I have a problem with linear constraints but in the objective function I want to have some linear terms along with a $x^2$ term. So it is like the following:
$$\min \sum \limits _i \sum \limits _j (a[i,j] + b[i,j]^2)$$
$ a[i,j] $ and $ b[i,j]$ are integer decision variables.
All my constraints are linear. So my question is, how I can linearize the quadratic objective function and convert the problem to a linear one.
I had an idea to convert it to piecewise linear function, with derivatives at specific points, but I am looking for another solution, as my problem includes many binary and integer variables, and adding more it will make it more complex and slow to solve.