Code for solving the constrained LASSO problem
min_{U} | U |_1 s.t | y_i - V' * u_i |_2^2 <= T_i, i = 1, 2, \dots, p
for some p by n data matrix Y, with rows y_i, which we hypothesize factors as Y = U*V', with U and V having inner dimension k.
The problem is solved by solving the KKT conditions, which we solve using ADMM on the Lagrangian and binary search on the dual variables (since it suffices to satisfy complementary slackness, and the problem is convex and assumed strictly feasible). The Lagrangian minimization problem factors into p separate vector LASSO problems.