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<p>In this paper, we consider an optimization problem in which the objective is the difference of two <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10957_2025_2755_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(\varPhi _c\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>Φ</mi> <mi>c</mi> </msub> </math></EquationSource> </InlineEquation>-convex functions and the constraints include a set constraint and a cone constraint. We first introduce three new constraint qualifications regarding the <i>c</i>-subdifferential and the epigraph properties of the involved functions. Under the new constraint qualifications, we provide some sufficient conditions for optimality conditions to hold. Similarly, strong and total Lagrange dualities for the optimization problem involving the difference of two <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10957_2025_2755_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(\varPhi _c\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>Φ</mi> <mi>c</mi> </msub> </math></EquationSource> </InlineEquation>-convex functions are also given.</p>

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Optimality Conditions and Lagrange Dualities for Optimization Problems Involving \(\varPhi _c\)-convex Functions

  • 期刊论文

  • 发表日期：

    2025年7月5日

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  • Ling-Li Hu,
  • Dong-Hui Fang

摘要

In this paper, we consider an optimization problem in which the objective is the difference of two \(\varPhi _c\) Φ c -convex functions and the constraints include a set constraint and a cone constraint. We first introduce three new constraint qualifications regarding the c-subdifferential and the epigraph properties of the involved functions. Under the new constraint qualifications, we provide some sufficient conditions for optimality conditions to hold. Similarly, strong and total Lagrange dualities for the optimization problem involving the difference of two \(\varPhi _c\) Φ c -convex functions are also given.