Optimization: Dual and Lagrangian

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Consider the optimization problem $P$: $$\underset{x\in {\mathbb{R}}^d}{min}f(x)\text{ where }\mathrm{\forall }i\in I,c_i(x)\ge 0\wedge \mathrm{\forall }j\in J,h_j(x)=0$$ The corresponding Lagrangian is $$L(x,\lambda ,\mu )=f(x)-{\lambda }^{T}c(x)-{\mu }^{T}h(x)$$ Define $g$ as $$g(\lambda ,\mu )=\underset{x\in {\mathbb{R}}^d}{min}L(x,\lambda ,\mu )$$ Let $D$ be this optimization problem: $$\underset{\lambda ,\mu }{max}g(\lambda ,\mu )\text{ where }g(\lambda ,\mu )\ne -\mathrm{\infty }\wedge \lambda \ge 0$$ Then $D$ is said to be the dual of $P$.

Dependency for:

1. Optimization: weak duality
2. Dual of a linear program

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