Papers

This paper studies a continuous-time general-sum non-cooperative game with linear costs, positive linear system dynamics, and elementwise linear input constraints. In the finite-horizon case, we present a verification theorem characterizing feedback Nash equilibria, in terms of absolutely continuous solutions of a coupled system of vector-valued ordinary differential equations, realized by time-varying feedback laws. Unlike linear-quadratic differential games, whose Riccati-based equilibria scale quadratically with the state dimension, this formulation scales linearly. However, the resulting piecewise-constant feedback saturates between its constraint bounds rather than varying smoothly, and additional mathematical challenges arise when characterizing the solutions of the differential equations, which are generally discontinuous due to the switching nature of the feedback gains. In this work, we study the case where switching occurs only at isolated time instants. In the infinite-horizon case, under stabilizability assumptions, the equilibrium is characterized by coupled vector-valued algebraic equations. For this game, we propose iterative methods to compute both finite and infinite-horizon equilibria. The approach is illustrated through a large-scale pollution game.

Alkaline water electrolysis (AWE) units supplied by insulated gate bipolar transistor rectifiers (IGBT-Rs) may experience oscil-lations caused by coupling between rectifier control and electro-lyzer (ELZ) dynamics. Because this risk varies with unit loading and power allocation, production-oriented dispatch may place a multi-ELZ renewable power-to-hydrogen (ReP2H) plant near or exceed its stability boundary. This paper proposes a stability-oriented framework for control design and plant production dis-patch. A three-port admittance model links the ac port, dc link, and electrolysis stack. Unit-level dc-port analysis quantifies the effects of loading, temperature, Buck bandwidth, and dc-link capacitance, while plant-level aggregation evaluates how unit commitment and power allocation affect stability. Results show that higher loading reduces stability, whereas larger dc-link ca-pacitance and higher Buck bandwidth improve it. Under the same plant loading, different power allocations result in different plant-level stability margins, with balanced allocation generally providing a larger margin than concentrated allocation. The plant-level model thus distinguishes the stability margins of ad-missible schedules. Hardware-in-the-loop (HIL) tests validate these trends and the proposed redistribution rule. The resulting operating regions and dispatch rules can be used to screen unit commitment and power allocation decisions in plant production scheduling.

We study integro-differential sweeping processes of Volterra type in a separable Hilbert space, in which the velocity is governed by the normal cone to a prox-regular moving set and is driven by an outer set-valued perturbation together with a history-dependent integral term that endows the dynamics with memory. The perturbation is assumed measurable, with closed convex values, of linear growth, and upper semicontinuous from the strong to the weak topology. We study the existence of absolutely continuous solutions under either of two alternative compactness hypotheses: ball-compactness of the moving sets, or a measure-of-noncompactness condition on the perturbation. After a reduction of the constrained dynamics to an unconstrained differential inclusion and uniform a priori bounds on the state and its velocity, existence follows from a fixed-point theorem for set-valued maps with contractible values. The memory of the process makes this contractibility delicate, and we obtain it through a continuation argument that propagates the history of the dynamics. As applications, we solve a quasistatic frictionless viscoelastic contact problem with long memory and a spatially distributed bioeconomic fishery model with ecological memory, both featuring uncertain set-valued forcing and a moving constraint set that is not ball-compact.

This article presents a novel algorithmic framework for solving a broad class of nonlinear inclusion problems involving non-monotone and Lipschitz continuous operators. Our primary motivation originates from the study of adjoint mismatch problems, which frequently arise in inverse problems and data science when the adjoint of a linear measurement operator is replaced by an approximation. Because this approximation may inherently destroy classical monotonicity properties, standard convergence guarantees for splitting methods may no longer apply. To address this, we consider a Nonlinear Forward-Backward algorithm and rigorously establish its convergence without assuming monotonicity. By leveraging a warped resolvent formulation and semimonotonicity assumptions, we derive explicit conditions ensuring both weak convergence and, under stronger assumptions, R-linear convergence. Furthermore, our theoretical analysis yields new convergence guarantees for several popular methods, including the Condat-Vũ and Forward-Half-Reflected-Backward algorithms. Finally, we illustrate our theoretical findings and demonstrate the applicability of our approach through numerical experiments in signal recovery.

