Abstract
In this paper, we present the Wolfe’s reduced gradient method for multiobjective (multicriteria) optimization. We precisely deal with the problem of minimizing nonlinear objectives under linear constraints and propose a reduced Jacobian method, namely a reduced gradient-like method that does not scalarize those programs. As long as there are nondominated solutions, the principle is to determine a direction that decreases all goals at the same time to achieve one of them. Following the reduction strategy, only a reduced search direction is to be found. We show that this latter can be obtained by solving a simple differentiable and convex program at each iteration. Moreover, this method is conceived to recover both the discontinuous and continuous schemes of Wolfe for the single-objective programs. The resulting algorithm is proved to be (globally) convergent to a Pareto KKT-stationary (Pareto critical) point under classical hypotheses and a multiobjective Armijo line search condition. Finally, experiment results over test problems show a net performance of the proposed algorithm and its superiority against a classical scalarization approach, both in the quality of the approximated Pareto front and in the computational effort.
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01 December 2018
The contents of the paper [1] were submitted for publication in J Optim Theory Appl (JOTA) on 22 December 2016, reviewed and approved by the Editors-in-Chief.
Notes
For \(y, y' \in {\mathbb {R}}^r\), \(y<y'\) (resp. \(y \le y'\)) means \(y_i<y'_i\) (resp. \(y_i \le y'_i\)) for all \(i \in \{1,\ldots ,r\}\); obviously, \(y \lneq y'\) means \(y \le y'\) and \(y\ne y'\).
Pointed means \(C\cap -C = \{0\}\), \(\mathrm{int}C\) stands for the topological interior of C, and, \(y \le _C y'\) means \(y-y'\in -C\).
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The authors are thankful for the valuable suggestions of the editor Dr. Franco Giannessi and the reviewers that helped improve the quality of the paper.
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Maghri, M.E., Elboulqe, Y. Reduced Jacobian Method. J Optim Theory Appl 179, 917–943 (2018). https://doi.org/10.1007/s10957-018-1362-x
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DOI: https://doi.org/10.1007/s10957-018-1362-x
Keywords
- Multiobjective optimization
- Nonlinear programming
- Pareto optima
- KKT-stationarity
- Descent direction
- Reduced gradient method