Une méthode de région de confiance avec ensemble actif pour l'optimisation non linéaire sans dérivées avec contraintes de bornes appliquée à des problèmes aérodynamiques bruités

L'optimisation sans dérivées (OSD) a connu un regain d'intérêt ces dernières années, principalement motivée par le besoin croissant de résoudre les problèmes d'optimisation définis par des fonctions dont les valeurs sont calculées par simulation (par exemple, la conception technique, la restauration d'images médicales ou de nappes phréatiques). Ces dernières années, un certain nombre de méthodes d'optimisation sans dérivée ont été développées et en particulier des méthodes fondées sur un modèle de région de confiance se sont avérées obtenir de bons résultats. Dans cette thèse, nous présentons un nouvel algorithme de région de confiance, basé sur l'interpolation, qui se montre efficace et globalement convergent (en ce sens que sa convergence vers un point stationnaire est garantie depuis tout point de départ arbitraire). Le nouvel algorithme repose sur la technique d'auto-correction de la géométrie proposé par Scheinberg and Toint (2010). Dans leur théorie, ils ont fait avancer la compréhension du rôle de la géométrie dans les méthodes d'OSD à base de modèles. Dans notre travail, nous avons pu améliorer considérablement l'efficacité de leur méthode, tout en maintenant ses bonnes propriétés de convergence. De plus, nous examinons l'influence de différents types de modèles d'interpolation sur les performances du nouvel algorithme. Nous avons en outre étendu cette méthode pour prendre en compte les contraintes de borne par l'application d'une stratégie d'activation. Considérer une méthode avec ensemble actif pour l'optimisation basée sur des modèles d'interpolation donne la possibilité d'économiser une quantité importante d'évaluations de fonctions. Il permet de maintenir les ensembles d'interpolation plus petits tout en poursuivant l'optimisation dans des sous-espaces de dimension inférieure. L'algorithme résultant montre un comportement numérique très compétitif. Nous présentons des résultats sur un ensemble de problèmes-tests issu de la collection CUTEr et comparons notre méthode à des algorithmes de référence appartenant à différentes classes de méthodes d'OSD. Pour réaliser des expériences numériques qui intègrent le bruit, nous créons un ensemble de cas-tests bruités en ajoutant des perturbations à l'ensemble des problèmes sans bruit. Le choix des problèmes bruités a été guidé par le désir d'imiter les problèmes d'optimisation basés sur la simulation. Enfin, nous présentons des résultats sur une application réelle d'un problème de conception de forme d'une aile fourni par Airbus. ABSTRACT : Derivative-free optimization (DFO) has enjoyed renewed interest over the past years, mostly motivated by the ever growing need to solve optimization problems defined by functions whose values are computed by simulation (e.g. engineering design, medical image restoration or groundwater supply). In the last few years, a number of derivative-free optimization methods have been developed and especially model-based trust-region methods have been shown to perform well. In this thesis, we present a new interpolation-based trust-region algorithm which shows to be efficient and globally convergent (in the sense that its convergence is guaranteed to a stationary point from arbitrary starting points). The new algorithm relies on the technique of self-correcting geometry proposed by Scheinberg and Toint [128] in 2009. In their theory, they advanced the understanding of the role of geometry in model-based DFO methods, in our work, we improve the efficiency of their method while maintaining its good theoretical convergence properties. We further examine the influence of different types of interpolation models on the performance of the new algorithm. Furthermore, we extended this method to handle bound constraints by applying an activeset strategy. Considering an active-set method in bound-constrained model-based optimization creates the opportunity of saving a substantial amount of function evaluations. It allows to maintain smaller interpolation sets while proceeding optimization in lower dimensional subspaces. The resulting algorithm is shown to be numerically highly competitive. We present results on a test set of smooth problems from the CUTEr collection and compare to well-known state-of-the-art packages from different classes of DFO methods. To report numerical experiments incorporating noise, we create a test set of noisy problems by adding perturbations to the set of smooth problems. The choice of noisy problems was guided by a desire to mimic simulation-based optimization problems. Finally, we will present results on a real-life application of a wing-shape design problem provided by Airbus. optimisation sans dérivées, région de confiance, contraintes de borne, fonctions bruitées

A Sequential Quadratic Programming Algorithm for Equality-Constrained Optimization without Derivatives

In this paper, we present a new model-based trust-region derivative-free optimization algorithm which can handle nonlinear equality constraints by applying a sequential quadratic programming (SQP) approach. The SQP methodology is one of the best known and most efficient frameworks to solve equality-constrained optimization problems in gradient-based optimization. Our derivative-free optimization (DFO) algorithm constructs local polynomial interpolation-based models of the objective and constraint functions and computes steps by solving QP sub-problems inside a region using the standard trust-region methodology. As it is crucial for such model-based methods to maintain a good geometry of the set of interpolation points, our algorithm exploits a self-correcting property of the interpolation set geometry. To deal with the trust-region constraint which is intrinsic to the approach of self-correcting geometry, the method of Byrd and Omojokun is applied. Numerical experiments are carried out on a set of test problems from the CUTEr library and on a simulation-based engineering design problem

A practical Trust-Region SQP algorithm for equality- and bound-constrained optimization without derivatives

In the last few years, a number of derivative-free optimization methods have been developed and especially model-based trust-region methods have been shown to perform well. Here, we present a new interpolation-based trust-region algorithm which can handle nonlinear and nonconvex optimization problems involving equality constraints and simple bounds on the variables. Our new algorithm is an extension of the algorithm BCDFO which handles bound constraints by an active-set method and has shown to be very competitive. It relies also on the technique of self-correcting geometry proposed by Scheinberg and Toint. The objective and constraint functions are approximated by polynomials of varying degree (linear or quadratic). The equality constraints are handled by a trust-region SQP approach, where each SQP step is decomposed into normal and tangential components. Special care must be taken in case an iterate is infeasible with respect to the models of the derivative-free constraints. Globalization is handled by using an Augmented Lagrangian penalty function as the merit function. We present numerical results on a test set of equality-constrained problems from the CUTEr problem collection and on a real-life application from engineering design in space craft development

