Working principles,behavior, and performance of MOEAs on MNK-landscapes

This work studies the working principles, behavior, and performance of multiobjective evolutionary algorithms (MOEAs) on multiobjective epistatic fitness functions with discrete binary search spaces by using MNK-landscapes. First, we analyze the structure and some of the properties of MNK-landscapes under a multiobjective perspective by using enumeration on small landscapes. Then, we focus on the performance and behavior of MOEAs on large landscapes. We organize our study around selection, drift, mutation, and recombination, the four major and intertwined processes that drive adaptive evolution over fitness landscapes. This work clearly shows pros and cons of the main features of MOEAs, gives a valuable guide for the practitioner on how to set up his/her algorithm, enhance MOEAs, and presents useful insights on how to design more robust and efficient MOEAs.     

Optimización mixta de estructuras de transporte de energía: aplicación del algoritmo de recocido simulado

A general methodology to optimize the weight of power transmission structures is presented in this article. This methodology is based on the simulated annealing algorithm defined by Kirkpatrick in the early ‘80s. This algorithm consists of a stochastic approach that allows to explore and analyze solutions that do not improve the objective function in order to develop a better exploration of the design region and to obtain the global optimum. The proposed algorithm allows to consider the discrete behavior of the sectional variables for each element and the continuous behavior of the general geometry variables. Thus, an optimization methodology that can deal with a mixed optimization problem and includes both continuum and discrete design variables is developed. In addition, it does not require to study all the possible design combinations defined by discrete design variables. The algorithm proposed usually requires to develop a large number of simulations (structural analysis in this case) in practical applications. Thus, the authors have developed first order Taylor expansions and the first order sensitivity analysis involved in order to reduce the CPU time required. Exterior penalty functions have been also included to deal with the design constraints. Thus, the general methodology proposed allows to optimize real power transmission structures in acceptable CPU time.     

Bilinear Generating Functions for Orthogonal Polynomials

Using realizations of the positive discrete series representations of the Lie algebra su(1,1) in terms of Meixner—Pollaczek polynomials, the action of su(1,1) on Poisson kernels of these polynomials is considered. In the tensor product of two such representations, two sets of eigenfunctions of a certain operator can be considered and they are shown to be related through continuous Hahn polynomials. As a result, a bilinear generating function for continuous Hahn polynomials is obtained involving the Poisson kernel of Meixner—Pollaczek polynomials; this result is also known as the Burchnall—Chaundy formula. For the positive discrete series representations of the quantized universal enveloping algebra U _q (su(1,1)) a similar analysis is performed and leads to a bilinear generating function for Askey—Wilson polynomials involving the Poisson kernel of Al-Salam and Chihara polynomials. July 6, 1997. Date accepted: September 23, 1998.     

Macroscopic finite element analyses of discretematerials based on directly evaluated micro‐macro transitions

We consider a homogenized macro‐continuum with locally attached microstructure of granules and derive specific micromacro transitions by a consistent transfer of discrete micro‐variables to field variables on a continuous macrostructure. Displacements and rotational constraints are imposed on the granules on the defined boundary frame of the microstructure. The constraints for linear displacements and uniform tractions on the surface yield upper and lower bound characteristics for periodic boundary conditions with regard to the aggregate stiffness. Secondly, we perform two‐scale analyses where we link simulations on the macro‐ and the microscales. Therein, coupled boundary‐value problems are solved on both scales. The macroscopic homogeneous problem is solved by a finite element method where the material model is implemented using the directly evaluated micro‐macro transitions on the basis of the discrete microstructures. Finally, a model problem is investigated to clarify the proposed method. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Fuzzy classifier identification using decision tree and multiobjective evolutionary algorithms

This paper presents a hybrid method for identification of Pareto-optimal fuzzy classifiers (FCs). In contrast to many existing methods, the initial population for multiobjective evolutionary algorithms (MOEAs) is neither created randomly nor a priori knowledge is required. Instead, it is created by the proposed two-step initialization method. First, a decision tree (DT) created by C4.5 algorithm is transformed into an FC. Therefore, relevant variables are selected and initial partition of input space is performed. Then, the rest of the population is created by randomly replacing some parameters of the initial FC, such that, the initial population is widely spread. That improves the convergence of MOEAs into the correct Pareto front. The initial population is optimized by NSGA-II algorithm and a set of Pareto-optimal FCs representing the trade-off between accuracy and interpretability is obtained. The method does not require any a priori knowledge of the number of fuzzy sets, distribution of fuzzy sets or the number of relevant variables. They are all determined by it. Performance of the obtained FCs is validated by six benchmark data sets from the literature. The obtained results are compared to a recently published paper [H. Ishibuchi, Y. Nojima, Analysis of interpretability-accuracy tradeoff of fuzzy systems by multiobjective fuzzy genetics-based machine learning, International Journal of Approximate Reasoning 44 (1) (2007) 4–31] and the benefits of our method are clearly shown.     

