Scaled-PAKKT sequential optimality condition for multiobjective problems and its application to an Augmented Lagrangian method
Based on the recently introduced Scaled Positive Approximate Karush–Kuhn–Tucker condition for single objective problems, we derive a sequential necessary optimality condition for multiobjective problems with equality and inequality constraints as well as additional abstract set constraints. These ne...
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| Published in: | Computational optimization and applications Vol. 89; no. 3; pp. 769 - 803 |
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| Main Authors: | , , , |
| Format: | Journal Article |
| Language: | English |
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Springer US
01.12.2024
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| ISSN: | 0926-6003, 1573-2894 |
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| Abstract | Based on the recently introduced Scaled Positive Approximate Karush–Kuhn–Tucker condition for single objective problems, we derive a sequential necessary optimality condition for multiobjective problems with equality and inequality constraints as well as additional abstract set constraints. These necessary sequential optimality conditions for multiobjective problems are subject to the same requirements as ordinary (pointwise) optimization conditions: we show that the updated Scaled Positive Approximate Karush–Kuhn–Tucker condition is necessary for a local weak Pareto point to the problem. Furthermore, we propose a variant of the classical Augmented Lagrangian method for multiobjective problems. Our theoretical framework does not require any scalarization. We also discuss the convergence properties of our algorithm with regard to feasibility and global optimality without any convexity assumption. Finally, some numerical results are given to illustrate the practical viability of the method. |
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| AbstractList | Based on the recently introduced Scaled Positive Approximate Karush–Kuhn–Tucker condition for single objective problems, we derive a sequential necessary optimality condition for multiobjective problems with equality and inequality constraints as well as additional abstract set constraints. These necessary sequential optimality conditions for multiobjective problems are subject to the same requirements as ordinary (pointwise) optimization conditions: we show that the updated Scaled Positive Approximate Karush–Kuhn–Tucker condition is necessary for a local weak Pareto point to the problem. Furthermore, we propose a variant of the classical Augmented Lagrangian method for multiobjective problems. Our theoretical framework does not require any scalarization. We also discuss the convergence properties of our algorithm with regard to feasibility and global optimality without any convexity assumption. Finally, some numerical results are given to illustrate the practical viability of the method. |
| Author | Schuverdt, M. L. Fazzio, N. S. Carrizo, G. A. Sánchez, M. D. |
| Author_xml | – sequence: 1 givenname: G. A. surname: Carrizo fullname: Carrizo, G. A. organization: Department of Mathematics, National University of the South (UNS) – sequence: 2 givenname: N. S. surname: Fazzio fullname: Fazzio, N. S. organization: CONICET-CMaLP, Department of Mathematics, FCE, National University of La Plata – sequence: 3 givenname: M. D. surname: Sánchez fullname: Sánchez, M. D. organization: Department of Mathematics, FCE, University of La Plata – sequence: 4 givenname: M. L. orcidid: 0000-0002-3296-7895 surname: Schuverdt fullname: Schuverdt, M. L. email: schuverd@mate.unlp.edu.ar organization: CONICET-CMaLP, Department of Mathematics, FCE, National University of La Plata |
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| Keywords | Augmented Lagrangian methods Global convergence Nonlinear programming Sequential optimality conditions |
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| SubjectTerms | Convex and Discrete Geometry Management Science Mathematics Mathematics and Statistics Operations Research Operations Research/Decision Theory Optimization Statistics |
| Title | Scaled-PAKKT sequential optimality condition for multiobjective problems and its application to an Augmented Lagrangian method |
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