Papers

We introduce Envelopt, a globally convergent iterative framework for a broad class of structured optimization problems where a smooth objective is augmented by a nonsmooth convex regularizer composed with a smooth mapping, and the variables are subject to general smooth constraints. All smooth functions may be nonconvex. The method is akin to an augmented-Lagrangian method in which partial minimization with respect to a lifting variable results in smooth subproblems involving the Moreau envelope of the nonsmooth regularizer, and the original constraints are retained explicitly. Only the proximal operator of the regularizer is required. Subproblems may be solved with off-the-shelf smooth optimization solvers. We state global convergence properties, establish that feasible limit points are asymptotically stationary, and develop an infeasibility detection mechanism. We derive worst-case iteration complexity bounds when the penalty parameter is and is not bounded away from zero. The framework subsumes the classical augmented Lagrangian method and accommodates important extensions, including stabilized formulations for degenerate problems, exact penalty methods, and conic constraints. We provide a Julia implementation, Envelopt.jl, as part of the JuliaSmoothOptimizers ecosystem. Numerical experiments with low-rank matrix completion, semidefinite programming, complementarity-constrained optimization, and nonconvex regularizers demonstrate the effectiveness and versatility of Envelopt.

Social dynamical networks significantly influence contemporary digital landscapes, affecting realms from social activism to public policy formulation. This paper investigates the use of multi-agent systems (MAS) to monitor and analyze these networks. Firstly, we propose a single-time-scale distributed inference model designed to effectively manage challenges such as latency and agent failure. Secondly, we provide sufficient conditions that ensure the stability of the proposed scheme. Notably, the observer gain design remains effective regardless of time delays. Thirdly, we develop a computationally efficient recovery mechanism for agent failures that relies on employing graph-theoretic approaches to restore network observability by replacing failed agents (due to sensing failure or unbounded delays, i.e., packet drops) by implementing computationally efficient graph-theoretic methods to assign observationally equivalent agent counterparts. Lastly, we illustrate the proposed scheme through a pedagogical example and real-world network applications.

In signaling games where a sender and a receiver have misaligned criteria, equilibrium behavior may lead to fully revealing, quantized, or randomized policies. Notably, the first arises in statistical decision theory and classical communication theoretic problems involving a fully aligned sensor and receiver, the second arises in Nash theoretic simultaneous signaling games, and the last may appear in Stackelberg type (leader-follower) Bayesian signaling games. In this paper, we investigate the Bayesian persuasion problem involving a receiver that tries to estimate the source. We show that for certain payoff structures, the equilibrium solution is such that a source observation is mapped to distinct messages with nonzero probabilities. More specifically, we completely characterize conditions under which the sender requires randomization for the Bayesian persuasion problem involving general sources with finite cardinality. In particular, regardless of whether the equilibrium solution under a deterministic policy restriction is fully revealing, quantized or noninformative, there exists a randomized sender policy that improves the sender's payoff under certain conditions characterized in the paper. Moreover, we provide an algorithmic procedure to obtain the Bayesian persuasion solution, where the algorithm compares the payoffs with finitely many posterior probability combinations. We also consider fully aligned and completely misaligned payoff structures, where the solutions respectively involve a fully revealing sender and a noninformative sender. Then, we unify these results by proving that if the sender's expected payoff with respect to posterior distributions is continuous, then the equilibrium solution involves either a fully revealing sender or a noninformative sender.

