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Control barrier functions (CBFs) offer an efficient framework for designing real-time safe controllers. However, CBF-based controllers can be short-sighted, resulting in poor performance, a behaviour which is aggravated in uncertain conditions. This motivated research on safety filters based on model predictive control (MPC) and its stochastic variant. MPC deals with safety constraints in a direct manner, however, its computational demands grow with the prediction horizon length. We propose a safety formulation that solves a finite horizon optimization problem at each time instance like MPC, but rather than explicitly imposing constraints along the prediction horizon, we enforce probabilistic safety constraints by means of CBFs only at the first step of the horizon. The probabilistic CBF constraints are transformed in a finite number of deterministic CBF constraints via the scenario based methodology. Capitalizing on results on scenario based MPC, we provide distribution-free, \emph{a priori} guarantees on the system's closed loop expected safety violation frequency. We demonstrate our results through a case study on unmanned aerial vehicle collision-free position swapping, and provide a numerical comparison with recent stochastic CBF formulations., Comment: To be published in: The 63rd IEEE Conference on Decision and Control (CDC-2024 Milano, Italy)

We consider a moving target that we seek to learn from samples. Our results extend randomized techniques developed in control and optimization for a constant target to the case where the target is changing. We derive a novel bound on the number of samples that are required to construct a probably approximately correct (PAC) estimate of the target. Furthermore, when the moving target is a convex polytope, we provide a constructive method of generating the PAC estimate using a mixed integer linear program (MILP). The proposed method is demonstrated on an application to autonomous emergency braking., Comment: 13 pages, 7 figures

In this paper, we propose a quadratic programming-based filter for safe and stable controller design, via a Control Barrier Function (CBF) and a Control Lyapunov Function (CLF). Our method guarantees safety and local asymptotic stability without the need for an asymptotically stabilizing control law. Feasibility of the proposed program is ensured under a mild regularity condition, termed relaxed compatibility between the CLF and CBF. The resulting optimal control law is guaranteed to be locally Lipschitz continuous. We also analyze the closed-loop behaviour by characterizing the equilibrium points, and verifying that there are no equilibrium points in the interior of the control invariant set except at the origin. For a polynomial system and a semi-algebraic safe set, we provide a sum-of-squares program to design a relaxed compatible pair of CLF and CBF. The proposed approach is compared with other methods in the literature using numerical examples, exhibits superior filter performance and guarantees safety and local stability.

Decision making and learning in the presence of uncertainty has attracted significant attention in view of the increasing need to achieve robust and reliable operations. In the case where uncertainty stems from the presence of adversarial attacks this need is becoming more prominent. In this paper we focus on linear and nonlinear classification problems and propose a novel adversarial training method for robust classifiers, inspired by Support Vector Machine (SVM) margins. We view robustness under a data driven lens, and derive finite sample complexity bounds for both linear and non-linear classifiers in binary and multi-class scenarios. Notably, our bounds match natural classifiers' complexity. Our algorithm minimizes a worst-case surrogate loss using Linear Programming (LP) and Second Order Cone Programming (SOCP) for linear and non-linear models. Numerical experiments on the benchmark MNIST and CIFAR10 datasets show our approach's comparable performance to state-of-the-art methods, without needing adversarial examples during training. Our work offers a comprehensive framework for enhancing binary linear and non-linear classifier robustness, embedding robustness in learning under the presence of adversaries.

We study the problem of co-designing control barrier functions (CBF) and linear state feedback controllers for continuous-time linear systems. We achieve this by means of a single semi-definite optimization program. Our formulation can handle mixed-relative degree problems without requiring an explicit safe controller. Different L-norm based input limitations can be introduced as convex constraints in the proposed program. We demonstrate our results on an omni-directional car numerical example., Comment: manuscript submitted to TAC

We study coalitional games with exogenous uncertainty in the coalition value, in which each agent is allowed to have private samples of the uncertainty. As a consequence, the agents may have a different perception of stability of the grand coalition. In this context, we propose a novel methodology to study the out-of-sample coalitional rationality of allocations in the set of stable allocations (i.e., the core). Our analysis builds on the framework of probably approximately correct learning. Initially, we state a priori and a posteriori guarantees for the entire core. Furthermore, we provide a distributed algorithm to compute a compression set that determines the generalization properties of the a posteriori statements. We then refine our probabilistic robustness bounds by specialising the analysis to a single payoff allocation, taking, also in this case, both a priori and a posteriori approaches. Finally, we consider a relaxed $\zeta$-core to include nearby allocations and also address the case of empty core. For this case, probabilistic statements are given on the eventual stability of allocations in the $\zeta$-core.

