pplication of the Euler algorithm in the study of dynamic complexity in cooperation mechanisms

Los dilemas sociales son conflictos de racionalidad en donde individuos racionales terminan en situaciones de estancamiento e irracionalidad no previstas. En la literatura se ha demostrado que la cooperación puede ser de utilidad para enfrentar dichos dilemas, pero aùn son necesarias herramientas para entender su complejidad. Este artículo presenta cómo el algo- ritmo de Euler puede ser una herramienta útil para el estudio de la complejidad dinámica en mecanismos de cooperación.Social dilemmas are situations where people face conflicts of rationality in which all per- sons could receive the worse outputs. Cooperation can be use to solve such dilemmas but it presents complexity. This papers presents a proposal to use algorithms to perform sensitivity analysis for studying cooperation mechanisms and suggests this approach to understand the dynamic complexity of cooperation

Aplicación del algoritmo de Euler en el estudio de la complejidad dinámica en mecanismos de cooperación

Social dilemmas are situations where people face conflicts of rationality in which all per- sons could receive the worse outputs. Cooperation can be use to solve such dilemmas but it presents complexity. This papers presents a proposal to use algorithms to perform sensitivity analysis for studying cooperation mechanisms and suggests this approach to understand the dynamic complexity of cooperation.Los dilemas sociales son conflictos de racionalidad en donde individuos racionales terminan en situaciones de estancamiento e irracionalidad no previstas. En la literatura se ha demostrado que la cooperación puede ser de utilidad para enfrentar dichos dilemas, pero aùn son necesarias herramientas para entender su complejidad. Este artículo presenta cómo el algo- ritmo de Euler puede ser una herramienta útil para el estudio de la complejidad dinámica en mecanismos de cooperación

Cooperation and Competition Strategies in Multi-objective Shape Optimization - Application to Low-boom/Low-drag Supersonic Business Jet

International audienceCooperation and competition are natural laws that regulate the interactions between agents in numerous physical, or social phenomena. By analogy, we transpose these laws to devise e cient multi-objective algorithms applied to shape optimization problems involving two or more disciplines. Two e cient strategies are presented in this paper: a multiple gradient descent algorithm (MGDA) and a Nash game strategy based on an original split of territories between disciplines. MGDA is a multi-objective extension of the steepest descent method. The use of a gradient-based algorithm that exploits the cooperation principle aims at reducing the number of iterations required for classical multi-objective evolutionary algorithms to converge to a Pareto optimal design. On the other hand side, the Nash game strategy is well adapted to typical aeronautical optimization problems, when, after having optimized a preponderant or fragile discipline (e.g. aerodynamics), by the minimization of a primary objective-function, one then wishes to reduce a secondary objective-function, representative of another discipline, in a process that avoids degrading excessively the original optimum. Presently, the combination of the two approaches is exploited, in a method that explores the entire Pareto front. Both algorithms are rst analyzed on analytical test cases to demonstrate their main features and then applied to the optimum-shape design of a low-boom/low-drag supersonic business jet design problem. Indeed, sonic boom is one of the main limiting factors to the development of civil supersonic transportation. As the driving design for low-boom is not compliant with the low-drag one, our goal is to provide a trade-o between aerodynamics and acoustics. Thus Nash games are adopted to de ne a low-boom con guration close to aerodynamic optimality w.r.t. wave drag

NEUMANN-DIRICHLET NASH STRATEGIES FOR THE SOLUTION OF ELLIPTIC CAUCHY PROBLEMS

International audienceWe consider the Cauchy problem for an elliptic operator, formulated as a Nash game. The overspecified Cauchy data are split between two players: the first player solves the elliptic equation with the Dirichlet part of the Cauchy data prescribed over the accessible boundary and a variable Neumann condition (which we call first player's strategy) prescribed over the inaccessible part of the boundary. The second player makes use correspondingly of the Neumann part of the Cauchy data, with a variable Dirichlet condition prescribed over the inaccessible part of the boundary. The first player then minimizes the gap related to the nonused Neumann part of the Cauchy data, and so does the second player with a corresponding Dirichlet gap. The two costs are coupled through a difference term. We prove that there always exists a unique Nash equilibrium, which turns out to be the reconstructed data when the Cauchy problem has a solution. We also prove that the completion Nash game has a stable solution with respect to noisy data. Some numerical two- and three-dimensional experiments are provided to illustrate the efficiency and stability of our algorithm

COOPERATION AND COMPETITION IN MULTIDISCIPLINARY OPTIMIZATION Application to the aero-structural aircraft wing shape 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 smoot

Multiobjective Design Optimization using Nash Games

International audienceIn the area of pure numerical simulation of multidisciplinary coupled systems, the computational cost to evaluate a configuration may be very high. A fortiori, in multi- disciplinary optimization, one is led to evaluate a number of different configurations to iterate on the design parameters. This observation motivates the search for the most in- novative and computationally efficient approaches in all the sectors of the computational chain : at the level of the solvers (using a hierarchy of physical models), the meshes and geometrical parameterizations for shape, or shape deformation, the implementation (on a sequential or parallel architecture; grid computing), and the optimizers (deterministic or semi-stochastic, or hybrid; synchronous, or asynchronous). In the present approach, we concentrate on situations typically involving a small number of disciplines assumed to be strongly antagonistic, and a relatively moderate number of related objective functions. However, our objective functions are functionals, that is, PDE-constrained, and thus costly to evaluate. The aerodynamic and structural optimization of an aircraft configuration is a prototype of such a context, when these disciplines have been reduced to a few major objectives. This is the case when, implicitly, many subsystems are taken into account by local optimizations. Our developments are focused on the question of approximating the Pareto set in cases of strongly-conflicting disciplines. For this purpose, a general computational technique is proposed, guided by a form of sensitivity analysis, with the additional objective to be more economical than standard evolutionary approaches