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Vector critical points and generalized quasi-efficient solutions in nonsmooth multi-objective programming.

Zhen Wang1, Ru Li1, Guolin Yu1

  • 1Institute of Applied Mathematics, North Minzu University, Yinchuan, Ningxia 750021 P.R. China.

Journal of Inequalities and Applications
|August 29, 2017
PubMed
Summary
This summary is machine-generated.

This study introduces higher-order approximate invex functions and generalized Jacobians for multi-objective programming. It establishes that solutions to vector variational-like inequalities correspond to generalized quasi-efficient solutions.

Keywords:
approximate invexitymulti-objective programmingquasi-efficiencyvector critical pointvector variational-like inequality

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Area of Science:

  • Optimization Theory
  • Vector Calculus
  • Mathematical Programming

Background:

  • Multi-objective programming problems often involve complex, nonsmooth functions.
  • Existing efficiency concepts may not fully capture the nuances of higher-order approximations.
  • Generalized Jacobian and invexity concepts are crucial for analyzing such problems.

Purpose of the Study:

  • To introduce and investigate extended approximately invex vector-valued functions of higher order.
  • To propose notions of higher-order (weak) quasi-efficiency for multi-objective programming.
  • To establish connections between generalized vector variational-like inequalities and quasi-efficient solutions.

Main Methods:

  • Introduction of generalized Jacobian for vector-valued functions.
  • Development of higher-order approximate invexity conditions.
  • Proof techniques linking variational inequalities to quasi-efficiency criteria.

Main Results:

  • Existence of extended approximately invex vector-valued functions is demonstrated.
  • Solutions to generalized vector variational-like inequalities are shown to be generalized quasi-efficient solutions.
  • Equivalent conditions are derived, linking vector critical points to higher-order weak quasi-efficiency.

Conclusions:

  • The introduced framework provides a robust method for analyzing nonsmooth multi-objective programming.
  • Higher-order approximate invexity assumptions are sufficient for establishing solution equivalences.
  • The study advances the understanding of efficiency concepts in generalized optimization settings.