Monotone Mapping
A mapping \( \phi \) is monotone if for all \( x,y \in \text{dom}\phi\),\[(x-y)^T(\phi(x)-\phi(y))\geq0\] 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.