Single-Period Optimizer Project Card

The Problem

maximize    αᵀw - γ( eᵀΣ_F e + Σ d_i²w_i² ) - κ‖w - w_prev‖₁
subject to  1ᵀw = 1,   0 ≤ w ≤ 0.40,   βᵀw ≤ 0.80        (e = Fᵀw)

stage-1 data:  α = [.09,.06,.04,.03]   β = [1.5,1,.5,0]   σ_f = .20
               idio vol = [.10,.08,.06,.02]   γ = 5   w_prev = .25 each   κ = 0

The KKT Test Suite (convex ⇒ passing = proven optimal)

1. primal feasibility:   1ᵀw = 1;  0 ≤ w ≤ cap;  βᵀw ≤ 0.80
2. dual signs:           λ, η↓, η↑ ≥ 0        (ν free — sign convention varies)
3. comp. slackness:      λ(0.80 - βᵀw) = 0;   η↓_i w_i = 0;   η↑_i(cap - w_i) = 0
4. stationarity:         -α + 2γΣw + ν1 + λβ - η↓ + η↑ = 0   (per asset)
   with κ > 0, parked names satisfy 0 ∈ (push) + κ[-1,1] instead

Debug Table

symptom                                      likely cause
stationarity off by SAME amount, all assets  budget dual sign flipped
λ > 0 but wall slack                        wrong constraint's dual pasted
one asset fails stationarity                 its bound dual missing (check w_i = 0 or cap)
DCP error on the cost term                   maximizing +κ‖·‖₁ (Ref 0015 padding bug)
solver says infeasible                       caps sum below 1? wall below min possible β?

Definition of Done

[] stage 1 passes all KKT checks         [] κ=.02 parks a name at w_prev (say which)
[] λ(wall) sweep is a step function      [] wall dual = after-toll breakeven (0016)
[] saw the DCP error once, on purpose

Use this page with Lesson 0019 (skeleton + live KKT checker), Reference 0007 (CVXPY template), and Reference 0004 (the conditions themselves).