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2 changes: 1 addition & 1 deletion chapter_optimization/convexity.md
Original file line number Diff line number Diff line change
Expand Up @@ -215,7 +215,7 @@ $$f(\lambda x + (1-\lambda) x') \leq \lambda f(x) + (1-\lambda) f(x') \leq b.$$
即对于所有$\mathbf{x} \in \mathbb{R}^n$,$\mathbf{x}^\top \mathbf{H} \mathbf{x} \geq 0$.
例如,函数$f(\mathbf{x}) = \frac{1}{2} \|\mathbf{x}\|^2$是凸的,因为$\nabla^2 f = \mathbf{1}$,即其导数是单位矩阵。

更正式地讲,$f$为凸函数,当且仅当任意二次可微一维函数$f: \mathbb{R}^n \rightarrow \mathbb{R}$是凸的。
更正式地讲,$f$为凸函数,当且仅当任意二次可微一维函数$f: \mathbb{R} \rightarrow \mathbb{R}$是凸的。
对于任意二次可微多维函数$f: \mathbb{R}^{n} \rightarrow \mathbb{R}$,
它是凸的当且仅当它的Hessian$\nabla^2f\succeq 0$。

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