Developing innovative system architectures increasingly relies on advanced modeling and optimization techniques to frame the architecting process and define the corresponding computational problems. For complex System-of-Systems (SoS), high-fidelity multiphysics and multidisciplinary simulations are essential for capturing detailed behaviors. However, their computational expense and the risk of evaluation failures make direct optimization challenging. To overcome these limitations, surrogate-based approaches, like Bayesian optimization, have emerged as effective tools for managing expensive, black-box simulation tasks. This work introduces a hierarchical Bayesian optimization framework that leverages Gaussian process meta-modeling to handle discrete architectural choices, conditional dependencies, and heterogeneous design variables inherent to SoS problems. Results show that the hierarchical formulation improves search efficiency and robustness compared to conventional surrogate-based methods, enabling the exploration of large and structurally diverse design spaces with limited simulation budgets. We apply the approach to an aircraft-based multi-agent system for wildfire suppression, a use case developed within the EU-funded COLOSSUS project that illustrates how SoS principles can coordinate heterogeneous aerial platforms with complementary roles, supporting both sustainable mobility and emergency response missions. Our framework provides a scalable methodology for SoS architecting and model exploration, offering transferable insights for applications in aviation, sustainable mobility, and resilience-oriented system design. By combining hierarchical representations with surrogate-based optimization, this work is among the first practical demonstrations of hierarchical Bayesian optimization applied to real-world SoS problems, advancing both methodology and practice.

Q-learning with linear function approximation can be unstable because an arbitrary approximation architecture need not preserve the Bellman contraction. We develop a stable infinite-dimensional linear function approximation framework for Q-learning from a single Markovian behavior-policy trajectory. The learning variable is a coefficient field $\theta\in C(\mathbb L)$ on a compact latent metric space $(\mathbb L,\rho)$. The framework uses a reconstruction operator that maps $\theta$ to a continuous Q-function and a compression operator that maps Bellman updates back to latent coordinates. Nonexpansiveness of both operators induces a contractive latent Bellman map on $C(\mathbb L)$, with a unique fixed point $\theta^*$ whose reconstruction approximates the optimal Q-function up to representation error. We propose two stochastic approximation (SA) algorithms and establish their sup-norm convergence bounds with a leading term of order $\widetilde O(n^{-1/2})$. The infinite-dimensional formulation provides a powerful abstraction for identifying the structures that govern statistical difficulty. Smoothness of the compression map in $\rho$ is inherited by $\theta^*$ and the SA iterates, allowing uniform estimation errors to be controlled through covering numbers of $(\mathbb L,\rho)$ rather than the dimension of $C(\mathbb L)$. Remarkably, the SA algorithms we propose are agnostic to the choice of $\rho$, and thus can automatically adapt to both the smoothness and the geometry. We further illustrate the framework through Q-measure-learning with linear density approximation and output-layer neural weight training under a frozen pretrained network.

