Papers

Quadratic Sum-Of-Squares (QSOS) optimization problems appear in system identification and machine learning, but standard Schur-complement and second-order cone liftings enlarge conic dimensions and create computational bottlenecks for interior-point methods. This paper introduces a lifting-free regularization that preserves the original conic structure by adding a norm penalty to SOS variables, yielding closed-form primal updates and an unconstrained, concave dual with Lipschitz-continuous gradient. Accelerated first-order methods efficiently maximize this dual, and convergence analysis shows non-asymptotic recovery of the solution. Numerical experiments on constrained regression problems show the proposed method can be 40% faster than existing solvers such as SCS and handle larger problems than MOSEK, with memory scaling only in the number of equality constraints.

The paper addresses the problem of finding the global minimum of a polynomial on a coordinate parallelepiped or on certain other sets with a simple structure. To solve the problem, the Variable Elimination Algorithm for polynomial optimization problems described in the author's previous works is used. It should be noted that the proposed method can be of practical significance only for problems of low dimensionality and for polynomials of not very high degree. This is due to the fact that, when implementing the VEA, a large number of alternative problems may arise concerning the determination of real roots of polynomials, and these polynomials may have very high degrees. Although, in theory, the Variable Elimination Algorithm is applicable to any polynomial optimization problems, there are often practically relevant problems that can be successfully solved using the proposed method.

Air and missile defense, naval force protection, and critical-infrastructure security require rapid allocation of scarce effectors against multiple incoming threats when neutralization is uncertain and observed only after a firing round. We formulate this defensive-allocation problem as a Markov decision process in which a commander assigns a fixed amount of effector capacity each round to heterogeneous threats under three operational objectives: minimizing expected threat-clearance time, maximizing the probability of clearance by a deadline, and maximizing effective assignments before a deadline. Rather than expensive full dynamic optimization, we study simple time-oblivious policies suited to fast implementation. Fair allocation, which ignores threat difficulty and spreads fire evenly, is highly effective: it is optimal for all three objectives under homogeneous threats or low-capacity engagements. Moreover, when effector capacity scales at least linearly with the number of threats, its threat-clearance time trails that of the optimal policy by at most a constant number of rounds. We further develop threat-difficulty-aware greedy policies for each objective, including a constant-factor guarantee for effective assignment maximization. Numerically, greedy policies are near-optimal across heterogeneous instances, while fair allocation remains a principled choice when threat difficulties are unknown.

Forecast value depends not only on accuracy but also on the information structure available to the operating model. We study a diagnostic current-endpoint-only forecast for offshore-wind thermal firming: at hour t, the controller observes the current net-demand state z_t and one perfect future target z_{t+h}, but not the intermediate path or earlier endpoint messages. Unlike rolling path forecasts, these endpoint information sets are not nested in h. We embed the signal in a cyclostationary MDP using quantile Fourier regression states estimated from ISO New England load and offshore wind data, and solve annual state-action-frequency LPs for h = 1, . . . , 6. A one-hour endpoint forecast reduces annual firming cost by 8.07%, while a six-hour endpoint reduces it by 2.28%. The decreasing profile shows that a single farther endpoint is less actionable for a one-step ramping decision, without implying that longer rolling forecasts are less valuable.

We study an infinite horizon stochastic optimal control problem by means of the associated Hamilton-Jacobi-Bellman equation. The problem we are studying has the particularity of having a discount factor which can take both signs, depending on the value of the state.

In an abstract functional framework, we investigate the equivalence between two fundamental properties: weak observability and energy decay associated with certain classes of dissipative operators. This analysis is motivated by the study of evolution equations and their long-time behavior. Our main result establishes that under appropriate assumptions, these two notions are not only closely related but in fact equivalent. The proof relies on recent advances in the theory of weak observability, particularly those presented in \cite{AT1}, where new techniques have been developed to quantify the weak observability inequalities. These tools allow us to bridge the gap between qualitative decay properties and spectral estimates, offering a unified perspective on stability and control in abstract dynamical systems.

