# Split of Territories in Concurrent Optimization

1 OPALE - Optimization and control, numerical algorithms and integration of complex multidiscipline systems governed by PDE
CRISAM - Inria Sophia Antipolis - Méditerranée , JAD - Laboratoire Jean Alexandre Dieudonné : UMR6621
Abstract : We propose a methodology for the numerical treatment of a concurrent optimization problem in which two criteria are considered, one, $J_A$, being more critical than the second, $J_B$. After completion of the parametric, possibly--constrained minimization of the single, primary functional $J_A$ alone, approximations of the functional gradient and Hessian matrix, as well as $K$ constraint gradients, are assumed to be available or calculated using {\em meta-models}. Then, the entire parametric space (a subset of $\RR^{n+1}$) is split into two supplementary subspaces on the basis of a criterion related to the second variation. The construction is such that from the initial convergence point of the primary--functional minimization in full dimension, infinitesimal perturbations of the parameters lying in the second subspace, of specified dimension $p \leq n+1-K$, potentially cause the least degradation to the value of the primary functional. The latter subspace is elected as the support of the parameterization strategy of the secondary functional, $J_B$, in a concurrent optimization realized by an algorithm simulating a Nash game between players associated with the two functionals respectively. We prove a second result indicating that the original global optimum point of the primary problem in full dimension is Pareto-optimal for a trivial concurrent problem. This latter result permits us to define a continuum of Nash equilibrium points originating from the initial single-criterion optimum, in which the designer could potentially make a rational election of operating point. Thirdly, the initial single-criterion optimum is found to be robust. A simple minimization problem involving quadratic criteria is treated explicitly to demonstrate these properties in both cases of a linear and a nonlinear constraint. Lastly we note that the hierarchy introduced between the criteria applies to the split of parameters in preparation of a Nash game. The bias is therefore different in nature from the one that a Stackelberg-type game would introduce.
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Cited literature [13 references]

https://hal.inria.fr/inria-00127194
Contributor : Jean-Antoine Désidéri <>
Submitted on : Friday, November 2, 2007 - 1:20:46 PM
Last modification on : Tuesday, December 8, 2020 - 10:21:39 AM
Long-term archiving on: : Friday, September 24, 2010 - 2:01:20 PM

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• HAL Id : inria-00127194, version 6

### Citation

Jean-Antoine Désidéri. Split of Territories in Concurrent Optimization. [Research Report] RR-6108, INRIA. 2007, pp.34. ⟨inria-00127194v6⟩

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