Shaul Salomon

The Gradient Free Directed Search Method as Local Search within Multi-Objective Evolutionary Algorithms

Recently, the Directed Search Method has been proposed as a point-wise iterative search procedure that allows to steer the search, in any direction given in objective space, of a multi-objective optimization problem. While the original version requires the objectives’ gradients, we consider here a possible modification that allows to realize the method without gradient information. This makes the novel algorithm in particular interesting for hybridization with set oriented search procedures, such as multi-objective evolutionary algorithms.

PSA – A New Scalable Space Partition Based Selection Algorithm for MOEAs

It has generally been acknowledged that both proximity to the Pareto front and a certain diversity along the front should be targeted when using evolutionary algorithms to evolve solutions to multi-objective optimization problems. Although many evolutionary algorithms are equipped with mechanisms to achieve both targets, most give priority to proximity over diversity. This priority is embedded within the algorithms through the selection of solutions to the elite population based on the concept of dominance. Although the current study does not change this embedded preference, it does utilize an improved diversity preservation mechanism that is based on a recently introduced partitioning algorithm for function selection. It is shown that this partitioning allows for the selection of a well-diversified set out of an arbitrary given set. Further, when embedded into an evolutionary search, this procedure significantly enhances the exploitation of diversity. The procedure is demonstrated on commonly used test cases for up to five objectives. The potential for further improving evolutionary algorithms through the use of the partitioning algorithm is highlighted.

PSA Based Multi Objective Evolutionary Algorithms

It has generally been acknowledged that both proximity to the Pareto front and a certain diversity along the front, should be targeted when using evolutionary multiobjective optimization. Recently, a new partitioning mechanism, the Part and Select Algorithm (PSA), has been introduced. It was shown that this partitioning allows for the selection of a well-diversified set out of an arbitrary given set, while maintaining low computational cost. When embedded into an evolutionary search (NSGA-II), the PSA has significantly enhanced the exploitation of diversity. In this paper, the ability of the PSA to enhance evolutionary multiobjective algorithms (EMOAs) is further investigated. Two research directions are explored here. The first one deals with the integration of the PSA within an EMOA with a novel strategy. Contrary to most EMOAs, that give a higher priority to proximity over diversity, this new strategy promotes the balance between the two. The suggested algorithm allows some dominated solutions to survive, if they contribute to diversity. It is shown that such an approach substantially reduces the risk of the algorithm to fail in finding the Pareto front. The second research direction explores the use of the PSA as an archiving selection mechanism, to improve the averaged Hausdorff distance obtained by existing EMOAs. It is shown that the integration of the PSA into NSGA-II-I and Δ p -EMOA as an archiving mechanism leads to algorithms that are superior to base EMOAS on problems with disconnected Pareto fronts.

Gearbox design for uncertain load requirements using active robust optimization

Design and optimization of gear transmissions have been intensively studied, but surprisingly the robustness of the resulting optimal design to uncertain loads has never been considered. Active Robust (AR) optimization is a methodology to design products that attain robustness to uncertain or changing environmental conditions through adaptation. In this study the AR methodology is utilized to optimize the number of transmissions, as well as their gearing ratios, for an uncertain load demand. The problem is formulated as a bi-objective optimization problem where the objectives are to satisfy the load demand in the most energy efficient manner and to minimize production cost. The results show that this approach can find a set of robust designs, revealing a trade-off between energy efficiency and production cost. This can serve as a useful decision-making tool for the gearbox design process, as well as for other applications.

Optimization of Adaptation - A Multi-objective Approach for Optimizing Changes to Design Parameters

Dynamic optimization problems require constant tracking of the optimum. A solution for such a problem has to be adjustable in order to remain optimal as the optimum changes. The manner of changing design parameters to predefined values is dealt with in the field of control. Common control approaches do not consider the optimality of the design, in terms of the objective function, while adjusting to the new solution. This study highlights the issue of the optimality of adaptation, and defines a new optimization problem – ”Optimization of Adaptation”. It is a multiobjective problem that considers the cost of the adaptation and the optimality while the adaptation takes place. An evolutionary algorithm is proposed in order to solve this problem, and it is demonstrated, first, with an academic example, and then with a real life application of a robotic arm control.

