Augmented Lagrangian Constraint Handling for CMA-ES---Case of a Single Linear Constraint

International audienceWe consider the problem of minimizing a function f subject to a single inequality constraint g(x) <= 0, in a black-box scenario. We present a co-variance matrix adaptation evolution strategy using an adaptive augmented La-grangian method to handle the constraint. We show that our algorithm is an instance of a general framework that allows to build an adaptive constraint handling algorithm from a general randomized adaptive algorithm for unconstrained optimization. We assess the performance of our algorithm on a set of linearly constrained functions, including convex quadratic and ill-conditioned functions, and observe linear convergence to the optimum

Adaptive Ranking Based Constraint Handling for Explicitly Constrained Black-Box Optimization

A novel explicit constraint handling technique for the covariance matrix adaptation evolution strategy (CMA-ES) is proposed. The proposed constraint handling exhibits two invariance properties. One is the invariance to arbitrary element-wise increasing transformation of the objective and constraint functions. The other is the invariance to arbitrary affine transformation of the search space. The proposed technique virtually transforms a constrained optimization problem into an unconstrained optimization problem by considering an adaptive weighted sum of the ranking of the objective function values and the ranking of the constraint violations that are measured by the Mahalanobis distance between each candidate solution to its projection onto the boundary of the constraints. Simulation results are presented and show that the CMA-ES with the proposed constraint handling exhibits the affine invariance and performs similarly to the CMA-ES on unconstrained counterparts.Comment: 9 page

Analysis of the (μ/μI,λ)(\mu/\mu_I,\lambda)-CSA-ES with Repair by Projection Applied to a Conically Constrained Problem

Theoretical analyses of evolution strategies are indispensable for gaining a deep understanding of their inner workings. For constrained problems, rather simple problems are of interest in the current research. This work presents a theoretical analysis of a multi-recombinative evolution strategy with cumulative step size adaptation applied to a conically constrained linear optimization problem. The state of the strategy is modeled by random variables and a stochastic iterative mapping is introduced. For the analytical treatment, fluctuations are neglected and the mean value iterative system is considered. Non-linear difference equations are derived based on one-generation progress rates. Based on that, expressions for the steady state of the mean value iterative system are derived. By comparison with real algorithm runs, it is shown that for the considered assumptions, the theoretical derivations are able to predict the dynamics and the steady state values of the real runs.Comment: This is a PREPRINT of an article that has been accepted for publication in the journal MIT Press Evolutionary Computation (ECJ). 25 pages + supplementary material. The work was supported by the Austrian Science Fund FWF under grant P29651-N3

On a smoothed penalty-based algorithm for global optimization

This paper presents a coercive smoothed penalty framework for nonsmooth and nonconvex constrained global optimization problems. The properties of the smoothed penalty function are derived. Convergence to an ε -global minimizer is proved. At each iteration k, the framework requires the ε(k) -global minimizer of a subproblem, where ε(k)→ε . We show that the subproblem may be solved by well-known stochastic metaheuristics, as well as by the artificial fish swarm (AFS) algorithm. In the limit, the AFS algorithm convergence to an ε(k) -global minimum of the real-valued smoothed penalty function is guaranteed with probability one, using the limiting behavior of Markov chains. In this context, we show that the transition probability of the Markov chain produced by the AFS algorithm, when generating a population where the best fitness is in the ε(k)-neighborhood of the global minimum, is one when this property holds in the current population, and is strictly bounded from zero when the property does not hold. Preliminary numerical experiments show that the presented penalty algorithm based on the coercive smoothed penalty gives very competitive results when compared with other penalty-based methods.The authors would like to thank two anonymous referees for their valuable comments and suggestions to improve the paper. This work has been supported by COMPETE: POCI-01-0145-FEDER-007043 and FCT - Fundac¸ao para a Ci ˜ encia e Tecnologia within the projects UID/CEC/00319/2013 and ˆ UID/MAT/00013/2013.info:eu-repo/semantics/publishedVersio

A merit function approach for evolution strategies

In this paper, we extend a class of globally convergent evolution strategies to handle general constrained optimization problems. The proposed framework handles quantifiable relaxable constraints using a merit function approach combined with a specific restoration procedure. The unrelaxable constraints, when present, can be treated either by using the extreme barrier function or through a projection approach. Under reasonable assumptions, the introduced extension guarantees to the regarded class of evolution strategies global convergence properties for first order stationary constraints. Numerical experiments are carried out on a set of problems from the CUTEst collection as well as on known global optimization problems