In a real Hilbert space $\mathcal H$, we study the following inertial differential inclusion $$ \ddot x(t)+\gamma\dot x(t)+\partial\phi(\dot x(t)) +\nabla f(x(t)+\beta\dot x(t))\ni0, $$ where $\gamma>0$ is the viscous damping coefficient, and $\phi$ is a convex potential with a sharp minimum at the origin that models the dry friction damping (typically $\phi=r\Vert\cdot\Vert$ where $r>0$ is the dry-friction parameter). The function $f$ represents the smooth potential to be minimized, and the shifted-gradient evaluation $\nabla f(x(t)+\beta\dot x(t))$ is known as the implicit Hessian-driven model. Here $\beta\geq 0$ represents the corresponding Hessian-driven parameter. Both the explicit and the implicit Hessian-driven dynamics are know to attenuate the oscillations that occurs in inertial systems. Our contribution concerns the quantitative analysis of this continuous dynamic and its temporal discretization counter-part under the action of these three combined dampings: viscous damping, dry friction, and implicit Hessian-driven damping. We establish a global well-posedness, an exact Lyapunov analysis adapted to the implicit Hessian-driven damping, finite length of the trajectory and its strong convergence to an approximate critical point $x_\infty$ of $f$ satisfying: $-\nabla f(x_\infty)\in\partial \phi(0)$. We show finite-time stabilization under a strict interior condition on the terminal force. We also compare the explicit and the implicit Hessian-driven dynamics and show that their trajectories differ by $O(\beta^2)$ on finite horizons. A temporal semi-implicit discretization of the dynamic above leads to a proximal implicit Hessian-driven algorithm based on one shifted-gradient evaluation. We derive its discrete Lyapunov analysis, asymptotic convergence and, under a strict terminal margin, finite convergence of the discrete iterates.

Many studies on finite-horizon stochastic optimal control, including covariance steering, parameterize control policies as state-history-affine. This parameterization enables a convex reformulation, thereby yielding a tractable solution method. However, the necessity of dependence on previous states has not been well established. Is this dependence necessary, or merely an artifact of the convex reformulation? We show that it is an artifact that can be removed losslessly. Given an optimal solution of the state-history-affine formulation, we construct a deterministic Markovian policy which is affine in the current state. We show that, even for the covariance steering problem with a broad class of commonly used state and control safety constraints, the synthesized Markovian policy almost surely produces the same control actions as the history-dependent policy and therefore the same state trajectories, cost, and moments. Thus, every optimum of the history-dependent formulation admits a lossless Markovian transformation. Geometrically, the history-dependent formulation lifts the policy space for convexity, and its optimal solution can be projected back to the Markovian policy space. We extend the analysis to output feedback and a convex upper-bounding surrogate for value-at-risk costs.

In this paper, we propose an extension of the well-known Robinson implicit function theorem for generalized equations from the upper Lipschitzian case to the upper Hölderian case. Explicit exponents dependence between the generalized equation and its linearization is determined. Applications to ball constrained least squares problems, including linear least squares and separable nonlinear least squares, are studied. In particular, we establish that the solution mapping of ball constrained linear least squares under linear perturbation is locally upper Hölder continuous with exponent $1/3$, which is of independent interest. Ultilizing the upper Hölderian version of the implicit function theorem, we show the local upper Hölderian of the solution mapping of parametric ball constrained linear least squares.

The Barzilai-Borwein (BB) method is an efficient gradient-based approach for unconstrained optimization that approximates spectral information of the Hessian matrix to capture curvature at low computational cost. In this paper, we extend the BB stepsize strategy to composite nonconvex optimization problems consisting of a smooth nonconvex term and a proper closed convex term, and propose an adaptive Barzilai-Borwein proximal gradient method for nonconvex optimization (AdaBBNC). The proposed method incorporates a flexible BB-based curvature estimate into the proximal gradient framework to enhance adaptability in nonconvex settings. Under mild assumptions, we establish that AdaBBNC achieves the optimal iteration complexity of $\mathcal{O}(\epsilon^{-2})$ for finding an $\epsilon$-stationary point, without requiring any prior knowledge of the global Lipschitz constant. Numerical experiments demonstrate the effectiveness and robustness of the proposed method. Compared with recent parameter-free and line-search-free adaptive proximal gradient methods, AdaBBNC exhibits more aggressive yet stable behavior in ill-conditioned optimization problems.