Fast calculation of multiobjective probability of improvement and expected improvement criteria for Pareto optimization

The use of surrogate based optimization (SBO) is widely spread in engineering design to reduce the number of computational expensive simulations. However, “real-world” problems often consist of multiple, conflicting objectives leading to a set of competitive solutions (the Pareto front). The objectives are often aggregated into a single cost function to reduce the computational cost, though a better approach is to use multiobjective optimization methods to directly identify a set of Pareto-optimal solutions, which can be used by the designer to make more efficient design decisions (instead of weighting and aggregating the costs upfront). Most of the work in multiobjective optimization is focused on multiobjective evolutionary algorithms (MOEAs). While MOEAs are well-suited to handle large, intractable design spaces, they typically require thousands of expensive simulations, which is prohibitively expensive for the problems under study. Therefore, the use of surrogate models in multiobjective optimization, denoted as multiobjective surrogate-based optimization, may prove to be even more worthwhile than SBO methods to expedite the optimization of computational expensive systems. In this paper, the authors propose the efficient multiobjective optimization (EMO) algorithm which uses Kriging models and multiobjective versions of the probability of improvement and expected improvement criteria to identify the Pareto front with a minimal number of expensive simulations. The EMO algorithm is applied on multiple standard benchmark problems and compared against the well-known NSGA-II, SPEA2 and SMS-EMOA multiobjective optimization methods.     

Continuous GRASP with a local active-set method for bound-constrained global optimization

Global optimization seeks a minimum or maximum of a multimodal function over a discrete or continuous domain. In this paper, we propose a hybrid heuristic—based on the CGRASP and GENCAN methods—for finding approximate solutions for continuous global optimization problems subject to box constraints. Experimental results illustrate the relative effectiveness of CGRASP–GENCAN on a set of benchmark multimodal test functions.     

Stochastic Methods for Practical Global Optimization

Engineering design problems often involve global optimization of functions that are supplied as black box functions. These functions may be nonconvex, nondifferentiable and even discontinuous. In addition, the decision variables may be a combination of discrete and continuous variables. The functions are usually computationally expensive, and may involve finite element methods. An engineering example of this type of problem is to minimize the weight of a structure, while limiting strain to be below a certain threshold. This type of global optimization problem is very difficult to solve, yet design engineers must find some solution to their problem – even if it is a suboptimal one. Sometimes the most difficult part of the problem is finding any feasible solution. Stochastic methods, including sequential random search and simulated annealing, are finding many applications to this type of practical global optimization problem. Improving Hit-and-Run (IHR) is a sequential random search method that has been successfully used in several engineering design applications, such as the optimal design of composite structures. A motivation to IHR is discussed as well as several enhancements. The enhancements include allowing both continuous and discrete variables in the problem formulation. This has many practical advantages, because design variables often involve a mixture of continuous and discrete values. IHR and several variations have been applied to the composites design problem. Some of this practical experience is discussed.     

New simulation-based frameworks for multi-objective reliability-based design optimization of structures

In the present study, two new simulation-based frameworks are proposed for multi-objective reliability-based design optimization (MORBDO). The first is based on hybrid non-dominated sorting weighted simulation method (NSWSM) in conjunction with iterative local searches that is efficient for continuous MORBDO problems. According to NSWSM, uniform samples are generated within the design space and, then, the set of feasible samples are separated. Thereafter, the non-dominated sorting operator is employed to extract the approximated Pareto front. The iterative local sample generation is then performed in order to enhance the accuracy, diversity, and increase the extent of non-dominated solutions. In the second framework, a pseudo-double loop algorithm is presented based on hybrid weighted simulation method (WSM) and the Non-dominated Sorting Genetic Algorithm II (NSGA-II) that is efficient for problems including both discrete and continuous variables. According to hybrid WSM-NSGA-II, proper non-dominated solutions are produced in each generation of NSGA-II and, subsequently, WSM evaluates the reliability level of each candidate solution until the algorithm converges to the true Pareto solutions. The valuable characteristic of presented approaches is that only one simulation run is required for WSM during entire optimization process, even if solutions for different levels of reliability be desired. Illustrative examples indicate that NSWSM with the proposed local search strategy is more efficient for small dimension continuous problems. However, WSM-NSGA-II outperforms NSWSM in terms of solutions quality and computational efficiency, specifically for discrete MORBDOs. Employing global optimizer in WSM-NSGA-II provided more accurate results with lower samples than NSWSM.     

