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A competitive algorithm for two-objective optimization : Nash game with territory splitting

Abstract : This contribution pertains to PDE-constrained multi-objective optimization, with a par- ticular 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. A strategy has been proposed [3] for the treatment of two-discipline optimization problems in which one discipline, the primary discipline, is preponderant, or fragile. 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 sub- spaces 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 equilib- rium solutions originating from the original single-discipline optimum point by smooth continuation, thus introducing competition gradually. This approach is first demonstrated over a test-case of aero-structural aircraft wing shape opti- mization, in which the eigen-split-based optimization reveals clearly superior [2] [4]. A significant reduction of 8 % of the structural criterion was realized while maintaining the flowfield configuration close to optimality (drag increase ¡ 3 %), by an automatic procedure of orthogonal decomposition of the parameter space (see Figure 1). Other examples of optimum-shape design in compressible aerodynamics will be provided. The first example relates to the aerodynamic shape optimization of a supersonic business jet, in which the primary criterion is again wave drag under a lift constraint, whereas the secondary criterion is a measure of the sonic boom intensity for a supersonic business jet, and both criteria ought to be minimized. The second example is a two-point shape optimization of a helicopter rotor blade in which the primary criterion is the Figure of Merit in hover condition (to be maximized), whereas the secondary criterion is the power to be developed to maintain forward motion (to be minimized)
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Jean-Antoine Désidéri, Andrea Minelli, Enric Roca Leon. A competitive algorithm for two-objective optimization : Nash game with territory splitting. 4th Inverse Problems, Design and Optimization Symposium (IPDO-2013), Ecole des Mines d'Albi-Carmaux, Jun 2013, Albi, France. ⟨hal-00934636⟩

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