Meta-model-assisted MGDA for multi-objective functional optimization

A novel numerical method for multi-objective differentiable optimization, the Multiple-Gradient Descent Algorithm (MGDA), has been proposed in (Désidéri, 2012) to identify Pareto fronts. In MGDA, a direction of search for which the directional gradients of the objective functions are all negative, and often equal by construction (Désidéri, 2012), is identified and used in a steepest-descent-type iteration. The method converges to Pareto-optimal points. MGDA is here briefly reviewed to outline its principal theoretical properties and applied first to a classical mathematical test-case for illustration. The method is then extended encompass cases where the functional gradients are approximated via meta-models, as it is often the case in complex situations, and demonstrated on three optimum-shape design problems in compressible aerodynamics. The first problem is purely related to aerodynamic performance. It is a wing shape optimization exercise w.r.t. lift and drag in typical transonic cruise conditions. The second problem involves the aerodynamic performance and an environmental criterion: a supersonic glider configuration is optimized w.r.t. drag under lift constraint concurrently with a measure of the sonic-boom intensity at ground level. The third problem is related to an essential problematics in wing design: simultaneous drag and structural weight reduction. In all three cases, the meta-model-assisted MGDA succeeds in a few updates of the meta-model database to provide a correct description of the Pareto front, thus in a very economical way compared to a standard evolutionary algorithm used for this purpose.