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Second-Order Strong Optimality and Second-Order Duality for Nonsmooth Constrained Multiobjective Fractional Programming Problems

Jiawei Chen, Luyu Liu, Yibing Lv, Debdas Ghosh, Jen-Chih Yao

Abstract

This paper investigates constrained nonsmooth multiobjective fractional programming problem (NMFP) in real Banach spaces. It derives a quotient calculus rule for computing the first- and second-order Clarke derivatives of fractional functions involving locally Lipschitz functions. A novel second-order Abadie-type regularity condition is presented, defined with the help of the Clarke directional derivative and the Páles-Zeidan second-order directional derivative. We establish both first- and second-order strong necessary optimality conditions, which contain some new information on multipliers and imply the strong KKT necessary conditions, for a Borwein-type properly efficient solution of NMFP by utilizing generalized directional derivatives. Moreover, it derives second-order sufficient optimality conditions for NMFP under a second-order generalized convexity assumption. Additionally, we derive duality results between NMFP and its second-order dual problem under some appropriate conditions.

Second-Order Strong Optimality and Second-Order Duality for Nonsmooth Constrained Multiobjective Fractional Programming Problems

Abstract

This paper investigates constrained nonsmooth multiobjective fractional programming problem (NMFP) in real Banach spaces. It derives a quotient calculus rule for computing the first- and second-order Clarke derivatives of fractional functions involving locally Lipschitz functions. A novel second-order Abadie-type regularity condition is presented, defined with the help of the Clarke directional derivative and the Páles-Zeidan second-order directional derivative. We establish both first- and second-order strong necessary optimality conditions, which contain some new information on multipliers and imply the strong KKT necessary conditions, for a Borwein-type properly efficient solution of NMFP by utilizing generalized directional derivatives. Moreover, it derives second-order sufficient optimality conditions for NMFP under a second-order generalized convexity assumption. Additionally, we derive duality results between NMFP and its second-order dual problem under some appropriate conditions.
Paper Structure (6 sections, 14 theorems, 82 equations)

This paper contains 6 sections, 14 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Second-order strong KKT necessary conditions for NMFP
  4. Second-order KKT sufficient optimality conditions for NMFP
  5. Second-order duality
  6. Conclusions

Key Result

lemma 1

A feasible solution $x_0\in X$ is a (weak) Pareto efficient solution of NMFP if and only if it is a (weak) Pareto efficient solution of $s$-MFP, where $s :=\frac{f(x_0)}{F(x_0)}$.

Theorems & Definitions (51)

  • definition 1
  • lemma 1
  • proof
  • definition 2
  • remark 1
  • definition 3
  • definition 4
  • definition 5
  • definition 6
  • remark 2
  • ...and 41 more