This paper presents a first-order distributed algorithm for solving a convex semi-infinite program (SIP) over a time-varying network. In this setting, the objective function associated with the optimization problem is a summation of a set of functions, each held by one node in a network. The semi-infinite constraint, on the other hand, is known to all agents. The nodes collectively aim to solve the problem using local data about the objective and limited communication capabilities depending on the network topology. Our algorithm is built on three key ingredients: consensus step, gradient descent in the local objective, and local gradient descent iterations in the constraint at a node when the estimate violates the semi-infinite constraint. The algorithm is constructed, and its parameters are prescribed in such a way that the iterates held by each agent provably converge to an optimizer. That is, as the algorithm progresses, the estimates achieve consensus, and the constraint violation and the error in the optimal value are bounded above by vanishing terms. Simulation examples illustrate our results.

Recent studies on transportation networks have shown that real-time route guidance can inadvertently induce congestion or oscillatory traffic patterns. Nevertheless, such technologies also offer a promising opportunity to manage traffic non-intrusively by shaping the information delivered to users, thereby mitigating congestion and enhancing network stability. A key step toward this goal is to identify information signals that ensure the existence of an equilibrium with desirable stability and convergence properties. This challenge is particularly relevant when traffic density and routing dynamics evolve concurrently, as increasingly occurs with digital signaling and real-time navigation technologies. To address this, we analyze a parallel-path transportation network with a single origin-destination pair, incorporating joint traffic density and logit-based routing dynamics that evolve at the same timescale. We characterize a class of density-dependent traffic information that guarantees a unique equilibrium in the free-flow regime, ensures its asymptotic stability, and keeps traffic densities within the free-flow region for all time. The theoretical results are complemented by a numerical case study demonstrating how the framework can inform the design of traffic information that reduces total travel time without compromising credibility.

This article proposes an iterative distributionally robust model predictive control (MPC) scheme to solve a risk-constrained infinite-horizon optimal control problem. In each iteration, the algorithm generates a trajectory from the starting point to the desired equilibrium while ensuring safety encoded using risk constraints and improving either the cost or safety relative to previous iterations. At the end of each iteration, the visited states and observed samples of the uncertainty are stored and accumulated with the previous observations. For each iteration, the states stored previously are considered as terminal constraints of the MPC scheme, and samples obtained thus far are used to construct distributionally robust risk constraints. As iterations progress, more data is obtained and the environment is explored progressively to ensure better safety and cost optimality. We prove that the MPC scheme is recursively feasible and the trajectories converge asymptotically to the desired equilibrium while ensuring safety. We identify conditions under which the cost-to-go reduces as iterations progress. We provide computationally tractable reformulations of risk constraints for total variation and Wasserstein-based ambiguity sets with finitely supported distributions. A simulation example illustrates the application of our results in finding a risk-constrained path for two mobile robots facing an uncertain obstacle.

The transition to renewable energy is pivotal for achieving decarbonization goals in the electricity sector. The focus on low-carbon emissions extends beyond the power sector, emphasizing the need for efficient planning for other energy carriers like green hydrogen. However, the intermittency of renewable sources necessitates solutions for operational planning and supply security, making energy storage systems vital for grid services. Battery energy storage systems (BESS) play a key role in future power networks dominated by renewables, as a short-term buffer. This study explores the conditions under which integrating large-scale BESS with hydrogen facilities in the power network enhances overall system profitability. We analyze this by formulating a joint investment-operation decision-making process, modeled as a mixed-integer chance-constrained optimization problem. To address computational challenges, we implement a scalable decomposition-based solution method that enables tractable optimization. Through the optimal solutions, we analyze different incentive strategies and technology cost scenarios to accelerate the kick-off of hydrogen production. By addressing the challenges of intermittency, pricing mechanisms, and optimizing the synergy between BESS and hydrogen systems, this research provides insights for resilient and profitable renewable energy networks, highlighting that hydrogen selling price significantly influences investment decisions and that incentives can accelerate market entry and transition.

