Design of a motorcycle frame using neuroacceleration strategies in MOEAs

Designing a low-budget lightweight motorcycle frame with superior dynamic and mechanical properties is a complex engineering problem. This complexity is due in part to the presence of multiple design objectives—mass, structural stress and rigidity—, the high computational cost of the finite element (FE) simulations used to evaluate the objectives, and the nature of the design variables in the frame’s geometry (discrete and continuous). Therefore, this paper presents a neuroacceleration strategy for multiobjective evolutionary algorithms (MOEAs) based on the combined use of real (FE simulations) and approximate fitness function evaluations. The proposed approach accelerates convergence to the Pareto optimal front (POF) comprised of nondominated frame designs. The proposed MOEA uses a mixed genotype to encode discrete and continuous design variables, and a set of genetic operators applied according to the type of variable. The results show that the proposed neuro-accelerated MOEAs, NN-NSGA II and NN-MicroGA, improve upon the performance of their original counterparts, NSGA II and MicroGA. Thus, this neuroacceleration strategy is shown to be effective and probably applicable to other FE-based engineering design problems.     

Constructing a Pareto front approximation for decision making

An approach to constructing a Pareto front approximation to computationally expensive multiobjective optimization problems is developed. The approximation is constructed as a sub-complex of a Delaunay triangulation of a finite set of Pareto optimal outcomes to the problem. The approach is based on the concept of inherent nondominance. Rules for checking the inherent nondominance of complexes are developed and applying the rules is demonstrated with examples. The quality of the approximation is quantified with error estimates. Due to its properties, the Pareto front approximation works as a surrogate to the original problem for decision making with interactive methods.     

Online surrogate multiobjective optimization algorithm for contaminated groundwater remediation designs

This paper proposes an online surrogate model-assisted multiobjective optimization framework to identify optimal remediation strategies for groundwater contaminated with dense non-aqueous phase liquids. The optimization involves three objectives: minimizing the remediation cost and duration and maximizing the contamination removal rate. The proposed framework adopts a multiobjective feasibility-enhanced particle swarm optimization algorithm to solve the optimization model and uses an online surrogate model as a substitute for the time-consuming multiphase flow model for calculating contamination removal rates during the optimization process. The resulting approach allows decision makers to find a balance among the remediation cost, remediation duration and contamination removal rate for remediating contaminated groundwater. The new algorithm is compared with the nondominated sorting genetic algorithm II, which is an extensively applied and well-known algorithm. The results show that the Pareto solutions obtained by the new algorithm have greater diversity and stability than those obtained by the nondominated sorting genetic algorithm II, indicating that the new algorithm is more applicable than the nondominated sorting genetic algorithm II for optimizing remediation strategies for contaminated groundwater. Additionally, the surrogate model and Pareto optimal set obtained by the proposed framework are compared with those of the offline surrogate model-assisted multiobjective optimization framework. The results indicate that the surrogate model accuracy and Pareto front achieved by the proposed framework outperform those of the offline surrogate model-assisted optimization framework. Thus, we conclude that the proposed framework can effectively enhance the surrogate model accuracy and further extend the comprehensive performance of Pareto solutions.     

A fast Pareto genetic algorithm approach for solving expensive multiobjective optimization problems

We present a new multiobjective evolutionary algorithm (MOEA), called fast Pareto genetic algorithm (FastPGA), for the simultaneous optimization of multiple objectives where each solution evaluation is computationally- and/or financially-expensive. This is often the case when there are time or resource constraints involved in finding a solution. FastPGA utilizes a new ranking strategy that utilizes more information about Pareto dominance among solutions and niching relations. New genetic operators are employed to enhance the proposed algorithm’s performance in terms of convergence behavior and computational effort as rapid convergence is of utmost concern and highly desired when solving expensive multiobjective optimization problems (MOPs). Computational results for a number of test problems indicate that FastPGA is a promising approach. FastPGA yields similar performance to that of the improved nondominated sorting genetic algorithm (NSGA-II), a widely-accepted benchmark in the MOEA research community. However, FastPGA outperforms NSGA-II when only a small number of solution evaluations are permitted, as would be the case when solving expensive MOPs.     