We study exact representations of $\mathrm{MAX}_N(x)=\max{x_1,\ldots,x_N}$ using two-hidden-layer ReLU neural networks. This problem has been studied in recent years in an attempt to characterize the exact number of hidden layers required to represent continuous piecewise linear functions. The best lower bound is 2, while the current upper bound is logarithmic in $N$. It remains completely open if the right answer is a constant number of hidden layers (possibly even 2!) or not. In fact, a recent breakthrough was the representation of $\mathrm{MAX}_5$ as a two-hidden-layer ReLU function obtained in [Bakaev et al., 2026], and the case of $\mathrm{MAX}_N$ was stated as open for $N\geq 6$ in that paper. Using a careful computer assisted search, we obtain two-hidden-layer ReLU representations of $\mathrm{MAX}_5, \mathrm{MAX}_6, \mathrm{MAX}_7$, and $\mathrm{MAX}_8$. We obtain these by considering rational linear combinations of terms of the form $\max\{\sum_{r=1}^{s}\max(x_{a_r},x_{b_r}),\sum_{r=1}^{s}\max(x_{c_r},x_{d_r})\}$, where $a_r,b_r,c_r,d_r\in\{1,\ldots,N\}$. Each inner maximum of two coordinates can be computed in a first hidden layer, and the outer maximum of the two side-sums can be computed in a second hidden layer. Consequently, every finite linear combination of these terms has a two-hidden-layer ReLU realization. An identity for $\mathrm{MAX}_N$ in this form therefore gives an exact two-hidden-layer ReLU representation of $\mathrm{MAX}_N$. Very recently, two-hidden-layer representations of $\mathrm{MAX}_N$ of the above form were obtained for all $N\leq 10$ in [Ruess et al., 2026]. Our representations are different and were developed independently. While our techniques share most of the high-level ideas presented in [Ruess et al., 2026], there are also some minor differences which may be of interest for future research on this problem.

Mixed-Integer Linear Programming (MILP) is a fundamental tool for combinatorial optimization with extensive real-world applications. A central challenge is designing computationally efficient MILP formulations. Large Language Models (LLMs) offer new opportunities to automate the modeling process, from deriving formulations to strengthening them. Reliable automation requires robust methods for verifying that proposed formulations preserve the underlying optimization problem. However, existing approaches evaluate formulations numerically and fail to reason about general problem instances. We resolve this limitation by introducing a constructive definition of MILP reformulation that can be formalized in Lean and machine-checked. We develop FLARE (Formulation-Level Automated Reformulation Evaluation), a method that uses an LLM-based agent and the Lean proof assistant to verify proposed reformulations against a reference formulation. To evaluate our approach, we introduce FormulationBench, a challenging dataset of 20 problems and 109 formulations. FLARE outperforms existing methods, with 100% accuracy on the NP-hard subset of FormulationBench. Furthermore, FLARE produces a machine-checkable certificate for every reformulation it accepts. For cases where formal guarantees are not necessary, we introduce FLARE-NL, a fast and cheap LLM proxy that matches FLARE's accuracy but produces no certificate. These methods enable reliable verification in automated optimization modeling.

Accelerating variable renewable energy integration introduces substantial operational challenges to modern power grids. Frequent curtailment of surplus renewable generation highlights the pressing need for long-duration energy storage capable of shifting generation to high-demand periods while providing essential ancillary services and grid inertia. Socio-environmental constraints increasingly limit conventional hydropower development; however, closed-loop pumped-storage hydropower (PSH) presents a viable alternative because its off-river reservoirs avoid natural watercourses. This paper introduces HERA-S, a computational modeling framework developed by PSR (supported by EDF, CTG, Brookfield, and Rio Light) to streamline and standardize regional PSH site prospecting. HERA-S automates spatial screening, dam optimization, cost estimation, and socio-environmental impact scoring. Applied to a case study in Rio de Janeiro, Brazil, the framework employs an integer programming model to optimize dam-fill and excavation mass balances against inter-reservoir distance, significantly improving pre-feasibility planning agility.

