# Lagrangian Duality Complementary Slackness solution

If $$\alpha^*,\mu^*$$ is the solution of optimalization $$\max_{\alpha, \mu}\mathcal{L}(\omega, \xi, b, \alpha, \mu)$$ How I can show that "complementary slackness condition" $$\alpha_i^* = 0$$ whenever $$\mathrm{y}_i(w^T x_i + b ) - 1 + \xi_i > 0$$ similary for $$\mu_i^* = 0$$ whenever $$\xi_i > 0$$

Primal: $$\min_{\omega, \xi, b} \frac{1}{2} \omega^T\omega + C\sum_{i=1}^n\xi_i \\ \text{subject to} \\ \mathrm{y}_i(w^T x_i + b ) \geq 1 - \xi_i \\ \xi_i \geq 0$$

The Lagrangian form looks like this: $$\mathcal{L}(\omega, \xi, b, \alpha, \mu) = \frac{1}{2} \omega^T\omega + C\sum_{i=1}^n\xi_i -\sum_{i=1}^n(y_i(\omega^T x_i + b) - 1 + \xi_i) -\sum_{i=1}^n{\mu_i \xi_i} \quad \alpha, \mu \text{ are dual variables}$$

Thanks for your time and help guys. I appreciate.

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## migrated from stats.stackexchange.comFeb 28 '13 at 20:33

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guys, really? no one? :( –  Tatar Elemér Feb 28 '13 at 19:05
This isn't really a data analysis question. Math has agreed to look at it. Cheers! –  whuber Feb 28 '13 at 20:33