In that case, we obtain the objective function: One of the main advantages of using the squared loss is the fact that the corresponding objective function is jointly convex in y and w. Indeed, the objective is the composition of an affine mapping, defined by (y, w) → y − Xw, with a convex function, defined by u → u u.
the objective function is convex (see section 3.2.2 of Boyd and Vandenberghe (2004)).