We develop a norm-cone conjugation framework, generated by translated norm-cones \[ x\mapsto r-\alpha\|x-x_0\|, \qquad \alpha\geq 0. \] We show that this family generates the same abstract-convex class as the family of Lipschitz continuous concave functions, and exploit its explicit metric structure to obtain a concrete support geometry, a Fenchel–Moreau-type biconjugation theorem, and an associated norm-cone subdifferential. The main optimization consequence is a perturbation-duality theory based on partial norm-cone conjugation. Although the partial conjugate depends on a slope and a centre, the centre can be fixed at the nominal perturbation in the complete biconjugate expression without changing its value. Consequently, the effective dual problem is a scalar program whose only dual variable is the real slope $\alpha\geq0$, even when the primal and perturbation spaces are vector-valued or infinite-dimensional. The resulting dual value is the norm-cone biconjugate of the value function at the nominal perturbation, so that strong duality is characterized by exact pointwise biconjugation. For nonconvex conic inequality problems, exact distance penalization is characterized by norm-cone supportability of the natural value function: the exact penalty parameters form its norm-cone subdifferential, and the least exact parameter is the minimal support slope. Global error bounds, equivalently global metric subregularity, provide a variational-analytic sufficient condition for exactness and hence for strong norm-cone duality. An infinite-dimensional example shows that the results remain applicable with a nonconvex objective and a nonsolid ordering cone.

This work introduces a nonlinear Hager-Zhang conjugate gradient method for solving set optimization problems. The objective function under consideration is defined by a finite collection of continuously differentiable functions. Notably, the proposed approach imposes restrictions neither on the existence of a finite generator of the ordering cone nor on any regularity condition at the optimal solution. As a result, the proposed method holds considerable significance for both set optimization and vector optimization problems, with the latter serving as a special case of the former. The study begins by discussing Wolfe line search conditions using Drummond-Svaiter scalarization function. Thereafter, we establish the existence of a step length satisfying the Wolfe line search conditions along a descent direction. The Hager-Zhang scalar conjugate parameter is introduced to derive the search direction for the proposed method. It is established that the direction generated by the proposed method is a descent direction. The well-definedness of the proposed method is given. Furthermore, we discuss some important results and a Zoutendijk-like condition to ensure global convergence. Subsequently, the global convergence of the proposed method is established in an asymptotic manner. Finally, numerical experiments on various test problems validate the practical performance and effectiveness of the proposed technique.

Column generation (CG) is central to many large-scale optimization algorithms, including branch-price-and-cut methods for vehicle routing problems, but unstable dual solutions can substantially slow its convergence. Existing deep dual-optimal inequalities can reduce this instability by restricting the dual space. Their construction, however, typically relies on problem-specific exchange arguments that are difficult to establish for routing problems with capacity limits, time windows, and other resource constraints. We introduce learned pairwise deep dual-optimal inequalities (L-PDDOIs), a learning framework that predicts pairwise orderings between dual variables and incorporates their primal counterparts directly into the master problem. To construct training labels, the framework samples optimal dual solutions and selects pairwise order relations that hold simultaneously on a sufficiently large common subset of the samples. A classifier then assigns a score to each candidate relation. Because conflicts and redundancies among the predicted relations can impair performance, graph-based postprocessing filters and compresses the candidate set before deployment. We further introduce a recovery procedure that selectively relaxes learned inequalities and provides a certificate when the baseline CG bound has been restored. On the main test sets for the capacitated vehicle routing problem and the vehicle routing problem with time windows, direct deployment of L-PDDOIs reduces the geometric mean root CG time by 89.7% and 93.9%, respectively, while incurring mean bound losses of only 1.3% and 0.5%. The recovery procedure retains corresponding time reductions of 54.8% and 83.1%, respectively, while guaranteeing no loss in the CG bound.

Derivative-free optimization (DFO) addresses unconstrained problems $\min_{\x\in\RR^n} f(\x)$ where $f$ is accessed only through a zeroth-order oracle. Model-based trust-region methods construct underdetermined quadratic interpolation models from $\mathcal{O}(n)$ points and solve a KKT system to determine model parameters, costing $\mathcal{O}(m^3)$ operations and limiting parallel scalability. It is shown that the KKT matrix for the minimum Frobenius norm updating model depends entirely on inner products of shifted coordinates. Reflecting the interpolation set across a single coordinate axis preserves these inner products and changes only one row and column of the KKT matrix, inducing a rank-at-most-two perturbation whose inverse update via the Sherman-Morrison-Woodbury formula costs $\mathcal{O}(n^2)$ when $m=\mathcal{O}(n)$. The reflection is an isometry in centered Euclidean trust regions and preserves the poisedness constant of the interpolation set; together with standard fully linear model-management assumptions this supports first-order global convergence. The mechanism is embedded in a master-worker parallel algorithm with a Truncated Conjugate Gradient subproblem solver. Numerical results on 530 benchmark problems compare performance against an established DFO solver.

