i have a convex optimization problem to find $M$ as a $n\times n$ matrix and $x,y,z,s$ are data vectors and all are $n\times 1$:
$$\min_M f(M,x,y,s,z) $$ $$s.t: x^TMy+z^TMz-s^TMs <0$$ $$M>0$$
The problem is that the above constraint is not feasible at all for some data samples, and those instances of data should be assumed as outlayers and not to be considered in the optimization problem.
Then i will face another problem which is checking the feasibility of the constraint for each data sample batch {x,y,z,s} to see regardless of the objective function $f(.)$ the quadratic constraint can ever be negative or not.
So is there any way to find out that sort of outlayers in a simple way?
The constraint is feasible if $\frac{1}{2}(yx^T+xy^T) + zz^T - ss^T$ is negative semidefinite.
You are interested in $\inf_M\{ x^TMy + z^TMz - s^T M s : M > 0 \} < 0$. This is optimizing a linear function over the cone of positive definite matrices. There is much software out there that can solve this problem for you (sedumi, sdpa, sdpt3 just to name a few), but since this problem is unconstrained, the solution is trivial. Rewrite the objective to $tr((yx^T + zz^T - ss^T)M)$. Now if the matrix $\frac{1}{2}(yx^T+xy^T) + zz^T - ss^T$ is positive semidefinite, the $\inf$ is $0$, otherwise it's $-\infty$.