A cooperative algorithm for multi-objective optimization: multiple-gradient descent algorithm (MGDA)

Abstract : The Multiple-Gradient Descent Algorithm (MGDA) has been proposed and tested for the treatment of multi-objective differentiable optimization. Originally introduced in [1], the method has been tested and reformulated in [4]. Its efficacy to identify the Pareto front has been demonstrated in [5], in comparison with an evolutionary strategy. Recently, a variant, MGDA-II, has been proposed in which the descent direction is calculated by a direct procedure [3] based on a Gram-Schmidt orthogonalization process (GSP) with special normalization. This algorithm was tested in the context of a simulation by domain partitioning, as a technique to match the different interface components concurrently [2]. The experimentation revealed the importance of scaling, and a slightly modified normalization procedure was proposed (MGDA-IIb). Two variants have since been proposed. The first, MGDA-III, realizes two enhancements. Firstly, the GSP is conducted incompletely whenever a test reveals that the current estimate of the direction of search is adequate also w.r.t. the gradients not yet taken into account; this improvement simplifies the identification of the search direction when the gradients point roughly in the same direction, and makes the Fre ́chet derivative common to several objective-functions larger. Secondly, the order in which the different gradients are considered in the GSP is defined in a unique way devised to favor an incomplete GSP. In the second variant, MGDA-IV, the question of scaling is addressed when the Hessians are known. A variant is also proposed in which the Hessians are estimated by the Broyden-Fletcher- Goldfarb-Shanno (BFGS) formula. The method has been successfully applied to a classical test-case proposed by Fonseca [5]. Other examples of application of this method to optimum-shape design in aerodynamics were presented [6]. In this new contribution, the basic principle of the method is recalled. A meta-model-assisted extension is proposed and applied to the shape optimization of a generic supersonic aircraft configuration w.r.t. drag and sonic-boom reduction. This cooperative algorithm permits to identify points on the Pareto set associated with these two objective functions. From one such point, a competitive Nash game with adapted territory splitting can be initiated to identify a path in function space tangent to the Pareto front. Thus, the two approaches, cooperative and competitive algorithms, can be combined to generate quickly a set of designs in the vicinity of the Pareto front.
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Communication dans un congrès
O. Fudym, J.-L. Battaglia, G.S. Dulikravich et al. 4th Inverse Problems, Design and Optimization Symposium (IPDO-2013), Jun 2013, Albi, France. 2013
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Jean-Antoine Désidéri, Andrea Minelli, Adrien Zerbinati. A cooperative algorithm for multi-objective optimization: multiple-gradient descent algorithm (MGDA). O. Fudym, J.-L. Battaglia, G.S. Dulikravich et al. 4th Inverse Problems, Design and Optimization Symposium (IPDO-2013), Jun 2013, Albi, France. 2013. 〈hal-00934637〉

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