中科大-凸优化 笔记（lec41）-可微凸优化问题的罚函数形式

全部笔记的汇总贴（视频也有传送门）：中科大-凸优化

例：

⇒ L ( x , v ) = f 0 ( x ) + v T ( A x − b ) ⇒ g ( v ) = inf ⁡ x { f 0 ( x ) + v T ( A x − b ) }        v = α ( A x ~ − b ) g ( α ( A x ~ − b ) ) = inf ⁡ x { f 0 ( x ) + α ( A x ~ − b ) T ( A x ~ − b ) } = f 0 ( x ~ ) + α ∣ ∣ A x ~ − b ∣ ∣ 2 2    f 0 ( x ∗ ) = P ∗ = d ∗ ≥ g ( α ( A x ~ − b ) ) = f 0 ( x ~ ) + α ∣ ∣ A x ~ − b ∣ ∣ 2 2 ≥ f 0 ( x ~ ) 当 α = 0 时 ， arg max ⁡ f 0 ( x ) 当 α → + ∞ 时 ， f ( x ∗ ) = f ( x ~ ) \Rightarrow L(x,v)=f_0(x)+v^T(Ax-b)\\\Rightarrow g(v)=\inf_x\{f_0(x)+v^T(Ax-b)\}\;\;\;v=\alpha(A\tilde x-b)\\g(\alpha(A\tilde x-b))=\inf_x\{f_0(x)+\alpha(A\tilde x-b)^T(A\tilde x-b)\}\\=f_0(\tilde x)+\alpha||A\tilde x-b||^2_2\\\;\\f_0(x^*)=P^*=d^*\ge g(\alpha(A\tilde x-b))=f_0(\tilde x)+\alpha||A\tilde x-b||_2^2\ge f_0(\tilde x)\\当\alpha=0时，\argmax f_0(x)\\当\alpha\rightarrow+\infty时，f(x^*)=f(\tilde x) ⇒L(x,v)=f0​(x)+vT(Ax−b)⇒g(v)=xinf​{ f0​(x)+vT(Ax−b)}v=α(Ax~−b)g(α(Ax~−b))=xinf​{ f0​(x)+α(Ax~−b)T(Ax~−b)}=f0​(x~)+α∣∣Ax~−b∣∣22​f0​(x∗)=P∗=d∗≥g(α(Ax~−b))=f0​(x~)+α∣∣Ax~−b∣∣22​≥f0​(x~)当α=0时，argmaxf0​(x)当α→+∞时，f(x∗)=f(x~)

例：带线性不等式约束的可微凸优化问题

下一章传送门：中科大-凸优化 笔记（lec42）-log-barrier法