Exact Regularization of Convex Programs

The regularization of a convex program is exact if all solutions of the regularized problem are also solutions of the original problem for all values of the regularization parameter below some positive threshold. For a general convex program, we show that the regularization is exact if and only if a certain selection problem has a Lagrange multiplier. Moreover, the regularization parameter threshold is inversely related to the Lagrange multiplier. We use this result to generalize an exact regularization result of Ferris and Mangasarian [Appl. Math. Optim., 23 (1991), pp. 266–273] involving a linearized selection problem. We also use it to derive necessary and sufficient conditions for exact penalization, similar to those obtained by Bertsekas [Math. Programming, 9 (1975), pp. 87–99] and by Bertsekas, Nedić, and Ozdaglar [Convex Analysis and Optimization, Athena Scientific, Belmont, MA, 2003]. When the regularization is not exact, we derive error bounds on the distance from the regularized solution to the original solution set. We also show that existence of a “weak sharp minimum” is in some sense close to being necessary for exact regularization. We illustrate the main result with numerical experiments on the $\ell_1$ regularization of benchmark (degenerate) linear programs and semidefinite/second-order cone programs. The experiments demonstrate the usefulness of $\ell_1$ regularization in finding sparse solutions.