Optimization: Dual and Lagrangian
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Consider the optimization problem $P$: \[ \min_{x \in \mathbb{R}^d} f(x) \textrm{ where } \forall i \in I, c_i(x) \ge 0 \wedge \forall j \in J, h_j(x) = 0 \] The corresponding Lagrangian is \[ L(x, \lambda, \mu) = f(x) - \lambda^Tc(x) - \mu^Th(x) \] Define $g$ as \[ g(\lambda, \mu) = \min_{x \in \mathbb{R}^d} L(x, \lambda, \mu) \] Let $D$ be this optimization problem: \[ \max_{\lambda, \mu} g(\lambda, \mu) \textrm{ where } g(\lambda, \mu) \neq -\infty \wedge \lambda \ge 0 \] Then $D$ is said to be the dual of $P$.
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