An active-set trust-region method for bound-constrained nonlinear optimization without derivatives applied to noisy aerodynamic design problems

Derivative-free optimization (DFO) has enjoyed renewed interest over the past years, mostly motivated by the ever growing need to solve optimization problems defined by functions whose values are computed by simulation (e.g. engineering design, medical image restoration or groundwater supply). In the last few years, a number of derivative-free optimization methods have been developed and especially model-based trust-region methods have been shown to perform well. In this thesis, we present a new interpolation-based trust-region algorithm which shows to be efficient and globally convergent (in the sense that its convergence is guaranteed to a stationary point from arbitrary starting points). The new algorithm relies on the technique of self-correcting geometry proposed by Scheinberg and Toint [128] in 2009. In their theory, they advanced the understanding of the role of geometry in model-based DFO methods, in our work, we improve the efficiency of their method while maintaining its good theoretical convergence properties. We further examine the influence of different types of interpolation models on the performance of the new algorithm. Furthermore, we extended this method to handle bound constraints by applying an activeset strategy. Considering an active-set method in bound-constrained model-based optimization creates the opportunity of saving a substantial amount of function evaluations. It allows to maintain smaller interpolation sets while proceeding optimization in lower dimensional subspaces. The resulting algorithm is shown to be numerically highly competitive. We present results on a test set of smooth problems from the CUTEr collection and compare to well-known state-of-the-art packages from different classes of DFO methods. To report numerical experiments incorporating noise, we create a test set of noisy problems by adding perturbations to the set of smooth problems. The choice of noisy problems was guided by a desire to mimic simulation-based optimization problems. Finally, we will present results on a real-life application of a wing-shape design problem provided by Airbus. optimisation sans dérivées, région de confiance, contraintes de borne, fonctions bruitées

A Sequential Quadratic Programming Algorithm for Equality-Constrained Optimization without Derivatives

In this paper, we present a new model-based trust-region derivative-free optimization algorithm which can handle nonlinear equality constraints by applying a sequential quadratic programming (SQP) approach. The SQP methodology is one of the best known and most efficient frameworks to solve equality-constrained optimization problems in gradient-based optimization. Our derivative-free optimization (DFO) algorithm constructs local polynomial interpolation-based models of the objective and constraint functions and computes steps by solving QP sub-problems inside a region using the standard trust-region methodology. As it is crucial for such model-based methods to maintain a good geometry of the set of interpolation points, our algorithm exploits a self-correcting property of the interpolation set geometry. To deal with the trust-region constraint which is intrinsic to the approach of self-correcting geometry, the method of Byrd and Omojokun is applied. Numerical experiments are carried out on a set of test problems from the CUTEr library and on a simulation-based engineering design problem

Constrained DFO for Multidisciplinary Design Optimization

The SpaceLiner is a concept study of a hypersonic airplane for ultra-fast passenger transport, developed at the German Aerospace Center (DLR). When designing spacecraft, one of the major issues is to devise its thermal protection systems needed during atmospheric re-entry. As the goal is to find the optimal preliminary SpaceLiner design which considers all main disciplines (geometry, aerodynamics, structural masses and thermal protection system), multidisciplinary design optimization (MDO) techniques have to be applied. In simulation-based engineering applications, oftentimes no derivatives are available. Thus, we would like to present our new optimization algorithm, a model-based trust-region SQP algorithm, to solve equality-constrained optimization problems without derivatives. Finally, we want to present the integration of the described optimization framework into the workflow management system RCE which provides a graphical user interface to couple simulation codes in a collaborative environment

A new Algorithm for Equality- and Bound-Constrained DFO

This poster will present a new algorithm for equality- and bound-constrained nonlinear optimization without derivatives. A trust-region SQP framework is used to handle the constraints. The algorithm can be viewed as an extension of the algorithm BCDFO as it also applies the technique of self-correcting geometry and an active-set strategy to handle bound constraints. We present numerical results on a test set of equality-constrained problems from the CUTEr problem collection

Different Second Order Approximations in a model-based SQP Trust-Region DFO Method

A trust-region SQP method for general nonlinear constrained optimization without derivatives is proposed. The trust-region step is computed by a Byrd-Omojokun-like approach. An active-set strategy is used to handle bound constraints and inequality constraints. The objective and constraint functions are approximated by local linear or quadratic interpolation. In this work, we want to examine different approximation techniques

A Sequential Quadratic Programming Algorithm for Equality-Constrained Optimization without Derivatives

In this paper, we present a new model-based trust-region derivative-free optimization algorithm which can handle nonlinear equality constraints by applying a sequential quadratic programming (SQP) approach. The SQP methodology is one of the best known and most efficient frameworks to solve equality-constrained optimization problems in gradient-based optimization. Our derivative-free optimization (DFO) algorithm constructs local polynomial interpolation-based models of the objective and constraint functions and computes steps by solving QP sub-problems inside a region using the standard trust-region methodology. As it is crucial for such model-based methods to maintain a good geometry of the set of interpolation points, our algorithm exploits a self-correcting property of the interpolation set geometry. To deal with the trust-region constraint which is intrinsic to the approach of self-correcting geometry, the method of Byrd and Omojokun is applied. Numerical experiments are carried out on a set of test problems from the CUTEr library and on a simulation-based engineering design problem