As opposed to classical converse Lyapunov theorems, finite-step converse results are constructive and offer a different starting point: for sufficiently large finite step ahead, say $M$, in an explicit sense, any scaled norm can serve as a converse finite-step Lyapunov function. As for interconnected discrete-time systems, similar lines of argument lead to “non-conservative” small-gain conditions. Motivated by this viewpoint, this paper develops a distributed model predictive control framework for constrained interconnected nonlinear discrete-time systems. Each subsystem solves one local optimization problem at each system time step instant by setting the local stage function in form of a local control finite-step like Lyapunov function, using time-aligned neighbor predictions, optimized state-constraint tightening radii, and a finite-step small-gain terminal inequality. Since received neighbor predictions need not equal the trajectories generated by future receding-horizon optimizations, the nominal converse certificates do not alone ensure recursive feasibility or stability. We therefore develop shift-compatible constraint margins, a local one-step terminal feasibility test for networks that are affine in control, and an analytical bound for the prediction and reoptimization mismatch. The resulting analysis gives recursive feasibility, constraint satisfaction, and a practical $M$-step Lyapunov estimate, with asymptotic convergence when the prediction and reoptimization mismatch bound tends to zero. For constrained linear networks, the conditions reduce to finite-dimensional matrix, QP, and SOCP tests. The framework is specialized to current sharing and terminal bus voltage safety under DC/DC power converters' operational constraints in a two-DGU DC microgrid evaluated on a small laboratory-scale prototype.

Reinforcement Learning (RL) has achieved tremendous success in recent years. However, the classical foundations of RL do not account for the risk sensitivity of the objective function, which is critical in various fields, including healthcare, finance, etc. A popular approach to incorporate risk sensitivity is to optimize a specific quantile of the cumulative reward distribution. However, exact quantile objectives are non-smooth and can change abruptly under small perturbations of the return distribution, making them difficult to optimize reliably when the transition model must be learned from data. Motivated by this instability, we develop UCB-BQRL, a model-based optimistic learning algorithm that maintains confidence sets for the transition kernel and plans using a lower-buffered quantile criterion. The buffered criterion smooths the exact quantile objective by averaging nearby lower quantiles, thereby improving stability under transition-estimation error. To compute the buffered-quantile policy at each episode, we introduce EVI-BQ, an exact dynamic-programming procedure. We establish a high-probability regret bound for UCB-BQRL, which up to logarithmic factors scales as $\mathcal{O}(\mathrm{e}^{\tau/\rho_\tau}+H^2\sqrt{SAT})$, where $\rho_\tau$ is denoted as the root-level left-plateau threshold, which is a problem-dependent constant. Further, we establish an information-theoretic lower bound of $\Omega(H/\rho_\tau\sqrt{AT})$ for the regret of any algorithm dealing with a quantile objective function. Finally, we prove that the exact point-quantile evaluation and exact lower-buffered quantile evaluation are PP-hard under polynomial-time Turing reductions, even for a fixed policy in a two-state, one-action finite-horizon MDP.

This paper studies the worst-case impact of constrained control inputs: an input seeks to maximize the average cost of some outputs, measured in the $L_2$ or $L_1$ norm, while remaining bounded in terms of other outputs. This problem template subsumes classical metrics such as the $H_\infty$ norm and the output-to-output gain, and arises in adversarial control, security assessment, and robust control. For linear time-invariant systems, we provide an exact semi-definite program (SDP) when there is a single constraint, and SDPs computing upper bounds when there are multiple constraints. We derive sufficient conditions, in terms of system zeros and relative degrees, under which the worst-case cost is unbounded, together with a constructive closed-loop modification that removes the unboundedness. From a security standpoint, unbounded values reveal structural limitations in detecting certain attack inputs. For positive systems, we provide scalable formulations whose complexity grows linearly in the state dimension: a scalable SDP for quadratic costs, and an exact linear program for linear costs. The results extend to nonlinear polynomial systems via a sum-of-squares program. We illustrate the results with numerical examples.

This paper investigate the conforming virtual element method for the optimal control problem governed by the linear poroelasticity equation. Both distributed control and Neumann boundary controls are considered with the optimize-then-discretize approach. We establish the well-posedness of the discrete state and adjoint problems in three-field formulation by deriving the first-order optimality condition. The optimal a priori error estimates are developed through a novel poroelastic projection for both control problems, and these estimates are uniform with respect to the relevant physical parameters of the model problem. Moreover, the numerical experiments are carried out using the primal dual active set strategy, and demonstrate the support for the derived theoretical results.

