Michael Emmerich

SMS-EMOA : Multiobjective selection based on dominated hypervolume

Abstract The hypervolume measure (or S metric) is a frequently applied quality measure for comparing the results of evolutionary multiobjective optimisation algorithms (EMOA). The new idea is to aim explicitly for the maximisation of the dominated hypervolume within the optimisation process. A steady-state EMOA is proposed that features a selection operator based on the hypervolume measure combined with the concept of non-dominated sorting. The algorithm’s population evolves to a well-distributed set of solutions, thereby focussing on interesting regions of the Pareto front. The performance of the devised S metric selection EMOA (SMS-EMOA) is compared to state-of-the-art methods on two- and three-objective benchmark suites as well as on aeronautical real-world applications.

An EMO algorithm using the hypervolume measure as selection criterion

The hypervolume measure is one of the most frequently applied measures for comparing the results of evolutionary multiobjective optimization algorithms (EMOA). The idea to use this measure for selection is self-evident. A steady-state EMOA will be devised, that combines concepts of non-dominated sorting with a selection operator based on the hypervolume measure. The algorithm computes a well distributed set of solutions with bounded size thereby focussing on interesting regions of the Pareto front(s). By means of standard benchmark problems the algorithm will be compared to other well established EMOA. The results show that our new algorithm achieves good convergence to the Pareto front and outperforms standard methods in the hypervolume covered. We also studied the applicability of the new approach in the important field of design optimization. In order to reduce the number of time consuming precise function evaluations, the algorithm will be supported by approximate function evaluations based on Kriging metamodels. First results on an airfoil redesign problem indicate a good performance of this approach, especially if the computation of a small, bounded number of well-distributed solutions is desired.

Single- and multiobjective evolutionary optimization assisted by Gaussian random field metamodels

This paper presents and analyzes in detail an efficient search method based on evolutionary algorithms (EA) assisted by local Gaussian random field metamodels (GRFM). It is created for the use in optimization problems with one (or many) computationally expensive evaluation function(s). The role of GRFM is to predict objective function values for new candidate solutions by exploiting information recorded during previous evaluations. Moreover, GRFM are able to provide estimates of the confidence of their predictions. Predictions and their confidence intervals predicted by GRFM are used by the metamodel assisted EA. It selects the promising members in each generation and carries out exact, costly evaluations only for them. The extensive use of the uncertainty information of predictions for screening the candidate solutions makes it possible to significantly reduce the computational cost of singleand multiobjective EA. This is adequately demonstrated in this paper by means of mathematical test cases and a multipoint airfoil design in aerodynamics

Metamodel-Assisted Evolution Strategies

This paper presents various Metamodel-Assisted Evolution Strategies which reduce the computational cost of optimisation problems involving time-consuming function evaluations. The metamodel is built using previously evaluated solutions in the search space and utilized to predict the fitness of new candidate solutions. In addition to previous works by the authors, the new metamodel takes also into account the error associated with each prediction, by correlating neighboring points in the search space. A mathematical problem and the problem of designing an optimal airfoil shape under viscous flow considerations have been worked out. Both demonstrate the noticeable gain in computational time one might expect from the use of metamodels in Evolution Strategies.

Adaptive niche radii and niche shapes approaches for niching with the cma-es

While the motivation and usefulness of niching methods is beyond doubt, the relaxation of assumptions and limitations concerning the hypothetical search landscape is much needed if niching is to be valid in a broader range of applications. Upon the introduction of radii-based niching methods with derandomized evolution strategies (ES), the purpose of this study is to address the so-called niche radius problem. A new concept of an adaptive individual niche radius is applied to niching with the covariance matrix adaptation evolution strategy (CMA-ES). Two approaches are considered. The first approach couples the radius to the step size mechanism, while the second approach employs the Mahalanobis distance metric with the covariance matrix mechanism for the distance calculation, for obtaining niches with more complex geometrical shapes. The proposed approaches are described in detail, and then tested on high-dimensional artificial landscapes at several levels of difficulty. They are shown to be robust and to achieve satisfying results.

On expected-improvement criteria for model-based multi-objective optimization

Surrogate models, as used for the Design and Analysis of Computer Experiments (DACE), can significantly reduce the resources necessary in cases of expensive evaluations. They provide a prediction of the objective and of the corresponding uncertainty, which can then be combined to a figure of merit for a sequential optimization. In singleobjective optimization, the expected improvement (EI) has proven to provide a combination that balances successfully between local and global search. Thus, it has recently been adapted to evolutionary multi-objective optimization (EMO) in different ways. In this paper, we provide an overview of the existing EI extensions for EMO and propose new formulations of the EI based on the hypervolume. We set up a list of necessary and desirable properties, which is used to reveal the strengths and weaknesses of the criteria by both theoretical and experimental analyses.

