You want to solve for 4 parameters, a,b,c,E. They all enter the problem linearly.
However, since the set of equality constraints
will not by solved exactly in general, you have two competing goals, which you have not said how they will be resolved. So if there are multiple elements in those vectors, no single value for {a,b,c,E} exists to satisfy all equations at once.
So, you MIGHT choose to define this as a linear least squares problem subject to bound constraints on a,b,c, but that ignores the goal that E is minimized. (May E be negative?)
This is effectively a homogeneous linear system in 4 unknowns, subject to bound constraints on {a,b,c}, as well as an additional requirement on E.
1. If your vectors have length greater than 4 then there will in general be NO exact solution.
2. If your vectors have length exactly 4, then depending on the vectors, there may be infinitely many solutions, or there will be no solution other than that all the unknowns are 0. Of course, in that case, it would be impossible for a solution to satisfy the requirement that each of {a,b,c}>1.
3. If there are less than 4 elements in each vector, or if the system is rank deficient, then there will usually be infinitely many solutions and one would choose a solution that minimizes E, subject to the bounds.
So, much will depend on the length of your vectors, and the vectors themselves.