In this paper, we aim to design a distributed approximate algorithm for seeking Nash equilibria of an aggregative game. Due to the local set constraints of each player, projectionbased algorithms have been widely employed for solving such problems ac
tually. Since it may be quite hard to get the exact projection in practice, we utilize inscribed polyhedrons to approximate local set constraints, which yields a related approximate game model. We first prove that the Nash equilibrium of the approximate game is the $epsilon$-Nash equilibrium of the original game, and then propose a distributed algorithm to seek the $epsilon$-Nash equilibrium, where the projection is then of a standard form in quadratic programming. With the help of the existing developed methods for solving quadratic programming, we show the convergence of the proposed algorithm, and also discuss the computational cost issue related to the approximation. Furthermore, based on the exponential convergence of the algorithm, we estimate the approximation accuracy related to $epsilon$. Additionally, we investigate the computational cost saved by approximation on numerical examples.
To solve distributed optimization efficiently with various constraints and nonsmooth functions, we propose a distributed mirror descent algorithm with embedded Bregman damping, as a generalization of conventional distributed projection-based algorith
ms. In fact, our continuous-time algorithm well inherits good capabilities of mirror descent approaches to rapidly compute explicit solutions to the problems with some specific constraint structures. Moreover, we rigorously prove the convergence of our algorithm, along with the boundedness of the trajectory and the accuracy of the solution.
In this paper, we employ a hypergame framework to analyze the single-leader-multiple-followers (SLMF) Stackelberg security game with two typical misinformed situations: misperception and deception. We provide a stability criterion with the help of hy
per Nash equilibrium (HNE) to analyze both strategic stability and cognitive stability of equilibria in SLMF games with misinformation. To this end, we find mild stable conditions such that the equilibria with misperception and deception can derive HNE. Moreover, we analyze the robustness of the equilibria to reveal whether the players have the ability to keep their profits.
We consider continuous-time dynamics for distributed optimization with set constraints in the note. To handle the computational complexity of projection-based dynamics due to solving a general quadratic optimization subproblem with projection, we pro
pose a distributed projection-free dynamics by employing the Frank-Wolfe method, also known as the conditional gradient algorithm. The process searches a feasible descent direction with solving an alternative linear optimization instead of a quadratic one. To make the algorithm implementable over weight-balanced digraphs, we design one dynamics for the consensus of local decision variables and another dynamics of auxiliary variables to track the global gradient. Then we prove the convergence of the dynamical systems to the optimal solution, and provide detailed numerical comparisons with both projection-based dynamics and other distributed projection-free algorithms.
