GENERIC OPTIMIZATION STRATEGIES
Concerning "global strategies", and mostly in the context of shape optimization, our research effort has been focused on two major items~:

• Multi-level parametric optimization strategies


• Algorithms for Multi-Objective Optimization.

Multi-level parametric optimization strategies have been proposed for CPU-demanding single-objective optimization. Our construction relies on a formal analogy between the nested discrete subspaces utilized to parameterize a shape deformation within an {\em optimization}, and the subspaces associated with nested meshes used in a {\em simulation} by the nowadays-classical multigrid method. Our first algorithms were devised for shape optimization, and involved estimates over coarse, medium and fine parameterizations alternatively [1] Significant gains in efficiency have been achieved in drag-reduction problems, in particular when combining this technique with PSO-simplex hybridization to treat the multimodality, and parallel computing to exploit modern computer architectures. Similar techniques were applied in the design of antennas [2], and formally analyzed [3] and [4].
This hierarchical concept has later been extended to non-geometrical parameters by using an abstract embedding based on the eigenmode decomposition of the Hessian matrix of the cost function [5-6] This approach has been successfully applied to inverse problems and to the multidisciplinary optimization of a supersonic business jet using simplified algebraic models [7-8].

Algorithms for Multi-Objective Optimization have been devised with two principal contributions~:

• competitive approaches by Nash games and territory splitting


• cooperative approaches by Mutiple-Gradient Descent Algorithm (MGDA).

Our approach to {\em competitive optimization} is based on a particular construction of {\em Nash games}, relying on a {\em split of territory} in the assignment of individual strategies. A methodology has been 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 has been 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 has been demonstrated originally over a test-case of aero-structural aircraft wing shape optimization, in which the eigensplit-based optimization had revealed clearly superior.

The theoretical foundations and first application to the aerodynamic-structural wing-shape design have been discussed in [9-12]. Additionally, the same strategy is currently being applied with success at ONERA/DAAP by A. Minelli for a the shape optimization of a generic supersonic-jet configuration w.r.t. drag and sonic-boom reduction, and by E. Roca Leon for a Helicopter rotor blade optimization w.r.t. drag reduction in hover conditions concurrently with the reduction of necessary power to be developed for forward motion (unsteady flow).

We have also investigated methods where the territory splitting problem was formulated as an allocation game. For general two-player Nash games, each player has a probability vector $P=(p_i)$ of using the variable $x_i$ of a design variable $x$ as her {\em not exclusive} strategy. Once $P$ distributions are chosen, a Nash game is played. Supervisor players play then with the $P$ tables to converge to an equilibrium closest to the Pareto Front [13-14].

Aside of this, we have also applied the formulation of Nash games to contexts where two players cooperate, that is, attempt to minimize the same objective function. We refer to such games as "virtual Nash games". These algorithms are very effective to avoid local minima and converge quickly to the global minimum, when the problem is highly multimodal and is characterized by a variable separation property. This approach has been used for instance to optimize simultaneously global and local shape parameters for a business aircraft [15].

These methodological developments and case-studies have recently been gathered in a joint article solicited by the AIAA for a Special Issue on Computational Intelligence, currently being reviewed.

In a way somewhat complementary to the above competitive approaches, {\em cooperative algorithms} have been devised as methods of differentiable optimization to identify Pareto sets, in fairly general situations. Our approach is based on a simple lemma of convex analysis [16]~: in the convex hull of the gradients, there exists a unique vector of minimal norm, $\omega$; if it is nonzero, the vector $\omega$ is a descent direction common to all criteria; otherwise, the current design point is Pareto-optimal. This result led us to generalize the classical steepest-descent algorithm by using the vector $\omega$ as search direction. We refer to the new algorithm as the multiple-gradient descent algorithm (MGDA). The MGDA yields 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. This general formulation has fostered several connected studies whose results are now summarized~:

• Original formulation [17] (version 2, 2009), [12] and [18];
• Experimentations with test cases and comparison with an evolutionary strategy [19];
• Meta-model-assisted MGDA and application to fluid flow problems [20];
• A similar method by a direct construction [21];
• Application as a coordination strategy in a domain-decomposition method, and importance of scaling [22];
• Application of MGDA to two-objective optimization in two-dimensional elasticity using exact gradient of compliance through isogeometric approximation;
• A comparative summary of variants [23]; the current method relies on a well-ordered, incomplete Gram-Schmidt process applied to gradients scales by Hessians;
• The notion of Pareto-stationarity revisited, and formal proof of Pareto-optimality generalized to arbitrary numbers of objective functions and design variables [17] (version 3, 2012).

[1] Chapter 1.7: Optimisation de forme paramétrique multiniveau, J.-A. Désidéri, R. Duvigneau, B. Abou El Majd, and J. Zhao, Optimisation Multidisciplinaire en Mécanique 1: démarche de conception, stratégies collaboratives et concourantes, multiniveau de modèles et de paramètres, sous la direction de Rajan Filomeno Coelho, Piotr Breitkopf, Hermes Science Publications-Lavoisier, 2009.

