Lipschitz continuity of solution multifunctions of extended $\ell_1$ regularization problems

The Lasso and the basis pursuit in compressed sensing and machine learning are convex optimization problems with three parameters: the regularization scalar, the observation vector and the data matrix. Relative to the first two parameters, we obtain the Lipschitz continuity of the solution multifunction on its convex domain. When the data matrix of the Lasso also perturbs, where non-polyhedral structure may display, we obtain full characterizations for the Lipschitz continuity of the solution multifunction on the product of a compact and convex set in the space of first two parameters and a neighborhood of the fixed data matrix. Moreover for the solution multifunction of the Lasso, we show that the Lipschitz continuity implies its single-valuedness. Our analysis is based on polyhedron theory, a sufficient condition that ensures the Lipschitz continuity of a polyhedral multifunction with a convex domain, and an explicit representation of the solution multifunction, where the latter is a consequence of the Lipschitz continuity of the solution multifunction relative to the first two parameters.