Synthesising verifiably correct controllers for dynamical systems is crucial for safety-critical problems. To achieve this, it is important to account for uncertainty in a robust manner, while at the same time it is often of interest to avoid being overly conservative with the view of achieving a better cost. We propose a method for verifiably safe policy synthesis for a class of finite state models, under the presence of structural uncertainty. In particular, we consider uncertain parametric Markov decision processes (upMDPs), a special class of Markov decision processes, with parameterised transition functions, where such parameters are drawn from a (potentially) unknown distribution. Our framework leverages recent advancements in the so-called scenario approach theory, where we represent the uncertainty by means of scenarios, and provide guarantees on synthesised policies satisfying probabilistic computation tree logic (PCTL) formulae. We consider several common benchmarks/problems and compare our work to recent developments for verifying upMDPs., Comment: 10 pages, accepted for oral presentation at L4DC

In this paper, we evaluate the different fully homomorphic encryption schemes, propose an implementation, and numerically analyze the applicability of gradient descent algorithms to solve quadratic programming in a homomorphic encryption setup. The limit on the multiplication depth of homomorphic encryption circuits is a major challenge for iterative procedures such as gradient descent algorithms. Our analysis not only quantifies these limitations on prototype examples, thus serving as a benchmark for future investigations, but also highlights additional trade-offs like the ones pertaining the choice of gradient descent or accelerated gradient descent methods, opening the road for the use of homomorphic encryption techniques in iterative procedures widely used in optimization based control. In addition, we argue that, among the available homomorphic encryption schemes, the one adopted in this work, namely CKKS, is the only suitable scheme for implementing gradient descent algorithms. The choice of the appropriate step size is crucial to the convergence of the procedure. The paper shows firsthand the feasibility of homomorphically encrypted gradient descent algorithms.

In this paper we focus on noncooperative games with uncertain constraints coupling the agents' decisions. We consider a setting where bounded deviations of agents' decisions from the equilibrium are possible, and uncertain constraints are inferred from data. Building upon recent advances in the so called scenario approach, we propose a randomised algorithm that returns a nominal equilibrium such that a pre-specified bound on the probability of violation for yet unseen constraints is satisfied for an entire region of admissible deviations surrounding it, thus supporting neighbourhoods of equilibria with probabilistic feasibility certificates. For the case in which the game admits a potential function, whose minimum coincides with the social welfare optimum of the population, the proposed algorithmic scheme opens the road to achieve a trade-off between the guaranteed feasibility levels of the region surrounding the nominal equilibrium, and its system-level efficiency. Detailed numerical simulations corroborate our theoretical results.

We consider multi-agent coalitional games with uncertainty in the coalitional values. We provide a novel methodology to study the stability of the grand coalition in the case where each coalition constructs ambiguity sets for the (possibly) unknown probability distribution of the uncertainty. As a less conservative solution concept compared to worst-case approaches for coalitional stability, we consider a stochastic version of the so-called core set, i.e., the expected value core. Unfortunately, without exact knowledge of the probability distribution, the evaluation of the expected value core is an extremely challenging task. Hence, we propose the concept of distributionaly robust (DR) core. Leveraging tools from data-driven DR optimization under the Wasserstein distance, we provide finite-sample guarantees that any allocation which lies in the DR core is also stable with respect to the true probability distribution. Furthermore, we show that as the number of samples grows unbounded, the DR core converges almost surely to the true expected value core. We dedicate the last section to the computational tractability of finding an allocation in the DR core., Comment: Accepted for publication at the IEEE Conference on Decision and Control 2023

We propose distributed iterative algorithms for safe control design and safety verification for networked multi-agent systems. These algorithms rely on distributing a control barrier function (CBF) related quadratic programming (QP) problem. The proposed distributed algorithm addresses infeasibility issues of existing schemes by dynamically allocating auxiliary variables across iterations. The resulting control input is guaranteed to be optimal, and renders the system safe. Furthermore, a truncated algorithm is proposed to facilitate computational implementation. The performance of the truncated algorithm is evaluated using a distributed safety verification algorithm. The algorithm quantifies safety for a multi-agent system probabilistically, using a certain locally Lipschitz continuous feedback controller by means of CBFs. Both upper and lower bounds on the probability of safety are obtained using the so called scenario approach. Both the scenario sampling and safety verification procedures are fully distributed. The efficacy of our algorithms is demonstrated by an example on multi-robot collision avoidance., Comment: manuscript submitted to Automatica

