Characterizing nonconvex boundaries via scalarization

Abstract

We present a unified approach for characterizing the boundary of a possibly nonconvex domain. Motivated by the well-known Pascoletti--Serafini method of scalarization, we recast the boundary characterization as a multi-criteria optimization problem with respect to a local partial order induced by a spherical cone with varying orient. Such an approach enables us to trace the whole boundary and can be considered a general dual representation for arbitrary (nonconvex) sets satisfying an exterior cone condition. We prove the equivalence between the geometrical boundary and the scalarization-implied boundary, particularly in the case of Euclidean spaces and two infinite-dimensional spaces for practical interest. By reformulating each scalarized problem as a parameterized constrained optimization problem, we shall develop a corresponding numerical scheme for the proposed approach. Some related applications are also discussed.