# Formulating objective function & constrain such that the eigenvalue is always positive

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HAMMED OBASEKORE on 30 Aug 2018
Kindly assist, I am trying to tune some parameters (4) such that the eigenvalues are positive
i.e eig(Q)>0 eig(P)>0
Please note: I am not concern about minimizing or maximizing the objective function, just want the tuning paramenters x(1), x(2), x(3) and x(4) that will make eig(Q) and eig(P) positive. dimension of both eig(Q) & eig(P) is 3x1
Thanks in anticipation.
HAMMED OBASEKORE on 31 Aug 2018
Edited: HAMMED OBASEKORE on 31 Aug 2018
Thanks for the prompt assistance;
I feel by sharing the relation of the parameters, it will pinpoint my request
0<=x1<=inf
-inf<x2<inf
-inf<x3<inf
-inf<x4<inf such that;
eig(P) > 0
eig(Q) > 0

Christine Tobler on 4 Sep 2018
Hi Hammed,
Your problem looks like a smooth nonlinear optimization problem: I believe the smallest eigenvalue of a linear combination of matrices is always a smooth function.
I know you don't care which solution that satisfies the equation you get, but the easiest way to approach this might be to use optimization. For example, use the function fmincon, for constrained nonlinear multi-variable optimization. You could use one of the min(eig(...)) > 0 equations as a constraint, and try to maximize over the other one; or use min(min(eig(P)), min(eig(Q)))| as the optimization function.
Alternatively, it might be simpler to just make a plot of min(eig(P)) and min(eig(Q)), depending on (some of) the variables, and try to visually find a spot where these are positive.
HAMMED OBASEKORE on 4 Sep 2018
Thanks alot for the very helpful response.
fmincon
fsolve
and
ga
but all the instances my formulation of constrain and objective function were probably wrong.
But this works fine for my problem
• You could use one of the min(eig(...)) > 0 equations as a constraint, and try to maximize over the other one;*
Best Regards once again.