All unconstrained strongly convex problems are weakly simplicial
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A multi-objective optimization problem is C^r weakly simplicial if there exists a C^r 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 C^r problems are C^r-1 weakly simplicial for 1≤ r ≤∞. In this paper, we show that all unconstrained strongly convex problems are C^0 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 Bézier simplex fitting method, which accelerates hyper-parameter search.
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