Multiplicative Approximations, Optimal Hypervolume Distributions, and the Choice of the Reference Point

Many optimization problems arising in applications have to consider several objective functions at the same time. Evolutionary algorithms seem to be a very natural choice for dealing with multi-objective problems as the population of such an algorithm can be used to represent the trade-offs with respect to the given objective functions. In this paper, we contribute to the theoretical understanding of evolutionary algorithms for multi-objective problems. We consider indicator-based algorithms whose goal is to maximize the hypervolume for a given problem by distributing {\mu} points on the Pareto front. To gain new theoretical insights into the behavior of hypervolume-based algorithms we compare their optimization goal to the goal of achieving an optimal multiplicative approximation ratio. Our studies are carried out for different Pareto front shapes of bi-objective problems. For the class of linear fronts and a class of convex fronts, we prove that maximizing the hypervolume gives the best possible approximation ratio when assuming that the extreme points have to be included in both distributions of the points on the Pareto front. Furthermore, we investigate the choice of the reference point on the approximation behavior of hypervolume-based approaches and examine Pareto fronts of different shapes by numerical calculations

Hypervolume based metaheuristics for multiobjective optimization

The purpose of multiobjective optimization is to find solutions that are optimal regarding several goals. In the branch of vector or Pareto optimization all these goals are considered to be of equal importance, so that compromise solutions that cannot be improved regarding one goal without deteriorating in another are Paretooptimal. A variety of quality measures exist to evaluate approximations of the Paretooptimal set generated by optimizers, wherein the hypervolume is the most significant one, making the hypervolume calculation a core problem of multiobjective optimization. This thesis tackles that challenge by providing a new hypervolume algorithm from computational geometry and analyzing the problem’s computational complexity. Evolutionary multiobjective optimization algorithms (EMOA) are state-of-the-art methods for Pareto optimization, wherein the hypervolume-based algorithms belong to the most powerful ones, among them the popular SMS-EMOA. After its promising capabilities have already been demonstrated in first studies, this thesis is dedicated to deeper understand the underlying optimization process of the SMS-EMOA and similar algorithms, in order to specify their performance characteristics. Theoretical analyses are accomplished as far as possible with established and newly developed tools. Beyond the limitations of rigorous scrutiny, insights are gained via thorough experimental investigation. All considered problems are continuous, whereas the algorithms are as well applicable to discrete problems. More precisely, the following topics are concerned. The process of approaching the Pareto-optimal set of points is characterized by the convergence speed, which is analyzed for a general framework of EA with hypervolume selection on several classes of bi-objective problems. The results are achieved by a newly developed concept of linking single and multiobjective optimization. The optimization on the Pareto front, that is turning the population into a set with maximal hypervolume, is considered separately, focusing on the question under which circumstances the steady-state selection of exchanging only one population member suffices to reach a global optimum. We answer this question for different bi-objective problem classes. In a benchmarking on so-called many-objective problems of more than three objectives, the qualification of the SMS-EMOA is demonstrated in comparison to other EMOA, while also studying their cause of failure. Within the mentioned examinations, the choice of the hypervolume’s reference point receives special consideration by exposing its influence. Beyond the study of the SMS-EMOA with default setup, it is analyzed to what extent the performance can be improved by parameter tuning of the EMOA anent to certain problems, focusing on the influence of variation operators. Lastly, an optimization algorithm based on the gradient of the hypervolume is developed and hybridized with the SMS-EMOA

Metaheuristic optimization of power and energy systems: underlying principles and main issues of the 'rush to heuristics'

In the power and energy systems area, a progressive increase of literature contributions containing applications of metaheuristic algorithms is occurring. In many cases, these applications are merely aimed at proposing the testing of an existing metaheuristic algorithm on a specific problem, claiming that the proposed method is better than other methods based on weak comparisons. This 'rush to heuristics' does not happen in the evolutionary computation domain, where the rules for setting up rigorous comparisons are stricter, but are typical of the domains of application of the metaheuristics. This paper considers the applications to power and energy systems, and aims at providing a comprehensive view of the main issues concerning the use of metaheuristics for global optimization problems. A set of underlying principles that characterize the metaheuristic algorithms is presented. The customization of metaheuristic algorithms to fit the constraints of specific problems is discussed. Some weaknesses and pitfalls found in literature contributions are identified, and specific guidelines are provided on how to prepare sound contributions on the application of metaheuristic algorithms to specific problems

