Cooperation and competition in multidisciplinary optimization

Abstract

This article aims to contribute to numerical strategies for PDE-constrained multiobjective optimization, with a particular emphasis on CPU-demanding computational applications in which the different criteria to be minimized (or reduced) originate from different physical disciplines that share the same set of design variables. Merits and shortcuts of the most-commonly used algorithms to identify, or approximate, the Pareto set are reviewed, prior to focusing on the approach by Nash games. A strategy is proposed for the treatment of two-discipline optimization problems in which one discipline, the primary discipline, is preponderant, or fragile. Then, it is recommended to identify, in a first step, the optimum of this discipline alone using the whole set of design variables. Then, an orthogonal basis is constructed based on the evaluation at convergence of the Hessian matrix of the primary criterion and constraint gradients. This basis is used to split the working design space into two supplementary subspaces to be assigned, in a second step, to two virtual players in competition in an adapted Nash game, devised to reduce a secondary criterion while causing the least degradation to the first. The formulation is proved to potentially provide a set of Nash equilibrium solutions originating from the original single-discipline optimum point by smooth continuation, thus introducing competition gradually. This approach is demonstrated over a testcase of aero-structural aircraft wing shape optimization, in which the eigen-split-based optimization reveals clearly superior. Thereafter, a result of convex analysis is established for a general unconstrained multiobjective problem in which all the gradients are assumed to be known. This results provides a descent direction common to all criteria, and adapting the classical steepest-descent algorithm by using this direction, a new algorithm is defined referred to as the multiple-gradient descent algorithm (MGDA). The MGDA realizes a phase of cooperative optimization yielding to a point on the Pareto set, at which a competitive optimization phase can possibly be launched on the basis of the local eigenstructure of the different Hessian matrices.