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.

Adaptive Grouping Brain Storm Optimization for Multimodal Optimization Problems

Multimodal optimization problem exists in multiple global and many local optimal solutions. The difficulty of solving these problems is finding as many local optimal peaks as possible on the premise of ensuring global optimal precision. This article presents adaptive grouping brainstorm optimization (AGBSO) for solving these problems. In this article, adaptive grouping strategy is proposed for achieving adaptive grouping without providing any prior knowledge by users. For enhancing the diversity and accuracy of the optimal algorithm, elite reservation strategy is proposed to put central particles into an elite pool, and peak detection strategy is proposed to delete particles far from optimal peaks in the elite pool. Finally, this article uses testing functions with different dimensions to compare the convergence, accuracy, and diversity of AGBSO with BSO. Experiments verify that AGBSO has great localization ability for local optimal solutions while ensuring the accuracy of the global optimal solutions.

Hybrid Multi-Population and Adaptive Search Range Strategy With Particle Swarm Optimization for Multimodal Optimization

This paper proposes an algorithm named hybrid multi-population and adaptive search range strategy with particle swarm optimization (ARPSO) for solving multimodal optimization problems. The main idea of the algorithm is to divide the global search space into multiple sub-populations searching in parallel and independently. For diversity increasing, each sub-population will continuously change the search area adaptively according to whether there are local optimal solutions in its search space and the position of the global optimal solution, and in each iteration, the optimal solution in this area will be reserved. For the purpose of accelerating convergence, at the global and local levels, when the global optimal solution or local optimal solution is found, the global search space and local search space will shrink toward the optimal solution. Experiments show that ARPSO has unique advantages for solving multi-dimensional problems, especially problems with only one global optimal solution but multiple local optimal solutions.

A Scalable, Black-box, Hybrid Genetic Algorithm for Continuous Multimodal Optimization in Moderate Dimensions

Optimization problems can be found in many areas of science and technology. Often, not only the global optimum, but also a (larger) number of near-optima are of interest. This gives rise to so-called multimodal optimization problems. In most of the cases, the number and quality of the optima is unknown and assumptions on the objective functions cannot be made. In this paper, we focus on continuous, unconstrained optimization in moderately high dimensional continuous spaces (<=10). We present a scalable algorithm with virtually no parameters, which performs well for general objective functions (non-convex, discontinuous). It is based on two well-established algorithms (CMA-ES, deterministic crowding). Novel elements of the algorithm are the detection of seed points for local searches and collision avoidance, both based on nearest neighbors, and a strategy for semi-sequential optimization to realize scalability. The performance of the proposed algorithm is numerically evaluated on the CEC2013 niching benchmark suite for 1-20 dimensional functions and a 9 dimensional real-world problem from constraint optimization in climate research. The algorithm shows good performance on the CEC2013 benchmarks and falls only short on higher dimensional and strongly inisotropic problems. In case of the climate related problem, the algorithm is able to find a high number (150) of optima, which are of relevance to climate research. The proposed algorithm does not require special configuration for the optimization problems considered in this paper, i.e. it shows good black-box behavior.