Distributed Nash Equilibrium Seeking with Limited Cost Function Knowledge via A Consensus-Based Gradient-Free Method

This paper considers a distributed Nash equilibrium seeking problem, where the players only have partial access to other players' actions, such as their neighbors' actions. Thus, the players are supposed to communicate with each other to estimate other players' actions. To solve the problem, a leader-following consensus gradient-free distributed Nash equilibrium seeking algorithm is proposed. This algorithm utilizes only the measurements of the player's local cost function without the knowledge of its explicit expression or the requirement on its smoothness. Hence, the algorithm is gradient-free during the entire updating process. Moreover, the analysis on the convergence of the Nash equilibrium is studied for the algorithm with both diminishing and constant step-sizes, respectively. Specifically, in the case of diminishing step-size, it is shown that the players' actions converge to the Nash equilibrium almost surely, while in the case of fixed step-size, the convergence to the neighborhood of the Nash equilibrium is achieved. The performance of the proposed algorithm is verified through numerical simulations

Nash Equilibrium Seeking in N-Coalition Games via a Gradient-Free Method

This paper studies an $N$-coalition non-cooperative game problem, where the players in the same coalition cooperatively minimize the sum of their local cost functions under a directed communication graph, while collectively acting as a virtual player to play a non-cooperative game with other coalitions. Moreover, it is assumed that the players have no access to the explicit functional form but only the function value of their local costs. To solve the problem, a discrete-time gradient-free Nash equilibrium seeking strategy, based on the gradient tracking method, is proposed. Specifically, a gradient estimator is developed locally based on Gaussian smoothing to estimate the partial gradients, and a gradient tracker is constructed locally to trace the average sum of the partial gradients among the players within the coalition. With a sufficiently small constant step-size, we show that all players' actions approximately converge to the Nash equilibrium at a geometric rate under a strongly monotone game mapping condition. Numerical simulations are conducted to verify the effectiveness of the proposed algorithm

Gradient-Free Distributed Optimization with Exact Convergence

In this paper, a gradient-free distributed algorithm is introduced to solve a set constrained optimization problem under a directed communication network. Specifically, at each time-step, the agents locally compute a so-called pseudo-gradient to guide the updates of the decision variables, which can be applied in the fields where the gradient information is unknown, not available or non-existent. A surplus-based method is adopted to remove the doubly stochastic requirement on the weighting matrix, which enables the implementation of the algorithm in graphs having no associated doubly stochastic weighting matrix. For the convergence results, the proposed algorithm is able to obtain the exact convergence to the optimal value with any positive, non-summable and non-increasing step-sizes. Furthermore, when the step-size is also square-summable, the proposed algorithm is guaranteed to achieve the exact convergence to an optimal solution. In addition to the standard convergence analysis, the convergence rate of the proposed algorithm with respect to different cases of step-sizes is investigated. Finally, the effectiveness of the proposed algorithm is verified through numerical simulations

Fully Distributed Nash Equilibrium Seeking in N-Cluster Games

Distributed optimization and Nash equilibrium (NE) seeking problems have drawn much attention in the control community recently. This paper studies a class of non-cooperative games, known as $N$-cluster game, which subsumes both cooperative and non-cooperative nature among multiple agents in the two problems: solving distributed optimization problem within the cluster, while playing a non-cooperative game across the clusters. Moreover, we consider a partial-decision information game setup, i.e., the agents do not have direct access to other agents' decisions, and hence need to communicate with each other through a directed graph whose associated adjacency matrix is assumed to be non-doubly stochastic. To solve the $N$-cluster game problem, we propose a fully distributed NE seeking algorithm by a synthesis of leader-following consensus and gradient tracking, where the leader-following consensus protocol is adopted to estimate the other agents' decisions and the gradient tracking method is employed to trace some weighted average of the gradient. Furthermore, the algorithm is equipped with uncoordinated constant step-sizes, which allows the agents to choose their own preferred step-sizes, instead of a uniform coordinated step-size. We prove that all agents' decisions converge linearly to their corresponding NE so long as the largest step-size and the heterogeneity of the step-size are small. We verify the derived results through a numerical example in a Cournot competition game