This paper introduces hierarchical optimization algorithms to solve, in a data-driven manner, an uncertain resource allocation problem over a two-layered tree network. In this optimization problem, the root of the tree aims to minimize the cost of procuring a certain resource, the demand for which is uncertain and originates at the leaves of the network. The demand is to be met with a certain probability which is represented as chance-constraints. We assume that data regarding the uncertain demand is available at each leaf of the network and we design a general framework of hierarchical optimization procedures where the chance-constrained allocation problem is solved using the available data with the constraint that the data is not transferred to the root. Our hierarchical procedure, termed the abstraction-allocation framework, is adapted to three data-driven algorithms for solving the chance-constrained problem: the scenario method, sample average approximation, and distributionally robust optimization. In our framework, the middle layer of the network facilitates the abstraction and allocation when information flows from leaves to the root and the other way around, respectively.

This paper considers a security constrained dispatch problem involving generation and line contingencies in the presence of the renewable generation. The uncertainty due to renewables is modeled using joint chance-constraint and the mismatch caused by contingencies and renewables are handled using reserves. We consider a distributionally robust approach to solve the chance-constrained program. We assume that samples of the uncertainty are available. Using them, we construct a set of distributions, termed ambiguity set, containing all distributions that are close to the empirical distribution under the Wasserstein metric. The chance constraint is imposed for all distributions in the ambiguity set to form the distributionally robust optimization problem. This problem is nonconvex and computationally heavy to solve exactly. We adopt a two-step approach to find an approximate solution. In the first step, we construct a polyhedral set in the space of uncertainty that contains enough mass under all distributions in the ambiguity set. This set is constructed by solving several two-dimensional distributionally robust problems. In the second step, we solve a linear robust optimization problem where the uncertain constraint is imposed for all uncertainty values lying in the polyhedral set. We demonstrate the scalability and robustness of our method using numerical experiments.

This paper considers Bayesian persuasion for routing games where information about the uncertain state of the network is provided by a traffic information system (TIS) using public signals. In this setup, the TIS commits to a signalling scheme and participants form a posterior belief about the state of the network based on prior beliefs and received signal. They subsequently select routes minimizing their individual expected travel time under their posterior beliefs, giving rise to a Wardrop equilibrium. We investigate how the TIS can infer the prior held by the participants by designing suitable signalling schemes, and observing the equilibrium flows under different signals. We provide an iterative algorithm that finds such a scheme in a finite number of steps.

This paper studies the problem of intervention design for steering the actions of noncooperative players in quadratic network games to the social optimum. The players choose their actions with the aim of maximizing their individual payoff functions, while a central regulator uses interventions to modify their marginal returns and maximize the social welfare function. This work builds on the key observation that the solution to the steering problem depends on the knowledge of the regulator on the players' parameters and the underlying network. We, therefore, consider different scenarios based on limited knowledge and propose suitable static, dynamic and adaptive intervention protocols. We formally prove convergence to the social optimum under the proposed mechanisms. We demonstrate our theoretical findings on a case study of Cournot competition with differentiated goods.

Apprenticeship learning is a framework in which an agent learns a policy to perform a given task in an environment using example trajectories provided by an expert. In the real world, one might have access to expert trajectories in different environments where system dynamics is different while the learning task is the same. For such scenarios, two types of learning objectives can be defined. One where the learned policy performs very well in one specific environment and another when it performs well across all environments. To balance these two objectives in a principled way, our work presents the cross apprenticeship learning (CAL) framework. This consists of an optimization problem where an optimal policy for each environment is sought while ensuring that all policies remain close to each other. This nearness is facilitated by one tuning parameter in the optimization problem. We derive properties of the optimizers of the problem as the tuning parameter varies. We identify conditions under which an agent prefers using the policy obtained from CAL over the traditional apprenticeship learning. Since the CAL problem is nonconvex, we provide a convex outer approximation. Finally, we demonstrate the attributes of our framework in the context of a navigation task in a windy gridworld environment.

This paper focuses on a class of variational inequalities (VIs), where the map defining the VI is given by the component-wise conditional value-at-risk (CVaR) of a random function. We focus on solving the VI using sample average approximation, where solutions of the VI are estimated with solutions of a sample average VI that uses empirical estimates of the CVaRs. We establish two properties for this scheme. First, under continuity of the random map and the uncertainty taking values in a bounded set, we prove asymptotic consistency, establishing almost sure convergence of the solution of the sample average problem to the true solution. Second, under the additional assumption of random functions being Lipschitz, we prove exponential convergence where the probability of the distance between an approximate solution and the true solution being smaller than any constant approaches unity exponentially fast. The exponential decay bound is refined for the case where random functions have a specific separable form in the decision variable and uncertainty. We adapt these results to the case of uncertain routing games and derive explicit sample guarantees for obtaining a CVaR-based Wardrop equilibria using the sample average procedure. We illustrate our theoretical findings by approximating the CVaR-based Wardrop equilibria for a modified Sioux Falls network.

