Multimodal Optimization with the Local Optimum Ranking 2 Algorithm

In optimization tasks, it is interesting to achieve a set of efficient solutions instead of one single output, in the case the best solution is not suitable. Many niching methods offer a diversified response, yet some important problems are common: (1) The most interesting solutions of each local optimum are not identified. Thus, the output is the overall population of solutions, which increases the work of the designer in verifying which solution is the most interesting. (2) Existing niching algorithms tend to distribute the solutions on the most promising regions, over-populating some local optima and sub-populating others, which leads to poor optimization.To solve these challenges, a novel niching method is presented, named local optimum ranking 2 (LOR2). This sorting methodology favors the exploration of a defined number of local optima and ranks each local population by objective value within each local optimum. Thus, is performed a multi-focus exploration, with an equalized number of solutions on each local optimum, while identifying which solutions are the local apices. To exemplify its application, the LOR2 algorithm is applied in the design optimization of a metallic cantilever beam. It achieves a set of efficient and diverse design configurations, offering both performance and diversity for structural design challenges.In addition, a second experiment describes how the algorithm can be applied to segment the domain of any function, into a mesh of similar sized or custom-sized elements. Thus, it can significantly simplify metamodels and reduce their computation time.

Multimodal Optimization with the Local Optimum Ranking 2 Algorithm

In optimization tasks, it is interesting to achieve a set of efficient solutions instead of one single output, in the case the best solution is not suitable. Many niching methods offer a diversified response, yet some important problems are common: (1) The most interesting solutions of each local optimum are not identified. Thus, the output is the overall population of solutions, which increases the work of the designer in verifying which solution is the most interesting. (2) Existing niching algorithms tend to distribute the solutions on the most promising regions, over-populating some local optima and sub-populating others, which leads to poor optimization.To solve these challenges, a novel niching method is presented, named local optimum ranking 2 (LOR2). This sorting methodology favors the exploration of a defined number of local optima and ranks each local population by objective value within each local optimum. Thus, is performed a multi-focus exploration, with an equalized number of solutions on each local optimum, while identifying which solutions are the local apices. To exemplify its application, the LOR2 algorithm is applied in the design optimization of a metallic cantilever beam. It achieves a set of efficient and diverse design configurations, offering both performance and diversity for structural design challenges.In addition, a second experiment describes how the algorithm can be applied to segment the domain of any function, into a mesh of similar sized or custom-sized elements. Thus, it can significantly simplify metamodels and reduce their computation time.

Multimodal Optimization with the Local Optimum Ranking 2 Algorithm

In optimization tasks, it is interesting to achieve a set of efficient solutions instead of one single output, in the case the best solution is not suitable. Many niching methods offer a diversified response, yet some important problems are common: (1) The most interesting solutions of each local optimum are not identified. Thus, the output is the overall population of solutions, which increases the work of the designer in verifying which solution is the most interesting. (2) Existing niching algorithms tend to distribute the solutions on the most promising regions, over-populating some local optima and sub-populating others, what leads to a poor optimization.To solve these challenges, a novel niching method is presented, named local optimum ranking 2. This sorting methodology favors the exploration of a defined number of local optima, and ranks each local population by objective value within each local optimum. Thus, is performed a multi-focus exploration, with an equalized number of solutions on each local optimum, while identifying which solutions are the local apices. Experimental results demonstrate the local optimum ranking 2 provides superior performance than other popular niching methods, for the selected test functions and global optimization algorithms. Also, its versatility is demonstrated in the several ways it can be combined with some of the most well-known methods.In a second experiment, the LOR2 algorithm is applied in the design optimization of a metallic cantilever beam. It is exemplified how the LOR2 algorithm can achieve a set of efficient and diverse design configurations, identifying which are the apices of each local optimum. Thus, the LOR2 facilitates multimodal optimization tasks, while offering both performance and diversity for design challenges.In addition, a third experiment describes how the algorithm can be applied to segment the domain of any function, with any type of input distribution or number of coordinates, into a mesh of similar sized or custom sized elements. Thus, it can segment a response surface named Kriging, significantly simplifying it and reducing computation time.