Advances and applications in high-dimensional heuristic optimization

“Applicable to most real-world decision scenarios, multiobjective optimization is an area of multicriteria decision-making that seeks to simultaneously optimize two or more conflicting objectives. In contrast to single-objective scenarios, nontrivial multiobjective optimization problems are characterized by a set of Pareto optimal solutions wherein no solution unanimously optimizes all objectives. Evolutionary algorithms have emerged as a standard approach to determine a set of these Pareto optimal solutions, from which a decision-maker can select a vetted alternative. While easy to implement and having demonstrated great efficacy, these evolutionary approaches have been criticized for their runtime complexity when dealing with many alternatives or a high number of objectives, effectively limiting the range of scenarios to which they may be applied. This research introduces mechanisms to improve the runtime complexity of many multiobjective evolutionary algorithms, achieving state-of-the-art performance, as compared to many prominent methods from the literature. Further, the investigations here presented demonstrate the capability of multiobjective evolutionary algorithms in a complex, large-scale optimization scenario. Showcasing the approach’s ability to intelligently generate well-performing solutions to a meaningful optimization problem. These investigations advance the concept of multiobjective evolutionary algorithms by addressing a key limitation and demonstrating their efficacy in a challenging real-world scenario. Through enhanced computational efficiency and exhibited specialized application, the utility of this powerful heuristic strategy is made more robust and evident”--Abstract, page iv

Many-Objective Hybrid Optimization Under Uncertainty With Applications

A novel method for solving many-objective optimization problems under uncertainty was developed. It is well known that no single optimization algorithm performs best for all problems. Therefore, the developed method, a many-objective hybrid optimizer (MOHO), uses five constitutive algorithms and actively switches between them throughout the optimization process allowing for robust optimization. MOHO monitors the progress made by each of the five algorithms and allows the best performing algorithm more attempts at finding the optimum. This removes the need for user input for selecting algorithm as the best performing algorithm is automatically selected thereby increasing the probability of converging to the optimum. An uncertainty quantification framework, based on sparse polynomial chaos expansion, to propagate the uncertainties in the input parameter to the objective functions was also developed and validated. Where the samples and analysis runs needed for standard polynomial chaos expansion increases exponentially with the dimensionality, the presented sparse polynomial chaos approach efficiently propagates the uncertainty with only a few samples, thereby greatly reducing the computational cost. The performance of MOHO was investigated on a total of 65 analytical test problems from the DTLZ and WFG test suite, for which the analytical solution is known. MOHO is also applied to two additional real-life cases of aerodynamic shape design of subsonic and hypersonic bodies. Aerodynamic shape optimization is often computationally expensive and is, therefore, a good test case to investigate MOHO`s ability to reduce the computational time through robust optimization and accelerated convergence. The subsonic design optimization had three objectives: maximize lift and minimize drag and moment. The hypersonic design optimization had two objectives: maximize volume and minimize drag. Two accelerated solvers based on fast multipole method and Newton impact theory are developed for simulating subsonic and hypersonic flows. The results show that MOHO performed, on average, better than all five remaining algorithms in 52% of the DTLZ+WFG problems. The results of robust optimization of a subsonic body and hypersonic bodies were in good agreement with theory. The MOHO developed is capable of solving many-objective, multi-objective and single objective, constrained and unconstrained optimization problems with and without uncertainty with little user input

Metaheuristic Optimization of Power and Energy Systems: Underlying Principles and Main Issues of the `Rush to Heuristics'

In the power and energy systems area, a progressive increase of literature contributions that contain applications of metaheuristic algorithms is occurring. In many cases, these applications are merely aimed at proposing the testing of an existing metaheuristic algorithm on a specific problem, claiming that the proposed method is better than other methods that are based on weak comparisons. This ‘rush to heuristics’ does not happen in the evolutionary computation domain, where the rules for setting up rigorous comparisons are stricter but are typical of the domains of application of the metaheuristics. This paper considers the applications to power and energy systems and aims at providing a comprehensive view of the main issues that concern the use of metaheuristics for global optimization problems. A set of underlying principles that characterize the metaheuristic algorithms is presented. The customization of metaheuristic algorithms to fit the constraints of specific problems is discussed. Some weaknesses and pitfalls that are found in literature contributions are identified, and specific guidelines are provided regarding how to prepare sound contributions on the application of metaheuristic algorithms to specific problems

Topological optimization for tailored designs of advection–diffusion-reaction porous reactors based on pore scale modeling and simulation: A PNM-NSGA framework

Alizadeh Mehrzad, Gostick Jeff, Suzuki Takahiro, et al. Topological optimization for tailored designs of advection–diffusion-reaction porous reactors based on pore scale modeling and simulation: A PNM-NSGA framework. Computers & Structures 301, 107452 (2024); https://doi.org/10.1016/j.compstruc.2024.107452