University of Science and Technology of China-Convex optimization notes (lec36)-KKT conditions

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原问题
$$\begin{gathered} \min f_0(x) \\ s.t.f_i\left(x\right)\le 0,i=1,\cdots ,m \\ {h}_i\left(x\right)=0,i=1,\cdots ,P \end{gathered}$$

对偶函数 g ( λ , v ) = inf ⁡ x { f 0 ( x ) + ∑ i = 1 m λ i f i ( x ) + ∑ i = 1 P v i h i ( x ) } g(\lambda,v)=\inf_x\{f_0(x)+\sum_{i=1}^m\lambda_if_i(x)+\sum_{i=1}^Pv_ih_i(x)\} g ( λ ,v )=xinf​{ f0​(x)+i=1∑m​λi​fi​(x)+i=1∑P​vi​hi​(x)}

Dual problem $$\begin{gathered} \max g(\lambda ,v) \\ s.t.\lambda \ge 0 \end{gathered}$$

Hypothesis

• $$P^{\ast }=d^{\ast }$$
• All functions are differentiable

$$\begin{gathered} x^{\ast },{\lambda }^{\ast },v^{\ast } \\ {f}_i(x^{\ast })\le 0,i=1,\cdots ,m \\ {h}_i\left(x^{\ast }\right)=0,i=1,\cdots ,P \\ {\lambda }^{\ast }\ge 0 \end{gathered}$$

KKT conditions

• primal feasibility
• dual feasibility
• complementary slackness
• stationary

If the original problem is a convex problem, each function is differentiable, and the dual gap is zero, then the KKT condition is a necessary and sufficient condition

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