Gradient-Free Nash Equilibrium Seeking in N-Cluster Games with Uncoordinated Constant Step-Sizes

In this paper, we consider a problem of simultaneous global cost minimization and Nash equilibrium seeking, which commonly exists in $N$-cluster non-cooperative games. Specifically, the agents in the same cluster collaborate to minimize a global cost function, being a summation of their individual cost functions, and jointly play a non-cooperative game with other clusters as players. For the problem settings, we suppose that the explicit analytical expressions of the agents' local cost functions are unknown, but the function values can be measured. We propose a gradient-free Nash equilibrium seeking algorithm by a synthesis of Gaussian smoothing techniques and gradient tracking. Furthermore, instead of using the uniform coordinated step-size, we allow the agents across different clusters to choose different constant step-sizes. When the largest step-size is sufficiently small, we prove a linear convergence of the agents' actions to a neighborhood of the unique Nash equilibrium under a strongly monotone game mapping condition, with the error gap being propotional to the largest step-size and the smoothing parameter. The performance of the proposed algorithm is validated by numerical simulations

Social Profit Optimization with Demand Response Management in Electricity Market: A Multi-timescale Leader-following Approach

In the electricity market, it is quite common that the market participants make "selfish" strategies to harvest the maximum profits for themselves, which may cause the social benefit loss and impair the sustainability of the society in the long term. Regarding this issue, in this work, we will discuss how the social profit can be improved through strategic demand response (DR) management. Specifically, we explore two interaction mechanisms in the market: Nash equilibrium (NE) and Stackelberg equilibrium (SE) among utility companies (UCs) and user-UC interactions, respectively. At the user side, each user determines the optimal energy-purchasing strategy to maximize its own profit. At the UC side, a governmental UC (g-UC) is considered, who aims to optimize the social profit of the market. Meanwhile, normal UCs play games to maximize their own profits. As a result, a basic leader-following problem among the UCs is formulated under the coordination of the independent system operator (ISO). Moreover, by using our proposed demand function amelioration (DFA) strategy, a multi-timescale leader-following problem is formulated. In this case, the maximal market efficiency can be achieved without changing the "selfish instinct" of normal UCs. In addition, by considering the local constraints for the UCs, two projection-based pricing algorithms are proposed for UCs, which can provide approximate optimal solutions for the resulting non-convex social profit optimization problems. The feasibility of the proposed algorithms is verified by using the concept of price of anarchy (PoA) in a multi-UC multi-user market model in the simulation.Comment: 33 pages, 15 figure

An Unsupervised Approach to Ultrasound Elastography with End-to-end Strain Regularisation

Quasi-static ultrasound elastography (USE) is an imaging modality that consists of determining a measure of deformation (i.e.strain) of soft tissue in response to an applied mechanical force. The strain is generally determined by estimating the displacement between successive ultrasound frames acquired before and after applying manual compression. The computational efficiency and accuracy of the displacement prediction, also known as time-delay estimation, are key challenges for real-time USE applications. In this paper, we present a novel deep-learning method for efficient time-delay estimation between ultrasound radio-frequency (RF) data. The proposed method consists of a convolutional neural network (CNN) that predicts a displacement field between a pair of pre- and post-compression ultrasound RF frames. The network is trained in an unsupervised way, by optimizing a similarity metric be-tween the reference and compressed image. We also introduce a new regularization term that preserves displacement continuity by directly optimizing the strain smoothness. We validated the performance of our method by using both ultrasound simulation and in vivo data on healthy volunteers. We also compared the performance of our method with a state-of-the-art method called OVERWIND [17]. Average contrast-to-noise ratio (CNR) and signal-to-noise ratio (SNR) of our method in 30 simulation and 3 in vivo image pairs are 7.70 and 6.95, 7 and 0.31, respectively. Our results suggest that our approach can effectively predict accurate strain images. The unsupervised aspect of our approach represents a great potential for the use of deep learning application for the analysis of clinical ultrasound data.Comment: Accepted at MICCAI 202