Evolution of Contours for Topology Optimization

Topology optimization is used to find a preliminary structural configuration that meets a predefined criterion. It involves optimizing both the external boundary and the distribution of the internal material within a structure. Usually, counters are used a posteriori to the topology optimization to further adapt the shape of the topology according to manufacturing needs. Here we suggest optimizing topologies by evolving counters. We consider both outer and inner counters to allow for holes in the structure. Due to the difficulty of defining a reliable measure for the differences among shapes, little research attention has been focused on simultaneously finding diverse sets of optimal topologies. Here, niching is implemented within a suggested evolutionary algorithm in order to find diverse topologies. The niching is then embedded within the algorithm through the use of our recently introduced partitioning algorithm. For this algorithm to be used, the topologies are represented as functions. Two examples are given to demonstrate the approach. These examples show that the algorithm evolves a set of diverse optimal topologies.

An Evolutionary Approach to Active Robust Multiobjective Optimisation

An Active Robust Optimisation Problem (AROP) aims at finding robust adaptable solutions, i.e. solutions that actively gain robustness to environmental changes through adaptation. Existing AROP studies have considered only a single performance objective. This study extends the Active Robust Optimisation methodology to deal with problems with more than one objective. Once multiple objectives are considered, the optimal performance for every uncertain parameter setting is a set of configurations, offering different trade-offs between the objectives. To evaluate and compare solutions to this type of problems, we suggest a robustness indicator that uses a scalarising function combining the main aims of multi-objective optimisation: proximity, diversity and pertinence. The Active Robust Multi-objective Optimisation Problem is formulated in this study, and an evolutionary algorithm that uses the hypervolume measure as a scalarasing function is suggested in order to solve it. Proof-of-concept results are demonstrated using a simplified gearbox optimisation problem for an uncertain load demand.

Transient responses’ optimization by means of set-based multi-objective evolution

In this article, a novel solution to multi-objective problems involving the optimization of transient responses is suggested. It is claimed that the common approach of treating such problems by introducing auxiliary objectives overlooks tradeoffs that should be presented to the decision makers. This means that, if at some time during the responses, one of the responses is optimal, it should not be overlooked. An evolutionary multi-objective algorithm is suggested in order to search for these optimal solutions. For this purpose, state-wise domination is utilized with a new crowding measure for ordered sets being suggested. The approach is tested on both artificial as well as on real life problems in order to explain the methodology and demonstrate its applicability and importance. The results indicate that, from an engineering point of view, the approach possesses several advantages over existing approaches. Moreover, the applications highlight the importance of set-based evolution.

Time-response-based evolutionary optimization

Solutions to engineering problems are often evaluated by considering their time responses; thus, each solution is associated with a function. To avoid optimizing the functions, such optimization is usually carried out by setting auxiliary objectives (e.g. minimal overshoot). Therefore, in order to find different optimal solutions, alternative auxiliary optimization objectives may have to be defined prior to optimization. In the current study, a new approach is suggested that avoids the need to define auxiliary objectives. An algorithm is suggested that enables the optimization of solutions according to their transient behaviours. For this optimization, the functions are sampled and the problem is posed as a multi-objective problem. The recently introduced algorithm NSGA-II-PSA is adopted and tailored to solve it. Mathematical as well as engineering problems are utilized to explain and demonstrate the approach and its applicability to real life problems. The results highlight the advantages of avoiding the definition of artificial objectives.

On the Effect of Scalarising Norm Choice in a ParEGO implementation

Computationally expensive simulations play an increasing role in engineering design, but their use in multi-objective optimization is heavily resource constrained. Specialist optimizers, such as ParEGO, exist for this setting, but little knowledge is available to guide their configuration. This paper uses a new implementation of ParEGO to examine three hypotheses relating to a key configuration parameter: choice of scalarising norm. Two hypotheses consider the theoretical trade-off between convergence speed and ability to capture an arbitrary Pareto front geometry. Experiments confirm these hypotheses in the bi-objective setting but the trade-off is largely unseen in many-objective settings. A third hypothesis considers the ability of dynamic norm scheduling schemes to overcome the trade-off. Experiments using a simple scheme offer partial support to the hypothesis in the bi-objective setting but no support in many-objective contexts. Norm scheduling is tentatively recommended for bi-objective problems for which the Pareto front geometry is concave or unknown.