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This paper presents and compares three topology optimization tools differing from the optimization algorithms they implement. The first two are respectively based on a local optimization algorithm, i.e. a Nelder-Mead method, and a global one, i.e. a genetic algorithm. The third uses a hybrid algorithm associating the local and the global algorithms used in the first two. Each of them implements the same material distribution formalism, i.e. a dynamic formalism based on Voronoi cells. The results obtained on a practical study case, the design of an electromagnet, show the better performance of the hybrid algorithm as well in terms of speed of the convergence as in terms of quality of the final solution and therefore the interest of coupling the exploration ability of the global search with the intensification ability of the local search.

We consider an evolutionary method applied to a topology optimization problem. We compare two material distribution formalisms (static vs. Voronoibased dynamic), and two sets of reproduction mechanisms (standard vs. topologyadapted). We test those four variants on both theoretical and practical test cases, to show that the Voronoi-based formalism combined with adapted reproduction mechanisms performs better and is less sensitive to its parameters.

Mathematical optimization encompasses both a rich and rapidly evolving body of fundamental theory, and a variety of exciting applications in science and engineering. The present book contains a careful selection of articles on recent advances in optimization theory, numerical methods, and their applications in engineering. It features in particular new methods and applications in the fields of optimal control, PDE-constrained optimization, nonlinear optimization, and convex optimization. The authors of this volume took part in the 14th Belgian-French-German Conference on Optimization (BFG09) organized in Leuven, Belgium, on September 14-18, 2009. The volume contains a selection of reviewed articles contributed by the conference speakers as well as three survey articles by plenary speakers and two papers authored by the winners of the best talk and best poster prizes awarded at BFG09. Researchers and graduate students in applied mathematics, computer science, and many branches of engineering will find in this book an interesting and useful collection of recent ideas on the methods and applications of optimization.

In this paper, we solve a class of convex infinite-dimensional optimization problems using a numerical approximation method that does not rely on discretization. Instead, we restrict the decision variable to a sequence of finite-dimensional linear subspaces of the original infinite-dimensional space and solve the corresponding finite-dimensional problems in a efficient way using structured convex optimization techniques. We prove that, under some reasonable assumptions, the sequence of these optimal values converges to the optimal value of the original infinite-dimensional problem and give an explicit description of the corresponding rate of convergence.

To perform parameter and shape optimization, an initial topology is required which affects the final solution. Topology optimization methods have the advantage to release this constraint. They are based on a splitting of the design space into cells, in which they attempt to distribute optimally some given materials. In this paper, the optimization is based on a discrete formulation of the Maxwell equations obtained thanks to a finite element model. The gradient of the objective function is computed immediately from this discrete form, which is more convenient than the adjoint variable method. A line-search method (steepest descent direction) is then applied. The step size is computed by a simple and quick algorithm, to achieve good and fast convergence. Several convexity issues were already highlighted in topology optimization. We explain how we can face some of them by performing the optimization on a variable linked to the permeability by a given mapping. This mapping actually affects the value of intermediate materials and must be selected carefully to avoid the algorithm to be trapped in some local minimizers. In order to illustrate the concepts presented along the paper, the method is applied for the topology optimization of a linear reluctant actuator.

In this study, we use nonnegative matrix factorization (NMF) and nonnegative matrix underapproximation (NMU) approaches to generate feature vectors that can be used to cluster Aviation Safety Reporting System (ASRS) documents obtained from the Distributed National ASAP Archive (DNAA). By preserving nonnegativity, both the NMF and NMU facilitate a sum-of-parts representation of the underlying term usage patterns in the ASRS document collection. Both the training and test sets of ASRS documents are parsed and then factored by both algorithms to produce a reduced-rank representations of the entire document space. The resulting feature and coefficient matrix factors are used to cluster ASRS documents so that the (known) associated anomalies of training documents are directly mapped to the feature vectors. Dominant features of test documents are then used to generate anomaly relevance scores for those documents.We demonstrate that the approximate solution obtained by NMU using Lagrangrian duality can lead to a better sum-of-parts representation and document classification accuracy.

We consider the single-facility location problem with mixed norms, i.e. the problem of minimizing the sum of the distances from a point to a set of fixed points in R, where each distance can be measured according to a different p-norm. We show how this problem can be expressed into a structured conic format by decomposing the nonlinear components of the objective into a series of constraints involving three-dimensional cones. Using the availability of a self-concordant barrier for these cones, we present a polynomial-time algorithm (a long-step path-following interior-point scheme) to solve the problem up to a given accuracy. Finally, we report computational results for this algorithm and compare with standard nonlinear optimization solvers applied to this problem.

In this paper, we propose new lower and upper bounds on the linear extension complexity of regular $n$-gons. Our bounds are based on the equivalence between the computation of (i) an extended formulation of size $r$ of a polytope $P$, and (ii) a rank-$r$ nonnegative factorization of a slack matrix of the polytope $P$. The lower bound is based on an improved bound for the rectangle covering number (also known as the boolean rank) of the slack matrix of the $n$-gons. The upper bound is a slight improvement of the result of Fiorini, Rothvoss and Tiwary [Extended Formulations for Polygons, Discrete Comput. Geom. 48(3), pp. 658-668, 2012]. The difference with their result is twofold: (i) our proof uses a purely algebraic argument while Fiorini et al. used a geometric argument, and (ii) we improve the base case allowing us to reduce their upper bound $2 \left\lceil \log_2(n) \right\rceil$ by one when $2^{k-1} < n \leq 2^{k-1}+2^{k-2}$ for some integer $k$. We conjecture that this new upper bound is tight, which is suggested by numerical experiments for small $n$. Moreover, this improved upper bound allows us to close the gap with the best known lower bound for certain regular $n$-gons (namely, $9 \leq n \leq 13$ and $21 \leq n \leq 24$) hence allowing for the first time to determine their extension complexity., 20 pages, 3 figures. New contribution: improved lower bound for the boolean rank of the slack matrices of n-gons

