lagrangian-duality #

Visualizes how the Lagrangian L(x,λ)=f(x)+λg(x) combines objective and constraints, how the dual function d(λ)=inf_x L(x,λ) produces a lower bound on the primal optimum (weak duality), and how dual ascent updates λ using the constraint residual g(x*(λ)) to maximize d(λ). The left plot shows f(x) and L(x,λ) for a changing multiplier, with horizontal lines for d(λ) and p*. The right panel displays the key equations and live values.

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⏮◀◀▶▶STEP0.25x1xZOOM

t=0s

practical uses #

01.Deriving dual problems to get computable lower bounds for minimization (certificates of suboptimality)
02.Designing algorithms like dual ascent / subgradient methods and projected updates for constrained optimization
03.Understanding KKT conditions and when strong duality lets you solve the primal via the dual

technical notes #

Uses a 1D convex quadratic objective with a single inequality constraint so L(x,λ) and d(λ) are explicit. The animation cycles through three stages and, in the final stage, runs a projected dual-ascent update λ←max(0,λ+α g(x*(λ)) dt). Rendering is grid-snapped for a blocky aesthetic and scaled via scale=Math.min(w,h)/baseSize.

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