Evolutionary multiobjective optimization : review, algorithms, and applications

Programa Doutoral em Engenharia Industrial e SistemasMany mathematical problems arising from diverse elds of human activity can be formulated as optimization problems. The majority of real-world optimization problems involve several and con icting objectives. Such problems are called multiobjective optimization problems (MOPs). The presence of multiple con icting objectives that have to be simultaneously optimized gives rise to a set of trade-o solutions, known as the Pareto optimal set. Since this set of solutions is crucial for e ective decision-making, which generally aims to improve the human condition, the availability of e cient optimization methods becomes indispensable. Recently, evolutionary algorithms (EAs) have become popular and successful in approximating the Pareto set. The population-based nature is the main feature that makes them especially attractive for dealing with MOPs. Due to the presence of two search spaces, operators able to e ciently perform the search in both the decision and objective spaces are required. Despite the wide variety of existing methods, a lot of open research issues in the design of multiobjective evolutionary algorithms (MOEAs) remains. This thesis investigates the use of evolutionary algorithms for solving multiobjective optimization problems. Innovative algorithms are developed studying new techniques for performing the search either in the decision or the objective space. Concerning the search in the decision space, the focus is on the combinations of traditional and evolutionary optimization methods. An issue related to the search in the objective space is studied in the context of many-objective optimization. Application of evolutionary algorithms is addressed solving two di erent real-world problems, which are modeled using multiobjective approaches. The problems arise from the mathematical modelling of the dengue disease transmission and a wastewater treatment plant design. The obtained results clearly show that multiobjective modelling is an e ective approach. The success in solving these challenging optimization problems highlights the practical relevance and robustness of the developed algorithms.Muitos problemas matemáticos que surgem nas diversas áreas da atividade humana podem ser formulados como problemas de otimização. A maioria dos problemas do mundo real envolve vários objetivos conflituosos. Tais problemas chamam-se problemas de otimização multiobjetivo. A presença de vários objetivos conflituosos, que têm de ser otimizados em simultâneo, dá origem a um conjunto de soluções de compromisso, conhecido como conjunto de soluções ótimas de Pareto. Uma vez que este conjunto de soluções é fundamental para uma tomada de decisão eficaz, cujo objetivo em geral é melhorar a condição humana, o desenvolvimento de métodos de otimização eficientes torna-se indispensável. Recentemente, os algoritmos evolucionários tornaram-se populares e bem-sucedidos na aproximação do conjunto de Pareto. A natureza populacional é a principal característica que os torna especialmente atraentes para lidar com problemas de otimização multiobjetivo. Devido à presença de dois espaços de procura, operadores capazes de realizar a procura de forma eficiente, tanto no espaço de decisão como no espaço dos objetivos, são necessários. Apesar da grande variedade de métodos existentes, várias questões de investigação permanecem em aberto na área do desenvolvimento de algoritmos evolucionários multiobjetivo. Esta tese investiga o uso de algoritmos evolucionários para a resolução de problemas de otimização multiobjetivo. São desenvolvidos algoritmos inovadores que estudam novas técnicas de procura, quer no espaço de decisão, quer no espaço dos objetivos. No que diz respeito à procura no espaço de decisão, o foco está na combinação de métodos de otimização tradicionais com algoritmos evolucionários. A questão relacionada com a procura no espaço dos objetivos é desenvolvida no contexto da otimização com muitos objetivos. A aplicação dos algoritmos evolucionários é abordada resolvendo dois problemas reais, que são modelados utilizando abordagens multiobjectivo. Os problemas resultam da modelação matemática da transmissão da doença do dengue e do desenho ótimo de estações de tratamento de águas residuais. O sucesso na resolução destes problemas de otimização constitui um desafio e destaca a relevância prática e robustez dos algoritmos desenvolvidos

Adaptive modeling strategy for constrained global optimization with application to aerodynamic wing design

Surrogate models are often used to reduce the cost of design optimization prob- lems that involve computationally costly models, such as computational fluid dynamics simulations. However, the number of evaluations required by sur- rogate models usually scales poorly with the number of design variables, and there is a need for both better constraint formulations and multimodal function handling. To address this issue, we developed a surrogate-based gradient-free optimization algorithm that can handle cases where the function evaluations are expensive, the computational budget is limited, the functions are multimodal, and the optimization problem includes nonlinear equality or inequality con- straints. The proposed algorithm—super efficient global optimization coupled with mixture of experts (SEGOMOE)—can tackle complex constrained design optimization problems through the use of an enrichment strategy based on a mixture of experts coupled with adaptive surrogate models. The performance of this approach was evaluated for analytic constrained and unconstrained prob- lems, as well as for a multimodal aerodynamic shape optimization problem with 17 design variables and an equality constraint. Our results showed that the method is efficient and that the optimum is much less dependent on the starting point than the conventional gradient-based optimization