Topology optimization is a method of conception which is able to define the topology, the form and the dimensions of a device with the aim of responding optimally to given design criteria. Initially reserved to the sizing of mechanics parts, this method is directed today towards the conception of more complexes devices as those encountered in applied electromagnetic. It is in this context that our work was performed. A topology optimization tool is made of the combination of an optimization algorithm and a material distribution formalism. In a first step, we compared different couplings of metaheuristic algorithms and material distribution formalisms. This comparison led us to choose as optimization tool for the problem under study, the combination of a genetic algorithm and a distribution of material by Voronoi cells. In a second step, we discussed how to improve the exploration and exploitation capabilities of this tool. We have, for this purpose, studied aspects related to the management of the size of the population and to the adaptation of the mechanisms of reproduction to the graphical nature of the problem. After this second step, we builded our optimization tool that we tested on study cases whose complexity is similar to that encountered at industrial showing its potential of to design electromechanical devices., L'optimisation topologique est une méthode de conception qui permet de définir de manière autonome la topologie, les formes et les dimensions d'un dispositif en vue de répondre de manière optimale à des critères de design. Initialement réservée au dimensionnement de pièces mécaniques, elle s'oriente aujourd’hui vers la conception de dispositifs plus complexes comme ceux rencontrés dans le domaine de l'électromécanique. C'est dans ce cadre que se situe notre travail. Un outil d'optimisation topologique étant formé de l'association d'un algorithme d'optimisation et d'un formalisme de distribution de matière, nous avons dans une première étape comparé différents couplages d'algorithmes métaheuristiques et de formalismes de distribution de matière en vue de choisir le couple qui semble le mieux adapté au problème traité. Cette comparaison nous a conduits à choisir comme outil d'optimisation l'association d'un algorithme génétique et d'une distribution de matière par cellules de Voronoï. Nous avons ensuite examiné comment améliorer les capacités d'exploration et d'exploitation de cet outil. Nous avons, à cet effet, étudié les aspects liés à la gestion de la taille de la population et à l'adaptation des mécanismes de reproduction au caractère graphique du problème. A l'issue de cette deuxième étape, nous avons finalisé un outil d'optimisation que nous avons testé sur des cas d'étude dont la complexité se rapproche de celle rencontrée au niveau industriel. Nous avons ainsi montré le potentiel de notre outil d'optimisation au niveau de la conception dans le cadre de l'électromécanique.

Topology optimization is a method of conception which is able to define the topology, the form and the dimensions of a device with the aim of responding optimally to given design criteria. Initially reserved to the sizing of mechanics parts, this method is directed today towards the conception of more complexes devices as those encountered in applied electromagnetic. It is in this context that our work was performed. A topology optimization tool is made of the combination of an optimization algorithm and a material distribution formalism. In a first step, we compared different couplings of metaheuristic algorithms and material distribution formalisms. This comparison led us to choose as optimization tool for the problem under study, the combination of a genetic algorithm and a distribution of material by Voronoi cells. In a second step, we discussed how to improve the exploration and exploitation capabilities of this tool. We have, for this purpose, studied aspects related to the management of the size of the population and to the adaptation of the mechanisms of reproduction to the graphical nature of the problem. After this second step, we builded our optimization tool that we tested on study cases whose complexity is similar to that encountered at industrial showing its potential of to design electromechanical devices., L'optimisation topologique est une méthode de conception qui permet de définir de manière autonome la topologie, les formes et les dimensions d'un dispositif en vue de répondre de manière optimale à des critères de design. Initialement réservée au dimensionnement de pièces mécaniques, elle s'oriente aujourd’hui vers la conception de dispositifs plus complexes comme ceux rencontrés dans le domaine de l'électromécanique. C'est dans ce cadre que se situe notre travail. Un outil d'optimisation topologique étant formé de l'association d'un algorithme d'optimisation et d'un formalisme de distribution de matière, nous avons dans une première étape comparé différents couplages d'algorithmes métaheuristiques et de formalismes de distribution de matière en vue de choisir le couple qui semble le mieux adapté au problème traité. Cette comparaison nous a conduits à choisir comme outil d'optimisation l'association d'un algorithme génétique et d'une distribution de matière par cellules de Voronoï. Nous avons ensuite examiné comment améliorer les capacités d'exploration et d'exploitation de cet outil. Nous avons, à cet effet, étudié les aspects liés à la gestion de la taille de la population et à l'adaptation des mécanismes de reproduction au caractère graphique du problème. A l'issue de cette deuxième étape, nous avons finalisé un outil d'optimisation que nous avons testé sur des cas d'étude dont la complexité se rapproche de celle rencontrée au niveau industriel. Nous avons ainsi montré le potentiel de notre outil d'optimisation au niveau de la conception dans le cadre de l'électromécanique.

International audience; We present a new family of sums of squares (SOS) relaxations to cones of positive polynomials. The SOS relaxations employed in the literature are cones of polynomials which can be represented as ratios, with an SOS as numerator and a fixed positive polynomial as denominator. We employ nonlinear transformations of the arguments instead. A fixed cone of positive polynomials, considered as a subset in an abstract coefficient space, corresponds to an infinite, partially ordered set of concrete cones of positive polynomials of different degrees and in a different number of variables. To each such concrete cone corresponds its own SOS cone, leading to a hierarchy of increasingly tighter SOS relaxations for the abstract cone.