 # Minimize function of matrix subject to matrix equality

newbie here, both to julia, JuMP, only intermediate at optimization.

I would like to solve a problem

``````   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?

What do you mean by minimizing a Matrix?

JuMP works very well for minimizing scalars. You can find many examples in JuMPTutorials.jl

You could use that to minimize some norm of the matrix.

There is experimental work on multi-objetive (https://github.com/JuliaOpt/JuMP.jl/issues/2099).

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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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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

``````    min f(A)     subject to     C = B*A
``````

where f() returns a scalar

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.

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?