Reduced Jacobian Method

Mounir El Maghri () and Youssef Elboulqe ()
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Mounir El Maghri: Hassan II University
Youssef Elboulqe: Hassan II University

Journal of Optimization Theory and Applications, 2018, vol. 179, issue 3, No 9, 917-943

Abstract: 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.

Keywords: Multiobjective optimization; Nonlinear programming; Pareto optima; KKT-stationarity; Descent direction; Reduced gradient method; 90C29; 90C30; 90C52 (search for similar items in EconPapers)
Date: 2018
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DOI: 10.1007/s10957-018-1362-x

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