Reduced Jacobian Method
- 6 Downloads
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.
KeywordsMultiobjective optimization Nonlinear programming Pareto optima KKT-stationarity Descent direction Reduced gradient method
Mathematics Subject Classification90C29 90C30 90C52
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.
- 2.Figueira, J., Greco, S., Ehrgott, M.: Multiple criteria decision analysis. In: State of the Art Surveys. Springer, New York (2005)Google Scholar
- 13.Wolfe, P.: Methods of nonlinear programming. In: Graves, R.L., Wolfe, P. (eds.) Recent Advances in Mathematical Programming, pp. 67–86. McGraw–Hill, New York (1963)Google Scholar
- 16.El Maghri, M.: A free reduced gradient scheme. Asian J. Math. Comput. Res. 9, 228–239 (2016)Google Scholar
- 23.Huard, P.: Convergence of the reduced gradient method. In: Mangasarian, O.L., Meyer, R.R., Robinson, S.M. (eds.) Nonlinear Programming, pp. 29–59. Academic Press, New York (1975)Google Scholar
- 26.Bolintinéanu, S., (new name: Bonnel, H.), El Maghri, M.: Pénalisation dans l’optimisation sur l’ensemble faiblement efficient. RAIRO Oper. Res. 31, 295–310 (1997)Google Scholar