We propose a distributed algorithm to solve a dynamic programming problem with multiple agents, where each agent has only partial knowledge of the state transition probabilities and costs. We provide consensus proofs for the presented algorithm and derive error bounds of the obtained value function with respect to what is considered as the "true solution" obtained from conventional value iteration. To minimize communication overhead between agents, state costs are aggregated and shared between agents only when the updated costs are expected to influence the solution of other agents significantly. We demonstrate the efficacy of the proposed distributed aggregation method to a large-scale urban traffic routing problem. Individual agents compute the fastest route to a common access point and share local congestion information with other agents allowing for fully distributed routing with minimal communication between agents., Comment: 6 pages, 4 figures

One of the fundamental problems in spacecraft trajectory design is finding the optimal transfer trajectory that minimizes the propellant consumption and transfer time simultaneously. We formulate this as a multi-objective optimal control (MOC) problem that involves optimizing over the initial or final state, subject to state constraints. Drawing on recent developments in reachability analysis subject to state constraints, we show that the proposed MOC problem can be stated as an optimization problem subject to a constraint that involves the sub-level set of the viscosity solution of a quasi-variational inequality. We then generalize this approach to account for more general optimal control problems in Bolza form. We relate these problems to the Pareto front of the developed multi-objective programs. The proposed approach is demonstrated on two low-thrust orbital transfer problems around a rotating asteroid., Comment: Paper published in European Journal of Control, 28 pages, 7 figures

A fundamental concern in progressing Airborne Wind Energy (AWE) operations towards commercial success, is guaranteeing that safety requirements placed on the systems are met. Due to the high dimensional complexity of AWE systems, however, formal mathematical robustness guarantees become difficult to compute. We draw on research from Hamilton-Jacobi (HJ) reachability analysis to compute the optimal control policy for tracking a flight path, while enforcing safety constraints on the system. In addition, the zero-sublevel set of the computed value function inherent in HJ reachability analysis indicates the backward reachable set, the set of states from which it is possible to safely drive the system into a target set within a given time without entering undesirable states. Furthermore, we derive a switching law, such that the safety controller can be used in conjunction with arbitrary least restrictive controllers to provide a safe hybrid control law. In such a setup, the safety controller is only activated when the system approaches the boundary of its maneuverability envelope. Such a hybrid control law is a notable improvement over existing robust control approaches that deteriorate performance by assuming the worst-case environmental and system behavior at all times. We illustrate our results via extensive simulation-based studies., Comment: 30 pages, 15 figures

We propose fully-distributed algorithms for Nash equilibrium seeking in aggregative games over networks. We first consider the case where local constraints are present and we design an algorithm combining, for each agent, (i) the projected pseudo-gradient descent and (ii) a tracking mechanism to locally reconstruct the aggregative variable. To handle coupling constraints arising in generalized settings, we propose another distributed algorithm based on (i) a recently emerged augmented primal-dual scheme and (ii) two tracking mechanisms to reconstruct, for each agent, both the aggregative variable and the coupling constraint satisfaction. Leveraging tools from singular perturbations analysis, we prove linear convergence to the Nash equilibrium for both schemes. Finally, we run extensive numerical simulations to confirm the effectiveness of our methods and compare them with state-of-the-art distributed equilibrium-seeking algorithms.

Safety is closely related to set invariance for dynamical systems. However, synthesizing a safe invariant set and at the same time synthesizing the associated safe controller still remains challenging. In this note we introduce a simple invariance-based method for linear systems with safety guarantee. The proposed method uses sum-of-squares programming., Comment: submitted to POEMA as an extended abstract

