I have the following problem which attempts to minimize the temperature deviation from a given set point in a home.
Minimize,
$ \sum_{t=1}^{N}{(T_{c}-T_{t})^{2}}$
where,
$ T_{0} =T_{s}$
$ T_{t} = T_{t-1} + \alpha(T_{t}^{a} - T_{t-1})+\beta x_{t},$ for $ t > 1 $
$ x_{t} \geq 0$
In this problem, $x_{t}$ is the variable. and $T_{s}$, $T_{c}$, $\alpha$, and $\beta$ are constant positives. $T^{a}_{t}$ is the $t$ th member of the $1\times N$ constant vector $T^{a}$ which is also positive.
I wan to know whether this is a convex optimization problem or not. It looks difficult and I don't know where to start. It's difficult to substitute and expand the problem but it looks like it gives a polynomial if I do. Since it has a squared term, the final polynomial could be an even degree polynomial (?) Does this mean the problem is a convex problem? Or is there any other way of checking this? Or is this not a convex problem?
Please help.
As pointed out in a comment by @LinAlg, this is a convex quadratic objective, subject to linear constraints, which are convex. Therefore, this is a convex Quadratic Programming problem (QP), for which there are many numerical solvers available under a variety of programming environments and optimization modeling packages. Unless N is gargantuan, this is an easy problem for a QP solver to solve.