Unconstrained optimization problem?

I have a control problem that requires (in addition to other linear constraints) finding optimal coefficients M (n \times n matrix) and a nth dimensional vector b such that

\det(I + xub^T - xM) \geq 0

for all x \in \mathbb{R}, where u = [1,1,\dots,1]^T

Is there any solver in Jump that accepts this sort of problems and solve it numerically? fmincon in MATLAB was able to accept it but the performance is poor. Plus, I couldn’t impose it for all x \in \mathbb{R} which I am not sure is or is not possible with any other solver. Any idea or suggestion?

Imposing a constraint for all x \in \mathbb{R} is not possible with a standard mathematical programming solver. Perhaps there is a linear algebra trick that would let you reformulate the constraint?

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What do you mean by “optimal” in your question? As you present the problem, I’d just pick any b and M such that ub’=M and call it a day.

yes but as I said in the post this is not the only constraint. There are other 50 linear constraints in terms of M and b.

I guess the problem that I am facing is what solver accept det in a constraint. Is there any way to optimize over an interval with fine grid like x = [-1000:0.0001:1000]. How to add that in JuMP?

Got it. Sorry I missed that. Let’s see if I understand you correctly now:

Let me call y the variables you have to optimise (b and M). Your problem is of the form

min_y f(y)
s.t. h_i(y) = 0
s.t. min_x g(x,y) \ge 0

(How do you do Latex here? - edited with Latex, now)

Interesting question. It turns out the general problem is of interest for my own work. So the dual problem will have a Lagrange multiplier function of x. I’ll have to think.

I am not knowledgeable about JuMP so I can’t help there.

Yes thank you. I should’ve been clearer in my post. You could latex by writing inside two dollar signs $like this$