Mathematics · Core
Lagrange Multiplier
A Lagrange multiplier is the shadow price of a constraint: how much the objective would improve if that constraint loosened by one unit.
Definition
To extremize f subject to g = 0, solve ∇f = λ ∇g. λ is the sensitivity of the optimal value to the constraint. Inequalities use KKT complementary slackness: slack constraints have λ = 0.
Why it matters
Risk-limit shadow prices tell you which limit is actually binding. A book with ten limits and one huge λ has one real constraint.
Case
A mean-variance book with a 2% tracking-error cap: the multiplier on TE is the P&L you left on the table per extra bp of TE.
How to read it
If a constraint is slack, ignore speeches about it. If λ is huge, the model is screaming that the limit, not the alpha, is the decision.
No query has been run yet. Which is tragically normal for most knowledge systems, but we are trying to evolve past decorative databases.