User Tools
world:kkt
Differences
This shows you the differences between two versions of the page.
| Both sides previous revision Previous revision | |||
|
world:kkt [2026/04/08 11:40] rdouc [Saddle points] |
world:kkt [2026/04/09 23:17] (current) rdouc |
||
|---|---|---|---|
| Line 45: | Line 45: | ||
| The rhs is called the <color red>//primal problem//</color>, while the lhs is referred to as the <color red>//dual problem//</color>. Since $x \mapsto \mcl(x,\lambda,\mu)$ is convex, the dual problem $\sup_{\lambda\geq 0,\mu}\inf_{x \in \Xset} \mcl(x,\lambda,\mu)$ is equivalent to | The rhs is called the <color red>//primal problem//</color>, while the lhs is referred to as the <color red>//dual problem//</color>. Since $x \mapsto \mcl(x,\lambda,\mu)$ is convex, the dual problem $\sup_{\lambda\geq 0,\mu}\inf_{x \in \Xset} \mcl(x,\lambda,\mu)$ is equivalent to | ||
| $$ | $$ | ||
| - | \sup \{\mcl(x,\lambda,\mu)\,:\lambda \geq 0, \mu \mbox{ and }\nabla_x \mcl(x,\lambda)=0\} | + | \sup \{\mcl(x,\lambda,\mu)\,:\lambda \geq 0, \mu \mbox{ and }\nabla_x \mcl(x,\lambda,\mu)=0\} |
| $$ | $$ | ||
| Line 172: | Line 172: | ||
| g_j(x) = a_j^T x - b_j, \quad j=1,\dots,m. | g_j(x) = a_j^T x - b_j, \quad j=1,\dots,m. | ||
| $$ | $$ | ||
| - | Without loss of generality, we assume that $(a_j)_{1\leq j \leq m}$ are \textbf{linearly independent}. | + | Without loss of generality, we assume that $(a_j)_{1\leq j \leq m}$ are **linearly independent**. |
| We define the set $U$ as | We define the set $U$ as | ||
| $$ | $$ | ||
world/kkt.1775641200.txt.gz ยท Last modified: 2026/04/08 11:40 by rdouc
Page Tools
