gen-in-3-4-sentences-explain-lagrangian-duality-state-any-key-fo-21dc

In 3-4 sentences, explain Lagrangian duality. State any key formula.

Lagrangian duality is a technique that transforms a constrained optimization problem (the primal) into an alternative problem (the dual) that provides bounds on the optimal value. The Lagrangian function is formed by adding weighted constraint violations to the objective: **L(x, λ, ν) = f(x) + λᵀg(x) + νᵀh(x)**, where λ ≥ 0 are multipliers for inequality constraints g(x) ≤ 0 and ν are multipliers for equality constraints h(x) = 0. The dual function **d(λ, ν) = inf_x L(x, λ, ν)** always provides a lower bound on the primal optimal value, and under certain conditions (strong duality), the primal and dual optimal values are equal.