We study the deterministic query complexity of minimizing a convex Lipschitz function over a $d$-dimensional Euclidean ball using only exact function values. At accuracy $\Theta(d^{-1/2})$, the previously applicable lower bound was $\Omega(d)$, inherited from the stronger full first-order oracle, while an upper bound from Protasov's value-only method requires $O(d^2\log^2 d)$ evaluations. By providing a lower bound of $\Omega(\,\frac{d^2}{\log(d+1)})$ on the oracle complexity in this setting, we thereby close this gap dating back to 1996, up to polylogarithmic factors. Furthermore, we are able to lift this result to the mixed-integer setting: Mixed-integer convex optimization with $d$ continuous and $n$ discrete variables using function values requires $\tilde{\Omega}(d^2\cdot 2^n)$ queries.

We study the existence of equilibria for state constrained, convex generalized Nash equilibrium problems (GNEPs) coupled with hyperbolic partial differential equations (PDEs). Analogous problems have been addressed for state constrained GNEPs coupled with elliptic and parabolic PDEs, respectively, but the problematic regularity of the set-valued constraint maps has been a barrier for development of an existence theory in the hyperbolic case. This is mainly due to compactness issues with the strategy-to-state maps. Beyond existence, we also provide first-order optimality conditions for a class of fairly general linear hyperbolic PDEs and show its relevance for applications such as the wave equation, advertising dynamics, and the linearized isothermal Euler system on networks.

Laser-based thermal processing, such as laser powder bed fusion, requires tight regulation of the peak surface temperature: heat accumulates where the moving source re-enters previously heated material, driving the temperature out of its process window and causing defects. High-fidelity thermal models capture this physics but are too slow for online optimization, which motivates fast, differentiable, and generalizable surrogates. We develop and validate a complete surrogate-based control pipeline that regulates the maximum surface temperature of a moving laser on a 304-stainless-steel substrate. We also determine conditions under which our surrogate can be trusted inside the control loop by probing its out-of-distribution limits. A key component of our surrogate is a multi-step deep operator network bespoke for moving sources: its branch subnetwork encodes the future power and trajectory (position and velocity) sequence, while its trunk encodes the current peak temperature and the temperature at the future laser locations, yielding a one-shot five-step prediction. By way of illustration, we use this surrogate as a smooth (algebraic-rectifier) nonlinear program inside a receding-horizon model predictive controller solved in CasADi/IPOPT. The surrogate forward pass is over thousand times faster than the equivalent finite-difference steps. We show that aggregate open-loop accuracy is necessary but not sufficient for control-readiness: two surrogates with near-identical offline error behave drastically differently in closed loop. A controlled two-ensemble data design reduces a 91 K path-corner underprediction failure to 1.4 K, and a calibrated one-sided constraint margin of 13 K yields zero violations of the true upper bound on all tested paths.

