A global method for mixed categorical optimization with catalogs

In this article, we propose an algorithmic framework for globally solving mixed problems with continuous variables and categorical variables whose properties are available from a catalog. It supports catalogs of arbitrary size and properties of arbitrary dimension, and does not require any modeling effort from the user. Our tree search approach, similar to spatial branch and bound methods, performs an exhaustive exploration of the range of the properties of the categorical variables ; branching, constraint programming and catalog lookup phases alternate to discard inconsistent values. A novel catalog-based contractor guarantees consistency between the categorical properties and the existing catalog items. This results in an intuitive generic approach that is exact and easy to implement. We demonstrate the validity of the approach on a numerical example in which a categorical variable is described by a two-dimensional property space

Hybridation d’algorithmes évolutionnaires et de méthodes d’intervalles pour l’optimisation de problèmes difficiles

Award : Prix math/info de l'académie des sciences de Toulouse 2015Reliable global optimization is dedicated to finding a global minimum in the presence of rounding errors. The only approaches for achieving a numerical proof of optimality in global optimization are interval-based methods that interleave branching of the search-space and pruning of the subdomains that cannot contain an optimal solution. The exhaustive interval branch and bound methods have been widely studied since the 1960s and have benefittedfrom the development of refutation methods and filtering algorithms, stemming from the interval analysis and interval constraint programming communities. It is of the utmost importance: i) to compute sharp enclosures of the objective function and the constraints on a given subdomain; ii) to find a good approximation (an upper bound) of the globalminimum.State-of-the-art solvers are generally integrative methods, that is they embed local optimization algorithms to compute a good upper bound of the global minimum over each subspace. In this document, we propose a cooperativeframework in which interval methods cooperate with evolutionary algorithms. The latter are stochastic algorithms in which a population of individuals (candidate solutions) iteratively evolves in the search-space to reach satisfactory solutions. Evolutionary algorithms, endowed with operators that help individuals escape from local minima, are particularly suited for difficult problems on which traditional methods struggle to converge.Within our cooperative solver Charibde, the evolutionary algorithm and the interval- based algorithm run in parallel and exchange bounds, solutions and search-space via message passing. A strategy combining a geometric exploration heuristic and a domain reduction operator prevents premature convergence toward local minima and prevents theevolutionary algorithm from exploring suboptimal or unfeasible subspaces. A comparison of Charibde with state-of-the-art solvers based on interval analysis (GlobSol, IBBA, Ibex) on a benchmark of difficult problems shows that Charibde converges faster by an order of magnitude. New optimality results are provided for five multimodal problems, for which few solutions were available in the literature. We present an aeronautical application in which conflict solving between aircraft is modeled by an universally quantified constrained optimization problem, and solved by specific interval contractors. Finally, we certify the optimality of the putative solution to the Lennard-Jones cluster problem for five atoms, an open problem in molecular dynamics.L’optimisation globale fiable est dédiée à la recherche d’un minimum global en présence d’erreurs d’arrondis. Les seules approches fournissant une preuve numérique d’optimalité sont des méthodes d’intervalles qui partitionnent l’espace de recherche et éliminent les sous-espaces qui ne peuvent contenir de solution optimale. Ces méthodes exhaustives, appelées branch and bound par intervalles, sont étudiées depuis les années 60 et ont récemment intégré des techniques de réfutation et de contraction, issues des communautés d’analyse par intervalles et de programmation par contraintes. Il est d’une importance cruciale de calculer i)un encadrement précis de la fonction objectif et des contraintes sur un sous-domaine; ii)une bonne approximation (un majorant) du minimum global. Les solveurs de pointe sont généralement des méthodes intégratives : ils invoquent sur chaque sous-domaine des algorithmes d’optimisation locale afin d’obtenir une bonne approximation du minimum global. Dans ce document, nous nous intéressons à un cadre coopératif combinant des méthodes d’intervalles et des algorithmes évolutionnaires. Ces derniers sont des algorithmes stochastiques faisant évoluer une population de solutions candidates (individus) dans l’espace de recherche de manière itérative, dans l’espoir de converger vers des solutions satisfaisantes. Les algorithmes évolutionnaires, dotés de mécanismes permettant de s’échapper des minima locaux, sont particulièrement adaptés à la résolution de problèmes difficiles pour lesquels les méthodes traditionnelles peinent à converger Au sein de notre solveur coopératif Charibde, l’algorithme évolutionnaire et l’algorithme sur intervalles exécutés en parallèle échangent bornes, solutions et espace de recherche par passage de messages. Une stratégie couplant une heuristique d’exploration géométrique et un opérateur de réduction de domaine empêche la convergence prématurée de la populationvers des minima locaux et évite à l’algorithme évolutionnaire d’explorer des sous-espaces sous-optimaux ou non réalisables. Une comparaison de Charibde avec des solveurs de pointe (GlobSol, IBBA, Ibex) sur une base de problèmes difficiles montre un gain de temps d’un ordre de grandeur. De nouveaux résultats optimaux sont fournis pour cinq problèmes multimodaux pour lesquels peu de solutions, même approchées, sont connues dans la littérature. Nous proposons une application aéronautique dans laquelle la résolution de conflits est modélisée par un problème d’optimisation sous contraintes universellement quantifiées, et résolue par des techniques d’intervalles spécifiques. Enfin, nous certifions l’optimalité de la meilleure solution connue pour le cluster de Lennard-Jones à cinq atomes, un problème ouvert en dynamique moléculaire