This paper considers a risk-constrained motion planning problem and aims to find the solution combining the concepts of iterative model predictive control (MPC) and data-driven distributionally robust (DR) risk-constrained optimization. In the iterative MPC, at each iteration, safe states visited and stored in the previous iterations are imposed as terminal constraints. Furthermore, samples collected during the iteration are used in the subsequent iterations to tune the ambiguity set of the DR constraints employed in the MPC. In this method, the MPC problem becomes computationally burdensome when the iteration number goes high. To overcome this challenge, the emphasis of this paper is to reduce the realtime computational effort using two approximations. First one involves clustering of data at the beginning of each iteration and modifying the ambiguity set for the MPC scheme so that safety guarantees still holds. The second approximation considers determining DR-safe regions at the start of iteration and constraining the state in the MPC scheme to such safe sets. We analyze the computational tractability of these approximations and present a simulation example that considers path planning in the presence of randomly moving obstacle.

This work explores the relationship between the set of Wardrop equilibria~(WE) of a routing game, the total demand of that game, and the occurrence of Braess's paradox~(BP). The BP formalizes the counter-intuitive fact that for some networks, removing a path from the network decreases congestion at WE. For a single origin-destination routing games with affine cost functions, the first part of this work provides tools for analyzing the evolution of the WE as the demand varies. It characterizes the piece-wise affine nature of this dependence by showing that the set of directions in which the WE can vary in each piece is the solution of a variational inequality problem. In the process we establish various properties of changes in the set of used and minimal-cost paths as demand varies. As a consequence of these characterizations, we derive a procedure to obtain the WE for all demands above a certain threshold. The second part of the paper deals with detecting the presence of BP in a network. We supply a number of sufficient conditions that reveal the presence of BP and that are computationally tractable. We also discuss a different perspective on BP, where we establish that a path causing BP at a particular demand must be strictly beneficial to the network at a lower demand. Several examples throughout this work illustrate and elaborate our findings.

This paper focuses on solving a data-driven distributionally robust optimization problem over a network of agents. The agents aim to minimize the worst-case expected cost computed over a Wasserstein ambiguity set that is centered at the empirical distribution. The samples of the uncertainty are distributed across the agents. Our approach consists of reformulating the problem as a semi-infinite program and then designing a distributed algorithm that solves a generic semi-infinite problem that has the same information structure as the reformulated problem. In particular, the decision variables consist of both local ones that agents are free to optimize over and global ones where they need to agree on. Our distributed algorithm is an iterative procedure that combines the notions of distributed ADMM and the cutting-surface method. We show that the iterates converge asymptotically to a solution of the distributionally robust problem to any pre-specified accuracy. Simulations illustrate our results.

We propose an approach to find the optimal value of a convex semi-infinite program (SIP) that involves identifying a finite set of relevant constraints by solving a finite-dimensional global maximization problem. One of the major advantages of our approach is that it admits a plug-and-play module where any suitable global optimization algorithm can be employed to obtain the optimal value of the SIP. As an example, we propose a simulated annealing based algorithm which is useful especially when the constraint index set is high-dimensional. A proof of convergence of the algorithm is included, and the performance and accuracy of the algorithm itself are illustrated on several benchmark SIPs lifted from the literature.

This paper considers a risk-constrained infinite-horizon optimal control problem and proposes to solve it in an iterative manner. Each iteration of the algorithm generates a trajectory from the starting point to the target equilibrium state by implementing a distributionally robust risk-constrained model predictive control (MPC) scheme. At each iteration, a set of safe states (that satisfy the risk-constraint with high probability) and a certain number of samples of the uncertainty governing the risk constraint are available. These states and samples are accumulated in previous iterations. The safe states are used as terminal constraint in the MPC scheme and samples are used to construct a set of distributions, termed ambiguity set, such that it contains the underlying distribution of the uncertainty with high probability. The risk-constraint in each iteration is required to hold for all distributions in the ambiguity set. We establish that the trajectories generated by our iterative procedure are feasible, safe, and converge asymptotically to the equilibrium. Simulation example illustrates our results for the case of finding a risk-constrained path for a mobile robot in the presence of an uncertain obstacle.