The purpose of this paper is to study the existence and convergence of $\phi$-best proximity points in Banach spaces. A strong convergence theorem is established by employing a shrinking projection algorithm designed to compute common $\phi$-best proximity points of a proximally weakly suppressive mapping. Finally, we address a projection scheme to solve split $\phi$-best proximity point and variational problem in a Banach space. These contributions provide new tools for solving nonlinear problems involving non-self mappings.

This paper studies exact controllability and L2-stabilizability of discrete-time backward-structured stochastic linear systems. For each prescribed finite horizon, equivalent characterizations of exact controllability are established in terms of the controllability Gramian, a Kalman-type rank condition, and the reachable subspace. Exact null controllability without a prescribed horizon is then characterized by a finite-dimensional rank condition, and a Hautus-type criterion is derived using positive semidefinite eigenmatrices of a positive operator. For L2-stabilizability, we establish a stabilizability decomposition and obtain a corresponding Hautus-type spectral criterion. As a byproduct, we show that exact null controllability implies L2-stabilizability. These results provide algebraic and spectral tests for the controllability and stabilizability of the considered systems.

We introduce the centered weak discrete Riemannian gradient (c-wDRG) framework for the unified analysis of optimization methods on Riemannian manifolds. The framework uses a center point to represent the relevant logarithmic differences in a common tangent space and covers Riemannian steepest descent, proximal point, proximal gradient, implicit midpoint, geodesic average-vector-field, Gonzalez, and Itoh–Abe methods. We derive c-wDRG certificates for these schemes and establish convergence bounds with explicit curvature-dependent step-size conditions. Under suitable c-wDRG parameter and step-size conditions, nonaccelerated schemes achieve \(O(k^{-1})\) or linear convergence of the objective gap. We further develop accelerated c-wDRG schemes with \(O(k^{-2})\) or accelerated linear convergence under the respective parameter conditions. Numerical experiments support the theoretical convergence results.

We consider the problem of when it is best to sell a risky asset in the framework of the distribution builder approach under the consideration of potential ruin. This approach allows investors to express their preferences directly as a desired target distribution without first specifying a risk aversion or utility function. Mathematically, the problem is closely related to the Skorokhod embedding problem, where the goal is to attain a given distribution by stopping a diffusion process. We work in a general framework of one-dimensional diffusion processes and extend existing results to include the possibility of ruin. We first provide a full characterization of the set of distributions that can be attained before ruin occurs. Then, we formulate two optimization problems that tackle the issue of what to do if the originally specified distribution is either not attainable or not optimal: finding the attainable distribution closest to an unattainable distribution and selecting an optimal attainable distribution under first-order stochastic dominance constraints if the originally specified distribution is attainable, but not optimal. We show existence and uniqueness of solutions to these constrained convex optimization problems in an extended $f$-divergence framework, provide an analytic characterization of solutions and give numerical examples.

A long-standing open question in quasi-Newton optimization asks whether the classical Davidon–Fletcher–Powell (DFP) method converges globally on uniformly convex objectives when all accepted steps satisfy the standard weak Wolfe conditions. We show that the answer is no, even under the standard strong Wolfe conditions. Fix $0<c_1<2/3$ and $2/3\le c_2<1$. We construct a function $f\in C^2(\mathbb{R}^2)$ such that $\frac{1}{2}I\preceq\nabla^2 f(x)\preceq\frac{3}{2}I$ for all $x\in\mathbb{R}^2$. We also choose a fixed positive definite initial inverse Hessian approximation and a sequence of positive step lengths. The classical DFP iteration is well defined, and all accepted steps satisfy the standard strong Wolfe conditions, but $|\nabla f(x_k)|$ converges to a positive constant. The global Hessian condition number is at most three. The construction uses an alternating two-step DFP sequence near a one-dimensional invariant center manifold. Along this sequence, the smaller eigenvalue of the inverse Hessian approximation tends to zero. The changes in the gradient norm between cycle starts are summable, but the total rotation of the associated eigenvectors is unbounded. The accumulation points of the DFP sequence form a circle. A uniform separation bound allows us to interpolate the prescribed function values and gradients. We add smooth functions with pairwise disjoint supports to a quadratic and keep the global Hessian bounds. An affine change of variables gives an identity-initialized example with problem-dependent Hessian bounds. An orthogonal direct sum extends the result to every dimension $n\ge 2$.