The existence of multiple irregular obstacles in the environment introduces nonconvex constraints into the optimization for motion planning, which makes the optimal control problem hard to handle. One efficient approach to address this issue is Successive Convex Approximation (SCA), where the nonconvex problem is convexified and solved successively. However, this approach still faces two main challenges: I) infeasibility, caused by linearisation about infeasible reference points; ii) high computational complexity incurred by multiple constraints, when solving the optimal control problem with a long planning horizon and multiple obstacles. To overcome these challanges, this paper proposes an energy efficient safetyaware control method for motion planning with low computing complexity and address these challenges. Specifically, a control barrier function-based linear quadratic regulator is formulated for the motion planning to guarantee safety and energy efficiency. Then, to avoid infeasibility, Backward Receding SCA (BRSCA) approach with a dynamic constraints-selection rule is proposed. Dynamic programming with primal-dual iteration is designed to decrease computational complexity. It is found that BRSCA is applicable to time-varying control limits. Numerical simulations and hardware experiments vevify the efficiency of BRSCA. Simulations demonstrates that BRSCA has a higher probability of finding feasible solutions, reduces the computation time by about 17.4% and the energy cost by about four times compared to other methods in the literature.

The task assignment problem is fundamental in combinatorial optimisation, aiming at allocating one or more tasks to a number of agents while minimizing the total cost or maximizing the overall assignment benefit. This problem is known to be computationally hard since it is usually formulated as a mixed-integer programming problem. In this paper, we consider a novel time-triggered dimension reduction algorithm (TTDRA). We propose convexification approaches to convexify both the constraints and the cost function for the general non-convex assignment problem. The computational speed is accelerated via our time-triggered dimension reduction scheme, where the triggering condition is designed based on the optimality tolerance and the convexity of the cost function. Optimality and computational efficiency are verified via numerical simulations on benchmark examples.

In this paper we consider the safety verification and safe controller synthesis problems for nonlinear control systems. The Control Barrier Certificates (CBC) approach is proposed as an extension to the Barrier certificates approach. Our approach can be used to characterize the control invariance of a given set in terms of safety of a general nonlinear control system subject to input constraints. From the point of view of controller design, the proposed method provides an approach to synthesize a safe control law that guarantees that the trajectories of the system starting from a given initial set do not enter an unsafe set. Unlike the related control Barrier functions approach, our formulation only considers the vector field within the tangent cone of the zero level set defined by the certificates, and is shown to be less conservative by means of numerical evidence. For polynomial systems with semi-algebraic initial and safe sets, CBCs and safe control laws can be synthesized using sum-of-squares decomposition and semi-definite programming. Examples demonstrate our method.

The control Barrier function approach has been widely used for safe controller synthesis. By solving an online convex quadratic programming problem, an optimal safe controller can be synthesized implicitly in state-space. Since the solution is unique, the mapping from state-space to control inputs is injective, thus enabling us to evaluate the underlying relationship. In this paper we aim at explicitly synthesizing a safe control law as a function of the state for nonlinear control-affine systems with limited control ability. We propose to transform the online quadratic programming problem into an offline parameterized optimisation problem which considers states as parameters. The obtained explicit safe controller is shown to be a piece-wise Lipschitz continuous function over the partitioned state space if the program is feasible. We address the infeasible cases by solving a parameterized adaptive control Barrier function-based quadratic programming problem. Extensive simulation results show the state-space partition and the controller properties.

Hamilton-Jacobi reachability methods for safety-critical control have been well studied, but the safety guarantees derived rely on the accuracy of the numerical computation. Thus, it is crucial to understand and account for any inaccuracies that occur due to uncertainty in the underlying dynamics and environment as well as the induced numerical errors. To this end, we propose a framework for modeling the error of the value function inherent in Hamilton-Jacobi reachability using a Gaussian process. The derived safety controller can be used in conjuncture with arbitrary controllers to provide a safe hybrid control law. The marginal likelihood of the Gaussian process then provides a confidence metric used to determine switches between a least restrictive controller and a safety controller. We test both the prediction as well as the correction capabilities of the presented method in a classical pursuit-evasion example., Comment: Paper published in 2022 European Control Conference (ECC), 6 pages, 4 figures

In this paper we consider multi-agent coalitional games with uncertain value functions for which we establish distribution-free guarantees on the probability of allocation stability, i.e., agents do not have incentives to defect from the grand coalition to form subcoalitions for unseen realizations of the uncertain parameter. In case the set of stable allocations, the so called core of the game, is empty, we propose a randomized relaxation of the core. We then show that those allocations that belong to this relaxed set can be accompanied by stability guarantees in a probably approximately correct fashion. Finally, numerical experiments corroborate our theoretical findings.

