Monotone Mapping

A mapping $\varphi$ is monotone if for all $x,y\in \text{dom}\varphi$,

$$(x-y{)}^T(\varphi (x)-\varphi (y))\ge 0$$

 

Reference:

• Boyd, S. P., & Vandenberghe, L. (2004). Convex optimization. Cambridge university press.
• Khanh, P., Luong, H. C., Mordukhovich, B., & Tran, D. (2024). Fundamental convergence analysis of sharpness-aware minimization. Advances in Neural Information Processing Systems, 37, 13149-13182.