On the Consequences of the "No Free Lunch" Theorem for Optimization on the Choice of an Appropriate MDO Architecture

Multidisciplinary design optimization (MDO) based on high- delity models is challenging due to the high computational cost of evaluating the objective and constraints. To choose the best MDO architecture, a trial-and-error approach is not possible due to the high cost of the overall optimization and complexity of the implementation. We propose to address this issue by developing a generic methodology that applies to any (potentially expensive) physical problem and generates a scalable approximation that can be quickly computed, for which the input and output dimensions may be set independently. This facilitates evaluation of MDO architectures for the original MDO problem by capturing its structure and behavior. The methodology is applied to two academic MDO test cases: the Supersonic Business Jet problem and the propane combustion problem. Well-known architectures (MDF, IDF and BLISS) are benchmarked on various instances to demonstrate the dependency between the performance of the architecture and the problem dimensions

Une preuve numérique d'optimalité pour le cluster de Lennard-Jones à cinq atomes

National audienceLe potentiel de Lennard-Jones est un modèle relativement réaliste décrivant les interactions (répulsion à courte distance et attraction à grande distance) entre deux atomes sphériques au sein d’un gaz rare

Certified Global Minima for a Benchmark of Difficult Optimization Problems

PreprintWe provide the global optimization community with new optimality proofs for 6 deceptive benchmark functions (5 bound-constrained functions and one nonlinearly constrained problem). These highly multimodal nonlinear test problems are among the most challenging benchmark functions for global optimization solvers; some have not been solved even with approximate methods. The global optima that we report have been numerically certified using Charibde (Vanaret et al., 2013), a hybrid algorithm that combines an Evolutionary Algorithm and interval-based methods. While metaheuristics generally solve large problems and provide sufficiently good solutions with limited computation capacity, exact methods are deemed unsuitable for difficult multimodal optimization problems. The achievement of new optimality results by Charibde demonstrates that reconciling stochastic algorithms and numerical analysis methods is a step forward into handling problems that were up to now considered unsolvable. We also provide a comparison with state-of-the-art solvers based on mathematical programming methods and population based metaheuristics, and show that Charibde, in addition to being reliable, is highly competitive with the best solvers on the given test functions

