In this article, mixed equilibrium problems (MEPs) in uncertain and dynamic environments are investigated by employing a multiagent system (MAS), where each agent only has access to its own bifunction, and can communicate with its immediate neighbors via a time-varying digraph. At each time, the goal of agents is to cooperatively find a point in the constraint set such that the sum of local bifunctions with a free variable is nonnegative. Different from existing works on MEPs, here the bifunctions are stochastic and time-varying, and only available to agents after decisions are made. To tackle this problem, an online distributed learning algorithm based on the mirror descent algorithm and the clipping strategy is proposed. The dynamic regret, whose offline benchmark is to find the solution at each time, is employed to measure the performance of the algorithm. Of particular interest is that the high probability bound of the regret is established under the heavy-tailed noise condition. The result shows that if the variation in the solution sequence is within a certain range, then the dynamic regret grows sublinearly with high probability. Finally, a simulation example is provided to corroborate the validity of the theoretical results.

In this article, the problem of distributed optimization subject to a convex set is studied by employing a multiagent system, where each agent only has access to a stochastic gradient of its own cost function, and exchanges local information with its neighbors through a time-varying digraph. To handle this problem, a stochastic distributed algorithm based on the clipped distributed strategy and the dual averaging strategy is proposed. Of particular interest follows that we study the high probability convergence of the stochastic distributed algorithm under the assumption that the stochastic gradient follows a heavy-tailed distribution. Heavy-tailed distributions often causes extreme values, which bring difficulties in achieving the high probability convergence. Under mild assumptions on the graph and cost functions, we prove that the difference between the cost function value generated by the algorithm and its minimum value converges with high probability. Finally, a simulation experiment is presented to demonstrate the effectiveness of our theoretical results.

In this paper, the problem of online distributed optimization with stochastic and nonconvex objective functions is studied by employing a multi-agent system. When making decisions, each agent only has access to a noisy gradient of its own objective function in the previous time and can only communicate with its immediate neighbors via a time-varying digraph. To handle this problem, an online distributed stochastic projection-free algorithm is proposed. Of particular interest is that the dynamic regrets are employed to measure the performance of the online algorithm. Existing works on online distributed algorithms involving stochastic gradients only provide the sublinearity results of regrets in expectation. Different from them, we study the high probability bounds of the dynamic regrets, i.e., the sublinear bounds of dynamic regrets are characterized by the natural logarithm of the failure probability's inverse. Under mild assumptions on the graph and objective functions, we prove that if the variations in both the objective function sequence and its gradient sequence grow within a certain rate, then the high probability bounds of the dynamic regrets grow sublinearly. Finally, a simulation example is carried out to demonstrate the effectiveness of our theoretical results. Index TermsOnline distributed optimization; Noncon-vex optimization; Stochastic gradients; Dynamic regrets; High probability.

In this article, the problem of distributed generalized Nash equilibrium (GNE) seeking in noncooperative games is investigated via multiagent networks, where each player aims to minimize his or her own cost function with a nonsmooth term. Each player's cost function and feasible action set in the noncooperative game are both determined by actions of others who may not be neighbors, as well as his/her own action. Particularly, feasible action sets are constrained by private convex inequalities and shared linear equations. Each player can only have access to his or her own cost function, private constraint, and a local block of shared constraints, and can only communicate with his or her neighbours via a digraph. To address this problem, a novel continuous-time distributed primal-dual algorithm involving Clarke's generalized gradient is proposed based on consensus algorithms and the primal-dual algorithm. Under mild assumptions on cost functions and graph, we prove that players' actions asymptotically converge to a GNE. Finally, a simulation is presented to demonstrate the effectiveness of our theoretical results.

In this brief, the problem of distributively solving a mixed equilibrium problem (EP) with multiple sets is investigated. A network of agents is employed to cooperatively find a point in the intersection of multiple convex sets ensuring that the sum of multiple bifunctions with a free variable is nonnegative. Each agent can only access information associated with its own bifunction and a local convex set. To solve this problem, a distributed algorithm involving a fixed step size is proposed by combining the mirror descent algorithm, the primal-dual algorithm, and the consensus algorithm. Under mild conditions on bifunctions and the graph, we prove that all agents' states asymptotically converge to a solution of the mixed EP. A numerical simulation example is provided for demonstrating the effectiveness of theoretical results.

In this paper, the distributed Nash equilibrium (NE) searching problem is investigated, where the feasible action sets are constrained by nonlinear inequalities and linear equations. Different from most of the existing investigations on distributed NE searching problems, we consider the case where both cost functions and feasible action sets depend on actions of all players, and each player can only have access to the information of its neighbors. To address this problem, a continuous-time distributed gradient-based projected algorithm is proposed, where a leader-following consensus algorithm is employed for each player to estimate actions of others. Under mild assumptions on cost functions and graphs, it is shown that players' actions asymptotically converge to a generalized NE. Simulation examples are presented to demonstrate the effectiveness of the theoretical results.