Biobjective Pareto Solution Mapping with Application in Group Lasso Regularization Path

We study the biobjective properly Pareto set, even when both objectives are nondifferentiable. We focus on convex problems that subsume the lasso regression and the group lasso. We define the biobjective Pareto solution mapping that maps any positive weighting scalar to the corresponding set of Pareto optimal solutions. The union of these sets for all positive weighting parameters is the properly Pareto set. Our main innovation is the discovery of a smooth dual problem via the augmented Lagrangian even though the original problem is not even differentiable (thus not smooth). We further obtain a Lipschitz continuous function, called the shadow path, that maps any positive weighting parameter to the corresponding dual multiplier. We design a pathfollowing algorithm to take advantage of the Lipschitz continuity and recover the properly Pareto set. Our new theory addresses the key challenges of nondifferentiability and nonuniqueness in, for instance, group lasso type of applications. Our algorithm shows a five-times speedup over a state-of-the-art benchmark when tracing regularization paths.