Reduced Jacobian Method
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Reduced Jacobian Method Mounir El Maghri1
· Youssef Elboulqe1
Received: 14 March 2018 / Accepted: 26 July 2018 © Springer Science+Business Media, LLC, part of Springer Nature 2018
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 Mathematics Subject Classification 90C29 · 90C30 · 90C52
1 Introduction Nowadays, in many areas of applications arising in decision-making problems, for instance in economics, management science, engineering, medicine (the literature on
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Mounir El Maghri [email protected] Youssef Elboulqe [email protected]
1
Department of Mathematics and Computer, Faculty of Sciences Aïn Chock, Hassan II University, BP. 5366, Casablanca, Morocco
123
Journal of Optimization Theory and Applications
this subject is abundant, e.g., [1]), practitioners are often confronted to optimization problems, which by nature are multiobjective. Such problems minimize several objectives simultaneously, and they cannot be described adequately by only one-objective function. For example, optimal single-objective solutions are often pursued at the expense of the much broader applicability of designs and solutions that satisfy multiple objectives. There is often a conflict between the objectives when a convex cone identifies a partial order, such as the natural ordering cone (nonnegative orthant). In general, there is no optimal solution minimizing the competing objective functions all at once. So the Pareto optima, also called efficient solutions, have to be considered. Taking account the principle, these solutions are incomparable and no single point can represent any of all the others. The problem becomes then to determine the set of Pareto points, or its image via the objectives, c
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