A multi-objective memetic algorithm based on decomposition for big optimization problems

When solving multi-objective optimization problems (MOPs) with big data, traditional multi-objective evolutionary algorithms (MOEAs) meet challenges because they demand high computational costs that cannot satisfy the demands of online data processing involving optimization. The gradient heuristic optimization methods show great potential in solving large scale numerical optimization problems with acceptable computational costs. However, some intrinsic limitations make them unsuitable for searching for the Pareto fronts. It is believed that the combination of these two types of methods can deal with big MOPs with less computational cost. The main contribution of this paper is that a multi-objective memetic algorithm based on decomposition for big optimization problems (MOMA/D-BigOpt) is proposed and a gradient-based local search operator is embedded in MOMA/D-BigOpt. In the experiments, MOMA/D-BigOpt is tested on the multi-objective big optimization problems with thousands of variables. We also combine the local search operator with other widely used MOEAs to verify its effectiveness. The experimental results show that the proposed algorithm outperforms MOEAs without the gradient heuristic local search operator.     

Multi-start and path relinking methods to deal with multiobjective knapsack problems

This paper deals with a multiobjective combinatorial optimization problem called Extended Knapsack Problem. By applying multi-start search and path relinking we rapidly guide the search toward the most balanced zone of the Pareto-optimal front. The Pareto relation is applied in order to designate a subset of the best generated solutions to be the current efficient set of solutions. The max-min criterion with the Hamming distance is used as a measure of dissimilarity in order to find diverse solutions to be combined. The performance of our approach is compared with several state-of-the-art MOEAs for a suite test problems taken from the literature.     

On solving biquadratic optimization via semidefinite relaxation

A method called PAINT is introduced for computationally expensive multiobjective optimization problems. The method interpolates between a given set of Pareto optimal outcomes. The interpolation provided by the PAINT method implies a mixed integer linear surrogate problem for the original problem which can be optimized with any interactive method to make decisions concerning the original problem. When the scalarizations of the interactive method used do not introduce nonlinearity to the problem (which is true e.g., for the synchronous NIMBUS method), the scalarizations of the surrogate problem can be optimized with available mixed integer linear solvers. Thus, the use of the interactive method is fast with the surrogate problem even though the problem is computationally expensive. Numerical examples of applying the PAINT method for interpolation are included.     

Generalized Homotopy Approach to Multiobjective Optimization

This paper proposes a new generalized homotopy algorithm for the solution of multiobjective optimization problems with equality constraints. We consider the set of Pareto candidates as a differentiable manifold and construct a local chart which is fitted to the local geometry of this Pareto manifold. New Pareto candidates are generated by evaluating the local chart numerically. The method is capable of solving multiobjective optimization problems with an arbitrary number k of objectives, makes it possible to generate all types of Pareto optimal solutions, and is able to produce a homogeneous discretization of the Pareto set. The paper gives a necessary and sufficient condition for the set of Pareto candidates to form a (k-1)-dimensional differentiable manifold, provides the numerical details of the proposed algorithm, and applies the method to two multiobjective sample problems.     

Reduced Jacobian Method

In this paper, we present the Wolfe’s reduced gradient method for multiobjective (multicriteria) optimization. We precisely deal with the problem of minimizing nonlinear objectives under linear constraints and propose a reduced Jacobian method, namely a reduced gradient-like method that does not scalarize those programs. As long as there are nondominated solutions, the principle is to determine a direction that decreases all goals at the same time to achieve one of them. Following the reduction strategy, only a reduced search direction is to be found. We show that this latter can be obtained by solving a simple differentiable and convex program at each iteration. Moreover, this method is conceived to recover both the discontinuous and continuous schemes of Wolfe for the single-objective programs. The resulting algorithm is proved to be (globally) convergent to a Pareto KKT-stationary (Pareto critical) point under classical hypotheses and a multiobjective Armijo line search condition. Finally, experiment results over test problems show a net performance of the proposed algorithm and its superiority against a classical scalarization approach, both in the quality of the approximated Pareto front and in the computational effort.     

