Stationarity Finds the Dual

The One-Step Skill

You already know the active/inactive test: if the unconstrained target is past the cap, the solution sits on the cap and the dual is positive. The next skill is computing how positive.

Use this toy score for the weight in one asset:

score(w) = -(w - 0.80)^2
constraint: w <= 0.70
solution: w* = 0.70

The score still wants to move right at w* = 0.70. The cap says no. The dual variable measures the pressure needed to cancel that blocked push.

Derive the Pressure

The derivative of the score is:

score'(w) = -2(w - 0.80)

At the constrained solution:

score'(0.70) = -2(0.70 - 0.80)
             = 0.20

For a maximization problem with an upper bound w <= cap, stationarity says:

objective push - constraint pressure = 0
score'(w*) - lambda = 0
lambda = score'(w*) = 0.20

Try the Lab

Move the unconstrained target and cap. When the solution is on the cap, the displayed slope is the upper-bound dual. When the target is inside the cap, the solution is unconstrained and the dual falls to zero.

Portfolio Reading

In a real Markowitz optimizer, the gradient has more terms: expected return, risk penalty, transaction cost, and maybe factor exposure. Stationarity is the same idea. At the solution, every active constraint contributes enough dual pressure to cancel any remaining objective push in forbidden directions.

This is why solver duals are useful diagnostics. A high dual on an asset cap means the model's local tradeoff wants more of that asset, after accounting for the objective and all other constraints.

Retrieval Practice

Primary Source

For the formal version, read the KKT sections in Boyd and Vandenberghe's Convex Optimization. To see the same idea in code, use the official CVXPY quadratic-program example, which shows dual values on constraints. Keep the stationarity reference open while reviewing this lesson.