Gromov-Wasserstein (GW) compares distributions through relations within each space. This pointwise comparison can be too sensitive in one-to-many settings, where several target outcomes refine one source state and their mean carries the geometry of interest. We introduce a weak GW framework that compares source relations with relations between the target conditional laws induced by a coupling. For inner-product relations, we retain the conditional means $m_\pi(x)=\mathbb{E}_\pi[Y\mid X=x]$. The resulting barycentric weak inner-product GW (wIGW) satisfies $\mathrm{wIGW}_{\mathrm{bar}}^2(\mu,\nu)=\inf_{\eta\preceq_{\mathrm{cx}}\nu}\mathrm{IGW}^2(\mu,\eta)$. Here $\eta\preceq_{\mathrm{cx}}\nu$ means that $\nu$ is a mean-preserving spread of $\eta$. Thus wIGW searches for an intermediate target geometry that can be refined into the prescribed target law without changing conditional means. Under finite second moments, minimizers exist and martingale gluing recovers an optimal coupling. With ridge regularization, moment duality gives an $A$-$B$ min-max problem whose inner step is weak optimal transport with a quadratic cost parameterized by $A$ and $B$; the outer problem optimizes these matrices. For finitely supported measures, we give an iterative algorithm. Under a quantitative ridge condition, the reduced problem is convex–concave, and the projected outer iteration satisfies an explicit contraction bound for inexact inner solves. Point cloud and graph feature refinement experiments illustrate how mean-preserving target refinements can have zero cost. A paired peripheral blood mononuclear cell (PBMC) multiome study evaluates atlas based cell type transfer through RNA/ATAC alignment in cell to cell and prototype to cell settings, with the prototype to cell setting representing the one-to-many case.

In this chapter, we review and discuss the regularity properties of spatially adaptive regularisation weight functions used in variational image reconstruction. Incorporating such weights into classical model-based regularisers, such as Total Variation (TV) and Total Generalised Variation (TGV), allows the regularisation strength to vary across the image and adapt to local image content. When appropriately estimated, these weights can thus significantly improve edge and detail preservation in the reconstructions. We review the existing theoretical literature on this topic for different regularity classes, including constant, continuous, and piecewise constant functions. Our discussion is motivated by recent work on hybrid image reconstruction methods that combine model-based regularisation with deep neural networks to learn highly adaptive regularisation weights. In particular, we discuss how the structural properties of these weights influence the reconstruction from both theoretical and practical perspectives. Through representative examples in image denoising and magnetic resonance imaging (MRI) reconstruction, we demonstrate that the learned weights are often of low regularity and can adapt not only to the image structure but also to the specific noise realisation. We conclude by highlighting several directions for future research on this topic.

Large language model multi-agent systems (LLM-MAS) rely on inter-agent communication to solve complex reasoning tasks, yet rigorous guarantees relating consensus performance to computational resources remain limited. Here, we present a novel way to characterize collective belief dynamics as a discrete-time switched system in which communication topologies have distinct consensus-contraction rates and token costs. By augmenting the belief dynamics with the remaining computational budget, we define a consensus safe set that jointly captures agreement and resource feasibility. We derive explicit bounds on consensus time and token expenditure and construct a consensus-budget certificate region guaranteeing finite-time convergence without resource exhaustion. We further establish conditions under which adaptive topology switching achieves a trade-off between convergence speed and communication cost relative to fixed-topology strategies. Numerical experiments and live LLM-MAS deployments show the predicted consensus-cost trade-offs, demonstrating how control-theoretic certificates can enable resource-aware coordination in AI systems.

The proliferation of distributed energy resources (DERs) in distribution grids enables the active coordination of these assets to reduce costs and enable cleaner operations. Realizing this potential requires solving multiphase AC optimal power flow (AC-OPF) quickly across varying loads, DER availabilities, and topology reconfigurations, at much greater speed and scale than conventional nonlinear solvers. Learning-based surrogates can offer millisecond inference, yet existing methods target largely balanced transmission systems and do not scale to the multiphase, unbalanced, and reconfigurable nature of distribution feeders at utility scale. We present the Penalty + Sequential Linearized Feasibility Seeking (SLFS) algorithm, a self-supervised learning framework for multiphase distribution AC-OPF under switch-induced topology changes. Penalty+SLFS requires no labeled optimal solutions and trains directly from the AC-OPF objective and constraints through a differentiable fixed-point power flow solver, avoiding expensive label generation and admitting robust training procedures. Topology changes are handled efficiently using Sherman-Morrison-Woodbury updates of the admittance-matrix inverse, while an M-step Jacobian approximation accelerates differentiation through the power flow solver. At inference, SLFS repairs any infeasible predictions, providing feasibility guarantees with low computational overhead. On IEEE feeders ranging from 13 to 8,500 nodes, Penalty+SLFS achieves negligible optimality gaps and near-zero constraint violations, delivers up to three orders of magnitude speedups over IPOPT, and remains robust under large distributional shifts, demonstrating a viable path toward real-time, topology-aware AC-OPF for large-scale distribution grids.