We consider a multi-agent optimal resource sharing problem that is represented by a linear program. The amount of resource to be shared is fixed, and agents belong to a population that is characterized probabilistically so as to allow heterogeneity among the agents. In this paper, we provide a characterization of the probability that the arrival of a new agent affects the resource share of other agents, which means that accommodating the new agent request at the detriment of the other agents allocation provides some payoff. This probability represents a sensitivity index for the optimal solution of a linear programming resource sharing problem when a new agent shows up, and it is of fundamental importance for a correct and profitable operation of the multi-agent system. Our developments build on the equivalence between the resource sharing problem and certain dual reformulations which can be interpreted as scenario programs with the number of scenarios corresponding to the number of agents in the primal problem. The recent "wait-and-judge" scenario approach is then used to obtain the sought sensitivity index. Our theoretical findings are demonstrated through a numerical example on optimal cargo aircraft loading., Comment: 14 pages, 3 figures

We propose a reachability approach for infinite and finite horizon multi-objective optimization problems for low-thrust spacecraft trajectory design. The main advantage of the proposed method is that the Pareto front can be efficiently constructed from the zero level set of the solution to a Hamilton-Jacobi-Bellman equation. We demonstrate the proposed method by applying it to a low-thrust spacecraft trajectory design problem. By deriving the analytic expression for the Hamiltonian and the optimal control policy, we are able to efficiently compute the backward reachable set and reconstruct the optimal trajectories. Furthermore, we show that any reconstructed trajectory will be guaranteed to be weakly Pareto optimal. The proposed method can be used as a benchmark for future research of applying reachability analysis to low-thrust spacecraft trajectory design., Comment: Extended version of the paper published in 2021 European Control Conference (ECC), 9 pages, 2 figures

We provide out-of-sample certificates on the controlled invariance property of a given set with respect to a class of black-box linear systems. Specifically, we consider linear time-invariant models whose state space matrices are known only to belong to a certain family due to a possibly inexact quantification of some parameters. By exploiting a set of realizations of those undetermined parameters, verifying the controlled invariance property of the given set amounts to a linear program, whose feasibility allows us to establish an a-posteriori probabilistic certificate on the controlled invariance property of such a set with respect to the nominal linear time-invariant dynamics. The proposed framework is applied to the control of a networked multi-agent system with unknown weighted graph.

We investigate the probabilistic feasibility of randomized solutions to two distinct classes of uncertain multi-agent optimization programs. We first assume that only the constraints of the program are affected by uncertainty, while the cost function is arbitrary. Leveraging recent a posteriori developments of the scenario approach, we provide probabilistic guarantees for all feasible solutions of the program under study. This result is particularly useful in cases where numerical difficulties related to the convergence of the solution-seeking algorithm hinder the exact quantification of the optimal solution. Furthermore, it can be applied to cases where the agents' incentives lead to a suboptimal solution, e.g., under a non-cooperative setting. We then focus on optimization programs where the cost function admits an aggregate representation and depends on uncertainty while constraints are deterministic. By exploiting the structure of the program under study and leveraging the so called support rank notion, we provide agent-independent robustness certificates for the optimal solution, i.e., the constructed bound on the probability of constraint violation does not depend on the number of agents, but only on the dimension of the agents' decision. This substantially reduces the number of samples required to achieve a certain level of probabilistic robustness as the number of agents increases. All robustness certificates provided in this paper are distribution-free and can be used alongside any optimization algorithm. Our theoretical results are accompanied by a numerical case study involving a charging control problem of a fleet of electric vehicles.

We consider the revenue management problem of finding profit-maximising prices for delivery time slots in the context of attended home delivery. This multi-stage optimal control problem admits a dynamic programming formulation that is intractable for realistic problem sizes due to the so-called "curse of dimensionality". Therefore, we study three approximate dynamic programming algorithms both from a control-theoretical perspective and in a parametric numerical case study. Our numerical analysis is based on real-world data, from which we generate multiple scenarios to stress-test the robustness of the pricing policies to errors in model parameter estimates. Our theoretical analysis and numerical benchmark tests show that one of these algorithms, namely gradient-bounded dynamic programming, dominates the others with respect to computation time and profit-generation capabilities of the delivery slot pricing policies that it generates. Finally, we show that uncertainty in the estimates of the model parameters further increases the profit-generation dominance of this approach., Comment: 13 pages, 7 figures

