Lagrangian Duality Solver

Lagrangian Duality Solver

Primal Problem

Minimize: \( f(x) = x^2 \)

Subject to: \( x \leq 1 \)

Lagrangian

\( L(x, \lambda) = x^2 + \lambda (x - 1) \)

Lagrange Multiplier (λ):

1.0

Dual Function

\( D(\lambda) = \min_x L(x, \lambda) \)

Optimal x: 0.5

Dual Value: -0.25

Visualization

Explanation

1. Primal Problem: Minimize \( x^2 \) s.t. \( x \leq 1 \).

2. Lagrangian: \( L(x, \lambda) = x^2 + \lambda (x - 1) \).

3. Dual Function: \( D(\lambda) = \min_x L(x, \lambda) \).

4. Strong Duality: If \( \lambda \geq 0 \), the dual gives a lower bound on the primal.