Published experience with neural surrogates in derivative-free optimisation is contradictory: the same family of models that cuts the evaluation count of one solver leaves another unchanged, or makes it worse. We show that the contradiction dissolves once three factors are stated, and that these, rather than the fit accuracy a training curve reports, are what delimit when a learned local model pays. Role: a surrogate that proposes candidates the true objective must still approve helps, while one that replaces a gradient the solver depends on hurts. Radius: a model fitted to an optimisation path is reliable only inside a bounded neighbourhood, and its error neither vanishes as that neighbourhood shrinks nor survives its growth. Room: a surrogate can only accelerate progress the base method is still able to make. We formalise radius-aware local generalisation, relate it to the classical fully linear condition, and test each factor with the surrogate class, training pipeline and base method held fixed. Over 117 benchmark instances safeguarded assistance raises the instances solved to high accuracy from 67 to 84 while gradient replacement lowers them to 65; removing the gradient term from the training loss cuts surrogate acceptance from 0.703 to 0.148; and 1000 paired comparisons over ten noise levels show no noise threshold, only a base method that stops early. The same factors bound the gain: a model-based trust-region solver, which leaves little room, drops from 88 to 86 when the identical surrogate is attached, and released interpolation software stays ahead at 103, and on a Monte-Carlo inventory model repairing the acceptance interface is worth 10.40 cost units against 0.00 for the surrogate.

This article proposes a Caputo fractional accelerated gradient descent (CFAGD) method for unconstrained optimization problems that is applicable to both smooth and a class of non-smooth objective functions. The proposed approach incorporates an adaptive (e̱ta}k)-parameter, which is heuristically updated throughout the iterative process to improve the search direction. Furthermore, the method employs the Caputo fractional derivative together with the adaptive (e̱ta}^k)-parameter, thereby preserving the memory characteristics associated with non-integer-order derivatives. A suitable step-size is selected via an inexact line-search technique based on the Armijo condition. The central idea is to scale the step-size by a positive parameter to improve the behavior of the iterates as they approach an optimal point, thereby generating a descent sequence. Under strong convexity and bounded Hessian assumptions, linear convergence of the proposed method is established. Numerical validations, including neural-network-based examples, further indicate that the CFAGD method can achieve faster and more stable performance than competing approaches.

Linear programming (LP) formulations offer a conceptually elegant approach to infinite-horizon, model-free nonlinear optimal control in continuous spaces. However, in addition to the curse of dimensionality, their practical use is limited by the difficulty of consistently obtaining bounded solutions. In this work, we use moment-matching techniques to derive sufficient boundedness conditions in terms of the available dataset and the cost vector of the LP. Moreover, we discuss practical design methods for nonlinear systems and polynomial features.

Approximate Linear Programming (ALP) is widely used for large-scale Markov Decision Processes (MDPs), but its performance can be sensitive to the choice of state-relevance weights, which are typically selected heuristically. Performance bounds suggest aligning these weights with the discounted occupancy measure of the induced policy, and existing primal approaches address this through repeated greedy-policy construction. Nonetheless, they lack convergence guarantees and are computationally expensive. We propose a dual-based method that uses projected occupancy information from the ALP dual solution to construct a smooth stochastic policy and update the state-relevance weights, which avoids separate greedy-action calculations. We establish conditions under which the weights match the discounted occupancy of the induced policy and prove uniqueness and global convergence under appropriate smoothing. We also derive an a posteriori policy-loss bound that separates error from the weighted Bellman residual, occupancy mismatch, and stochastic-versus-greedy disagreement. Experiments on classical queueing and multi-priority scheduling problems show that the proposed approach reduces sensitivity to fixed weights and achieves comparable or better policy quality than primal updates at lower computational cost. Finally, we show that adaptive weighting is most valuable when the basis functions are sufficiently expressive for occupancy information to influence the resulting policy.

