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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2512.16141 |
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| _version_ | 1866912772968677376 |
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| author | Arefizadeh, Sina Nedić, Angelia |
| author_facet | Arefizadeh, Sina Nedić, Angelia |
| contents | 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 $F_K^{\rm nor}(\cdot)$ is norm coercive over a set $K$, and the generalized Jacobian of the normal mapping has a full rank at points $x$ where $F_K^{\rm nor}(x)\ne0$, then the VI$(K,F)$ has a solution. We then investigate conditions on the mapping $F(\cdot)$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_16141 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Existence of Solutions for Non-monotone VIs and Implications for Games Arefizadeh, Sina Nedić, Angelia Optimization and Control 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 $F_K^{\rm nor}(\cdot)$ is norm coercive over a set $K$, and the generalized Jacobian of the normal mapping has a full rank at points $x$ where $F_K^{\rm nor}(x)\ne0$, then the VI$(K,F)$ has a solution. We then investigate conditions on the mapping $F(\cdot)$ 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. |
| title | Existence of Solutions for Non-monotone VIs and Implications for Games |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2512.16141 |