Neural combinatorial optimization solvers generate many candidate solutions per instance and report the best one found, using the same sample budget for every instance regardless of difficulty. A companion study showed that reallocating a fixed budget toward harder instances can improve solution quality, but that the standard way of measuring this improvement is biased: deciding an allocation and evaluating it on the same data can manufacture an apparent gain even when none exists. This left open what property of a workload determines whether reallocation is worth doing, and whether a policy that spends part of the budget to decide how to allocate the rest still pays once that cost is counted. This paper answers both questions through pre-registered confirmatory experiments – analysis and verdict criteria fixed before data collection – across three independently trained solvers and two ways of constructing harder workloads on the traveling salesman problem. Within the workloads we study, the deciding property is how varied the instances within a workload are in difficulty, not how difficult the workload is on average: a uniformly easy or uniformly hard workload offers little room for reallocation, while a mixed workload offers substantial room. A budget-aware policy that pays for its own information about instance difficulty recovers most, though not all, of the improvement available when that information is assumed free. Every experiment was independently recomputed from its written specification, and every correction to an earlier version – including two that weakened the paper's own claims – is reported with the direction it moved the conclusion. The paper offers a specific empirical answer and a template for verifying that answer is not an artifact of how it was measured.

We show that for any real, connected, semi-simple matrix group, its Lie algebra of algebraic vector fields can be generated by three complete vector fields.

Trajectory optimization computes dynamically feasible motions that enable autonomous systems to accomplish complex tasks while satisfying operational and environmental constraints. This tutorial presents OpenSCvx, an open-source Python framework that bridges the gap between high-level problem specification and efficient numerical optimization. Rather than requiring users to derive solver-specific mathematical formulations, OpenSCvx provides a symbolic modeling interface that automatically constructs and solves trajectory optimization problems from modular descriptions of objectives, dynamics, and constraints. Beyond simplifying problem formulation, OpenSCvx supports (i) continuous-time constraint modeling, (ii) temporal and logical specifications, (iii) automatic vectorization for scalable and batched optimization, and (iv) a modular architecture that enables new algorithms, models, and solver backends to be incorporated with minimal effort. These capabilities allow researchers and practitioners to rapidly prototype, solve, and extend state-of-the-art trajectory optimization methods.

The revenue maximization problem in service systems with strategic customers is a central problem in the thread of Rational Queueing literature. The standard modeling assumption is that customers act independently and, therefore, the situation is modeled as a game among the customers and the service provider. In the present work, we focus on the case where the customers are grouped and controlled by their leader and the situation becomes a game between the customers' leader and the service provider. We examine the effect of customers' cooperation for the join-or-balk dilemma at the unobservable single-server Markovian queue and present the impact on the social welfare and service provider's revenue.

