{"ID":2827370,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2512.16141","arxiv_id":"2512.16141","title":"Existence of Solutions for Non-monotone VIs and Implications for Games","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 $F_K^{\\rm nor}(\\cdot)$ 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.","short_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 $F_K^{\\rm nor}(\\cdot)$ is norm coercive over a set $K$, and the generalized Jacobian of the normal mapping has a full rank at point...","url_abs":"https://arxiv.org/abs/2512.16141","url_pdf":"https://arxiv.org/pdf/2512.16141v1","authors":"[\"Sina Arefizadeh\",\"Angelia Nedić\"]","published":"2025-12-18T03:54:31Z","proceeding":"math.OC","tasks":"[\"math.OC\"]","methods":"[]","has_code":false}