<p>In this study, we introduce the notion of a higher-order strongly E-convex function of order <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\varvec{\sigma }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">σ</mi> </mrow> </math></EquationSource> </InlineEquation> as a generalization of the E-convex function and higher-order strongly convex function. We address nonconvex multiobjective fractional programming problems involving E-differentiable functions <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\varvec{(MFP_E)}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo mathvariant="bold" stretchy="false">(</mo> <mi mathvariant="bold-italic">M</mi> <mi mathvariant="bold-italic">F</mi> <msub> <mi mathvariant="bold-italic">P</mi> <mi mathvariant="bold-italic">E</mi> </msub> <mo mathvariant="bold" stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. We establish E-Karush-Kuhn-Tucker sufficient optimality conditions for nonsmooth vector optimization problems under higher-order E-convexity hypothesis. Further, we formulate a multiobjective Schaible-type dual problem involving E-differentiable functions <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\varvec{(MSD_E)}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo mathvariant="bold" stretchy="false">(</mo> <mi mathvariant="bold-italic">M</mi> <mi mathvariant="bold-italic">S</mi> <msub> <mi mathvariant="bold-italic">D</mi> <mi mathvariant="bold-italic">E</mi> </msub> <mo mathvariant="bold" stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\varvec{(MFP_E)}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo mathvariant="bold" stretchy="false">(</mo> <mi mathvariant="bold-italic">M</mi> <mi mathvariant="bold-italic">F</mi> <msub> <mi mathvariant="bold-italic">P</mi> <mi mathvariant="bold-italic">E</mi> </msub> <mo mathvariant="bold" stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and establish duality results between <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\varvec{(MFP_E)}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo mathvariant="bold" stretchy="false">(</mo> <mi mathvariant="bold-italic">M</mi> <mi mathvariant="bold-italic">F</mi> <msub> <mi mathvariant="bold-italic">P</mi> <mi mathvariant="bold-italic">E</mi> </msub> <mo mathvariant="bold" stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and corresponding <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\varvec{(MSD_E)}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo mathvariant="bold" stretchy="false">(</mo> <mi mathvariant="bold-italic">M</mi> <mi mathvariant="bold-italic">S</mi> <msub> <mi mathvariant="bold-italic">D</mi> <mi mathvariant="bold-italic">E</mi> </msub> <mo mathvariant="bold" stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> under the assumption of higher-order strongly E-convexity hypothesis.</p>

On the Optimality and Duality in Nonsmooth Multiobjective Fractional Optimization Problems via Higher-Order E-Convexity

期刊论文
发表日期：

2026年3月6日

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S. K. Mishra,
D. Singh,
B. C. Joshi,
Pankaj Kumar

摘要

In this study, we introduce the notion of a higher-order strongly E-convex function of order $\varvec{\sigma }$ $σ$ as a generalization of the E-convex function and higher-order strongly convex function. We address nonconvex multiobjective fractional programming problems involving E-differentiable functions $\varvec{(MFP_E)}$ $(M F P_{E})$ . We establish E-Karush-Kuhn-Tucker sufficient optimality conditions for nonsmooth vector optimization problems under higher-order E-convexity hypothesis. Further, we formulate a multiobjective Schaible-type dual problem involving E-differentiable functions $\varvec{(MSD_E)}$ $(M S D_{E})$ for $\varvec{(MFP_E)}$ $(M F P_{E})$ and establish duality results between $\varvec{(MFP_E)}$ $(M F P_{E})$ and corresponding $\varvec{(MSD_E)}$ $(M S D_{E})$ under the assumption of higher-order strongly E-convexity hypothesis.