Weighted integrated observability inequalities for heat equations usually involve a small-time factor of the form $e^{-\gamma/t}$. We prove that this scale is not an artefact of Carleman or spectral methods: it is forced by the geometry of the observation set. Let $A$ be a nonnegative self-adjoint operator on sections of a finite-rank Euclidean vector bundle over a doubling metric measure space, satisfying ultracontractivity, Davies-Gaffney estimates (equivalently, finite speed of propagation for the wave equation) and a pointwise local Weyl law. If a weighted integrated observability inequality holds on a measurable set $\omega$, for a fixed horizon $T\in(0,+\infty]$ and an admissible weight $h$, then, for every $0<\kappa<\frac{1}{2}$, $$ h(t)\leq A_{T,\kappa}\exp\left(-\kappa\frac{\mathcal{L}(\omega)^2}{t}\right),\qquad 0<t<T, $$ where $\mathcal{L}(\omega)$ is the essential maximal distance to $\omega$, replaced by any finite radius when $\mathcal{L}(\omega)=+\infty$. Thus, for $h(t)=e^{-\gamma/t}$, necessarily $\gamma\geq\mathcal{L}(\omega)^2/2$. This settles, with the optimal threshold, the maximal-distance lower bound for the infinite-time constant left open in earlier work. In the control-norm convention, the fast-control rate is at least $\mathcal{L}(\omega)^2/4$, recovering Miller's bound. The proof rests on the spectral packet $(\cosh(r\sqrt A)-1)e^{-tA}$. A pointwise Weyl law gives its sharp lower growth, while finite propagation speed and a weak-kernel Kannai transmutation formula make it exponentially small on $\omega$. Without kernel continuity or compact resolvent, we develop pointwise spectral measures and weak wave kernels. The framework covers Laplace-type operators on compact Riemannian manifolds, coupled heat systems, Schrödinger operators on $\mathbb{R}^d$, equiregular sub-Laplacians and Grushin models, and $\delta'$-coupled Laplacians on metric graphs.

We introduce EuclideanDistanceDegree, a Macaulay2 package that implements symbolic and numerical methods for computing Euclidean Distance (ED) degrees. The package includes symbolic methods based on minors and conormal varieties as well as numerical methods for unit and generic ED degrees using tools from numerical algebraic geometry. We illustrate the package functionality with a range of examples in the paper and in the accompanying GitHub repository.

Demand responsiveness is an important consideration in public transport line planning, as network design and service quality influence passenger demand. However, accounting for this interaction further complicates an already challenging combinatorial optimization problem. To address this challenge, we propose a scalable Adaptive Large Neighborhood Search (ALNS) algorithm for large-scale line planning with endogenous demand. The algorithm jointly optimizes lines and frequencies while accounting for passenger mode choice, passenger assignment, and vehicle capacities. Candidate lines are generated dynamically throughout the search, and solutions are evaluated using an embedded evaluation procedure for passenger assignment and demand estimation, together with a dedicated local search procedure for frequency optimization. The proposed methodology is evaluated on the public transport network of Odense, Denmark, comprising approximately 1,800 origin-destination pairs. Computational results demonstrate the applicability of the approach to realistic, large-scale instances. The optimized networks concentrate resources on fewer, higher-frequency services, reducing average headways from approximately 41 minutes to 6.8-13 minutes while substantially increasing public transport ridership. Furthermore, the results show that network design is highly sensitive to assumptions regarding passenger behavior, highlighting the importance of carefully calibrated demand models when incorporating demand responsiveness into line planning.

We introduce and analyze Darcy flow through a saturated porous medium subject to bilateral constraints on the normal flux across a portion of the boundary. The problem is posed as the maximization of a velocity-based dual concave energy over a convex subset of $H(\mathrm{div};\Omega)$; Fenchel duality identifies a pressure-based predual formulation, yields strong duality, and provides convex optimality conditions with a complementarity structure on the constrained boundary. The primal–dual gap satisfies an a posteriori error identity, free of generic constants, valid for arbitrary admissible approximations. The duality structure is inherited by a Raviart–Thomas/Crouzeix–Raviart discretization, from which we derive a discrete error identity and a priori error decay rates under fractional regularity assumptions on the solution and the flux bounds. Numerical experiments, including adaptive refinement driven by localized primal–dual gap indicators, support the theory.

Solvent-based carbon capture can reduce CO2 emissions resulting from a continued reliance on fossil power plants for firm power. These capture processes remove CO2 from flue gases via a solvent. Careful design via process systems optimization can limit the overall cost of carbon capture, which is both capital- and energy intensive. As dispatchable power plants operate to meet varying load demand, the design process needs to account for varying operating points. However, optimizing the design over multiple operating points yields high computational complexity, which is why designs are often based on a single operating point in practice. Here, we identify optimal carbon capture process designs via stochastic optimization, reducing computational complexity through a data-driven approach-to-equilibrium model of the absorption and desorption processes. We represent variable flue gas conditions based on part-load operation data of a representative coal power plant. Accounting for this variability in the design substantially reduces equipment size and total plant cost by 6-9 % at the expense higher operating costs, yielding a reduction in total cost of carbon capture by 0.7-1.7 %. Given the capital intensity of carbon capture, variability of flue gas conditions therefore should be considered at the design stage, particularly if capture is deployed on plants subject to load following.

