How about we omit a linear term of QP(Quadratic Programming) problem?

[WestsideMS]

In our model predictive control (MPC) problem, the cost function was simply composed of the square term of the state variable and the square term of the input variable with weights.
When rewriting this cost function as a Quadratic Programming (QP) problem, I would like to ask a question here.

What would happen if we simply put the Hessian matrix (the second term of QP), as the square term of the state variable and the square term of the input variable instead, setting the relationship between each state variable as constraints?

ex) cost function = w1 * sum of x_state + w2 * sum of input_state -> Hessian matrix is only w1 * sum of x_state or w2 * sum of input_state
(in this problem, we need not use a linear part of the QP stand form because there are only square terms)

This is a different way than how people typically implement MPC.
What are the problems compared to inserting a matrix consisting of the product of a large matrix created using state prediction, which is the basic philosophy of MPC, and a large weight matrix, as the second term of the "Hessian" matrix?

This means that we need to set equality constraints for the relationship between each state. Does this give the solver a harsher environment for calculations, and does this make it difficult to find the optimal solution?