We consider stochastic dynamic programming problems with high-dimensional, discrete state-spaces and finite, discrete-time horizons that prohibit direct computation of the value function from a given Bellman equation for all states and time steps due to the "curse of dimensionality". For the case where the value function of the dynamic program is concave extensible and submodular in its state-space, we present a new algorithm that computes deterministic upper and stochastic lower bounds of the value function in the realm of dual dynamic programming. We show that the proposed algorithm terminates after a finite number of iterations. Furthermore, we derive probabilistic guarantees on the value accumulated under the associated policy for a single realisation of the dynamic program and for the expectation of this value. Finally, we demonstrate the efficacy of our approach on a high-dimensional numerical example from delivery slot pricing in attended home delivery., Comment: 12 pages, 4 figures. arXiv admin note: substantial text overlap with arXiv:2005.11213

We consider dynamic programming problems with finite, discrete-time horizons and prohibitively high-dimensional, discrete state-spaces for direct computation of the value function from the Bellman equation. For the case that the value function of the dynamic program is concave extensible and submodular in its state-space, we present a new algorithm that computes deterministic upper and stochastic lower bounds of the value function similar to dual dynamic programming. We then show that the proposed algorithm terminates after a finite number of iterations. Finally, we demonstrate the efficacy of our approach on a high-dimensional numerical example from delivery slot pricing in attended home delivery., Comment: 6 pages, 2 figures, accepted for IFAC World Congress 2020

We address the problem of assessing the robustness of the equilibria in uncertain, multi-agent games. Specifically, we focus on generalized Nash equilibrium problems in aggregative form subject to linear coupling constraints affected by uncertainty with a possibly unknown probability distribution. Within a data-driven context, we apply the scenario approach paradigm to provide a-posteriori feasibility certificates for the entire set of generalized Nash equilibria of the game. Then, we show that assessing the violation probability of such set merely requires to enumerate the constraints that ``shape'' it. For the class of aggregative games, this results in solving a feasibility problem on each active facet of the feasibility region, for which we propose a semi-decentralized algorithm. We demonstrate our theoretical results by means of an academic example.

We develop a data-driven approach to the computation of a-posteriori feasibility certificates to the solution sets of variational inequalities affected by uncertainty. Specifically, we focus on instances of variational inequalities with a deterministic mapping and an uncertain feasibility set, and represent uncertainty by means of scenarios. Building upon recent advances in the scenario approach literature, we quantify the robustness properties of the entire set of solutions of a variational inequality, with feasibility set constructed using the scenario approach, against a new unseen realization of the uncertainty. Our results extend existing results that typically impose an assumption that the solution set is a singleton and require certain non-degeneracy properties, and thereby offer probabilistic feasibility guarantees to any feasible solution. We show that assessing the violation probability of an entire set of solutions, rather than of a singleton, requires enumeration of the support constraints that "shape" this set. Additionally, we propose a general procedure to enumerate the support constraints that does not require a closed form description of the solution set, which is unlikely to be available. We show that robust game theory problems can be modelling via uncertain variational inequalities, and illustrate our theoretical results through extensive numerical simulations on a case study involving an electric vehicle charging coordination problem.

We revisit the so-called sampling and discarding approach used to quantify the probability of constraint violation of a solution to convex scenario programs when some of the original samples are allowed to be discarded. Motivated by two scenario programs that possess analytic solutions and the fact that the existing bound for scenario programs with discarded constraints is not tight, we analyze a removal scheme that consists of a cascade of optimization problems, where at each step we remove a superset of the active constraints. By relying on results from compression learning theory, we show that such a removal scheme leads to less conservative bounds for the probability of constraint violation than the existing ones. We also show that the proposed bound is tight by characterizing a class of optimization problems that achieves the given upper bound. The performance improvement of the proposed methodology is illustrated by an example that involves a resource sharing linear program.

In this paper a distribution-free methodology is presented for providing robustness guarantees for Nash equilibria (NE) of multi-agent games. Leveraging recent a posteriori developments of the so called scenario approach (Campi et al., 2018), we provide probabilistic guarantees for feasibility problems with polytopic constraints. This result is then used in the context of multi-agent games, allowing to provide robustness certificates for constraint violation of any NE of a given game. Our guarantees can be used alongside any NE seeking algorithm that returns some equilibrium solution. Finally, by exploiting the structure of our problem, we circumvent the need of employing computationally prohibitive algorithms to find an irreducible support subsample, a concept at the core of the scenario approach. Our theoretical results are accompanied by simulation studies that investigate the robustness of the solutions of two different problems, namely, a 2-dimensional feasibility problem and an electric vehicle (EV) charging control problem.