In this paper we deal with parametric, linear-quadratic optimal control problems in which the solution can be uniquely characterized by the optimal final time adjoint state. As a motivating example, we establish theoretical results showing that for distributed control of the heat equation, the manifold of final time adjoints over the parameter space exhibits a slow decay of its Kolmogorov width if this was already the case for the parameter-dependent target states. Traditional linear reduced-order models would thus require a large reduced space in order to guarantee a sufficient accuracy, making them inefficient in this application. To overcome the limitation of linear models, we discuss a nonlinear surrogate based on U-Nets that maps parametric fields to approximate final time adjoints. We show that a suitable a posteriori error estimator remains applicable to the U-Net approximation and can be used to certify the surrogate results. Through two extensive numerical experiments, we show the potential of the U-Net surrogate and compare it with several linear and nonlinear methods from the literature. The results show that the U-Net consistently achieves the highest accuracy among the methods considered while requiring significantly fewer training samples.

We study online control of a known linear dynamical system with adversarial costs and bounded disturbances, measuring regret against a general class of benchmark policies. We introduce counterfactual tracking, which separates the challenge of learning from the challenge of controlling the system. An online learner builds a reference trajectory by selecting or averaging the trajectories that the benchmark policies would have generated under the realized costs and disturbances, and a corrective law steers the system toward that reference. Charging each change in the reference its recovery cost (the cost of steering the system onto the new reference) reduces the problem to online learning with switching costs. Conversely, under additional natural assumptions, we show that this reduction is tight: the two problems have the same minimax regret up to a system-dependent factor, uniformly over horizons and policy classes. The reduction gives sharp regret guarantees for policy classes beyond standard finite-memory parameterizations. For a class of $N$ possibly nonlinear or history-dependent policies, it achieves $O(\sqrt{T\log N})$ regret over $T$ rounds, provided their trajectories remain within a bounded distance of one another and recovery costs are bounded. For the full $\ell_1$ ball of disturbance-response controllers, it achieves $O(\sqrt{T\log T})$ regret, which is minimax optimal in $T$, without assuming a common decay rate for disturbance effects. The framework also improves the best known regret bounds for linear state-feedback policies.

Data assimilation estimates the state of a dynamical system from model forecasts and incoming observations. Many observation mechanisms, however, are many-to-one, implicit, non-smooth, or accessible only through simulation, and need not provide the residual structures or likelihood guidance required by existing ensemble filters. We introduce implicit data assimilation, in which the analysis law is defined as an energy tilt of the forecast distribution. We then propose the Ensemble Controlled-flow Filter (EnCF), which realizes this update through a stochastic controlled flow and learns the observation-dependent control by adjoint matching from terminal energy gradients. For simulator-defined observations, EnCF-LF learns a surrogate conditional energy from samples and applies the same controlled-flow solver. We prove ideal exactness, derive a one-step error decomposition, and establish non-accumulation of local errors under filter stability. Numerical results show that Kalman-type filters remain preferable for smooth additive-Gaussian observations, while the proposed methods are better suited to non-Gaussian, many-to-one, multimodal, and implicit observation models.

Strict complementarity of a primal-dual pair of optimal solutions is fundamental in the numerical analysis of semidefinite programs (SDPs). Strict complementarity drives the convergence behavior of interior point algorithms. When it fails, pathological examples show a striking gap between two error measures of approximate solutions. The first of these is the forward or "true" error, i.e., the distance to the optimal solution set. The second is the less useful backward error, measured by the constraint violation. We first characterize the lack of strict complementarity in SDPs via a simple normal form. Our normal form has three key features: (i) it is obtained using elementary row operations and rotations; (ii) it makes the lack of strict complementarity evident; and (iii) it lets us construct any such SDP by a simple algorithm. A variant of our generating algorithm allows us to construct any SDP that fails strict complementarity but satisfies Slater's condition on both the primal and dual sides. Thus, we parametrize the data of all SDPs that lack strict complementarity in a manner similar to how the Jordan normal form parametrizes square matrices with given eigenvalue structure. Next, we precisely characterize when the singularity degree of an SDP equals the number of constraints – a result that underlies our generating algorithms and that we believe is of independent interest. We construct and share a set of SDPs that lack strict complementarity and present a detailed computational study. We find that forward-backward error gaps are quite common: in many small SDPs (with matrix order $\leq 20$), the forward ("true") error exceeds the backward error by up to seven orders of magnitude. Further, in several data sets the forward and backward errors are inversely correlated. In other words, the worse the "true" forward error is, the harder it is to detect.

