<p>By means of the augmented higher-order radial epiderivatives, we investigate higher-order optimality conditions of Benson proper efficient and weakly efficient solutions for a class of uncertain nonsmooth nonconvex vector optimization problems with set and cone constraints. We introduce a new version of the higher-order augmented proper (weak) radial epiderivatives of profile mappings and establish some their basic characterizations. Simultaneously, we construct several fundamental sum calculation formulas for the augmented higher-order radial sets. Some applications to the uncertain vector optimization problem <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(UVOP (\mathcal {U}),\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>U</mi> <mi>V</mi> <mi>O</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">U</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> where the decision variable of problem functions is given on real normal space, are derived. As a consequence, higher-order necessary and sufficient optimality conditions in terms of the augmented higher-order weak radial epiderivatives and radial sets of Benson proper (weakly) efficient solutions for <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(UVOP (\mathcal {U})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>U</mi> <mi>V</mi> <mi>O</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">U</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> without any convexity are established. Some illustrative examples are also given to validate our results numerically.</p>

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Higher-order optimality of Benson proper efficient and weakly efficient solutions for robust vector optimization problems involving set and cone constraints

  • Dinh Dieu Hang,
  • Tran Van Su

摘要

By means of the augmented higher-order radial epiderivatives, we investigate higher-order optimality conditions of Benson proper efficient and weakly efficient solutions for a class of uncertain nonsmooth nonconvex vector optimization problems with set and cone constraints. We introduce a new version of the higher-order augmented proper (weak) radial epiderivatives of profile mappings and establish some their basic characterizations. Simultaneously, we construct several fundamental sum calculation formulas for the augmented higher-order radial sets. Some applications to the uncertain vector optimization problem \(UVOP (\mathcal {U}),\) U V O P ( U ) , where the decision variable of problem functions is given on real normal space, are derived. As a consequence, higher-order necessary and sufficient optimality conditions in terms of the augmented higher-order weak radial epiderivatives and radial sets of Benson proper (weakly) efficient solutions for \(UVOP (\mathcal {U})\) U V O P ( U ) without any convexity are established. Some illustrative examples are also given to validate our results numerically.