construct an endpoint map with no regular values, disproving the sub-Riemannian Sard conjecture. Remarkably, in our construction the minimizing Sard property remains valid.

Mixed-integer programming (MIP) is a cornerstone in applied optimization, both in industry and academia. Recently, there has been increased attention to finding strong primal solutions quickly. This is reflected, for example, in the development of the NVIDIA cuOpt solver and, most recently, in the new MIPFEAS benchmark, which has a tight time limit of 600 seconds and evaluates solvers based on how quickly they find high-quality primal solutions. This article introduces a MIP portfolio parallelization scheme, focusing on efficiently exchanging information between its workers. We present two implementations of this scheme: one built directly into the open-source MIP solver SCIP, and an external one, which we call ReXi. ReXi is currently the fastest non-commercial solver in the MIPFEAS benchmark, followed by the SCIP-integrated implementation. Moreover, we present new versions of both implementations that considerably outperform their predecessors on the MIPFEAS benchmark.

Consider the stabilizable finite-dimensional linear control system $\dot x=Ax+Bu$. For a prescribed decay rate $\delta>0$, we define the optimal stabilization prefactor $$ \hat{C}(\delta) := \inf_{K\in\mathbb{R}^{m\times n}} \sup_{t\ge0}e^{\delta t}\|e^{(A+BK)t}\|. $$ We determine its asymptotic behavior as $\delta$ approaches the right endpoint of the achievable decay-rate range. If $(A,B)$ is controllable and $\mu$ is its largest controllability index, then $\hat{C}(\delta)\asymp\delta^{\mu-1}$ as $\delta\to+\infty$; whereas if $(A,B)$ is stabilizable but not controllable, then $\hat{C}(\delta)\asymp(\delta_*-\delta)^{-(q-1)}$ as $\delta\uparrow\delta_*$, where $\delta_*$ is the supremal achievable decay rate and $q$ is the largest size of a Jordan block of the uncontrollable part associated with the spectral boundary $\operatorname{Re}\lambda=-\delta_*$. Thus, in both cases, the endpoint asymptotic order of $\hat{C}(\delta)$ is determined by the corresponding structural invariant—$\mu$ in the controllable case and $q$ in the noncontrollable case—and conversely this order recovers that invariant.

This work provides a learning approach to solving finite-horizon Markov decision processes (MDPs) when the underlying model of a given finite MDP is unknown to the decision maker. We transform the adaptive multistage sampling (AMS) algorithm into a sampling-free algorithm, called "adaptive multistage rollout (AMR)," for estimating the optimal value at an initial state when only the state set and the action set are known. AMR emulates the backward induction as in AMS but in a reinforcement learning (RL) setting. At each iteration, AMR generates a non-stationary policy to be used for exploration and rolls out the policy in order to obtain a single trajectory of experiences and traces it backwards in a non-recursive way while doing relevant updates only at visited states and for actions taken at the visited states. We show that AMR is asymptotically optimal such that the sequence of the expected absolute errors approaches zero and its convergence rate depends on the number of visits to each reachable state at each stage from the initial state, essentially transforming the result of AMS into the RL setting.

The flatness of the under-actuated unmanned surface vessel with circular hull-shape (“the hovercraft") has long been known. We resolve the long-standing open problem: whether the generic under-actuated 6-degrees-of-freedom (dof) unmanned surface vessel (gUSV) admits a feedback linearization. Lie symmetries of the gUSV are used to construct a dynamic feedback linearization, explicit flat outputs and flat parametrization, for all hydrodynamic parameter values. In this generic case the dynamic compensator has order 6, in contrast to the hovercraft vessel whose dynamic compensator has order 4.

It is known that any convex cone in a general vector space is the positive cone of a preorder compatible with the vector space, and that any convex set is an affine slice of a convex cone. We therefore treat convex sets order-theoretically, as affine slices of positive cones. We find that faces are sets of strong minima, and we generalize functionals to oriented maps valued in arbitrary ordered vector spaces. Oriented maps expose every face, and an oriented affine map $f$ separates $C$ from $D$ when $f(D) \le 0 \le f(C)$ element-wise. This generalization of hyperplane separation recovers mathematical objects that are "local", including the generated face at a point and the normal cone. The kernels of separating oriented maps form a lattice of separating affine flats whose bottom element is the joint supporting subspace (JSS), recently introduced in finite dimensions for qualification-free convex analysis results. We define the JSS in general vector spaces for two convex sets $C, D$ as the affine slice of the cross-lineality $\mathrm{lin}(K-G)$, where $K,G$ are the homogenization cones of $C, D$. We show that in finite dimensions, bilateral facial reduction corresponds to a descent in the lattice of separating affine flats, which we formalize as lexicographic products of oriented maps. The lexicographic characterization of faces follows as a corollary. By making no assumption other than convexity, our approach yields general results: we show that the face of a difference of two convex sets is a difference of faces, dropping assumptions of compactness, finite dimensionality, and exposedness. We also show that any face of an intersection of two convex sets is an intersection of faces, resolving open problem 5.10 by Weis in "A note on faces of convex sets", dropping the assumption of non-empty intrinsic core.