Papers

Stochastic inverse variational inequalities (SIVIs) arise in applications in which equilibrium responses are observed under uncertainty, such as inverse road pricing and network equilibrium control. Existing methods typically rely on co-coercivity or strong monotonicity, while general monotone SIVIs remain less understood. We propose a regularized projection algorithm that combines Tikhonov regularization with increasing batch sizes. Under monotonicity and Lipschitz continuity, we prove almost sure boundedness of the iterates and almost sure convergence of their distance to the SIVI solution set. We further establish, to the best of our knowledge, the first explicit nonasymptotic rate of $O(T^{-1/2})$ for the expected squared residual under general monotonicity. This yields $O(\epsilon^{-2})$ iterations and $O(\epsilon^{-4-2\delta})$ stochastic oracle calls, for any $\delta>0$, to obtain an $\epsilon$-solution in expected squared residual. A deterministic variant attains the same iteration complexity using $O(\epsilon^{-2})$ exact operator evaluations. Numerical experiments illustrate the proposed methods on monotone SIVI problems.

We continue the study of the natural polytope $\mathcal{D}$ in $\mathbb{R}^{n+d}$ associated with the disjunction of a set of $n+1$ polytopes in $\mathbb{R}^d$, managed by $n$ binary variables. Already $\mathcal{D}$ had been characterized for arbitrary $n\geq 1$ and (i) $d\in\{1,2\}$, and (ii) for a broad generalization of hyper-rectangles. In both cases, the complete characterization employs full optimal big-M lifting. Here, we give a complete description of $\mathcal{D}$ for the case of $n=1$ and arbitrary $d$, when the (two) polytopes are arbitrary generalized cross polytopes. Furthermore, we characterize when our complete description employs only optimal big-M lifting. For $n>1$, we generalize the family of facet-describing inequalities used for $n=1$. Finally, we carry out some computational experiments demonstrating the value of our theoretical results.

This paper presents a cooperative distributed model predictive control (MPC) scheme for nonlinear continuous-time systems. The centralized optimal control problem is solved asynchronously via a fixed number of sensitivity-based distributed programming (SBDP) iterations. The proposed scheme requires only neighbor-to-neighbor communication and no synchronization between agents during optimization. Under nominal MPC stability and bounded information delay, local exponential stability is established for a sufficiently large number of per-agent SBDP iterations. Numerical and hardware-in-the-loop results on both Ethernet and Wi-Fi demonstrate the benefits of an asynchronous execution, reducing execution times by over 60% while maintaining comparable closed loop performance.

Unmanned underwater vehicles (UUV) integrate ocean forecasts with path planning algorithms in order to identify energy- or time-minimizing paths that enable mission completion. Typically, a well-defined deterministic ocean forecast is assumed to be available for path planning; however, in practice, different ocean forecasts can disagree. In this paper, we extend previous work on deterministic optimal path planning to identify optimal mean-time paths when presented with an ensemble of possible ocean forecasts. In particular, we formulate a system of time-independent Hamilton-Jacobi partial differential equations that incorporates forecast uncertainty and yields the optimal mean reachability travel time and the necessary controls to find the associated optimal path. An efficient numerical solution of this system of PDEs is obtained through an extension of the Fast Sweeping Method; verification and benchmarking results are provided. Additional numerical examples illustrate the impact uncertainty can have on the optimal path; in particular, these results demonstrate that the vehicle's optimal path can deviate significantly from the deterministic optimal paths associated with the individual ensemble members.

Optimal power flow (OPF) solved by an interior-point method (IPM) requires repeatedly solving Newton linear systems. When variational quantum linear solvers (VQLS) are used, each IPM iteration involves an additional nested inner variational optimization loop, which can significantly slow the overall quantum-assisted IPM convergence. To address this challenge, this paper proposes a dual-level trainable quantum IPM framework for OPF that leverages early solver-generated trajectories rather than relying on single-point prediction. The key observation is that early IPM iterates provide informative primal-dual, slack, and barrier-variable evolution about the path to optimality, while early VQLS parameter updates provide useful information about the later variational search. At the quantum-solver level, a trainable parameter model uses a short prefix of the VQLS parameter trajectory to project the remaining variational search toward a lower-cost region. At the OPF-solver level, a second trainable model uses early primal-dual IPM iterates to project a later central path state, which is restored to an admissible point before IPM refinement continues. Simulation studies show that the proposed approach reduces the number of variational updates by up to $95\%$ while maintaining OPF objective values close to the classical IPM reference. A 2-bus demonstration on real quantum hardware is also included to validate the implementation of the proposed workflow.

