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KKT conditions for optimization with generalized invex fuzzy functions

Ville Rinne, Yury Nikulin, Marko M. Mäkelä

TL;DR

The classical KKT framework is extended to settings in which the objective and constraint functions are nonsmooth, vector-valued, and fuzzy-valued, and satisfy various generalized invexity conditions such as V-invexity, V-pseudoinvexity, and Vquasiinvexity, ensuring (weak) Pareto optimality in fuzzy environments.

Abstract

This paper explores optimality conditions in optimization problems involving generalized invex fuzzy functions. We extend the classical KKT framework to settings in which the objective and constraint functions are nonsmooth, vector-valued, and fuzzy-valued, and satisfy various generalized invexity conditions such as V-invexity, V-pseudoinvexity, and Vquasiinvexity. After reviewing key concepts from nonsmooth analysis and multiobjective optimization, we derive new KKT-type conditions under weaker assumptions than classical convexity, ensuring (weak) Pareto optimality in fuzzy environments. Our results unify and generalize earlier work by Antczak and Mishra as well as demonstrate the power of generalized invexity in establishing optimality without requiring differentiability nor convexity. Several illustrative examples are included to demonstrate the applicability of the developed theory.

KKT conditions for optimization with generalized invex fuzzy functions

TL;DR

The classical KKT framework is extended to settings in which the objective and constraint functions are nonsmooth, vector-valued, and fuzzy-valued, and satisfy various generalized invexity conditions such as V-invexity, V-pseudoinvexity, and Vquasiinvexity, ensuring (weak) Pareto optimality in fuzzy environments.

Abstract

This paper explores optimality conditions in optimization problems involving generalized invex fuzzy functions. We extend the classical KKT framework to settings in which the objective and constraint functions are nonsmooth, vector-valued, and fuzzy-valued, and satisfy various generalized invexity conditions such as V-invexity, V-pseudoinvexity, and Vquasiinvexity. After reviewing key concepts from nonsmooth analysis and multiobjective optimization, we derive new KKT-type conditions under weaker assumptions than classical convexity, ensuring (weak) Pareto optimality in fuzzy environments. Our results unify and generalize earlier work by Antczak and Mishra as well as demonstrate the power of generalized invexity in establishing optimality without requiring differentiability nor convexity. Several illustrative examples are included to demonstrate the applicability of the developed theory.
Paper Structure (17 sections, 16 theorems, 170 equations, 1 figure)

This paper contains 17 sections, 16 theorems, 170 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Single‑valued invex functions
  4. Vector-valued functions
  5. Vector minimization and Pareto optimality
  6. Geoffrion proper efficiency
  7. V-invex
  8. V-pseudoinvex and V-quasiinvex functions
  9. Fuzzy optimization problems
  10. Fuzzy numbers and fuzzy–valued functions
  11. Fuzzy‑valued optimization
  12. KKT optimality conditions
  13. KKT with invexity
  14. KKT with V-pseudoinvexity and V-quasiinvexity
  15. KKT with pseudoinvexity and quasiinvexity
  16. ...and 2 more sections

Key Result

Theorem 3.3

Rei Let $f:X_{0}\to\mathbb{R}$ be LLC and, for each $\boldsymbol u\in X_{0}$ and $\boldsymbol x\in X_{0}$, suppose the convex cone is closed. Then $f$ is invex on $X_{0}$iff every stationary point ($\boldsymbol 0\in\partial_{C}f(\boldsymbol u)$) is a global minimizer of $f$ on $X_{0}$.

Figures (1)

  • Figure 1: Triangular fuzzy number $\widetilde{u}=(u_1,u_2,u_3)$ and its $\alpha$–level set $[\widetilde{u}]^{\alpha}$.

Theorems & Definitions (58)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Remark 2.6
  • Definition 3.1
  • Example 3.2
  • Theorem 3.3
  • Remark 3.4
  • ...and 48 more