{
  "id": "2406.08147",
  "version": "v1",
  "published": "2024-06-12T12:37:14.000Z",
  "updated": "2024-06-12T12:37:14.000Z",
  "title": "A New Linear Programming Approach and a New Backtracking Strategy for Multiple-Gradient Descent in Multi-Objective Optimization",
  "authors": [
    "Francesco Della Santa"
  ],
  "categories": [
    "math.OC",
    "cs.NA",
    "math.NA"
  ],
  "abstract": "In this work, the author presents a novel method for finding descent directions shared by two or more differentiable functions defined on the same unconstrained domain space. Then, the author illustrates an alternative Multiple-Gradient Descent procedure for Multi-Objective Optimization problems that is based on this new method. In particular, the proposed method consists in finding the shared descent direction solving a relatively cheap Linear Programming (LP) problem, where the LP's objective function and the constraints are defined by the gradients of the objective functions of the Multi-Objective Optimization problem. More precisely, the formulation of the LP problem is such that, if a shared descent direction does not exist for the objective functions, but a non-ascent direction for all the objectives does, the LP problem returns the latter. Moreover, the author defines a new backtracking strategy for Multiple-Gradient Descent methods such that, if the proposed LP is used for computing the direction, the ability to reach and/or explore the Pareto set and the Pareto front is improved. A theoretical analysis of the properties of the new methods is performed, and tests on classic Multi-Objective Optimization problems are proposed to assess their goodness.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-06-12T12:37:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "90C29",
      "65K05",
      "90C26"
    ],
    "keywords": [
      "multiple-gradient descent",
      "linear programming approach",
      "backtracking strategy",
      "multi-objective optimization problem",
      "objective function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}