Lagrange Multipliers and KKT Conditions

Motivation: why this matters in quant finance

Constrained portfolios obey budget constraints, leverage caps, factor-neutrality rules, and box limits. Lagrange multipliers and KKT conditions turn those restrictions into equations and shadow prices.

The informal idea

A multiplier measures how much the objective would improve if a constraint were relaxed. KKT combines stationarity, feasibility, complementary slackness, and sign restrictions.

Formal definitions

For equality constraints $g_i(x)=0$, $\mathcal{L}(x,\lambda)=f(x)+\sum_i\lambda_i g_i(x)$. With inequalities $h_j(x)\le0$, KKT adds $\nu_j\ge0$ and $\nu_jh_j(x)=0$.

Key properties

Stationarity balances objective and constraints

At the optimum, the objective gradient lies in the span of active constraint gradients.

Complementary slackness identifies active constraints

A non-binding inequality has zero multiplier.

Convexity makes KKT sufficient

Under regularity conditions, KKT certifies global optima for convex problems.

Worked example

For minimum variance with $\mathbf{1}^\top\mathbf{w}=1$, $\mathcal{L}=\mathbf{w}^\top\Sigma\mathbf{w}+\lambda(\mathbf{1}^\top\mathbf{w}-1)$. Stationarity gives $2\Sigma\mathbf{w}+\lambda\mathbf{1}=0$.

Common confusions and pitfalls

"A multiplier is an arbitrary trick." It is the shadow value of relaxing a constraint.

"All constraints need positive multipliers." Only active inequalities can have non-zero multipliers.

"KKT always proves a global optimum." Sufficiency needs convexity and regularity.

Where this goes next

• Quadratic Programming: turns KKT systems into solver-ready problems.
• Convex Duality: interprets multipliers as dual variables.
• Mean-Variance Optimisation: uses these conditions.