We deal with the problem of energy management in buildings subject to uncertain occupancy. To this end, we formulate this as a finite horizon optimization program and optimize with respect to the windows' blinds position, radiator and cooling flux. Aiming at a schedule which is robust with respect to uncertain occupancy levels while avoiding imposing arbitrary assumptions on the underlying probability distribution of the uncertainty, we follow a data driven paradigm. In particular, we apply an incremental scenario approach methodology that has been recently proposed in the literature to our energy management formulation. To demonstrate the efficacy of the proposed implementation we provide a detailed numerical analysis on a stylized building and compare it with respect to a deterministic design and the standard scenario approach typically encountered in the literature. We show that our schedule is not agnostic with respect to uncertainty as deterministic approaches, while it requires fewer scenarios with respect to the standard scenario approach, thus resulting in a less conservative performance.

We study the dynamic programming approach to revenue management in the context of attended home delivery. We draw on results from dynamic programming theory for Markov decision problems to show that the underlying Bellman operator has a unique fixed point. We then provide a closed-form expression for the resulting fixed point and show that it admits a natural interpretation. Moreover, we also show that -- under certain technical assumptions -- the value function, which has a discrete domain and a continuous codomain, admits a continuous extension, which is a finite-valued, concave function of its state variables, at every time step. This result opens the road for achieving scalable implementations of the proposed formulation in future work, as it allows making informed choices of basis functions in an approximate dynamic programming context. We illustrate our findings on a simple numerical example and provide suggestions on how our results can be exploited to obtain closer approximations of the exact value function., Comment: 30 pages, 2 figures. arXiv admin note: substantial text overlap with arXiv:1903.06647

We consider a multi-agent setting with agents exchanging information over a possibly time-varying network, aiming at minimising a separable objective function subject to constraints. To achieve this objective we propose a novel subgradient averaging algorithm that allows for non-differentiable objective functions and different constraint sets per agent. Allowing different constraints per agent simultaneously with a time-varying communication network constitutes a distinctive feature of our approach, extending existing results on distributed subgradient methods. To highlight the necessity of dealing with a different constraint set within a distributed optimisation context, we analyse a problem instance where an existing algorithm does not exhibit a convergent behaviour if adapted to account for different constraint sets. For our proposed iterative scheme we show asymptotic convergence of the iterates to a minimum of the underlying optimisation problem for step sizes of the form $ \frac{\eta}{k+1} $, $ \eta > 0 $. We also analyse this scheme under a step size choice of $ \frac{\eta}{\sqrt{k+1}} $, $ \eta > 0 $, and establish a convergence rate of $ \mathcal{O}(\frac{\ln k}{\sqrt{k}}) $ in objective value. To demonstrate the efficacy of the proposed method, we investigate a robust regression problem and an $ \ell_2 $ regression problem with regularisation.

We consider a multi-agent noncooperative game with agents' objective functions being affected by uncertainty. Following a data driven paradigm, we represent uncertainty by means of scenarios and seek a robust Nash equilibrium solution. We treat the Nash equilibrium computation problem within the realm of probably approximately correct (PAC) learning. Building upon recent developments in scenario-based optimization, we accompany the computed Nash equilibrium with a priori and a posteriori probabilistic robustness certificates, providing confidence that the computed equilibrium remains unaffected (in probabilistic terms) when a new uncertainty realization is encountered. For a wide class of games, we also show that the computation of the so called compression set - a key concept in scenario-based optimization - can be directly obtained as a byproduct of the proposed solution methodology. Finally, we illustrate how to overcome differentiability issues, arising due to the introduction of scenarios, and compute a Nash equilibrium solution in a decentralized manner. We demonstrate the efficacy of the proposed approach on an electric vehicle charging control problem., Comment: Preprint submitted to IEEE Transactions on Automatic Control

We study the approximate dynamic programming approach to revenue management in the context of attended home delivery. We draw on results from dynamic programming theory for Markov decision problems, convex optimisation and discrete convex analysis to show that the underlying dynamic programming operator has a unique fixed point. Moreover, we also show that -- under certain assumptions -- for all time steps in the dynamic program, the value function admits a continuous extension, which is a finite-valued, concave function of its state variables. This result opens the road for achieving scalable implementations of the proposed formulation, as it allows making informed choices of basis functions in an approximate dynamic programming context. We illustrate our findings using a simple numerical example and conclude with suggestions on how our results can be exploited in future work to obtain closer approximations of the value function., Comment: 16 pages, 3 figures, accepted for ECC 2019