Dynamic congestion pricing is an important tool for managing congestion and coordinating distributed energy resources in active distribution networks. However, scalable mechanisms that preserve participant autonomy remain computationally challenging because the operator-resource interaction is naturally bilevel. This paper develops a convex-analytic framework in which a distribution system operator computes dynamic congestion-price adders, while decentralized energy hubs schedule flexible demand, storage, local generation, renewable curtailment, and grid import/export. Unlike conventional single-level reformulations that replace lower-level problems by Karush-Kuhn-Tucker (KKT) conditions, complementarity constraints, and big-M linearizations, the proposed model represents follower feasibility and optimality through a Fenchel-Young equality involving the convex conjugate of an extended follower objective. The remaining bilinear price-response term is handled through a penalized difference-of-convex reformulation and sequential convex approximation. The method solves continuous convex subproblems and avoids the constraint-wise complementarity and branch-and-bound scaling of mixed-integer KKT reformulations; its main computational drivers are price-response dimension and conjugate evaluation rather than binary encodings of follower inequalities. On augmented IEEE 13- and 34-node feeders, it reduces congestion by 96.89% and 96.45%, respectively, approaches centralized full-information dispatch, certifies price-response consistency to numerical precision, and yields lower residual congestion than time-limited KKT incumbents within the computational budget.

This paper addresses the efficient solution of hierarchical Bayesian inverse problems with a high- or infinite-dimensional parameter field and a moderate number of hyperparameters. We focus on a class of problems in which the parameter-to-observable mapping is a linear PDE, so that, for fixed hyperparameters, the problem becomes conditionally Gaussian. Marginalizing the hyperparameters entails repeated evaluation of their marginal density, which in turn entails repeated large-scale log-determinant ratio computations and maximum a posteriori (MAP) estimations. To address these computational challenges, we introduce a scalable framework that relies on generalized low-rank approximations of the update from the prior to the posterior precision matrix. Though such a framework is state-of-the-art in non-hierarchical settings, in the case of nonlinear prior hyperparameters such as covariance length scales, a direct application to hierarchical problems is inefficient. We propose two amortized variants of this framework, comparing their theoretical properties and computational complexity to the direct method. In numerical experiments, we evaluate their performance on an advection-diffusion initial condition inverse problem in two and three spatial dimensions, with hyperparameters in both the prior and the noise covariance. We find that our amortized methods achieve a factor of 30-45 speedup for 100 marginal density evaluations relative to the direct method in the 3D problem.

This paper studies finite-horizon optimal stopping problems for semilinear stochastic evolution equations in real, separable Hilbert spaces, together with their associated parabolic variational inequalities. We prove continuous differentiability of the value function in infinite dimensions, thereby obtaining a smooth-fit principle for the corresponding optimal stopping problem. The analysis has two parts. First, we prove existence and uniqueness, in a suitable weighted class, of a mild solution to the variational inequality and identify it with the optimal stopping value function. We also establish local spatial Lipschitz continuity of the value function by probabilistic methods, without requiring any smoothing property of the underlying transition semigroup. Second, under a global regularizing assumption on this semigroup, we prove higher-order spatial regularity: the value function is continuously Fréchet differentiable and its gradient is locally Hölder continuous. The abstract results are then applied to a stochastic heat equation with additive noise.

Uniformly weighted divergence preferences (UWDP) introduced in Maccheroni et al. (2006) are an important class of risk-averse preferences that contain as a special case the monotone mean–variance utility. UWDP are characterised by the lowest expected value of an act in $L^\infty$ under an adversarially chosen probability measure combined with the divergence of this measure. Our main result provides an alternative, computationally friendlier formula, which establishes in full generality that UWDP are the translation-invariant hull of state-independent expected utility over $L^0$. Some consequences of the new representation are studied.