This paper presents an iterative way of computing a control algorithm with the aim of enabling reference tracking for an unknown nonlinear system. The method consists of three blocks: iterative learning control (ILC), robust model predictive control (MPC), and a linear approximation of the Koopman operator. The method proceeds in iterations, where at the end of an iteration, two steps are performed. First, the trajectories of the system obtained from previous iterations are used to build the linear approximation of the Koopman operator. Second, the linear model is used to compute the ILC signal. While these steps are executed in an offline manner, during an iteration, the control actions are computed online using the robust tubebased MPC. The tubes are defined by constraint tightening sets that compensate for the discrepancy between the true dynamics and its linear approximation. We demonstrate our method on the reference tracking for a 4 tank system.

This paper considers the problem of steering the aggregative behavior of a population of noncooperative price-taking agents towards a desired behavior. Different from conventional pricing schemes where the price is fully available for design, we consider the scenario where a system regulator broadcasts a price prediction signal that can be different from the actual price incurred by the agents. The resulting reliability issues are taken into account by including trust dynamics in our model, implying that the agents will not blindly follow the signal sent by the regulator, but rather follow it based on the history of its accuracy, i.e, its deviation from the actual price. We present several nudge mechanisms to generate suitable price prediction signals that are able to steer the aggregative behavior of the agents to stationary as well as temporal desired aggregative behaviors. We provide analytical convergence guarantees for the resulting multi-components models. In particular, we prove that the proposed nudge mechanisms earn and maintain full trust of the agents, and the aggregative behavior converges to the desired one. The analytical results are complemented by a numerical case study of coordinated charging of plug-in electric vehicles.

We consider the problem of look-ahead economic dispatch (LAED) with uncertain renewable energy generation. The goal of this problem is to minimize the cost of conventional energy generation subject to uncertain operational constraints. The risk of violating these constraints must be below a given threshold for a family of probability distributions with characteristics similar to observed past data or predictions. We present two data-driven approaches based on two novel mathematical reformulations of this distributionally robust decision problem. The first one is a tractable convex program in which the uncertain constraints are defined via the distributionally robust conditional-value-at-risk. The second one is a scalable robust optimization program that yields an approximate distributionally robust chance-constrained LAED. Numerical experiments on the IEEE-39 bus system with real solar production data and forecasts illustrate the effectiveness of these approaches. We discuss how system operators should tune these techniques in order to seek the desired robustness-performance trade-off and we compare their computational scalability.

This paper considers Bayesian persuasion for routing games where information about the uncertain state of the network is provided by a traffic information system (TIS) using public signals. In this setup, the TIS commits to a signalling scheme and participants form a posterior belief about the state of the network based on prior beliefs and the received signal. They subsequently select routes minimizing their individual expected travel time under their posterior beliefs, giving rise to a Wardrop equilibrium. We investigate how the TIS can infer the prior beliefs held by the participants by designing suitable signalling schemes, and observing the equilibrium flows under different signals. We show that under mild conditions a signalling scheme that allows for exact inference of the prior exists. We then provide an iterative algorithm that finds such a scheme in a finite number of steps. We show that schemes designed by our algorithm are robust, in the sense that they can still identify the prior after a small enough perturbation. We also investigate the case where the population is divided among multiple priors, and give conditions under which the fraction associated to each prior can be identified. Several examples illustrate our results.

We consider the problem of steering the actions of noncooperative players in quadratic network games to the social optimum. To this end, a central regulator modifies the marginal returns of the players, while the players’ strategies are determined by continuous pseudo-gradient dynamics. Depending on the available information on the players parameters and network quantities, suitable static and dynamic intervention protocols are devised that maximize the social welfare at steady-state. We show that adaptive interventions can compensate for the lack of knowledge on network topology and coupling weights. Numerical examples are provided to demonstrate the effectiveness of the proposed interventions.