# Minimize function of matrix subject to matrix equality
**URL:** https://discourse.julialang.org/t/minimize-function-of-matrix-subject-to-matrix-equality/38701
**Category:** Optimization (Mathematical)
**Created:** [May 3, 2020, 8:46pm UTC](https://discourse.julialang.org/t/minimize-function-of-matrix-subject-to-matrix-equality/38701 "2020-05-03T20:46:49Z")
**Posts on this page:** 6
**Page:** 1
### Author:  [@uncertainly](https://discourse.julialang.org/u/uncertainly)
#### Post date: [May 3, 2020, 8:46pm UTC](https://discourse.julialang.org/t/minimize-function-of-matrix-subject-to-matrix-equality/38701/1 "2020-05-03T20:46:49Z")
newbie here, both to julia, JuMP, only intermediate at optimization.
I would like to solve a problem
```julia
min(A) subject to C = B*A
```
where `A,B,C` are all matrices.
Is it possible to express this in JuMP? Any hints please?
---
### Author:  [@joaquimg](https://discourse.julialang.org/u/joaquimg)
#### Post date: [May 3, 2020, 9:13pm UTC](https://discourse.julialang.org/t/minimize-function-of-matrix-subject-to-matrix-equality/38701/2 "2020-05-03T21:13:50Z")
What do you mean by minimizing a Matrix?
JuMP works very well for minimizing scalars. You can find many examples in [JuMPTutorials.jl](https://github.com/JuliaOpt/JuMPTutorials.jl)
You could use that to minimize some norm of the matrix.
There is experimental work on multi-objetive ([Multiobjective support in JuMP · Issue #2099 · jump-dev/JuMP.jl · GitHub](https://github.com/JuliaOpt/JuMP.jl/issues/2099)).
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### Author:  [@stevengj](https://discourse.julialang.org/u/stevengj)
#### Post date: [May 4, 2020, 11:40am UTC](https://discourse.julialang.org/t/minimize-function-of-matrix-subject-to-matrix-equality/38701/3 "2020-05-04T11:40:42Z")
I think you mean `min f(A)` for some function `f(A)`?
If the matrix `B` is small enough to solve explicitly for its null space `N = nullspace(B)`, then you can completely eliminate this constraint. If `X = B \ C` is a particular solution, then write `A = X + N*Z` where `Z` is an unknown matrix. You then have an _unconstrained_ minimization problem: minimize `f(X + N*Z)` over `Z`.
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### Author:  [@uncertainly](https://discourse.julialang.org/u/uncertainly)
#### Post date: [May 5, 2020, 2:40am UTC](https://discourse.julialang.org/t/minimize-function-of-matrix-subject-to-matrix-equality/38701/4 "2020-05-05T02:40:42Z")
err, that was a typo! First post and I was paying more attention to how to post than to what I wrote.
yes I meant
```julia
min f(A) subject to C = B*A
```
where f() returns a scalar
---
### Author:  [@uncertainly](https://discourse.julialang.org/u/uncertainly)
#### Post date: [May 5, 2020, 2:41am UTC](https://discourse.julialang.org/t/minimize-function-of-matrix-subject-to-matrix-equality/38701/5 "2020-05-05T02:41:38Z")
Is it possible to solve the proposed constrained problem?
Any hints on what the syntax would be?
The problem is that I think the particular f() no longer makes sense if applied to the sum X+N\*Z, it has specific meaning for the matrix A.
---
### Author:  [@joaquimg](https://discourse.julialang.org/u/joaquimg)
#### Post date: [May 5, 2020, 3:06am UTC](https://discourse.julialang.org/t/minimize-function-of-matrix-subject-to-matrix-equality/38701/6 "2020-05-05T03:06:02Z")
note that `X+N*Z` is a matrix with the same dimensions of `A` so there should be no problem computing `f` of that.
can you say something else about `f`?
you only have equalities and no inequalities?