Unknown linear time-varying (LTV) systems require the control policy to adapt from online closed-loop data as dynamics evolve. Existing methods usually update the policy by solving a one-shot optimization problem, which can be computationally demanding and sensitive to noisy model estimates. In this paper, we propose a policy gradient adaptive control (PGAC) method for LTV system control with unknown model parameters. Specifically, PGAC integrates online policy optimization into feedback by updating the state-feedback policy with one-step gradient descent of the linear quadratic regulator cost at each time instant. This incremental update is computationally light and naturally limits policy variations caused by noisy data. To explicitly compute the policy gradient online, we estimate local models from recent closed-loop trajectories using normalized sliding-window least-squares. We provide stability and convergence certificates of PGAC for two classes of LTV systems. For slowly time-varying systems, we prove that the closed-loop system achieves practical exponential stability without a dwell-time condition. For piecewise-constant LTV systems, we establish practical stability through a dwell-time contraction argument. We also provide average frozen-time optimality-gap bounds of the policy sequence for both classes. Finally, we validate the effectiveness of our method via numerical case studies of both LTV and nonlinear systems.

Let $(T,\Sigma,\mu)$ be a complete, $\sigma$-finite measure space, let $Y,Z$ be a separable Banach spaces, and let $S : T \rightrightarrows Z$ be a measurable multifunction with nonempty closed values. We study the relation between the Clarke tangent cone to the decomposable set $\mathsf{Sel}_p(S):=\{x \in L^p(T,Z) : x(t) \in S(t) \hspace{0.1cm} \text{a.e.}\}$ and the $L^p$-selections of the pointwise Clarke tangent cones. We prove that $$ \widehat{T}_{\mathsf{Sel}_p(S)}(x) = \{v \in L^p(T,Z) : v(t) \in \widehat{T}_{S(t)}(x(t)) \hspace{0.1cm} \text{a.e.} \} \hspace{0.2cm} \text{for any $p \in [1, \infty)$}, $$ More generally, we consider this problem in $L^p(T,Y) \times L^r(T,Z), p,r \in [1,\infty),$ and denote $S : T \rightrightarrows Y \times Z$ and $\mathsf{Sel}_{p,r}(S):=\{(x,y) \in L^p(T,Y) \times L^r(T,Z) : (x(t),y(t)) \in S(t) \hspace{0.1cm} \text{a.e.}\},$ The global-to-pointwise inclusion is shown to hold for all exponents unconditionally, whereas the reverse inclusion is obtained from an equi-integrable correction condition. Consequently, if $p,r$ are finite it holds $$\widehat{T}_{\mathsf{Sel}_{p,r}(S)}(x,y)=\mathsf{Sel}_{p,r}\left\{t \mapsto \widehat{T}_{S(t)}(x(t),y(t))\right\} \hspace{0.2cm} \text{for any $(x,y) \in \mathsf{Sel}_{p,r}(S).$}$$ Whereas, if $(p,r)=(1,\infty)$ the equality holds if $\mathsf{Sel}_{\infty,1}(S)$ satisfies a variational correction condition (V). Condition shown to hold whenever $S(t):=Gr F(t,\cdot),$ where $F : T \times Y \rightrightarrows Z$ is measurable in $t$ and $F(t,y) \subset F(t,x)+ \ell(t) ||y-x|| \overline{\textbf{B}}_{Z}.$ Applications are given to nonsmooth optimization with pointwise constraints, graphs of Nemytskii operators, and a multiplier rule for nonconvex integral programs is derived.

This paper addresses the inverse optimal control (IOC) problem for stochastic linear systems subject to both Brownian motion and Poisson jumps, using an inverse reinforcement learning (IRL) framework. Given a target feedback gain from an expert, the objective is to identify an equivalent cost functional-specifically, the set of all cost weights-that yields this same gain. To solve this problem when system dynamics are unknown, we propose two model-free, off-policy IRL algorithms that operate entirely from data, circumventing the need to solve the generalized algebraic Riccati equation or compute the cost weights analytically. The first is an inverse Q-learning algorithm that constructs data-driven equations from expert demonstrations to compute the Q-function matrix, with equivalent cost weights updated algebraically and without requiring additional trajectory data. The second is a model-free off-policy inverse policy iteration algorithm that leverages data collected under an initial stabilizing policy, offering a complementary approach suited to different data availability scenarios. Crucially, by decoupling the data-collection behavior policies from the policies being iteratively updated, both algorithms can learn equivalent cost weights from sufficiently excited trajectories without identifying the system dynamics or jump intensity. Numerical simulations validate the effectiveness of the proposed methods.

