Dual is scarcity
High dual means the constraint is blocking a valuable local move.
Reference 0010
Active-Set Sensitivity
A dual is a local sensitivity. It changes when scarcity changes, and it resets when the active set changes.
Core Rule
minimization with upper cap: if cap binds, dual = blocked improvement from relaxing cap if cap is loose, dual = 0
Lesson 0007 Objective
f(w_A) = 0.15 w_A^2 - 0.30 w_A + 0.08 f'(w_A) = 0.30 w_A - 0.30 unconstrained target: f'(w_A) = 0 w_A = 1.00
Dual Decay
while cap_A < 1.00:
upper dual = -f'(cap_A)
= 0.30(1 - cap_A)
cap_A = 0.70 -> dual 0.09
cap_A = 0.80 -> dual 0.06
cap_A = 0.90 -> dual 0.03
cap_A = 1.00 -> dual 0.00
Interpretation
Scarcity decays
As the cap approaches the unconstrained target, the blocked move becomes less valuable.
Zero dual
When the cap stops blocking the optimizer, the cap has no local shadow price.
Active-set change
When binding constraints change, old duals stop describing the new local problem.
Practical Use
Use a dual for: small perturbations near the solved problem ranking currently scarce constraints debugging which model limits are shaping the optimum Do not use one old dual for: large policy changes moves past a kink or active-set change claims about real-world alpha outside the model
Use this page with Lesson 0010, Reference 0009, and Reference 0008.