A fast steady-state <Emphasis Type="Italic">ε</Emphasis>-dominance multi-objective evolutionary algorithm

Multi-objective evolutionary algorithms (MOEAs) have become an increasingly popular tool for design and optimization tasks in real-world applications. Most of the popular baseline algorithms are pivoted on the use of Pareto-ranking (that is empirically inefficient) to improve the convergence to the Pareto front of a multi-objective optimization problem. This paper proposes a new ε-dominance MOEA (EDMOEA) which adopts pair-comparison selection and steady-state replacement instead of the Pareto-ranking. The proposed algorithm is an elitist algorithm with a new preservation technique of population diversity based on the ε-dominance relation. It is demonstrated that superior results could be obtained by the EDMOEA compared with other algorithms: NSGA-II, SPEA2, IBEA, ε-MOEA, PESA and PESA-II on test problems. The EDMOEA is able to converge to the Pareto optimal set much faster especially on the ZDT test functions with a large number of decision variables.     

A two-stage hybrid optimization for honeycomb-type cellular structures under out-of-plane dynamic impact

Honeycomb structures with better balance between lightweight and crashworthiness have aroused growing attentions. However, structural parameters design by traditional optimization algorithm in small design space is not sufficient to significantly enhance the specific energy absorption (SEA) with the lower peak acceleration (a_max). In this paper, a two-stage hybrid optimization for honeycomb-type cellular parameters is proposed to achieve rapid positioning of design space and significantly increase crashworthiness in a larger variable domain under out-of-plane dynamic impact. In stage I, a Taguchi-based grey correlation discrete optimization, combining Taguchi analysis, grey relational analysis, analysis of variance (ANOVA) with grey entropy measurement, is performed to determine the initial optimal value with a higher robustness and the significant influence variables. In stage II, a multi-objective design technique, namely non-nominated sorting genetic algorithm II based on surrogated model, is adopted to maximize the SEA and minimize the a_max in a relatively small design domain. And it is found that the proposed two-stage hybrid method can broaden the optimal design space compared to that of traditional method attributable to its center point positioned by stage I. And the final optimization based on the proposed strategy is superior to the original structure, i.e., the SEA is increased by 47.55% and the a_max is decreased by 80.8%. Therefore, the proposed algorithm can also be used to solve other more complicated engineering problems in a large design space with insightful design data.     

Analysis of the dynamics of local error control via a piecewise continuous residual

Positive results are obtained about the effect of local error control in numerical simulations of ordinary differential equations. The results are cast in terms of the local error tolerance. Under theassumption that a local error control strategy is successful, it is shown that a continuous interpolant through the numerical solution exists that satisfies the differential equation to within a small, piecewise continuous, residual. The assumption is known to hold for thematlab ode23 algorithm [10] when applied to a variety of problems. Using the smallness of the residual, it follows that at any finite time the continuous interpolant converges to the true solution as the error tolerance tends to zero. By studying the perturbed differential equation it is also possible to prove discrete analogs of the long-time dynamical properties of the equation—dissipative, contractive and gradient systems are analysed in this way. Supported by the Engineering and Physical Sciences Research Council under grants GR/H94634 and GR/K80228. Supported by the Office of Naval Research under grant N00014-92-J-1876 and by the National Science Foundation under grant DMS-9201727.     

An evolutionary programming approach to mixed-variable optimization problems

Many engineering optimization problems frequently encounter discrete variables as well as continuous variables and the presence of nonlinear discrete variables considerably adds to the solution complexity. Very few of the existing methods can find a globally optimal solution when the objective functions are non-convex and non-differentiable. In this paper, we present a mixed-variable evolutionary programming (MVEP) technique for solving these nonlinear optimization problems which contain integer, discrete, zero-one and continuous variables. The MVEP provides an improvement in global search reliability in a mixed-variable space and converges steadily to a good solution. An approach to handle various kinds of variables and constraints is discussed. Some examples of mixed-variable optimization problems in the literature are tested, which demonstrate that the proposed approach is superior to current methods for finding the best solution, in terms of both solution quality and algorithm robustness.     

Multiobjective evolutionary algorithms for complex portfolio optimization problems

This paper investigates the ability of Multiobjective Evolutionary Algorithms (MOEAs), namely the Non-dominated Sorting Genetic Algorithm II (NSGA-II), Pareto Envelope-based Selection Algorithm (PESA) and Strength Pareto Evolutionary Algorithm 2 (SPEA2), for solving complex portfolio optimization problems. The portfolio optimization problem is a typical bi-objective optimization problem with objectives the reward that should be maximized and the risk that should be minimized. While reward is commonly measured by the portfolio’s expected return, various risk measures have been proposed that try to better reflect a portfolio’s riskiness or to simplify the problem to be solved with exact optimization techniques efficiently. However, some risk measures generate additional complexities, since they are non-convex, non-differentiable functions. In addition, constraints imposed by the practitioners introduce further difficulties since they transform the search space into a non-convex region. The results show that MOEAs, in general, are efficient and reliable strategies for this kind of problems, and their performance is independent of the risk function used.     

