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Sublinear Scalarizations for Proper and Approximate Proper Efficient Points in Nonconvex Vector Optimization

Fernando García-Castaño, Miguel Ángel Melguizo-Padial, G. Parzanese

TL;DR

The paper addresses nonconvex vector optimization by unifying multiple notions of proper efficiency through $Q$-minimality under a separation property called $SSP$ (strict separation property). It proves that, under $SSP$, a $Q$-minimal point is the minimum of a fixed sublinear function, enabling explicit scalarizations for Benson, Henig, and related approximate efficient points without assuming convexity or boundedness. It then derives necessary and, under additional hypotheses, sufficient scalarization conditions for Benson proper efficiency and, in turn, for Henig global and tangential Borwein points, while extending the method to approximate Benson and Henig notions via $D(\varepsilon)$-dilations and associated cone constructs. The results provide a robust, nonconvex-oriented scalarization framework with potential applications where traditional convexity assumptions fail.

Abstract

We show that under a separation property, a $\mathcal{Q}$-minimal point in a normed space is the minimum of a given sublinear function. This fact provides sufficient conditions, via scalarization, for nine types of proper efficient points; establishing a characterization in the particular case of Benson proper efficient points. We also obtain necessary and sufficient conditions in terms of scalarization for approximate Benson and Henig proper efficient points. The separation property we handle is a variation of another known property and our scalarization results do not require convexity or boundedness assumptions.

Sublinear Scalarizations for Proper and Approximate Proper Efficient Points in Nonconvex Vector Optimization

TL;DR

The paper addresses nonconvex vector optimization by unifying multiple notions of proper efficiency through -minimality under a separation property called (strict separation property). It proves that, under , a -minimal point is the minimum of a fixed sublinear function, enabling explicit scalarizations for Benson, Henig, and related approximate efficient points without assuming convexity or boundedness. It then derives necessary and, under additional hypotheses, sufficient scalarization conditions for Benson proper efficiency and, in turn, for Henig global and tangential Borwein points, while extending the method to approximate Benson and Henig notions via -dilations and associated cone constructs. The results provide a robust, nonconvex-oriented scalarization framework with potential applications where traditional convexity assumptions fail.

Abstract

We show that under a separation property, a -minimal point in a normed space is the minimum of a given sublinear function. This fact provides sufficient conditions, via scalarization, for nine types of proper efficient points; establishing a characterization in the particular case of Benson proper efficient points. We also obtain necessary and sufficient conditions in terms of scalarization for approximate Benson and Henig proper efficient points. The separation property we handle is a variation of another known property and our scalarization results do not require convexity or boundedness assumptions.
Paper Structure (5 sections, 17 theorems, 3 equations)

This paper contains 5 sections, 17 theorems, 3 equations.

Table of Contents

  1. Introduction
  2. Notation and previous definitions
  3. Separation theorems and scalarization for Q-minimal points
  4. Scalarization for approximate proper efficient points
  5. Conclusions

Key Result

Theorem 3.1

Let $X$ be a normed space and ${{\mathcal{C}}}$, ${{\mathcal{K}}}$ cones on $X$. The following assertions are equivalent.

Theorems & Definitions (37)

  • Definition 1
  • Definition 2
  • Remark 1
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • Theorem 3.3
  • proof
  • Corollary 3.4
  • proof
  • ...and 27 more