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Characterizing nonconvex boundaries via scalarization

Jin Ma, Weixuan Xia, Jianfeng Zhang

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.

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.

Paper Structure

This paper contains 12 sections, 7 theorems, 71 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Boundary characterization by way of scalarization
  4. Further analysis of boundary characterization
  5. Dimensionality reduction
  6. An alternative formulation
  7. A numerical example
  8. Extension to infinite-dimensional spaces
  9. Discussion on related applications
  10. Direct applications: Multiplicity in optimization and games
  11. New connections: Constrained reinforcement learning
  12. Concluding remarks

Key Result

Theorem 3.3

Under Assumption as:1, it holds that $\partial D=\partial_{\rm s}D$.

Figures (5)

  • Figure 1: Exemplary nonconvex domains in 2D (I)
  • Figure 2: Exemplary nonconvex domains in 2D (II)
  • Figure 4: Approximations of scalarization-implied boundary
  • Figure : (spherical cone)
  • Figure : (spherical cone)

Theorems & Definitions (20)

  • Definition 2.1
  • Remark 3.2
  • Theorem 3.3
  • proof
  • Theorem 4.1
  • proof
  • Remark 4.2
  • Remark 4.3
  • Corollary 4.4
  • proof
  • ...and 10 more