Entropy regularization is widely used in continuous-time reinforcement learning (RL) to reduce sensitivity to environmental perturbations, yet its robustness benefits lack a rigorous theoretical foundation. This paper establishes the first robustness guarantees for entropy-regularized continuous-time Markov decision processes. We show that maximizing an entropy-regularized objective yields a lower bound on a worst-case robust RL problem with joint reward and transition perturbations. We analytically characterize the induced robust sets and prove that they expand monotonically with the regularization strength, justifying the empirical observation that stronger entropy improves robustness. In contrast to prior discrete-time analyses, our results remove the intractable state-distribution entropy term and provide guarantees invariant to action frequency. Experiments on queueing network control and market making confirm our theory, showing that entropy-regularized policies outperform greedy and $\epsilon$-greedy baselines under dynamics perturbations.

We investigate quantitative rapid stabilization for the one-dimensional Allen–Cahn equation and develop a quantitative modal decomposition approach that makes explicit the dependence of the feedback laws and stabilization costs on the prescribed decay rate. We construct an explicit feedback law on the finite-dimensional unstable modes via Ackermann's formula. The explicit structure of the feedback allows us to derive quantitative low-frequency estimates, which, combined with the frequency Lyapunov method, yield quantitative stabilization estimates. Together with the stabilization framework of [37], the resulting estimates can be adapted to a broader class of one-dimensional parabolic models. We further construct piecewise feedback laws that yield the null controllability with control costs and finite-time stabilization.

Many image-processing problems can be formulated as composite cardinality optimization (CCOP), whose objective is the sum of two convex terms and a cardinality function composed with a linear operator. The composite cardinality term creates major algorithmic challenges: the separability of the cardinality function is lost and convergence analysis often requires surjectivity-type assumptions on the linear operators. To overcome these challenges, we consider the stationary dual formulation of CCOP, which has more favorable structure consisting of two convex terms and a separable cardinality function. In this paper, we develop an efficient proximal point algorithm (PPA) to solve the stationary dual problem. The efficiency of our PPA stems from two aspects. Firstly, the key step of its subproblem solver minimizes a smooth convex function over a low-dimensional subspace by the classic semismooth Newton algorithm (SNA), which has global convergence and local superlinear rate under suitable conditions. Secondly, implementable inexact criteria are proposed for early termination of the SNA subroutine. These components form the basic framework of our inexact PPA. Under suitable conditions, it enjoys global convergence and local linear convergence rate. In particular, we provide examples in which the convergence assumptions are automatically satisfied. Finally, the SNA subroutine is incorporated into our inexact PPA to solve jump-sparse signal recovery and computed tomography (CT) image restoration. Numerical results demonstrate the time efficiency and solution accuracy of our proposed method.

We study a nonlinear size-structured transport equation where the endogenous scalar output $E(t)=\int_{l_0}^{l_m}\chi(l)x(t,l)\,dl$ feeds back into velocity and mortality. This principal-coefficient feedback precludes a semilinear perturbation framework. Freezing the feedback path yields a non-autonomous linear evolution, reducing the closed-loop problem to a scalar Volterra fixed point $E=\mathcal K(E)$. Mass balance provides an intrinsic feedback interval, while a Bielecki-norm contraction ensures unique nonnegative weak solutions. Stationary equilibria satisfy a scalar closure equation $E=\Phi(E)$. We prove uniqueness below the sharp margin $1-\Phi'(E)>0$ and identify $\Phi'(E)=1$ as a nondegenerate fold threshold. Linearization yields a finite-memory renewal equation with characteristic equation $\mathcal E(\lambda)=1$, whose root set determines the feedback spectrum and stability. Finally, the stationary harvesting adjoint reduces to a rank-one perturbation formula. At zero discount, we establish the identity $\mathcal E(0)=\Phi'(E^*)=B(0)$, linking closure resonance, spectral crossing, and adjoint loop gain.