[2] Méthodes Hiérarchiques pour l'Optimisation Géométrique de Structures Rayonnantes, B. Chaigne, PhD thesis, Université de Nice Sophia Antipolis, E.D. Sciences Fondamentales et Appliquées, Octobre 2009.

[3] Convergence of a two-level ideal algorithm for a parametric shape inverse model problem, Benoit Chaigne and Jean-Antoine Désidéri, Inverse Problems in Science and Engineering, 19(3):363 -- 393, 2011.

[4] Convergence of a two-level ideal algorithm for a parametric shape optimization model problem, B. Chaigne and J.-A. Désidéri, Research Report 7068, INRIA, October 2009. (version 2).

[5] Multiparameter shape optimization, Abderrahmane Benzaoui and Regis Duvigneau, Multidisciplinary Design Optimization in Computational Mechanics, Piotr Breitkopf and Rajan Filomeno Coelho, editors, ISTE - Wiley, April 2010.

[6] Efficient hierarchical optimization using an algebraic multilevel approach, Abderrahmane Benzaoui and Regis Duvigneau, Research Report 6974, INRIA, June 2009.

[7] Multilevel optimization based on spectral decomposition of the hessian - application to the multidisciplinary optimization of a supersonic business jet, Abderrahmane Benzaoui and Regis Duvigneau, EUROGEN Evolutionary Methods for Design, Optimization and Control, June 2009.

[8] Méthode algébrique pour l'optimisation multiniveau - application \`a l'optimisation d'un avion d'affaire supersonique, Abderrahmane Benzaoui and Regis Duvigneau, Giens'09 Colloque National de Calcul des Structures, May 2009.

[9] Chapter 1.5: Partage de territoire en ingénierie concourante, J.-A. Désidéri, Optimisation Multidisciplinaire en Mécanique 1: démarche de conception, stratégies collaboratives et concourantes, multiniveau de modèles et de paramètres, sous la direction de Rajan Filomeno Coelho, Piotr Breitkopf, Hermes Science Publications-Lavoisier, 2009.

[10] Two-discipline optimization, J.-A. Désidéri, Multidisciplinary Design Optimization in Computational Mechanics, Piotr Breitkopf and Rajan Filomeno Coelho, editors, 287--320, ISTE and John Wiley, April 2010.

[11] Optimisation de forme fluide-structure par un jeu de Nash, B. Abou El Majd, J.-A. Désidéri, and A. Habbal, Revue Africaine de la Recherche en Informatique et Mathématiques Appliquées, 13:3--15, 2010.

[12] Cooperation and Competition in Multidisciplinary Optimization - Application to the aero-structural aircraft wing shape optimization, J.-A. Désidéri, Computational Optimization and Applications, 52(1):29--68, 2012.

[13] Split of an optimization variable in game theory, Rajae Aboulaich, Abderrahmane Habbal, and Noureddine Moussaid, Mathematical Modelling of Natural Phenomena, 5(7):122--127, 2010.

[14] Optimisation multicritère. Une approche par partage des variables, Rajae Aboulaich, Abderrahmane Habbal, and Noureddine Moussaid, Revue Africaine de la Recherche en Informatique et Mathématiques Appliquées, 13:77--89, 2010.

[15] Wing shape optimization using FFD and twist parameterization, D. Chauhan, Praveen Chandrashekarappa, and Regis Duvigneau, 12th Aerospace Society of India CFD Symposium, Bangalore, India, August 2010.

[16] Coupling Local and Global Shape Optimization in Aerodynamic Design, Régis Duvigneau, Research Report 7684, INRIA, July 2011.

[17] Multiple-gradient descent algorithm (MGDA), J.-A. Désidéri, Research Report 6953 (version 2), INRIA, 2009. (Version 3: Novembre 2012.)

[18] Multiple-gradient descent algorithm (MGDA) for multiobjective optimization Algorithme de descente \`a gradients multiples pour l'optimisation multiobjectif, J.-A. Désidéri, Comptes Rendus de l'Académie des Sciences, Tome 350, Fascicule 5-6, 313-318, March 2012.

[19] Comparison between two multi objective optimization algorithms : PAES and MGDA. Testing MGDA on Kriging metamodels, Adrien Zerbinati, Jean-Antoine Désidéri, and Régis Duvigneau, Numerical Methods for Differential Equations, Optimization, and Technological Problems, Computational Methods in Applied Sciences, volume 27, T. Tuovinen S. Repin, T. Tiihonen, editors, 237--252, Springer Dordrecht, January 2013.

[20] Application of Metamodel-Assisted Multiple-Gradient Descent Algorithm (MGDA) to Air-Cooling Duct Shape Optimization, Adrien Zerbinati, Jean-Antoine Désidéri, and Régis Duvigneau, ECCOMAS - European Congress on Computational Methods in Applied Sciences and Engineering - 2012, Vienna, Austria, September 2012.

[21] MGDA II: A direct method for calculating a descent direction common to several criteria, J.-A. Désidéri, Research Report 7922, INRIA, April 2012.

[22] Application of MGDA to domain partitioning, J.-A. Désidéri, Research Report 7968, INRIA, May 2012.

[23] MGDA Variants for Multi-Objective Optimization, J.-A. Désidéri, Research Report 8068, INRIA, September 2012.