We address the optimal design of a large scale multi-agent system where each agent has discrete and/or continuous decision variables that need to be set so as to optimize the sum of linear local cost functions, in presence of linear local and global constraints. The problem reduces to a Mixed Integer Linear Program (MILP) that is here addressed according to a decentralized iterative scheme based on dual decomposition, where each agent determines its decision vector by solving a smaller MILP involving its local cost function and constraint given some dual variable, whereas a central unit enforces the global coupling constraint by updating the dual variable based on the tentative primal solutions of all agents. An appropriate tightening of the coupling constraint through iterations allows to obtain a solution that is feasible for the original MILP. The proposed approach is inspired by a recent method to the MILP approximate solution via dual decomposition and constraint tightening, and presents the advantage of guaranteeing feasibility in finite-time and providing better performance guarantees. The two approaches are compared on a numerical example on plug-in electric vehicles optimal charging., Comment: 10 pages, 2 figures, submitted to Automatica

We consider the problem of optimal charging of plug-in electric vehicles (PEVs). We treat this problem as a multi-agent game, where vehicles/agents are heterogeneous since they are subject to possibly different constraints. Under the assumption that electricity price is affine in total demand, we show that, for any finite number of heterogeneous agents, the PEV charging control game admits a unique Nash equilibrium, which is the optimizer of an auxiliary minimization program. We are also able to quantify the asymptotic behaviour of the price of anarchy for this class of games. More precisely, we prove that if the parameters defining the constraints of each vehicle are drawn randomly from a given distribution, then, the value of the game converges almost surely to the optimum of the cooperative problem counterpart as the number of agents tends to infinity. In the case of a discrete probability distribution, we provide a systematic way to abstract agents in homogeneous groups and show that, as the number of agents tends to infinity, the value of the game tends to a deterministic quantity.

This paper deals with energy management in a district where multiple buildings can communicate over a time-varying network and aim at optimizing the use of shared resources like storage systems. We focus on building cooling, and propose an iterative, distributed algorithm that accounts for information privacy, since buildings are not required to disclose information about their individual utility functions and constraint sets encoding, e.g., their consumption profiles, and overcomes the communication and computational challenges imposed by centralized management paradigms. Our approach relies on a methodology based on proximal minimization that has recently appeared in the literature. Motivated by the structure of the considered energy management optimization program, we provide a theoretical extension of this novel methodology, that is applicable to any problem that exhibits such structural properties. The efficacy of the resulting energy management algorithm is illustrated by means of a detailed simulation based study, considering different network topologies., Comment: 13 pages, 10 figures, submitted to IEEE Transactions on Control Systems Technology

We consider a multi-agent system where each agent has its own estimate of a given quantity and the goal is to reach consensus on the average. To this purpose, we propose a distributed consensus algorithm that guarantees convergence to the average in a finite number of iterations. The algorithm is tailored to ring networks with bidirectional pairwise communications. If the number of agents $m$ is even, say $m=2n$, then, the number of iterations needed is equal to $n$, which in this case is the diameter of the network, whereas the number of iterations grows to $3n$ if the number of agents is odd and equal to $m=2n+1$., Comment: 6 pages, 3 figures, submitted to IEEE Transactions on Control of Network Systems

We study distributed optimization in a cooperative multi-agent setting, where agents have to agree on the usage of shared resources and can communicate via a time-varying network to this purpose. Each agent has its own decision variables that should be set so as to minimize its individual objective function subject to local constraints. Resource sharing is modeled via coupling constraints that involve the non-positivity of the sum of agents' individual functions, each one depending on the decision variables of one single agent. We propose a novel distributed algorithm to minimize the sum of the agents' objective functions subject to both local and coupling constraints, where dual decomposition and proximal minimization are combined in an iterative scheme. Notably, privacy of information is guaranteed since only the dual optimization variables associated with the coupling constraints are exchanged by the agents. Under convexity assumptions, jointly with suitable connectivity properties of the communication network, we are able to prove that agents reach consensus to some optimal solution of the centralized dual problem counterpart, while primal variables converge to the set of optimizers of the centralized primal problem. The efficacy of the proposed approach is demonstrated on a plug-in electric vehicles charging problem., Comment: 13 pages, 3 figures, submitted to Automatica