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Interval-Valued Optimization Problems for Strongly LU-E-Invex and Strongly LU-E-Preinvex Functions

Tauheed, Akhlad Iqbal, Amir Suhail

Abstract

In this paper, we introduce and explore the concepts of strongly LU-E-preinvex (SLUEP), pseudo strongly LU-E-preinvex (PSLUEP) and strongly LU-E-invex (SLUEI) functions. To illustrate and validate these definitions, we provide several non-trivial examples. Additionally, we extend the idea of strongly-G invex sets to the context of interval-valued functions. The epigraph of a SLUEP function is derived, and a relationship between SLUEP and PSLUEP functions have been explored. A key contribution of this work is the identification of a significant connection between weakly-strongly E-invex functions and SLUEP functions. As an application, we consider a nonlinear programming problem involving SLUEP functions. Under certain conditions, we prove that a local minimum of the problem is also a global minimum. Moreover, the sufficiency of Karush-Kuhn-Tucker (KKT) optimality conditions by considering the objective and constraint functions are SLUEI and SEI respectively. The theoretical results are validated through illustrative examples and counterexamples.

Interval-Valued Optimization Problems for Strongly LU-E-Invex and Strongly LU-E-Preinvex Functions

Abstract

In this paper, we introduce and explore the concepts of strongly LU-E-preinvex (SLUEP), pseudo strongly LU-E-preinvex (PSLUEP) and strongly LU-E-invex (SLUEI) functions. To illustrate and validate these definitions, we provide several non-trivial examples. Additionally, we extend the idea of strongly-G invex sets to the context of interval-valued functions. The epigraph of a SLUEP function is derived, and a relationship between SLUEP and PSLUEP functions have been explored. A key contribution of this work is the identification of a significant connection between weakly-strongly E-invex functions and SLUEP functions. As an application, we consider a nonlinear programming problem involving SLUEP functions. Under certain conditions, we prove that a local minimum of the problem is also a global minimum. Moreover, the sufficiency of Karush-Kuhn-Tucker (KKT) optimality conditions by considering the objective and constraint functions are SLUEI and SEI respectively. The theoretical results are validated through illustrative examples and counterexamples.
Paper Structure (5 sections, 19 theorems, 157 equations, 5 figures)

This paper contains 5 sections, 19 theorems, 157 equations, 5 figures.

Table of Contents

  1. introduction
  2. preliminaries
  3. main results
  4. SLUEP programming problem
  5. Conclusion

Key Result

Proposition 2.1

Stefanini2009$\forall$$\mathcal{ U}_1, \mathcal{ U}_2 \in \mathcal{I}(\mathbb{R})$, $\mathcal{ U}_1 \ominus_{gH} \mathcal{ U}_2$ always exists and $\mathcal{ U}_1 \ominus_{gH} \mathcal{ U}_2 \in \mathcal{I}(\mathbb{R})$. $\mathcal{ U}_1 \ominus_{gH} \mathcal{ U}_2 \preceq 0$ if and only if $\mathcal

Figures (5)

  • Figure 1: $\tilde{h}$ is SLU$\mathcal{E}$P but not SSLU$\mathcal{\mathcal{E}}$P with respect to the mappings $\mathcal{E}$ and $\Psi$ given in Example \ref{['example3.1']}.
  • Figure 2: $\tilde{h}$ is SSLU$\mathcal{E}$P but not SLU$\mathcal{E}$P with respect to the mappings $\mathcal{E}$ and $\Psi$ given in Example \ref{['example3.2']}.
  • Figure 3: The function $\tilde{h}$ is an SLU$\mathcal{E}$P as well as PSLU$\mathcal{E}$P with respect to the mappings $\mathcal{E}$ and $\Psi$ given in Example \ref{['example3.3']}.
  • Figure 4: The function $\tilde{h}$ is SLU$\mathcal{E}$I but not weakly S$\mathcal{E}$I with respect to the mappings $\mathcal{E}$ and $\Psi$ given in Example \ref{['example3.6']}.
  • Figure 5: $\zeta^* = ln(2)$ is a non-dominated solution of $(\rm P1^*)$

Theorems & Definitions (61)

  • Definition 2.1
  • Proposition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Proposition 2.2
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • ...and 51 more