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Existence Results and KKT Optimality Conditions for Generalized Quasiconvex Functions

M. H. Alizadeh, F. Lara

TL;DR

The paper introduces $e$-quasiconvexity, a generalized convexity framework that unifies quasiconvex and $e$-convex functions and includes Lipschitz functions. It develops existence and compactness results for minimizers using generalized asymptotic functions, and proves sufficiency of KKT conditions for differentiable problems with $e$-quasiconvex constraints, extending classical results to nonconvex settings. The authors provide a robust theoretical foundation with detailed properties, an explicit minimal error function, and illustrative nonconvex examples to demonstrate applicability where standard convexity-based results fail. This work broadens the optimization toolkit for nonconvex problems by enabling rigorous solution-existence guarantees and tractable optimality conditions under generalized convexity assumptions, with potential impact on variational inequalities and constrained optimization.

Abstract

We studied a new notion of generalized convex functions called $e$-quasi\-con\-ve\-xi\-ty, which encompasses both quasiconvex and $e$-convex functions, including all Lipschitz functions. By extending the standard properties of quasiconvex functions to $e$-quasiconvex functions, we establish sufficient conditions for the nonemptiness and compactness of the solution set when minimizing an $e$-quasiconvex function, leveraging generalized asymptotic functions, a result which remains applicable even when the set of minimizers is nonconvex. Furthermore, in the differentiable case, we ensure the sufficiency of the KKT optimality conditions when the constraint functions in the mathematical programming problems are $e$-quasiconvex. Finally, we illustrate our new results with several nonconvex (non-quasiconvex) examples.

Existence Results and KKT Optimality Conditions for Generalized Quasiconvex Functions

TL;DR

The paper introduces -quasiconvexity, a generalized convexity framework that unifies quasiconvex and -convex functions and includes Lipschitz functions. It develops existence and compactness results for minimizers using generalized asymptotic functions, and proves sufficiency of KKT conditions for differentiable problems with -quasiconvex constraints, extending classical results to nonconvex settings. The authors provide a robust theoretical foundation with detailed properties, an explicit minimal error function, and illustrative nonconvex examples to demonstrate applicability where standard convexity-based results fail. This work broadens the optimization toolkit for nonconvex problems by enabling rigorous solution-existence guarantees and tractable optimality conditions under generalized convexity assumptions, with potential impact on variational inequalities and constrained optimization.

Abstract

We studied a new notion of generalized convex functions called -quasi\-con\-ve\-xi\-ty, which encompasses both quasiconvex and -convex functions, including all Lipschitz functions. By extending the standard properties of quasiconvex functions to -quasiconvex functions, we establish sufficient conditions for the nonemptiness and compactness of the solution set when minimizing an -quasiconvex function, leveraging generalized asymptotic functions, a result which remains applicable even when the set of minimizers is nonconvex. Furthermore, in the differentiable case, we ensure the sufficiency of the KKT optimality conditions when the constraint functions in the mathematical programming problems are -quasiconvex. Finally, we illustrate our new results with several nonconvex (non-quasiconvex) examples.
Paper Structure (13 sections, 11 theorems, 71 equations, 1 figure)

This paper contains 13 sections, 11 theorems, 71 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries and Basic Definitions
  3. Generalized Convexity
  4. Asymptotic Analysis
  5. Generalized Quasiconvex Functions
  6. Nonemptiness and Compactness of the Solution Set
  7. Sufficient KKT Optimality Conditions
  8. Conclusions
  9. Declarations
  10. Availability of supporting data
  11. Competing Interests
  12. Funding
  13. Acknowledgments

Key Result

Theorem 5

(Huang 24) Let $e: \mathbb{R}^{n} \times \mathbb{R}^{n} \rightarrow \mathbb{R} \cup \left\{ + \infty \right\}$ be an error bifunction such that ${\rm dom}\,f \times {\rm dom}\,f \subseteq {\rm dom}\,e$ and that for every $x, y \in {\rm dom}\,f$, $e(\cdot, y)$ is convex and $e(x, x) = 0$. Suppose tha Conversely, if relation eq(conv=mon) holds, then $f$ is a $2e$-quasiconvex function.

Figures (1)

  • Figure 1: Function $f$ in Example \ref{['exam:exist']}.

Theorems & Definitions (21)

  • Definition 1
  • Definition 2
  • Example 3
  • Remark 4
  • Theorem 5
  • Proposition 6
  • Proposition 7
  • Proposition 8
  • Example 9
  • Proposition 10
  • ...and 11 more