Stochastic convex optimization is a classical problem with well-understood guarantees under first-order feedback. In contrast, for zero-order optimization with noisy function evaluations, a logarithmic gap has persisted between known upper bounds and the $\Omega(1/\sqrt{T})$ lower bound, even in the one-dimensional case. In this work, we study the problem of minimizing a convex function $f : [0,1] \to [0,1]$ using a zero-order oracle with subGaussian noise. We propose a computationally efficient algorithm that achieves the optimal $O(1/\sqrt{T})$ convergence rate, matching the lower bound. The result closes the existing gap in one dimension, providing the first sharp rate guarantee in this setting.

We investigate biological pest control strategies for the sugarcane borer Diatraea saccharalis through the combined action of two parasitoid species: the egg parasitoid Trichogramma galloi and the larval parasitoid Cotesia flavipes. We describe the population dynamics with a six-dimensional host-parasitoid model in which host-parasitoid interactions are represented through a Holling Type II functional response, extending previous models by coupling egg and larval stage dynamics and incorporating parasitism saturation observed in laboratory experiments. We characterize the equilibrium structure of the model and analyze the local stability of the extinction equilibrium. Bifurcation analysis reveals that, for a wide range of Holling parameters, the pest population exceeds the economic damage threshold, motivating the design of active control strategies. We formulate and compare three biological control approaches: open-loop optimal control, State-Dependent Riccati Equation (SDRE) feedback control, and impulsive feedback control based on Lyapunov arguments. We perform numerical simulations to show that all three strategies successfully keep the pest population below the economic damage threshold. The impulsive strategy, in particular, achieves effective suppression with substantially fewer parasitoid releases than the continuous approaches, making it the most practically viable option for field implementation.

We study a Newton acceleration for sparse composite optimization on the Stiefel manifold. The main difficulty is geometric: the active manifold identified by the nonsmooth regularizer may fail to intersect the Stiefel manifold transversely, which obstructs a Riemannian Newton step on the identified manifold. In the transverse case, we prove local identification of the ManPG tangent proximal mapping. For nontransverse cases, we introduce an off-diagonally perturbed Stiefel family that generically restores the identification geometry while yielding an \(O(\|\Delta\|_F)\)-KKT guarantee for the original problem. We also derive verifiable support-level conditions for clean intersection, which cover nontransverse sparse patterns and yield the smooth moving local models used by the Newton correction. Based on these results, we propose MIX, a safeguarded ManPG/Newton-CG method on moving identified intersections. In the general clean-intersection setting, we prove global descent and KKT-residual guarantees for MIX. In the transverse or generically perturbed cases, if the sequence has an accumulation point satisfying certain regularity assumptions and the second-order sufficient condition (SOSC), then the full sequence converges to that point, with finite active-manifold identification and a local Q-superlinear rate. Numerical experiments on compressed modes and sparse PCA show that MIX substantially improves efficiency while preserving solution quality. Beyond the Stiefel manifold, we also outline how the safeguarded global-convergence mechanism of MIX extends to general smooth equality-constrained manifolds.

We consider a quiz show game in which a contestant is presented with a sequence of questions. Each time the contestant answers a question correctly, she receives a prize and proceeds to the next question; the probability of answering each question correctly is given. If the contestant answers a question incorrectly, she receives a consolation prize and the game ends. The contestant's problem of determining the optimal order in which to answer questions, for known fixed parameters, is a classic one studied in Kadane (1969), and admits a simple index-based solution. We consider game-theoretic versions of this problem in which a game show host can choose how to allocate a fixed prize budget. Our models are motivated by operational search problems in national security involving reconnaissance missions and inspecting for evidence of nuclear enrichment, as well as certain scheduling problems. We study three variants of the game, corresponding to different ways in which the host can distribute the prize money. For the first variant, we provide complete closed-form solutions, including equilibrium strategies and the value of the game. We reduce the second variant to a game solved in the literature. For the third variant, we obtain partial results by analyzing a more general game with a geometric structure.

