Lagrangian

336. Lagrangian

\begin{matrix} \nabla 𝑓 = 𝜆 \nabla 𝑔 \\ 𝑔 = 0 \end{matrix}

Constrained optimization problem:

\begin{aligned} min & 𝑓 (𝑥) \\ 𝑠 . 𝑡 . & 𝑔_{𝑖} (𝑥) = 0 \forall 𝑖 = 1, \dots, 𝑚 \\ ℎ_{𝑗} (𝑥) = 0 \forall 𝑗 = 1, \dots, 𝑝 \end{aligned}

Lagrangian $ℒ︀ (𝑥, 𝜆, 𝜇)$ :

ℒ︀ (𝑥, 𝜆, 𝜇) = \underset{objective function}{\underset{⏟}{𝑓 (𝑥)}} + \underset{equality constraints}{\underset{⏟}{\sum_{𝑖 = 1}^{𝑚} 𝜆_{𝑖} 𝑔_{𝑖} (𝑥)}} + \underset{inequality constraints}{\underset{⏟}{\sum_{𝑗 = 1}^{𝑝} 𝜇_{𝑗} ℎ_{𝑗} (𝑥)}}

Example