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.     

An empirical study on similarity-based mating for evolutionary multiobjective combinatorial optimization

We have already proposed a similarity-based mating scheme to recombine extreme and similar parents for evolutionary multiobjective optimization. In this paper, we examine the effect of the similarity-based mating scheme on the performance of evolutionary multiobjective optimization (EMO) algorithms. First we examine which is better between recombining similar or dissimilar parents. Next we examine the effect of biasing selection probabilities toward extreme solutions that are dissimilar from other solutions in each population. Then we examine the effect of dynamically changing the strength of this bias during the execution of EMO algorithms. Computational experiments are performed on a wide variety of test problems for multiobjective combinatorial optimization. Experimental results show that the performance of EMO algorithms can be improved by the similarity-based mating scheme for many test problems.     

The Pareto fitness genetic algorithm: Test function study

Evolutionary algorithms have shown some success in solving multiobjective optimization problems. The methods of fitness assignment are mainly based on the information about the dominance relation between individuals. We propose a Pareto fitness genetic algorithm (PFGA) in which we introduce a modified ranking procedure and a promising way of sharing; a new fitness function based on the rank of the individual and its density value is designed. This is considered as our main contribution. The performance of our algorithm is evaluated on six multiobjective benchmarks with different Pareto front features. Computational results (quality of the approximation of the Pareto optimal set and the number of fitness function evaluations) proving its efficiency are reported.     

A variational approach to define robustness for parametric multiobjective optimization problems

In contrast to classical optimization problems, in multiobjective optimization several objective functions are considered at the same time. For these problems, the solution is not a single optimum but a set of optimal compromises, the so-called Pareto set. In this work, we consider multiobjective optimization problems that additionally depend on an external parameter ${\lambda \in \mathbb{R}}$ , so-called parametric multiobjective optimization problems. The solution of such a problem is given by the λ-dependent Pareto set. In this work we give a new definition that allows to characterize λ-robust Pareto points, meaning points which hardly vary under the variation of the parameter λ. To describe this task mathematically, we make use of the classical calculus of variations. A system of differential algebraic equations will turn out to describe λ-robust solutions. For the numerical solution of these equations concepts of the discrete calculus of variations are used. The new robustness concept is illustrated by numerical examples.     

Multiobjective Optimization of the Flow Around a Cylinder Using Model Order Reduction

In this article an efficient numerical method to solve multiobjective optimization problems for fluid flow governed by the Navier Stokes equations is presented. In order to decrease the computational effort, a reduced order model is introduced using Proper Orthogonal Decomposition and a corresponding Galerkin Projection. A global, derivative free multiobjective optimization algorithm is applied to compute the Pareto set (i.e. the set of optimal compromises) for the concurrent objectives minimization of flow field fluctuations and control cost. The method is illustrated for a 2D flow around a cylinder at Re = 100. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Experiments with classification-based scalarizing functions in interactive multiobjective optimization

In multiobjective optimization methods, the multiple conflicting objectives are typically converted into a single objective optimization problem with the help of scalarizing functions and such functions may be constructed in many ways. We compare both theoretically and numerically the performance of three classification-based scalarizing functions and pay attention to how well they obey the classification information. In particular, we devote special interest to the differences the scalarizing functions have in the computational cost of guaranteeing Pareto optimality. It turns out that scalarizing functions with or without so-called augmentation terms have significant differences in this respect. We also collect a set of mostly nonlinear benchmark test problems that we use in the numerical comparisons.     

Hausdorff methods for approximating the convex Edgeworth–Pareto hull in integer problems with monotone objectives

Adaptive methods for the polyhedral approximation of the convex Edgeworth–Pareto hull in multiobjective monotone integer optimization problems are proposed and studied. For these methods, theoretical convergence rate estimates with respect to the number of vertices are obtained. The estimates coincide in order with those for filling and augmentation H-methods intended for the approximation of nonsmooth convex compact bodies.     