The embedding dimension of categorical predictors is usually selected through heuristic tuning, although it directly affects model complexity, approximation quality, and finite-sample generalization. This paper formulates embedding dimension selection as a constrained allocation problem. The main contribution is to show that embedding capacity can be allocated across heterogeneous categorical predictors according to an explicit approximation-estimation tradeoff. We characterize approximation error through the singular-value tail of the latent category representation, while estimation error increases with total embedding complexity. Under a fixed global embedding budget and a tractable approximation model, this leads to a closed-form allocation rule in which the dimension assigned to each predictor is proportional to the square root of its approximation value relative to its parameter cost. Simulation experiments support the proposed approximation-estimation interpretation and show that the allocation rule improves budget efficiency relative to standard uniform and cardinality-based heuristics, particularly when the budget is binding and predictor heterogeneity is substantial. A real-data healthcare application further shows improvements in predictive accuracy and probabilistic calibration. Overall, the results establish embedding-dimension allocation as a principled finite-sample optimization problem rather than a purely heuristic modeling choice.

Optimization problems with geometric constraints have a broad range of applications, including machine learning, finance, and control. A powerful algorithmic tool to resolve these geometric constraints are constraint dissolving methods. To this end, we propose a framework for constraint dissolving mappings for nonconvex geometric constraints. Leveraging these, we develop a constraint dissolving inexact penalty method to solve optimization problems with general set-membership constraints and possibly nonconvex geometric constraints. We establish the convergence of the proposed algorithm and prove that every feasible accumulation point is Mordukhovich stationary. Notably, we rely only on mild asymptotic Mordukhovich regularity, which is significantly weaker than the constraint qualifications adopted in the existing literature on constraint dissolving methods. Numerical experiments addressing classical equality-, complementarity-, sparsity-, and low-rank constrained optimization problems demonstrate that the proposed method is competitive with the safeguarded augmented Lagrangian method in terms of solution quality and significantly outperforms the penalty decomposition method.

This paper describes the electricity markets and operational processes in the Irish power system and translates them into a Mixed Integer Linear Programming (MILP) model. The model is designed to estimate renewable generation Curtailment and Constraint. A full mathematical formulation is presented and tested on both a simple example and a realistic model of the Irish transmission network. The results show that the proposed formulation works as intended, capturing curtailment and constraint effects, while also highlighting model limitations and directions for future improvement.

We show that the key optimization results of the classical Markowitz portfolio selection theory, originally formulated for variance as the risk measure, remain available in explicit closed form under a broader class of strictly convex quadratic risk measures. The proposed framework replaces the covariance matrix with an arbitrary symmetric positive definite matrix and allows additional linear and constant terms, thereby containing various models arising in transaction cost optimization, benchmark relative optimization, covariance regularization, and factor models. Closed-form formulas are obtained for the efficient frontier, the global minimum risk portfolio, the maximum Sharpe ratio portfolio, the Capital Market Curve, the tangency portfolio, and the maximum utility portfolio. In contrast to the classical Markowitz model, the tangency portfolio does not coincide with the maximum Sharpe ratio portfolio, revealing a new geometric phenomenon. A numerical example confirms the derived formulas.

