Towards efficient multiobjective optimization: multiobjective statistical criterions

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 equivalent 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 making those decisions 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 (MOSBO), may prove to be even more worthwhile than SBO methods to expedite the optimization process. In this paper, the authors propose the Efficient Multiobjective Optimization (EMO) algorithm which uses Kriging models and multiobjective versions of the expected improvement and probability of improvement criterions 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 and SPEA2 multiobjective optimization methods with promising results

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

Scalarizing Functions in Bayesian Multiobjective Optimization

Scalarizing functions have been widely used to convert a multiobjective optimization problem into a single objective optimization problem. However, their use in solving (computationally) expensive multi- and many-objective optimization problems in Bayesian multiobjective optimization is scarce. Scalarizing functions can play a crucial role on the quality and number of evaluations required when doing the optimization. In this article, we study and review 15 different scalarizing functions in the framework of Bayesian multiobjective optimization and build Gaussian process models (as surrogates, metamodels or emulators) on them. We use expected improvement as infill criterion (or acquisition function) to update the models. In particular, we compare different scalarizing functions and analyze their performance on several benchmark problems with different number of objectives to be optimized. The review and experiments on different functions provide useful insights when using and selecting a scalarizing function when using a Bayesian multiobjective optimization method

Evolutionary Multiobjective Optimization Driven by Generative Adversarial Networks (GANs)

Recently, increasing works have proposed to drive evolutionary algorithms using machine learning models. Usually, the performance of such model based evolutionary algorithms is highly dependent on the training qualities of the adopted models. Since it usually requires a certain amount of data (i.e. the candidate solutions generated by the algorithms) for model training, the performance deteriorates rapidly with the increase of the problem scales, due to the curse of dimensionality. To address this issue, we propose a multi-objective evolutionary algorithm driven by the generative adversarial networks (GANs). At each generation of the proposed algorithm, the parent solutions are first classified into real and fake samples to train the GANs; then the offspring solutions are sampled by the trained GANs. Thanks to the powerful generative ability of the GANs, our proposed algorithm is capable of generating promising offspring solutions in high-dimensional decision space with limited training data. The proposed algorithm is tested on 10 benchmark problems with up to 200 decision variables. Experimental results on these test problems demonstrate the effectiveness of the proposed algorithm

Which Surrogate Works for Empirical Performance Modelling? A Case Study with Differential Evolution

It is not uncommon that meta-heuristic algorithms contain some intrinsic parameters, the optimal configuration of which is crucial for achieving their peak performance. However, evaluating the effectiveness of a configuration is expensive, as it involves many costly runs of the target algorithm. Perhaps surprisingly, it is possible to build a cheap-to-evaluate surrogate that models the algorithm's empirical performance as a function of its parameters. Such surrogates constitute an important building block for understanding algorithm performance, algorithm portfolio/selection, and the automatic algorithm configuration. In principle, many off-the-shelf machine learning techniques can be used to build surrogates. In this paper, we take the differential evolution (DE) as the baseline algorithm for proof-of-concept study. Regression models are trained to model the DE's empirical performance given a parameter configuration. In particular, we evaluate and compare four popular regression algorithms both in terms of how well they predict the empirical performance with respect to a particular parameter configuration, and also how well they approximate the parameter versus the empirical performance landscapes

A constrained multi-objective surrogate-based optimization algorithm

Surrogate models or metamodels are widely used in the realm of engineering for design optimization to minimize the number of computationally expensive simulations. Most practical problems often have conflicting objectives, which lead to a number of competing solutions which form a Pareto front. Multi-objective surrogate-based constrained optimization algorithms have been proposed in literature, but handling constraints directly is a relatively new research area. Most algorithms proposed to directly deal with multi-objective optimization have been evolutionary algorithms (Multi-Objective Evolutionary Algorithms -MOEAs). MOEAs can handle large design spaces but require a large number of simulations, which might be infeasible in practice, especially if the constraints are expensive. A multi-objective constrained optimization algorithm is presented in this paper which makes use of Kriging models, in conjunction with multi-objective probability of improvement (PoI) and probability of feasibility (PoF) criteria to drive the sample selection process economically. The efficacy of the proposed algorithm is demonstrated on an analytical benchmark function, and the algorithm is then used to solve a microwave filter design optimization problem

Automatic surrogate model type selection during the optimization of expensive black-box problems

The use of Surrogate Based Optimization (SBO) has become commonplace for optimizing expensive black-box simulation codes. A popular SBO method is the Efficient Global Optimization (EGO) approach. However, the performance of SBO methods critically depends on the quality of the guiding surrogate. In EGO the surrogate type is usually fixed to Kriging even though this may not be optimal for all problems. In this paper the authors propose to extend the well-known EGO method with an automatic surrogate model type selection framework that is able to dynamically select the best model type (including hybrid ensembles) depending on the data available so far. Hence, the expected improvement criterion will always be based on the best approximation available at each step of the optimization process. The approach is demonstrated on a structural optimization problem, i.e., reducing the stress on a truss-like structure. Results show that the proposed algorithm consequently finds better optimums than traditional kriging-based infill optimization