Multimodal Optimization with the Local Optimum Ranking 2 Algorithm

In optimization tasks, it is interesting to achieve a set of efficient solutions instead of one single output, in the case the best solution is not suitable. Many niching methods offer a diversified response, yet some important problems are common: (1) The most interesting solutions of each local optimum are not identified. Thus, the output is the overall population of solutions, which increases the work of the designer in verifying which solution is the most interesting. (2) Existing niching algorithms tend to distribute the solutions on the most promising regions, over-populating some local optima and sub-populating others, what leads to a poor optimization.To solve these challenges, a novel niching method is presented, named local optimum ranking 2. This sorting methodology favors the exploration of a defined number of local optima, and ranks each local population by objective value within each local optimum. Thus, is performed a multi-focus exploration, with an equalized number of solutions on each local optimum, while identifying which solutions are the local apices. Experimental results demonstrate the local optimum ranking 2 provides superior performance than other popular niching methods, for the selected test functions and global optimization algorithms. Also, its versatility is demonstrated in the several ways it can be combined with some of the most well-known methods.In a second experiment, the LOR2 algorithm is applied in the design optimization of a metallic cantilever beam. It is exemplified how the LOR2 algorithm can achieve a set of efficient and diverse design configurations, identifying which are the apices of each local optimum. Thus, the LOR2 facilitates multimodal optimization tasks, while offering both performance and diversity for design challenges.In addition, a third experiment describes how the algorithm can be applied to segment the domain of any function, with any type of input distribution or number of coordinates, into a mesh of similar sized or custom sized elements. Thus, it can segment a response surface named Kriging, significantly simplifying it and reducing computation time.

Runtime Analysis of Restricted Tournament Selection for Bimodal Optimisation

Niching methods have been developed to maintain the population diversity, to investigate many peaks in parallel and to reduce the effect of genetic drift. We present the first rigorous runtime analyses of restricted tournament selection (RTS), embedded in a (μ+1) EA, and analyse its effectiveness at finding both optima of the bimodal function TwoMax. In RTS, an offspring competes against the closest individual, with respect to some distance measure, amongst w (window size) population members (chosen uniformly at random with replacement), to encourage competition within the same niche. We prove that RTS finds both optima on TwoMax efficiently if the window size w is large enough. However, if w is too small, RTS fails to find both optima even in exponential time, with high probability. We further consider a variant of RTS selecting individuals for the tournament without replacement. It yields a more diverse tournament and is more effective at preventing one niche from taking over the other. However, this comes at the expense of a slower progress towards optima when a niche collapses to a single individual. Our theoretical results are accompanied by experimental studies that shed light on parameters not covered by the theoretical results and support a conjectured lower runtime bound.

Conditional Granger Causality and Genetic Algorithms in VAR Model Selection

Overcoming symmetry in combinatorial evolutionary algorithms is a challenge for existing niching methods. This research presents a genetic algorithm designed for the shrinkage of the coefficient matrix in vector autoregression (VAR) models, constructed on two pillars: conditional Granger causality and Lasso regression. Departing from a recent information theory proof that Granger causality and transfer entropy are equivalent, we propose a heuristic method for the identification of true structural dependencies in multivariate economic time series. Through rigorous testing, both empirically and through simulations, the present paper proves that genetic algorithms initialized with classical solutions are able to easily break the symmetry of random search and progress towards specific modeling.

Multimodal Optimization by Covariance Matrix Self-Adaptation Evolution Strategy with Repelling Subpopulations

During the recent decades, many niching methods have been proposed and empirically verified on some available test problems. They often rely on some particular assumptions associated with the distribution, shape, and size of the basins, which can seldom be made in practical optimization problems. This study utilizes several existing concepts and techniques, such as taboo points, normalized Mahalanobis distance, and the Ursem’s hill-valley function in order to develop a new tool for multimodal optimization, which does not make any of these assumptions. In the proposed method, several subpopulations explore the search space in parallel. Offspring of a subpopulation are forced to maintain a sufficient distance to the center of fitter subpopulations and the previously identified basins, which are marked as taboo points. The taboo points repel the subpopulation to prevent convergence to the same basin. A strategy to update the repelling power of the taboo points is proposed to address the challenge of basins of dissimilar size. The local shape of a basin is also approximated by the distribution of the subpopulation members converging to that basin. The proposed niching strategy is incorporated into the covariance matrix self-adaptation evolution strategy (CMSA-ES), a potent global optimization method. The resultant method, called the covariance matrix self-adaptation with repelling subpopulations (RS-CMSA), is assessed and compared to several state-of-the-art niching methods on a standard test suite for multimodal optimization. An organized procedure for parameter setting is followed which assumes a rough estimation of the desired/expected number of minima available. Performance sensitivity to the accuracy of this estimation is also studied by introducing the concept of robust mean peak ratio. Based on the numerical results using the available and the introduced performance measures, RS-CMSA emerges as the most successful method when robustness and efficiency are considered at the same time.