André H. Deutz

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.

Test problems based on Lamé superspheres

Pareto optimization methods are usually expected to find well-distributed approximations of Pareto fronts with basic geometry, such as smooth, convex and concave surfaces. In this contribution, test-problems are proposed for which the Pareto front is the intersection of a Lame supersphere with the positive Rn-orthant. Besides scalability in the number of objectives and decision variables, the proposed test problems are also scalable in a characteristic we introduce as resolvability of conflict, which is closely related to convexity/concavity, curvature and the position of knee-points of the Pareto fronts. As a very basic bi-objective problem we propose a generalization of Schaffers problem. We derive closed-form expressions for the efficient sets and the Pareto fronts, which are arcs of Lame supercircles. Adopting the bottom-up approach of test problem construction, as used for the DTLZ test-problem suite, we derive test problems of higher dimension that result in Pareto fronts of superspherical geometry. Geometrical properties of these test-problems, such as concavity and convexity and the position of knee-points are studied. Our focus is on geometrical properties that are useful for performance assessment, such as the dominated hypervolume measure of the Pareto fronts. The use of these test problems is exemplified with a case-study using the SMSEMOA, for which we study the distribution of solution points on different 3-D Pareto fronts.

Faster Exact Algorithms for Computing Expected Hypervolume Improvement

This paper is about computing the expected improvement of the hypervolume indicator given a Pareto front approximation and a predictive multivariate Gaussian distribution of a new candidate point. It is frequently used as an infill or prescreening criterion in multiobjective optimization with expensive function evaluations where predictions are provided by Kriging or Gaussian process surrogate models. The expected hypervolume improvement has good properties as an infill criterion, but exact algorithms for its computation have so far been very time consuming even for the two and three objective case. This paper introduces faster exact algorithms for computing the expected hypervolume improvement for independent Gaussian distributions. A new general computation scheme is introduced and a lower bound for the time complexity. By providing new algorithms, upper bounds for the time complexity for problems with two as well as three objectives are improved. For the 2-D case the time complexity bound is reduced from previously \(O(n^3 \log n)\) to \(O(n^2)\). For the 3-D case the new upper bound of \(O(n^3)\) is established; previously \(O(n^4 \log n)\). It is also shown how an efficient implementation of these new algorithms can lead to a further reduction of running time. Moreover it is shown how to process batches of multiple predictive distributions efficiently. The theoretical analysis is complemented by empirical speed comparisons of C++ implementations of the new algorithms to existing implementations of other exact algorithms.

A robust optimization approach using Kriging metamodels for robustness approximation in the CMA-ES

This paper presents a study for using Kriging metamodeling in combination with Covariance Matrix Adaptation Evolution Strategies (CMA-ES) to find robust solutions. A general, archive based, framework is proposed for integrating Kriging within CMA-ES, including a method to utilize the covariance matrix of the CMA-ES in a straightforward way to improve the accuracy of the Kriging predictions without introducing much additional computational cost. Moreover, it adopts an elegant way to select appropriate archive points for building a local metamodel. The study shows that this Kriging metamodeling scheme for finding robust solutions outperforms common, straightforward approaches and is very useful when there is a limited budget of function evaluations. Though using the covariance matrix can improve the prediction quality, it has no significant effect on the overall quality of the optimization results.

Time Complexity and Zeros of the Hypervolume Indicator Gradient Field

In multi-objective optimization the hypervolume indicator is a measure for the size of the space within a reference set that is dominated by a set of μ points. It is a common performance indicator for judging the quality of Pareto front approximations. As it does not require a-priori knowledge of the Pareto front it can also be used in a straightforward manner for guiding the search for finite approximations to the Pareto front in multi-objective optimization algorithm design.

On Quality Indicators for Black-Box Level Set Approximation

This chapter reviews indicators that can be used to compute the quality of approximations to level sets for black-box functions. Such problems occur, for instance, when finding sets of solutions to optimization problems or in solving nonlinear equation systems. After defining and motivating level set problems from a decision theoretic perspective, we discuss quality indicators that could be used to measure how well a set of points approximates a level set. We review simple indicators based on distance, indicators from biodiversity, and propose novel indicators based on the concept of Hausdorff distance. We study properties of these indicators with respect to continuity, spread, and monotonicity and also discuss computational complexity. Moreover, we study the use of these indicators in a simple indicatorbased evolutionary algorithm for level set approximation.

Cone-Based Hypervolume Indicators: Construction, Properties, and Efficient Computation

In this paper we discuss cone-based hypervolume indicators (CHI) that generalize the classical hypervolume indicator (HI) in Pareto optimization. A family of polyhedral cones with scalable opening angle γ is studied. These γ-cones can be efficiently constructed and have a number of favorable properties. It is shown that for γ-cones dominance can be checked efficiently and the CHI computation can be reduced to the computation of the HI in linear time with respect to the number of points μ in an approximation set. Besides, individual contributions to these can be computed using a similar transformation to the case of Pareto dominance cones.

A Multicriteria Generalization of Bayesian Global Optimization

This chapter discusses a generalization of the expected improvement used in Bayesian global optimization to the multicriteria optimization domain, where the goal is to find an approximation to the Pareto front. The expected hypervolume improvement (EHVI) measures improvement as the gain in dominated hypervolume relative to a given approximation to the Pareto front. We will review known properties of the EHVI, applications in practice and propose a new exact algorithm for computing EHVI. The new algorithm has asymptotically optimal time complexity O(nlogn). This improves existing computation schemes by a factor of n∕logn. It shows that this measure, at least for a small number of objective functions, is as fast as other simpler measures of multicriteria expected improvement that were considered in recent years.