We study a deterministic first-order mean field game model for human capital accumulation in which a continuum of agents invest in education and training to reach a certain qualification level in minimum time. The productivity of learning depends on both the individual's current level of human capital and the distribution of the population, modeling knowledge spillovers and social interactions. Learning is financed by the income generated by existing human capital, leading to a state constraint requiring the financial resources to remain nonnegative throughout the evolution. For a fixed population flow, we formulate the individual problem as a state-constrained minimum-time optimal control problem with time-dependent dynamics. We establish the existence of optimal controls and trajectories, analyze the associated value function through the Dynamic Programming Principle and a Hamilton–Jacobi equation, and characterize the optimal learning effort under suitable regularity and convexity assumptions. The optimal feedback induces a continuity equation describing the evolution of the population distribution. The resulting mean field game is formulated as a fixed-point problem for the population flow. By constructing a compact invariant set and applying Schauder's fixed-point theorem, we prove the existence of at least one equilibrium on every finite population horizon, while the individual minimum-time problem remains formulated on the infinite time horizon.

We prove the rank-one Lyapunov spectral identity recorded as Problem 9.1 in the 2004 collection of unsolved problems in mathematical systems and control theory. Let $P,Q,R$ solve the coupled discrete Lyapunov and Sylvester equations associated with $A$ and its rank-one update $A_2=A+vw^\top$. When the displayed inverses exist, we show that $P^{-1}RQ^{-1}R^\top$ and $(I+PQ)^{-1}$ have the same characteristic polynomial. A rank-one determinant factorization of the equation for $Q$ produces a scalar bilinear kernel. Evaluating it at the eigenvalues of $A$ and at their reciprocals gives $RQ^{-1}R^\top=BQ^{-1}B=P-BPB$, after which the two target matrices are the same two factors in opposite order. Polynomial continuation extends the identity from a nonempty open set of admissible systems to the full admissible domain and yields a determinant corollary without stability assumptions; when the spectra of $A$ and $A_2$ are disjoint, $Z=b(A)^{-1}P$ gives an explicit similarity. In the Schur-stable realization setting, the result recovers the associated principal-angle and past/future canonical-correlation spectra.

Profitability metrics, such as the single-product gross profit margin, constitute the standard toolbox to prioritize products in sales and operations planning (S&OP). These metrics leverage the premise that production lines can be committed independently, which endows them with a simple per-ton ranking that a planner can compute and use easily in practice. However, coupled plants, especially polymer plants whose parallel lines draw simultaneously on a single bulk feed, can fundamentally undermine this premise and mislead to value-destroying production plans. We propose a framework built on what we call the Augmented Gross Profit for Product Clusters (AGPPC), which converts the per-ton ranking metric into a per-coupled-hour ranking that carries a per-instance optimality certificate. We present the theory of AGPPC, which combines the co-production column concept with a fluid relaxation of the planning problem, alongside an integrated bilinear mixed integer program of the plant and a McCormick-linearized baseline that provides certified bounds for it, and we demonstrate its effectiveness on real operational data from an Indonesian polymer producer, where the coupling-aware metric alone raises operating profit by 7.6% and the full optimization by 28% over current practice.

We consider the absolute value equation (AVE) $Ax-|x|=b$ under residual monotonicity. We characterize this property through the symmetric part of the coefficient matrix, thereby allowing nonsymmetry, and propose a coordinate proximal predictor-corrector (CPPC) method based on an AVE-specific forward-proximal decomposition. An exact scalar proximal update on the largest proximal-residual coordinate generates a predictor point, while a positive-alignment identity certifies the ensuing full-residual separating-hyperplane correction without backtracking. With the current matrix product cached, each iteration requires one new full matrix-vector product. For a nonempty solution set, we prove Fejér monotonicity, whole-sequence convergence, and an $O(K^{-1/2})$ best-iterate residual bound. A positive monotonicity margin further ensures unique solvability for every right-hand side and global linear convergence. Numerical results identify regimes in which the reduced per-iteration work yields shorter solution times.

