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world:kkt
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world:kkt [2026/04/08 11:36] rdouc [Saddle points] |
world:kkt [2026/04/08 11:40] (current) rdouc [Saddle points] |
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| Finally, choosing $\lambda = 0$ and $\mu = 0$, we obtain | Finally, choosing $\lambda = 0$ and $\mu = 0$, we obtain | ||
| $$ | $$ | ||
| - | f(x^*) = \mcl(x^*, 0, 0) \le \mcl(x, \lambda^*, \mu^*) \le f(x), \quad \forall x \in \mcD, | + | f(x^*) = \mcl(x^*, 0, 0) \le \sup_{\lambda \geq 0, \mu} \mcl(x^*, \lambda, \mu) |
| + | = \inf_{x \in \Xset} \mcl(x, \lambda^*, \mu^*) \le \mcl(x, \lambda^*, \mu^*) \le f(x), \quad \forall x \in \mcD, | ||
| $$ | $$ | ||
| which shows that $x^*$ minimizes $f$ over $\mcD$ and that $\lambda_i^* h_i(x^*) = 0$ for all $i$ where the last identity follows from the above inequality with $x=x^*$. | which shows that $x^*$ minimizes $f$ over $\mcD$ and that $\lambda_i^* h_i(x^*) = 0$ for all $i$ where the last identity follows from the above inequality with $x=x^*$. | ||
world/kkt.txt · Last modified: 2026/04/08 11:40 by rdouc
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