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update strong-convex function
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zhimin-z committed Sep 25, 2024
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## 1.1.6 强凸函数

对于定义在凸集上的函数 $f(x)$,若其满足以下性质,则称其为强凸函数
$f(x)$为定义在凸集上的强凸函数,则对于任意$x,y \in dom(f)$,$\alpha \in [0,1]$,存在$\lambda \gt 0$,使得
$$
\begin{equation}
\forall x,y\in dom(f),\ \alpha\in[0,1],\ \exists \lambda \gt 0,\ \text{使得} \\
f(\alpha x + (1-\alpha)y) \leq \alpha f(x) + (1-\alpha)f(y) - \frac{\lambda}{2}\alpha(1-\alpha)\|x-y\|_2^2
\end{equation}
$$
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