A Nikaido Isoda-Based Hybrid Genetic Algorithm and Relaxation Method for Finding Nash Equilibrium

Nash Equilibrium (NE) plays a crucial role in game theory. The relaxation method in conjunction with the Nikaido–Isoda (NI) function, namely the NI-based relaxation method, has been widely applied to the determination of NE. Genetic Algorithm (GA) with adaptive penalty is introduced and incorporated in the original NI-based relaxation method. The GA enhances the capability in the optimization step for computing the optimum response function. The optimization of the non-convex and non-concave NI function is made possible by GA. The proposed method thus combines the advantageous feature of the GA in its optimization capability and that of the relaxation method in its implementation simplicity together. The applicability of the method is shown through the illustrative examples, including the generalized Nash Equilibrium problem with nonlinear payoff functions and coupled constraints, the game with multiple strategic variables for individual players, and the non-differentiable payoff functions. All test example results suggest the appropriate crossover and mutation rate to be 0.05 and 0.002 for use in GA. These numbers are closed to the recommended values by DeJong. The proposed method shows its capability of finding correct NEs in all test examples.

Identifying Oscillatory Hyperconnectivity and Hypoconnectivity Networks in Major Depression Using Coupled Tensor Decomposition

Previous researches demonstrate that major depression disorder (MDD) is associated with widespread network dysconnectivity, and the dynamics of functional connectivity networks are important to delineate the neural mechanisms of MDD. Cortical electroencephalography (EEG) oscillations act as coordinators to connect different brain regions, and various assemblies of oscillations can form different networks to support different cognitive tasks. Studies have demonstrated that the dysconnectivity of EEG oscillatory networks is related with MDD. In this study, we investigated the oscillatory hyperconnectivity and hypoconnectivity networks in MDD under a naturalistic and continuous stimuli condition of music listening. With the assumption that the healthy group and the MDD group share similar brain topology from the same stimuli and also retain individual brain topology for group differences, we applied the coupled nonnegative tensor decomposition algorithm on two adjacency tensors with the dimension of time × frequency × connectivity × subject, and imposed double-coupled constraints on spatial and spectral modes. The music-induced oscillatory networks were identified by a correlation analysis approach based on the permutation test between extracted temporal factors and musical features. We obtained three hyperconnectivity networks from the individual features of MDD and three hypoconnectivity networks from common features. The results demonstrated that the dysfunction of oscillation-modulated networks could affect the involvement in music perception for MDD patients. Those oscillatory dysconnectivity networks may provide promising references to reveal the pathoconnectomics of MDD and potential biomarkers for the diagnosis of MDD.

Exact Penalization of Generalized Nash Equilibrium Problems

This paper presents an exact penalization theory of the generalized Nash equilibrium problem (GNEP) that has its origin from the renowned Arrow–Debreu general economic equilibrium model. Whereas the latter model is the foundation of much of mathematical economics, the GNEP provides a mathematical model of multiagent noncooperative competition that has found many contemporary applications in diverse engineering domains. The most salient feature of the GNEP that distinguishes it from a standard noncooperative (Nash) game is that each player’s optimization problem contains constraints that couple all players’ decision variables. Extending results for stand-alone optimization problems, the penalization theory aims to convert the GNEP into a game of the standard kind without the coupled constraints, which is known to be more readily amenable to solution methods and analysis. Starting with an illustrative example to motivate the development, this paper focuses on two kinds of coupled constraints, shared (i.e., common) and finitely representable. Constraint residual functions and the associated error bound theory play an important role throughout the development.