A New Linear Programming Approach and a New Backtracking Strategy for Multiple-Gradient Descent in Multi-Objective Optimization
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866918275817930752 |
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| author | Della Santa, Francesco |
| author_facet | Della Santa, Francesco |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_08147 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A New Linear Programming Approach and a New Backtracking Strategy for Multiple-Gradient Descent in Multi-Objective Optimization Della Santa, Francesco Optimization and Control Numerical Analysis 90C29, 65K05, 90C26 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. |
| title | A New Linear Programming Approach and a New Backtracking Strategy for Multiple-Gradient Descent in Multi-Objective Optimization |
| topic | Optimization and Control Numerical Analysis 90C29, 65K05, 90C26 |
| url | https://arxiv.org/abs/2406.08147 |