我的这篇博文中介绍了增广拉格朗日函数及KKT条件
增广拉格朗日函数(The augmented Lagrangian)及其KKT条件
这篇文章中介绍了Lagrangian的KKT条件和投影形式
KKT条件和投影定理(Projection Theorem)
下面我来介绍一下增广拉格朗日函数KKT条件的投影形式
在Lagrangian KKT条件中，投影形式为
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\left\{ \begin{aligned} &P_\Omega(x-(\nabla f(x)+\nabla g(x)\lambda+\nabla h(x)\nu)) =x\\ &(\lambda+g(x))^+=\lambda\\ &h(x)=0 \end{aligned} \right.
⎩⎪⎨⎪⎧PΩ(x−(∇f(x)+∇g(x)λ+∇h(x)ν))=x(λ+g(x))+=λh(x)=0
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(\xi)^+=max\{0, \xi\} \\ P_\Omega(\xi)=\left\{\begin{aligned} &u_i,\qquad \xi>u_i\\&\xi,\qquad l_i\le\xi\le u_i\\&l_i,\qquad \xi<l_i\end{aligned}\right.
(ξ)+=max{0,ξ}PΩ(ξ)=⎩⎪⎨⎪⎧ui,ξ>uiξ,li≤ξ≤uili,ξ<li
对于增广Lagrangian函数的KKT条件：
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(\lambda+g(x))^+=\lambda\; and\; h(x)=0
(λ+g(x))+=λandh(x)=0 意味着
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\lambda=0\;and\;\nu=0
λ=0andν=0 因此，上式的第一个式子等价于
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P_\Omega(x-(\nabla f(x)+\nabla g(x)(\lambda+\lambda_a)+\nabla h(x)(\nu+\nu_a))) =x
PΩ(x−(∇f(x)+∇g(x)(λ+λa)+∇h(x)(ν+νa)))=x
所以The argumented lagrangian KTT条件的投影形式为
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\left\{ \begin{aligned} &P_\Omega(x-(\nabla f(x)+\nabla g(x)(\lambda+\lambda_a)+\nabla h(x)(\nu+\nu_a))) =x\\ &(\lambda+g(x))^+=\lambda\\ &h(x)=0 \end{aligned} \right.
⎩⎪⎨⎪⎧PΩ(x−(∇f(x)+∇g(x)(λ+λa)+∇h(x)(ν+νa)))=x(λ+g(x))+=λh(x)=0