Mathematical elasticity has a long history and a huge body of literature. Surprisingly, at its heart elasticity exhibits a rich and transparent structure that, to our knowledge, is rarely presented in an explicit and unified form. Using standard tools from functional analysis, this article reveals four orthogonal decompositions of Hilbert spaces that characterize the static equilibrium of an elastic body at small deformations, universally for arbitrary spatial dimension, arbitrary boundary conditions, and classical as well as data-driven formulations. Regarding data-driven continuum mechanics, it highlights the intrinsic structure that remains unchanged when constitutive laws are replaced by material data sets.

We consider solving an optimization instance in which the objective is quadratic and where the decision variable is a probability measure. Our class of problems are motivated by applications arising from optimal transport (with the Gromov-Wasserstein problem being a prominent example) as well as energy landscape minimization. Because the objective depends quadratically on the decision variable, our class of problems fall outside the standard modeling framework of the Generalized Moment Problems (which requires the objective to be linear). To this end, we propose a hierarchy of convex relaxations based on searching over probability measures over products of the base space. These have a natural interpretation with the moment Sum-of-squares hierarchy-a prominent framework for solving polynomial optimization instances, which we adapt to accommodate probability measures. A key conceptual contribution is to introduce a notion of positive-semidefiniteness that extends the usual notion over matrices. Under the assumption that the decision variables satisfy certain marginal constraints (as in the Kantorovich formulation of the optimal transport problem), we establish convergence of our hierarchy towards the globally optimal solution. Under the additional assumption that the objective is a polynomial, we propose a moment-SOS type hierarchy of finite dimensional semidefinite programs whose optimal solution converges to that of the original quadratic optimization over measures. We demonstrate our framework with numerical experiments. More generally, optimization over measures where the objective and/or constraint depends on the decision in a polynomial way is a fundamental problem. It is hoped that our work provides a road-map as to how the ideas of the SOS-ordinarily developed for polynomial optimization-may be applied to a broader class of non-linear problems involving measures.

This paper studies a class of multi-cluster aggregative games characterized by the coexistence of cooperation and competition, where each agent's cost function depends on its own strategy and the aggregate of all agents' strategies. To address the Nash equilibrium seeking problem for such games in the non-Euclidean setting, a distributed mirror descent algorithm with gradient extrapolation is proposed over time-varying intra-cluster and inter-cluster networks. The mirror descent framework employs a general Bregman divergence as the distance measure, providing greater flexibility than Euclidean-based methods, while gradient extrapolation exploits historical gradient information to improve convergence performance. Under the restricted strong monotonicity characterized by the Bregman divergence, the convergence of the proposed algorithm is established, and it achieves the $\mathcal{O}(1/k)$ convergence rate with the appropriately selected step-size and parameters. Finally, the effectiveness of the proposed algorithm is verified by an example on the demand response of energy systems.

This paper presents a topology optimization approach for the form finding of cable networks based on the force density method. The proposed framework combines the force density method with topology optimization based on continuous density variables to identify the equilibrated geometry and effective connectivity of cable structures simultaneously. Design variables are assigned to the cable network members and control their force densities through a SIMP interpolation, complemented by an explicit binary promoting term in the objective function. As a result, cable members vanish or remain active throughout the optimization process. The equilibrium configuration is obtained by enforcing the force density method equations, while the objective drives the optimized network as close as possible to a prescribed reference form. Volume constraints, passive boundary regions, and a minimum connectivity requirement at loaded joints are incorporated to yield physically meaningful and numerically stable designs. Several numerical examples based on cable net ground structures defined by horizontal, vertical, and diagonal members are used to study the proposed approach. A dedicated rounding study shows that the relaxed density variables converge to nearly discrete values at the end of the optimization process, with only a minor correction required after thresholding. A study of the volume fraction constraint further shows that reducing the permitted structural volume can trigger a qualitative transition in the optimized topology, from a redundant, densely braced configuration to a minimal load path. Finally, an assessment of solver performance shows that the proposed formulation scales reasonably well with problem size.