La première preuve d'optimalité pour le cluster de Lennard-Jones à cinq atomes

National audienceLe potentiel de Lennard-Jones est un modèle relativement réaliste décrivant les interactions entre deux atomes au sein d'un gaz rare. Déterminer la configuration la plus stable d'un cluster à N atomes revient à trouver les positions relatives des atomes qui minimisent l'énergie potentielle globale ; ce potentiel joue un rôle important dans le cadre des agrégats atomiques et les nanotechnologies. Le problème de cluster est NP-difficile et ouvert pour N > 4, et n'a jamais été résolu par des méthodes globales fiables. Nous proposons de résoudre le problème de cluster à cinq atomes de manière optimale avec des méthodes d'intervalles qui garantissent un encadrement du minimum global, même en présence d'arrondis. Notre modèle spatial permet d'éliminer certaines symétries du problème et de calculer des minorants plus précis dans le branch and bound par intervalles. Nous montrons que la meilleure solution connue du problème à cinq atomes est optimale, fournissons la configuration spatiale correspondante et comparons notre solveur fiable aux solveurs BARON et Couenne. Alors que notre solution est numériquement certifiée avec une précision de 10 −9 , les solutions de BARON et Couenne sont entachées d'erreurs numériques

A fast and reliable hybrid algorithm for numerical nonlinear global optimization

PreprintHighly nonlinear and ill-conditioned numerical optimization problems take their toll on the convergence of existing resolution methods. Stochastic methods such as Evolutionary Algorithms carry out an efficient exploration of the searchspace at low cost, but get often trapped in local minima and do not prove the optimality of the solution. Deterministic methods such as Interval Branch and Bound algorithms guarantee bounds on the solution, yet struggle to converge within a reasonable time on high-dimensional problems. The contribution of this paper is a hybrid algorithm in which a Differential Evolution algorithm and an Interval Branch and Contract algorithm cooperate. Bounds and solutions are exchanged through shared memory to accelerate the proof of optimality. It prevents premature convergence toward local optima and outperforms both deterministic and stochastic existing approaches. We demonstrate the efficiency of this algorithm on two currently unsolved problems: first by presenting new certified optimal results for the Michalewicz function for up to 75 dimensions and then by proving that the putative minimum of Lennard-Jones clusters of 5 atoms is optimal

A reliable hybrid solver for nonconvex optimization

International audienceNonconvex and highly multimodal optimization problems represent a challenge both for stochastic and deterministic global optimization methods. The former (metaheuristics) usually achieve satisfactory solutions but cannot guarantee global optimality, while the latter (generally based on a spatial branch and bound scheme [1], an exhaustive and non-uniform partitioning method) may struggle to converge toward a global minimum within reasonable time. The partitioning process is exponential in the number of variables, which prevents the resolution of large instances. The performances of the solvers even dramatically deteriorate when using reliable techniques, namely techniques that cope with rounding errors.In this paper, we present a fully reliable hybrid algorithm named Charibde (Cooperative Hybrid Algorithm using Reliable Interval-Based methods and Dierential Evolution) [2] that reconciles stochastic and deterministic techniques. An Evolutionary Algorithm (EA) cooperates with intervalbased techniques to accelerate convergence toward the global minimum and prove the optimality of the solution with user-defined precision. Charibde may be used to solve continuous, nonconvex, constrained or bound-constrained problems involving factorable functions

Preventing premature convergence and proving the optimality in evolutionary algorithms

http://ea2013.inria.fr//proceedings.pdfInternational audienceEvolutionary Algorithms (EA) usually carry out an efficient exploration of the search-space, but get often trapped in local minima and do not prove the optimality of the solution. Interval-based techniques, on the other hand, yield a numerical proof of optimality of the solution. However, they may fail to converge within a reasonable time due to their inability to quickly compute a good approximation of the global minimum and their exponential complexity. The contribution of this paper is a hybrid algorithm called Charibde in which a particular EA, Differential Evolution, cooperates with a Branch and Bound algorithm endowed with interval propagation techniques. It prevents premature convergence toward local optima and outperforms both deterministic and stochastic existing approaches. We demonstrate its efficiency on a benchmark of highly multimodal problems, for which we provide previously unknown global minima and certification of optimality