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<p>This technical note proves that, for a smooth vector optimization problem on a closed convex feasible set ordered by a pointed cone, the projected gradient direction depends continuously on the decision variable. Our argument is based on a simple and direct proof via a fixed-domain reformulation of the subproblem. We then give a necessary and sufficient dual characterization of this direction and show that its associated set-valued dual variable mapping is outer semicontinuous.</p>

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The projected gradient direction in vector optimization: continuity and dual characterization

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年2月13日

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  • N. Fazzio,
  • L. F. Prudente,
  • M. L. Schuverdt

摘要

This technical note proves that, for a smooth vector optimization problem on a closed convex feasible set ordered by a pointed cone, the projected gradient direction depends continuously on the decision variable. Our argument is based on a simple and direct proof via a fixed-domain reformulation of the subproblem. We then give a necessary and sufficient dual characterization of this direction and show that its associated set-valued dual variable mapping is outer semicontinuous.