Hypervolume-based expected improvement: Monotonicity properties and exact computation

The expected improvement (EI) is a well established criterion in Bayesian global optimization (BGO) and metamodel-assisted evolutionary computation, both applied in optimization with costly function evaluations. Recently, it has been adopted in different ways to multiobjective optimization. A promising approach to formulate the expected improvement in this context, is to base it on the hypervolume indicator. Given the Bayesian model of the optimization landscape, the EI in hypervolume computes the expected gain in attained hypervolume for a given input point. Although a formulation of this expected improvement is relatively straightforward, its computation and mathematical properties are still to be investigated. This paper will outline and derive an algorithm for the exact computation of the proposed hypervolume-based EI. Moreover, this paper establishes monotonicity properties of the expected improvement. In particular the effect of the predictive distributions variance on the hypervolume-based EI and elementary properties of the EI landscape are studied. The monotonicity properties will reveal regions where Pareto front approximations can be improved as well as underexplored regions that are favored by the hypervolume-based expected improvement. A first numerical example is included that illustrates the behavior of the hypervolume-based EI in the multiobjective BGO framework.

Gradient-based/evolutionary relay hybrid for computing pareto front approximations maximizing the S-metric

The problem of computing a good approximation set of the Pareto front of a multiobjective optimization problem can be recasted as the maximization of its S-metric value, which measures the dominated hypervolume. In this way, the S-metric has recently been applied in a variety of metaheuristics. In this work, a novel high-precision method for computing approximation sets of a Pareto front with maximal S-Metric is proposed as a high-level relay hybrid of an evolutionary algorithm and a gradient method, both guided by the S-metric. First, an evolutionary multiobjective optimizer moves the initial population close to the Pareto front. The gradient-based method takes this population as its starting point for computing a local maximal approximation set with respect to the S-metric. Thereby, the population is moved according to the gradient of the S-metric. This paper introduces expressions for computing the gradient of a set of points with respect to its S-metric on basis of the gradients of the objective functions. It discusses singularities where the gradient is vanishing or differentiability is one sided. To circumvent the problem of vanishing gradient components of the S-metric for dominated points in the population a penalty approach is introduced. In order to test the new hybrid algorithm, we compute the precise maximizer of the S-metric for a generalized Schaffer problem and show, empirically, that the relay hybrid strategy linearly converges to the precise optimum. In addition we provide first case studies of the hybrid method on complicated benchmark problems.

Metamodel Assisted Multiobjective Optimisation Strategies and their Application in Airfoil Design

In this paper various metamodel-assisted multiobjective evolutionary algorithms (M-MOEA) for optimisation with time-consuming function evaluations are proposed and studied. Gaussian field (Kriging) models fitted by results from previous evaluations are used in order to pre-screen candidate solutions and decide whether they should be rejected or evaluated precisely. The approximations provide upper and lower bound estimations for the true function values. Three different rejection principles are proposed, discussed and integrated into recent MOEA variants (NSGA-II and ∈-MOEA). Experimental studies on a theoretical test case and in airfoil design demonstrate the improvements in diversity of solutions and convergence to the pareto fronts that can be achieved by using metamodels for pre-screening.

Design of Graph-Based Evolutionary Algorithms: A Case Study for Chemical Process Networks

This paper describes the adaptation of evolutionary algorithms (EAs) to the structural optimization of chemical engineering plants, using rigorous process simulation combined with realistic costing procedures to calculate target function values. To represent chemical engineering plants, a network representation with typed vertices and variable structure will be introduced. For this representation, we introduce a technique on how to create problem specific search operators and apply them in stochastic optimization procedures. The applicability of the approach is demonstrated by a reference example. The design of the algorithms will be oriented at the systematic framework of metricbased evolutionary algorithms (MBEAs). MBEAs are a special class of evolutionary algorithms, fulfilling certain guidelines for the design of search operators, whose benefits have been proven in theory and practice. MBEAs rely upon a suitable definition of a metric on the search space. The definition of a metric for the graph representation will be one of the main issues discussed in this paper. Although this article deals with the problem domain of chemical plant optimization, the algorithmic design can be easily transferred to similar network optimization problems. A useful distance measure for variable dimensionality search spaces is suggested.