Comparative analysis of multiobjective evolutionary algorithms for random and correlated instances of multiobjective d-dimensional knapsack problems

This study analyzes multiobjective d-dimensional knapsack problems (MOd-KP) within a comparative analysis of three multiobjective evolutionary algorithms (MOEAs): the ε-nondominated sorted genetic algorithm II (ε-NSGAII), the strength Pareto evolutionary algorithm 2 (SPEA2) and the ε-nondominated hierarchical Bayesian optimization algorithm (ε-hBOA). This study contributes new insights into the challenges posed by correlated instances of the MOd-KP that better capture the decision interdependencies often present in real world applications. A statistical performance analysis of the algorithms uses the unary ε-indicator, the hypervolume indicator and success rate plots to demonstrate their relative effectiveness, efficiency, and reliability for the MOd-KP instances analyzed. Our results indicate that the ε-hBOA achieves superior performance relative to ε-NSGAII and SPEA2 with increasing number of objectives, number of decisions, and correlative linkages between the two. Performance of the ε-hBOA suggests that probabilistic model building evolutionary algorithms have significant promise for expanding the size and scope of challenging multiobjective problems that can be explored.     

Extending quick hypervolume

We extend the functionality of the quick hypervolume (QHV) algorithm. Given a set of d-dimensional points this algorithm determines the hypervolume of the dominated space, a useful measure for multiobjective evolutionary algorithms (MOEAs). We extend QHV in two ways: adapt it to compute the exclusive hypervolume of each point, and speed it up with parallel computation, that adjusts nicely to the divide and conquer methodology of QHV. The resulting algorithms are faster and more informative sub-routines, which can be used for MOEAs with a large number of objectives.     

Influence of ensemble surrogate models and sampling strategy on the solution quality of algorithms for computationally expensive black-box global optimization problems

This paper examines the influence of two major aspects on the solution quality of surrogate model algorithms for computationally expensive black-box global optimization problems, namely the surrogate model choice and the method of iteratively selecting sample points. A random sampling strategy (algorithm SO-M-c) and a strategy where the minimum point of the response surface is used as new sample point (algorithm SO-M-s) are compared in numerical experiments. Various surrogate models and their combinations have been used within the SO-M-c and SO-M-s sampling frameworks. The Dempster–Shafer Theory approach used in the algorithm by Müller and Piché (J Glob Optim 51:79–104, 2011) has been used for combining the surrogate models. The algorithms are numerically compared on 13 deterministic literature test problems with 2–30 dimensions, an application problem that deals with groundwater bioremediation, and an application that arises in energy generation using tethered kites. NOMAD and the particle swarm pattern search algorithm (PSWARM), which are derivative-free optimization methods, have been included in the comparison. The algorithms have also been compared to a kriging method that uses the expected improvement as sampling strategy (FEI), which is similar to the Efficient Global Optimization (EGO) algorithm. Data and performance profiles show that surrogate model combinations containing the cubic radial basis function (RBF) model work best regardless of the sampling strategy, whereas using only a polynomial regression model should be avoided. Kriging and combinations including kriging perform in general worse than when RBF models are used. NOMAD, PSWARM, and FEI perform for most problems worse than SO-M-s and SO-M-c. Within the scope of this study a Matlab toolbox has been developed that allows the user to choose, among others, between various sampling strategies and surrogate models and their combinations. The open source toolbox is available from the authors upon request.     

Two-Sided Pareto Front Approximations

A new approach to derive Pareto front approximations with evolutionary computations is proposed here. At present, evolutionary multiobjective optimization algorithms derive a discrete approximation of the Pareto front (the set of objective maps of efficient solutions) by selecting feasible solutions such that their objective maps are close to the Pareto front. However, accuracy of such approximations is known only if the Pareto front is known, which makes their usefulness questionable. Here we propose to exploit also elements outside feasible sets to derive pairs of such Pareto front approximations that for each approximation pair the corresponding Pareto front lies, in a certain sense, in-between. Accuracies of Pareto front approximations by such pairs can be measured and controlled with respect to distance between elements of a pair. A rudimentary algorithm to derive pairs of Pareto front approximations is presented and the viability of the idea is verified on a limited number of test problems.