We characterize stability and its robustness for a class of boundary controlled, boundary observed hyperbolic partial differential equations. In particular, we show that asymptotic and exponential stability coincide for this class. Furthermore, we introduce the real and complex stability radii for which we give explicit formulas. In addition, we consider the dual system and its application to controllability and observability. Our main results are illustrated with two examples.

Scenario-based model predictive control (SBMPC) is a variant of model predictive control (MPC) that explicitly accounts for uncertainty by optimizing control actions over multiple predicted scenarios. However, its computational complexity increases rapidly with the number of scenarios and prediction horizon, limiting its applicability to real-time planning and control. This paper presents a learning-accelerated Alternating Direction Method of Multipliers (ADMM) algorithm for efficiently solving SBMPC problems by leveraging parallel computing and Moreau envelope learning, while maintaining high solution accuracy. We reformulate the SBMPC problems into consensus forms that can be decomposed via ADMM, separating the scenario-dependent dynamics from non-anticipativity constraints and enabling parallel updates across scenarios and time steps. Building on this decomposition, we utilize a learning-to-optimize scheme that leverages Moreau envelope learning of the cost function to accelerate the primal update in ADMM, thereby reducing computation time. The proposed framework is evaluated on a microgrid energy management problem subject to load and renewable generation uncertainties. Comparisons with IPOPT and MadNLP, two popular and modern nonlinear programming solvers, demonstrate substantial computational speedups while maintaining reliable closed-loop control performance.

We develop a tensor reduced-order modeling (TROM) framework for optimization-based inverse problems governed by parameter-dependent dynamical systems. The approach approximates the parameter-to-observation map directly in tensor-train (TT) format using TT-SVD or TT-Cross and integrates it into a regularized nonlinear least-squares formulation. Beyond accelerating forward evaluations, the low-rank tensor structure reformulates the inverse problem in reduced coordinates, assembles Gauss–Newton quantities without forming the full observation-space Jacobian, and minimizes the TROM objective over the discrete parameter grid. This tensor optimization provides either a stand-alone approximate solution or a data-informed initialization for a subsequent Gauss–Newton solve. We study an inverse heat-transfer problem in a heterogeneous medium, where the parameters describe the locations and radii of low-conductivity inclusions, and a FitzHugh–Nagumo parameter-estimation problem with a highly nonconvex landscape. Numerical experiments assess reduced-order model error, measurement noise, regularization, initialization, spatial discretization, and increasing parameter dimension. The results show that TROM reproduces full-order inversion at substantially reduced online cost. They also demonstrate that reduced-coordinate inversion, tensor-based optimization, and appropriate regularization improve robustness in higher-dimensional, noisy, and strongly nonconvex regimes. For the continuous TROM inverse problem, we develop an error-to-inversion analysis. Under local strong convexity of the regularized FOM objective and parametric smoothness of the FOM observation map, the error between the parameters recovered with the full-order model and TROM is bounded by controlled uniform errors in the surrogate map and its Jacobian, together with local FOM and curvature quantities.

In this paper, we study dual semismooth Newton (SSN) methods for degenerate polyhedral projection problems, where generalized Jacobians of the dual residual may remain singular even arbitrarily close to the solution set. Rather than regularizing these singular systems, we exploit the nonuniqueness of the dual representation. We introduce a primal–dual lifted projection-equivalent set that always possesses extreme points without additional structural assumptions on the polyhedron, and show that its extreme-point geometry identifies dual representatives at which nonsingular generalized Jacobians of the dual residual can be constructed. This geometry is further linked to a full-column-rank condition and a generalized weak strict Robinson constraint qualification, showing that the regularity required by the Newton step can be recovered rather than imposed a priori. We also establish displacement bounds that connect representative selection throughout the algorithm with the local Newton mechanism. Building on this variational framework, we develop an inexact dual SSN method with local superlinear convergence and a globalized version combining monotone representative selection with a Wolfe line search. The resulting method is globally convergent and eventually recovers the fast local rate. Numerical experiments on regularized optimal transport, battery-scheduling feasibility restoration, and occupation-measure projection demonstrate its robustness in highly degenerate settings.