# DiffOpt - differentiating through a QP/different from manual

**URL:** <https://discourse.julialang.org/t/diffopt-differentiating-through-a-qp-different-from-manual/118141>\
**Category:** Optimization (Mathematical)\
**Tags:** diffopt\
**Created:** [August 13, 2024, 2:57pm UTC](https://discourse.julialang.org/t/diffopt-differentiating-through-a-qp-different-from-manual/118141 "2024-08-13T14:57:37Z")\
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**Author:** ![blob](https://avatars.discourse-cdn.com/v4/letter/b/ebca7d/32.png) [@blob](https://discourse.julialang.org/u/blob)\
**Post date:** [August 14, 2024, 9:34am UTC](https://discourse.julialang.org/t/diffopt-differentiating-through-a-qp-different-from-manual/118141/4 "2024-08-14T09:34:30Z")

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Well, it’s difficult to argue with finite differences… The signs matter, especially when using the [function `dual` from JuMP](https://discourse.julialang.org/t/to-confirm-the-sign-of-the-dual-variable-on-a-constraint-of-jump-model/114292/2). For the future:

> **Code for Amos and Kolter (2017)**
>
> ```julia
> ######This uses duals from JuMP which can be negative. 
> ######However, Amos and Kolter (2017) assume λ≥0, so
> ######we need to adjust the signs
> ######Duals from JuMP are non-positive for ≤ constraints
> q = q
> G = G
> h = h
> Q = Q
> 
> LHM = [
> Q -transpose(G)
> -Diagonal(dual.(cons))*G -Diagonal(G*value.(x)-h) 
> ]
> LHMinv = inv(LHM)
> k1 = length(LHM[:,1])
> k2 = length(Diagonal(dual.(cons))[1,:])
> RHMb = [zeros(k1-k2,k2)
> -Diagonal(dual.(cons))] #Matrix(1.0I, size(h)[1])  
> RHMc = [-Matrix(1.0I, 2,2)
> zeros(k2,2)]
> 
> RHMc = [-Matrix(1.0I, 2,2)
> zeros(k2,2)]
> 
> dallb = LHMinv*RHMb ####both w and λ
> dallc = LHMinv*RHMc ####both w and λ
> 
> dwh = dallb[1:2,:] ####picking only w
> dwq = dallc[1:2,:] ####picking only w
> 
> dAm = zeros(size(G))
> dwA = []
> for i = 1:2
> dAm = zeros(size(G))
> dAm[i] = 1.0 ######Do for every element of G
> RHMA = [
> transpose(dAm).* dual.(cons)
> Diagonal(dual.(cons)).* dAm* value.(x)
> ]
> dallAtemp = LHMinv * RHMA
> dwAtemp=dallAtemp[1:2,1:end] ####picking only w
> push!(dwA, dwAtemp)
> end
> dwG = reduce(hcat,dwA)
> 
> @show dwh,dwq,dwG
> 
> ```

> **Code for Boot (1963)**
>
> ```julia
> x_s = Q^(-1)*(-q)-Q^(-1)*G'*(G*Q^(-1)*G')^(-1)*(G*Q^(-1)*(-q)-h)
> 
> dxih=Q^(-1)*G'*(G*Q^(-1)*G')^(-1)
> dxiq=(Q^(-1)-dxih*G*Q^(-1))
> vh = (G*Q^(-1)*G')^(-1)*(G*Q^(-1)*(-q)-h)
> dxiA1=-x_s[1]*dxih-vh[1]*dxiq[:,1] ####We use the expressions as they are derived by the authors
> dxiA2=-x_s[2]*dxih-vh[1]*dxiq[:,2]
> dxiG = hcat(dxiA1,dxiA2)
> 
> dxiq = -dxiq #####But we know that a=-q so we need to switch signs
> 
> @show dxih,dxiq,dxiG;
> 
> ```

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