We explore fast convergence of the Riemannian Frank–Wolfe method for smooth geodesically convex optimization over compact feasible sets. Hadamard manifolds are the main setting. On general complete manifolds, the analysis accounts for all feasible minimizing geodesics. We consider the open loop step-size $\eta_k=a/(k+a)$, which only uses the iteration index. Under a local Hölderian error bound and local length-normalized directional scaling, every $a>2$ gives the eventual rate $O(k^{-1/(1-\theta)})$ for $\theta\in(0,1/2]$. An interior-ball condition yields $O(k^{-2})$ for strongly geodesically convex objectives. Under an exact Riemannian scaling inequality and a uniform positive lower bound on the gradient norm, every $a\geq2$ gives $O(k^{-a})$ after an explicit threshold index. The same rate holds for the smallest Frank–Wolfe gap over the most recent half of the iterates. For geodesic balls of radius \(R<\pi/2\) in the unit sphere, we establish the scaling inequality with $\alpha_R=\tfrac12\cot R$, yielding $O(k^{-a})$ primal error and recent-window gap rates. We also analyze the standard gap-feedback short step under the local error-bound conditions, obtaining $O(k^{-1/(1-2\theta)})$ for $\theta<1/2$ and a linear rate for $\theta=1/2$. Numerical experiments illustrate the predicted rates and compare iteration-only and feedback-based step selection.

We study the computational complexity of the decision versions of three classic robust optimization problems: static robust optimization, two-stage (adjustable) robust optimization, and $K$-adaptability. We consider that the feasibility, uncertainty, and recourse sets are polyhedra or integer-programming-representable sets, the uncertainty is decision-independent or decision-dependent, and the feasible regions are bounded or unbounded. While these problems are well-established in the current literature, we give a systematic classification that places the resulting problems in $\mathsf{P}$, $\mathsf{NP}$, $\mathsf{coNP}$, and higher levels of the polynomial hierarchy ($\Sigma_2^p$ and $\Sigma_3^p$), and, once decision-dependent uncertainty introduces quadratic constraints, in the existential theory of the reals and its hierarchy ($\exists\mathbb{R}$, $\Sigma_2\mathbb{R}$, and $\Sigma_3\mathbb{R}$) or among the undecidable problems. Beyond hardness reductions, we pay particular attention to membership proofs, establishing polynomial-size certificates even though the sets considered generally contain vectors of exponential encoding length. As a by-product, we give an alternative proof that bilevel linear optimization lies in $\mathsf{NP}$, exploiting its connection to decision-dependent robust optimization. For $K$-adaptability, we relate the problem to its static and two-stage counterparts, showing that hardness grows monotonically with $K$ and that, for fully discrete decision-dependent instances, $K$-adaptability reduces back to the static problem.

In this paper, we study infinite-horizon deterministic linear-quadratic difference games with an output feedback information structure. We consider linear time-invariant dynamics and quadratic cost functionals defined over an infinite horizon. We first demonstrate that computing an output feedback Nash equilibrium (OF-NE) in difference games is challenging, even for low-dimensional games. To address this difficulty, we introduce an output feedback guaranteed cost equilibrium (OF-GCE) for difference games. In an OF-GCE, each player seeks a feedback strategy that guarantees its cost remains below a prescribed bound while satisfying an equilibrium condition. We derive necessary and sufficient conditions for the existence of an OF-GCE in terms of the solvability of a set of coupled bilinear matrix inequalities. We provide a linear matrix inequality-based iterative algorithm for synthesizing OF-GCE strategies. We further show that the state feedback GCE (SF-GCE) is a special case of the OF-GCE when players have complete state information. Numerical examples illustrate the effectiveness of the proposed approach.

The primary challenge of conducting PDE-constrained optimization for high-dimensional problems, such as kinetic equations, is the often prohibitive memory cost. Computing gradients using the adjoint state method would require the storage of the entire time history of the forward solution. For such problems, where the memory cost for storing a single instance of the forward solution can already be a limiting factor, this is clearly not feasible. In this paper, we propose a memory-efficient adjoint state method that compresses the forward and adjoint solution with a dynamical low-rank approximation (a model order reduction technique) and bypasses the need to store the entire forward solution by employing a time-reversible low-rank integrator. The dynamical low-rank approach introduces a number of challenges: reversibility can fail in the rank-deficient case and the low-rank trajectories can show chaotic behavior. In particular, the latter has a number of important consequences for the optimization problem. We address those challenges and show that our method can drastically reduce the memory requirement for gradient-based optimization of kinetic equations. In particular, we consider two examples from kinetic plasma physics: optimizing beam profiles to suppress a bump-on-tail instability and shaping a particle beam using external electric fields.

