I would like to linearize $x^2$ term in my objective function. I understand this can be solved using quadratic programming solver; however, for my use case linearization is necessary to convert it to MIP if possible. Any help is much appreciated.
There are two possibilities that come to mind, assuming that $x$ is a continuous variable. One is to do a piecewise-linear approximation (leading to an answer that is, well, approximate). The other is to replace $x^2$ with a new variable $z$ and then, on the fly, add constraints of the form $z \ge x_0^2 + 2x_0(x-x_0)$ when a solution is obtained containing $x_0$ and $z_0$ with $z_0 \lt x_0^2$. The latter method is potentially useful if the nature of the model precludes the solver choosing $z \gt x^2$. With CPLEX, you can add these constraints in a lazy constraint callback. With a solver that lacks the needed callback structure, you would be in a solve - cut - solve cycle until you got an optimal solution needing no further cuts.