Existence of Solutions for Non-monotone VIs and Implications for Games

2025/12/18 by Arefizadeh, Sina, Nedić, Angelia
#FOS: Mathematics #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.2512.16141

Abstract

In this paper, we study the existence of solutions in non-monotone variational inequalities (VIs) through the normal mapping properties. In particular, we show that when the normal mapping F_K^{\rm nor}(⋅) is norm coercive over a set K, and the generalized Jacobian of the normal mapping has a full rank at points x where F_K^{\rm nor}(x)≠0, then the VI(K,F) has a solution. We then investigate conditions on the mapping F(⋅) and its Jacobian that imply the full rank condition for the generalized Jacobian, such as the uniform P-function and the uniform P-matrix condition. Subsequently, we focus on VIs arising from games and interpret our main result in a game setting. Based on the P_Υ-matrix condition, we provide a sufficient condition for a game to have a Nash equilibrium. Additionally, through examples we show that our sufficient conditions can be used to assert the existence of a solution to a VI, or a quasi-Nash in a game, while the existing results relying on the uniform P-function property or the P_Υ-matrix condition cannot be employed.

Citations

• Non-Monotone Variational Inequalities
• United We Fall: On the Nash Equilibria of Multiplex and Multilayer Network Games
• Beyond Monotone Variational Inequalities: Solution Methods and Iteration Complexities
• A variational inequality framework for network games: Existence, uniqueness, convergence and sensitivity analysis
• A BEHAVIORAL MODEL OF RATIONAL CHOICE

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• A Class of Quasi-Variational Inequalities with Unbounded Constraint Maps: Existence Results and Applications
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• The Computational Complexity of Variational Inequalities and Applications in Game Theory