# Cannot multiply a quadratic expression by an affine expression

**URL:** <https://discourse.julialang.org/t/cannot-multiply-a-quadratic-expression-by-an-affine-expression/101240>\
**Category:** Optimization (Mathematical)\
**Created:** [July 6, 2023, 4:21am UTC](https://discourse.julialang.org/t/cannot-multiply-a-quadratic-expression-by-an-affine-expression/101240 "2023-07-06T04:21:54Z")\
**Posts on this page:** 7\
**Page:** 1

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**Author:** ![horvetz](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/horvetz/32/31993_2.png) [@horvetz](https://discourse.julialang.org/u/horvetz)\
**Post date:** [July 6, 2023, 4:21am UTC](https://discourse.julialang.org/t/cannot-multiply-a-quadratic-expression-by-an-affine-expression/101240/1 "2023-07-06T04:21:54Z")

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I am trying to define an expression P = Q’ \* R \* Q using @expression but it is not cooperating. I tried to split the matrix product and even use @NLexpression macro but no luck. Can someone suggest a way to define P?

> using LinearAlgebra  
> using JuMP  
> model = Model()  
> n = 5  
> @variable(model, R[1:n, 1:n] , Symmetric)  
> @variable(model, Q[1:n, 1:n] , Symmetric)
> 
> #method 1
> 
> # @expression(model, P , Q’ \* R \* Q)
> 
> #method 2
> 
> # @expression(model, P1, R \* Q)
> 
> # @expression(model, P2, Q’ \* P1 )
> 
> #method 3
> 
> # @expression(model, P1, R \* Q)
> 
> # @NLexpression(model, P2, Q’ \* P1 )
> 
> println(model)

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<div class="post-metadata">

**Author:** ![gdalle](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/gdalle/32/27854_2.png) [@gdalle](https://discourse.julialang.org/u/gdalle)\
**Post date:** [July 6, 2023, 5:41am UTC](https://discourse.julialang.org/t/cannot-multiply-a-quadratic-expression-by-an-affine-expression/101240/2 "2023-07-06T05:41:51Z")

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I think that’s because all nonlinear expressions [must be scalars](https://jump.dev/JuMP.jl/stable/manual/nlp/#Scalar-operations-only)?

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**Author:** ![horvetz](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/horvetz/32/31993_2.png) [@horvetz](https://discourse.julialang.org/u/horvetz)\
**Post date:** [July 6, 2023, 10:30pm UTC](https://discourse.julialang.org/t/cannot-multiply-a-quadratic-expression-by-an-affine-expression/101240/3 "2023-07-06T22:30:01Z")

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Yeah that’s what the error message says actually. Does that mean I can’t define matrix product at all?

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**Author:** ![odow](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/odow/32/28685_2.png) [@odow](https://discourse.julialang.org/u/odow)\
**Post date:** [July 7, 2023, 2:05am UTC](https://discourse.julialang.org/t/cannot-multiply-a-quadratic-expression-by-an-affine-expression/101240/4 "2023-07-07T02:05:16Z")

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> Does that mean I can’t define matrix product at all?

Correct. In the current version of JuMP, you cannot create expressions like `X * Y * Z` where `X`, `Y` and `Z` are arrays of JuMP variables.

One work-around is:

```julia
using JuMP, LinearAlgebra
n = 5
model = Model()
@variable(model, R[1:n, 1:n], Symmetric)
@variable(model, Q[1:n, 1:n], Symmetric)
@variable(model, RQ[1:n, 1:n], Symmetric)
@constraint(model, RQ .== R * Q)
@expression(model, P, Q' * QR)

```

This limitation is fixed in the work-in-progress nonlinear rewrite: [https://github.com/jump-dev/JuMP.jl/pull/3106](https://github.com/jump-dev/JuMP.jl/pull/3106).

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**Author:** ![horvetz](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/horvetz/32/31993_2.png) [@horvetz](https://discourse.julialang.org/u/horvetz)\
**Post date:** [July 7, 2023, 11:33pm UTC](https://discourse.julialang.org/t/cannot-multiply-a-quadratic-expression-by-an-affine-expression/101240/5 "2023-07-07T23:33:36Z")

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> [@odow](#):
>
> ```julia
> using JuMP, LinearAlgebra
> n = 5
> model = Model()
> @variable(model, R[1:n, 1:n], Symmetric)
> @variable(model, Q[1:n, 1:n], Symmetric)
> @variable(model, RQ[1:n, 1:n], Symmetric)
> @constraint(model, RQ .== R * Q)
> @expression(model, P, Q' * QR)
> 
> ```

