See the literature on exact penalty methods. I believe the short of it is yes, for a large class of problems this works, but the new problem will not be necessa

See the literature on exact penalty methods. I believe the short of it is yes, for a large class of problems this works, but the new problem will not be necessarily easier to solve. In the case of the abs(.) function, the nonlinearity at 0 makes subgradient methods slow, and the large constant in front of the abs(.) might prove numerically unstable.