All unconstrained strongly convex problems are weakly simplicial

A multi-objective optimization problem is Cr weakly simplicial if there exists a Cr surjection from a simplex onto the Pareto set/front such that the image of each subsimplex is the Pareto set/front of a subproblem, where 0≤ r≤ ∞. This property is helpful to compute a parametric-surface approximation of the entire Pareto set and Pareto front. It is known that all unconstrained strongly convex Cr problems are Cr-1 weakly simplicial for 1≤ r ≤ ∞. In this paper, we show that all unconstrained strongly convex problems are C0 weakly simplicial. The usefulness of this theorem is demonstrated in a sparse modeling application: we reformulate the elastic net as a non-differentiable multi-objective strongly convex problem and approximate its Pareto set (the set of all trained models with different hyper-parameters) and Pareto front (the set of performance metrics of the trained models) by using a B'ezier simplex fitting method, which accelerates hyper-parameter search.