Thanks for the help. Apparently the workaround still doesn’t work. I got the following error

> MethodError: no method matching \*(::Adjoint{VariableRef, Symmetric{VariableRef, Matrix{VariableRef}}}, ::Type{QR})  
> Closest candidates are:  
> \*(::Any, ::Any, ::Any, ::Any…) at /Volumes/Julia-1.7.2/Julia-1.7.app/Contents/Resources/julia/share/julia/base/operators.jl:655  
> \*(::Union{MathOptInterface.ScalarAffineFunction{T}, MathOptInterface.ScalarQuadraticFunction{T}, MathOptInterface.VectorAffineFunction{T}, MathOptInterface.VectorQuadraticFunction{T}}, ::T) where T at ~/.julia/packages/MathOptInterface/LqT2k/src/Utilities/functions.jl:3300  
> \*(::AbstractMatrix{T}, ::LinearAlgebra.AbstractRotation{S}) where {T, S} at /Volumes/Julia-1.7.2/Julia-1.7.app/Contents/Resources/julia/share/julia/stdlib/v1.7/LinearAlgebra/src/givens.jl:20  
> …
> 
> Stacktrace:  
> [1] operate(::typeof(\*), ::Adjoint{VariableRef, Symmetric{VariableRef, Matrix{VariableRef}}}, ::Type)  
> @ MutableArithmetics ~/.julia/packages/MutableArithmetics/Lnlkl/src/interface.jl:186  
> [2] operate(::typeof(MutableArithmetics.add\_mul), ::MutableArithmetics.Zero, ::Adjoint{VariableRef, Symmetric{VariableRef, Matrix{VariableRef}}}, ::Type)  
> @ MutableArithmetics ~/.julia/packages/MutableArithmetics/Lnlkl/src/rewrite.jl:40  
> [3] operate\_fallback!!(::MutableArithmetics.IsNotMutable, ::typeof(MutableArithmetics.add\_mul), ::MutableArithmetics.Zero, ::Adjoint{VariableRef, Symmetric{VariableRef, Matrix{VariableRef}}}, ::Type)  
> @ MutableArithmetics ~/.julia/packages/MutableArithmetics/Lnlkl/src/interface.jl:555  
> [4] operate!!(::typeof(MutableArithmetics.add\_mul), ::MutableArithmetics.Zero, ::Adjoint{VariableRef, Symmetric{VariableRef, Matrix{VariableRef}}}, ::Type)  
> @ MutableArithmetics ~/.julia/packages/MutableArithmetics/Lnlkl/src/rewrite.jl:94  
> [5] macro expansion  
> @ ~/.julia/packages/MutableArithmetics/Lnlkl/src/rewrite.jl:294 [inlined]  
> [6] macro expansion  
> @ ~/.julia/packages/JuMP/60Bnj/src/macros.jl:1352 [inlined]  
> [7] top-level scope  
> @ In[1]:8  
> [8] eval  
> @ ./boot.jl:373 [inlined]  
> [9] include\_string(mapexpr::typeof(REPL.softscope), mod::Module, code::String, filename::String)  
> @ Base ./loading.jl:1196

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<div class="post-metadata">

**Author:** ![odow](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/odow/32/28685_2.png) [@odow](https://discourse.julialang.org/u/odow)\
**Post date:** [July 7, 2023, 11:41pm UTC](https://discourse.julialang.org/t/cannot-multiply-a-quadratic-expression-by-an-affine-expression/101240/6 "2023-07-07T23:41:50Z")

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I made a typo. Should be RQ, not QR 🙂

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**Author:** ![horvetz](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/horvetz/32/31993_2.png) [@horvetz](https://discourse.julialang.org/u/horvetz)\
**Post date:** [July 7, 2023, 11:57pm UTC](https://discourse.julialang.org/t/cannot-multiply-a-quadratic-expression-by-an-affine-expression/101240/7 "2023-07-07T23:57:01Z")

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> [@odow](#):
>
> ```julia
> using JuMP, LinearAlgebra
> n = 5
> model = Model()
> @variable(model, R[1:n, 1:n], Symmetric)
> @variable(model, Q[1:n, 1:n], Symmetric)
> @variable(model, RQ[1:n, 1:n], Symmetric)
> @constraint(model, RQ .== R * Q)
> @expression(model, P, Q' * QR)
> 
> ```

Perfect thanks a lot!!