Whittle index policies offer a scalable method for restless multi-armed bandits, but under partial observability even determining the indifference subsidy at a single belief requires solving an infinite-horizon belief-state problem with no closed-form value function. Liu [10] addresses this difficulty by linearizing the unknown decision boundary, leading to a linear system and a closed-form approximate Whittle index. However, the resulting threshold uses only a one-step active–passive comparison and does not account for longer-horizon continuation values. We extend this framework to a \emph{$t$-step lookahead threshold policy}. For each subsidy $m$, the threshold is defined by the active-minus-passive advantage under $t$-step finite-horizon value iteration. At $t=1$, the threshold is $m$-independent and recovers the linear threshold of Liu [10]; for $t>1$, it becomes subsidy-dependent through the induced first-crossing structure and tracks the exact decision boundary more closely. The proposed algorithmic framework does not require indexability as an input and includes an indexability verification. Under the original Whittle indexability, we prove that the $t$-step approximate Whittle index converges geometrically to the exact Whittle index, \[ |\widehat W_t(\omega)-W(\omega)|=O(\beta^t). \] Numerically, all 2,715 tested three-state instances are verified with computable priority index functions according to the proposed criterion. The P95 index error decreases from $2.18\times10^{-2}$ at $t=1$ to $8.93\times10^{-4}$ at $t=8$. In an exact-comparable instance with $\beta=0.9999$, $t=2$ already recovers the exact Whittle-index ordering. Moderate-depth threshold policies also outperform the one-step baseline and remain close to the optimal dynamic-programming benchmark, while runtime grows mildly with $t$.

In this paper, we consider generalized Nash equilibrium (GNE) seeking in non-cooperative games with coupled constraint sets. Specifically, we aim to enforce safety for distributed GNE seeking, whereby the safety specifications are encoded in the coupled constraint set. To achieve this, we introduce the control barrier function (CBF) in the design of the GNE seeking dynamics. We design the dynamics for both full- and partial-information setting, where each player has knowledge of the decision information of all other players or only neighboring players, respectively. We justify the proposed dynamics by showing that the coupled constraint set is forward invariant, the equilibrium of the dynamics coincides with the exact GNE of the game, and the dynamics is asymptotically stable. Furthermore, we extend the approach to games where the agents are multi-integrators. Numerical simulations are provided to verify our results.

Controllability of nonlinear systems has been extensively studied by using the Lie algebra methods. Although local controllability can be proved, global controllability is in general hard to obtain even for bilinear systems. Recently, a new approach is developed to study controllability of bilinear systems by checking whether the reduced rank points corresponding to the bilinear terms form hyperplanes or hypersurfaces, so that near-controllability, in the global sense, can be algebraically verified once no hypersurface exists. In this paper, we show that there is a flaw in the algorithm for checking the reduced rank hypersurface germ condition (RRHGC) that some special cases are overlooked. We thus propose a new algorithm for testing the RRHGC based on a Darboux-polynomial characterization, which can be used to deal with all cases. An example is provided to demonstrate the proposed new algorithm.

This paper studies the existence of relaxed equilibria for finite-horizon continuous-time time-inconsistent mean field games. We work directly on the product space of relaxed feedback policies and population flows. The policy component is endowed with the stable topology of Young measures, while the population component is restricted to a compact convex set of Wasserstein-continuous flows with uniform moment and time-regularity bounds. For every policy-flow pair, we establish uniform Sobolev and Hölder estimates for the associated auxiliary value function and prove its stability under Young-measure convergence of policies and uniform Wasserstein convergence of population flows. We also establish continuity of the induced population-flow map. The latter requires a duality argument for the Fokker-Planck equations because Young-measure convergence yields only weak-$*$ convergence of the controlled drifts. We then construct a set-valued best-response/consistency map with nonempty compact convex values and apply the Kakutani-Fan-Glicksberg fixed-point theorem. The resulting fixed point satisfies both the equilibrium response condition of the intra-personal game and the mean field consistency condition, , thereby establishing the existence of a relaxed equilibrium