We develop spectral algorithms for selecting state-space partitions that define averaging kernels for finite, ergodic and reversible Markov chains. For a partition $\mathcal O$, the Gibbs kernel $G_{\mathcal O}$ resamples within the current block from the stationary conditional distribution; when this update is tractable, composing or mixing it with a baseline kernel $P$ can accelerate convergence. We select $\mathcal O$ by rounding the bottom nonconstant eigenfunctions of $P^2$, or the algebraically smallest eigenfunctions of $P$ for additive mixtures, using weighted $k$-means. For $F(\mathcal O)=\|G_{\mathcal O}P-\Pi\|_{F,\pi}^2$, we derive exact trace and normalized-cut representations and show that $F$ equals the Pearson $\chi^2$-mutual information between the initial block label and the state after one transition, giving this matrix objective a natural probabilistic interpretation. In the two-block case, a threshold sweep exactly solves the associated one-dimensional weighted two-means rounding problem. For general $k \geq 2$, weighted $k$-means rounds the bottom $(k-1)$-dimensional embedding, after which candidates are rescored by $F$; the rounding distortion is a distance between subspaces that yields spectral approximation bounds. We extend the framework to additive mixtures, finite-horizon objectives, and discounted infinite-horizon objectives. In contrast to classical normalized spectral clustering, which uses top nonconstant modes to find low-flow persistent clusters, our method uses bottom modes to favor large normalized cross-block flow and rapid loss of block-label information. Experiments on a controlled-spectrum graph, a mean-field Ising model, and Bayesian variable selection show notable per-iteration improvements in convergence and statistical estimation.

We develop a new direct accelerated Newton method for minimizing convex functions with Lipschitz continuous Hessian. The algorithm uses only primal variables and performs just one linear solve per iteration. With a simple predetermined choice of parameters, it achieves the global convergence rate of $O(1/k^3)$ in terms of the functional residual. To the best of our knowledge, this is the first second-order method for this problem class attaining this rate while relying solely on one linear system solve per iteration (without solving auxiliary nonlinear regularized subproblems, such as cubic regularization, performing nonlinear parameter searches, or using dual extragradient corrections). Our method can be implemented in a Hessian-free way, using an inexact linear system solver, while preserving the fast global rate. We further extend our construction to arbitrary geometry through Bregman divergence, and to composite optimization problems.

Pairwise reciprocal matrices are fundamental to the Analytic Hierarchy Process (AHP),a decision-making model. While the Direct Least Squares (DLS) method provides an intuitive mechanism for deriving priority vectors without complex transformations, it is susceptible to solution non-uniqueness. Under high levels of inconsistency, such as severe cyclic contradictions, the DLS optimization landscape becomes non-convex, yielding multiple distinct global minima. Consequently, priority rankings become unstable and critically dependent on initial algorithmic guesses. Furthermore, established prioritization operators (POs), including normalization techniques, the Eigenvector method, Singular Value Decomposition, Cosine Maximization, and the Pseudo-Inverse Gram Matrix (the closed-form solution of Weighted Least Squares), frequently generate disparate outcomes. To overcome these structural deficiencies, this paper introduces the Anchored Regularized Direct Least Squares (ARDLS) optimization model as a harmonizing framework. ARDLS integrates uniquely determined established POs as theoretical anchors within a regularization penalty. This integration systematically breaks mathematical symmetries and tilts the optimization landscape to guarantee convergence upon a single, unique global minimum. By minimizing the root mean square variance (RMSV) of the initial baseline vectors, ARDLS effectively unifies these divergent solutions. Comprehensive numerical experiments validate that the framework successfully fine-tune the solution of established POs by reducing RMSV while ensuring strict mathematical uniqueness. The practical utility of the method is further demonstrated through a numerical case study resolving an innovation fund dilemma in FinTech project selection. The proposed ARDLS approach offers a robust alternative to classical AHP across a wide range of decision-making domains.