Some Asymptotic Results for the Number of Generalized Records

In this paper, we study asymptotic properties (large deviations and functional central limit theorem) of generalized record processes built on a triangular array of continuous and exchangeable random variables. As an application of these results, the links with the Kendall's rank correlation statistic are studied and testing exchangeability is discussed. AMS 2000 Subject Classification Primary—60F10 Secondary—60F17, 62G10     

Nonparametric estimation of distributions with categorical and continuous data

In this paper we consider the problem of estimating an unknown joint distribution which is defined over mixed discrete and continuous variables. A nonparametric kernel approach is proposed with smoothing parameters obtained from the cross-validated minimization of the estimator's integrated squared error. We derive the rate of convergence of the cross-validated smoothing parameters to their ‘benchmark’ optimal values, and we also establish the asymptotic normality of the resulting nonparametric kernel density estimator. Monte Carlo simulations illustrate that the proposed estimator performs substantially better than the conventional nonparametric frequency estimator in a range of settings. The simulations also demonstrate that the proposed approach does not suffer from known limitations of the likelihood cross-validation method which breaks down with commonly used kernels when the continuous variables are drawn from fat-tailed distributions. An empirical application demonstrates that the proposed method can yield superior predictions relative to commonly used parametric models.     

Hamiltonian spatial structure for three-dimensional water waves in a moving frame of reference

Summary The governing equations for three-dimensional time-dependent water waves in a moving frame of reference are reformulated in terms of the energy and momentum flux. The novelty of this approach is that time-independent motions of the system—that is, motions that are steady in a moving frame of reference—satisfy a partial differential equation, which is shown to be Hamiltonian. The theory of Hamiltonian evolution equations (canonical variables, Poisson brackets, symplectic form, conservation laws) is applied to the spatial Hamiltonian system derived for pure gravity waves. The addition of surface tension changes the spatial Hamiltonian structure in such a way that the symplectic operator becomes degenerate, and the properties of this generalized Hamiltonian system are also studied. Hamiltonian bifurcation theory is applied to the linear spatial Hamiltonian system for capillary-gravity waves, showing how new waves can be found in this framework.     

A survey on multi-objective evolutionary algorithms for many-objective problems

Multi-objective evolutionary algorithms (MOEAs) are well-suited for solving several complex multi-objective problems with two or three objectives. However, as the number of conflicting objectives increases, the performance of most MOEAs is severely deteriorated. How to improve MOEAs’ performance when solving many-objective problems, i.e. problems with four or more conflicting objectives, is an important issue since a large number of this type of problems exists in science and engineering; thus, several researchers have proposed different alternatives. This paper presents a review of the use of MOEAs in many-objective problems describing the evolution of the field, the methods that were developed, as well as the main findings and open questions that need to be answered in order to continue shaping the field.     

Worst-case global optimization of black-box functions through Kriging and relaxation

A new algorithm is proposed to deal with the worst-case optimization of black-box functions evaluated through costly computer simulations. The input variables of these computer experiments are assumed to be of two types. Control variables must be tuned while environmental variables have an undesirable effect, to which the design of the control variables should be robust. The algorithm to be proposed searches for a minimax solution, i.e., values of the control variables that minimize the maximum of the objective function with respect to the environmental variables. The problem is particularly difficult when the control and environmental variables live in continuous spaces. Combining a relaxation procedure with Kriging-based optimization makes it possible to deal with the continuity of the variables and the fact that no analytical expression of the objective function is available in most real-case problems. Numerical experiments are conducted to assess the accuracy and efficiency of the algorithm, both on analytical test functions with known results and on an engineering application.     

A multiobjective immune algorithm based on a multiple-affinity model

This paper presents a new multiobjective immune algorithm based on a multiple-affinity model inspired by immune system (MAM-MOIA). The multiple-affinity model builds the relationship model among main entities and concepts in multiobjective problems (MOPs) and multiobjective evolutionary algorithms (MOEAs), including feasible solution, variable space, objective space, Pareto-optimal set, ranking and crowding distance. In the model, immune operators including clonal proliferation, hypermutation and immune suppression are designed to proliferate superior antibodies and suppress the inferiors. MAM-MOIA is compared with NSGA-II, SPEA2 and NNIA in solving the ZDT and DTLZ standard test problems. The experimental study based on three performance metrics including coverage of two sets, convergence and spacing proves that MAM-MOIA is effective for solving MOPs.