{
  "id": "2210.02852",
  "version": "v1",
  "published": "2022-10-04T04:28:23.000Z",
  "updated": "2022-10-04T04:28:23.000Z",
  "title": "Generalized Hukuhara Hadamard Derivative of Interval-valued Functions and Its Applications to Interval Optimization",
  "authors": [
    "Ram Surat Chauhan",
    "Debdas Ghosh",
    "Qamrul Hasan Ansari"
  ],
  "comment": "23 pages",
  "categories": [
    "math.OC"
  ],
  "abstract": "In this article, we study the notion of gH-Hadamard derivative for interval-valued functions (IVFs) and its applications to interval optimization problems (IOPs). It is shown that the existence of gH-Hadamard derivative implies the existence of gH-Frechet derivative and vise-versa. Further, it is proved that the existence of gH-Hadamard derivative implies the existence of gH-continuity of IVFs. We found that the composition of a Hadamard differentiable real-valued function and a gH-Hadamard differentiable IVF is gHHadamard differentiable. Further, for finite comparable IVF, we prove that the gH-Hadamard derivative of the maximum of all finite comparable IVFs is the maximum of their gH-Hadamard derivative. The proposed derivative is observed to be useful to check the convexity of an IVF and to characterize efficient points of an optimization problem with IVF. For a convex IVF, we prove that if at a point the gH-Hadamard derivative does not dominate to zero, then the point is an efficient point. Further, it is proved that at an efficient point, the gH-Hadamard derivative does not dominate zero and also contains zero. For constraint IOPs, we prove an extended Karush-Kuhn-Tucker condition by using the proposed derivative. The entire study is supported by suitable examples.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-10-04T04:28:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "26A24",
      "90C30",
      "65K05"
    ],
    "keywords": [
      "generalized hukuhara hadamard derivative",
      "interval-valued functions",
      "efficient point",
      "finite comparable ivf",
      "gh-hadamard derivative implies"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}