A Landscape-Aware Differential Evolution for Multimodal Optimization Problems

Multimodal optimization problem (MMOP) is one of the most common problems in engineering practices that requires multiple optimal solutions to be located simultaneously. An efficient algorithm for solving MMOPs should balance the diversity and convergence of the population, so that the global optimal solutions can be located as many as possible. However, most of existing algorithms are easy to be trapped into local peaks and cannot provide high-quality solutions. To better deal with MMOPs, considerations on the solution quality angle and the evolution stage angle are both taken into account in this paper and a multi-angle hierarchical differential evolution (MaHDE) algorithm is proposed. Firstly, a fitness hierarchical mutation (FHM) strategy is designed to balance the exploration and exploitation ability of different individuals. In the FHM strategy, the individuals are divided into two levels (i.e., low/high-level) according to their solution quality in the current niche. Then, the low/high-level individuals are applied to different guiding strategies. Secondly, a directed global search (DGS) strategy is introduced for the low-level individuals in the late evolution stage, which can improve the population diversity and provide these low-level individuals with the opportunity to re-search the global peaks. Thirdly, an elite local search (ELS) strategy is designed for the high-level individuals in the late evolution stage to refine their solution accuracy. Extensive experiments are developed to verify the performance of MaHDE on the widely used MMOPs test functions i.e., CEC’2013. Experimental results show that MaHDE generally outperforms the compared state-of-the-art multimodal algorithms.

In artificial intelligence, evolutionary algorithms (EAs) have shown to be effective and robust in solving difficult optimization problems. EAs are generic populationbased metaheuristic optimization algorithms. The mechanisms used in EAs are inspired by biological evolution: reproduction, mutation, recombination, and selection. The development of EAs can be classified into two categories: single objective and multi-objective optimization. In this thesis, both single objective and multi-objective evolutionary algorithms have been studied. For single objective optimization, various niching techniques are integrated with differential evolution (DE) and particle swarm optimization (PSO) for multi-modal optimization. Multi-modal optimization deals with optimization tasks that involve finding all or most of the global/local peaks in one single run. EAs in their original forms are usually designed for locating one single global solution. To promote and maintain formation of multiple stable subpopulations within a single population, we introduced a neighborhood mutation technique to enhance DE with ability of handling multi-modal problems. We also proposed a locally informed PSO to tackle multi-modal optimization. Beside these, several existing niching techniques from the literature were modified and improved by us.

Single-objective continuous optimization can be challenging, especially when dealing with multimodal problems. This work sheds light on the effects that multi-objective optimization may have in the single-objective space. For this purpose, we examine the inner mechanisms of the recently developed sophisticated local search procedure SOMOGSA. This method solves multimodal single-objective continuous optimization problems based on first expanding the problem with an additional objective (e.g., a sphere function) to the bi-objective domain and subsequently exploiting local structures of the resulting landscapes. Our study particularly focuses on the sensitivity of this multiobjectivization approach w.r.t. (1) the parametrization of the artificial second objective, as well as (2) the position of the initial starting points in the search space. As SOMOGSA is a modular framework for encapsulating local search, we integrate Nelder–Mead local search as optimizer in the respective module and compare the performance of the resulting hybrid local search to its original single-objective counterpart. We show that the SOMOGSA framework can significantly boost local search by multiobjectivization. Hence, combined with more sophisticated local search and metaheuristics, this may help solve highly multimodal optimization problems in the future.

Differential evolution based on strategy adaptation and deep reinforcement learning for multimodal optimization problems

Improved differential evolution algorithms for handling economic dispatch optimization with generator constraints

We compared two floating-point-encoded evolutionary algorithms (EA) – differential evolution (DE) and the generalized generation gap (G3) – using a set of problems with different characteristics. G3 is reported to offer superior performance with unimodal functions, which are, however, often solved more efficiently using derivative-based optimization for example and it is interesting to know, how these algorithms perform in multimodal global optimization problems. Our results suggest that G3 converges fast but is prone to converge prematurely rather than finding the global optimum in high-dimensional multimodal problems. DE, in contrast, appears to handle multimodal problems better but cannot match convergence speed of G3 in unimodal problems.

A revised definition about the task of multi-modal function optimization problem (called "rMOP"), which is to locate all optimal peaks including global and local optima, is presented. Then, a novel evolutionary algorithm aimed to rMOP with dynamic population (DPEA) is given. In DPEA, the initial population size is specified randomly. In the process of evolution, the size of population is tuned by a mechanism called "suppression" to delete crowded individuals and a process called "introduction of new individuals" to reinforce the global searching. Some experiments show that the population size of DPEA converges to the number of all peaks of the test function adaptively.

Differential evolution (DE) is undoubtedly one of the most powerful stochastic searching optimization algorithms. However, solving a specific problem using DE crucially depends on appropriately choosing of trial vector generation strategies and their associated control parameters. At the same time, multimodal optimization refers to locating not only one optimum but a set of optimal solutions. Niching is a useful technique to solve multi-modal optimization problems. Discovering multiple niches is the key capability of niching algorithms. In this paper, we propose a Strategy Adaptive Memetci Crowding DE (SAMCDE), which incorporate Crowding DE (CDE) with strategies and control parameter self-adaptation technique as well as fine search technique to handle multi-modal optimization problems. The algorithm is tested on 10 benchmark multi-modal functions and compared with the original CDE as well as several popular multimodal optimization algorithms in literature. As shown by the experimental results, the proposed algorithm is able to generate superior performance on the tested functions.

A flexible tolerance genetic algorithm for optimal problems with nonlinear equality constraints

Dynamic multimodal optimization problems (DMMOPs) represent the multimodal optimization problems that the optimal solution changes over time. Due to the wide application of DMMOPs in reality, some related algorithms have been proposed in recent years. Most existing algorithms employ a single dynamic response mechanism and embed it in existing multi-modal evolutionary algorithms. However, these algorithms often perform limited when environmental change involves multiple types, and they fail to consider utilizing historical information to assist static multimodal optimizers. To solve these issues, this paper proposes historical information-assisted dynamic response integration and adaptive niche methods (HIA-DRI-AN) for dynamic multi-modal optimization. In HIA-DRI-AN, an dynamic response integration method with adaptive adjustment mechanism is proposed for generating the initial population when the change happens. This method considers the change types of DMMOPs, and integrates targeted dynamic response mechanisms to respond to the different change types. Also, this method can adaptively self-adjust to balance the convergence and diversity of the initial population depending on the integrated response mechanism’s performance in historical environments. Furthermore, a niching adaptive division strategy is proposed to enhance the performance of the static optimizer. The strategy dynamically divides niches based on the integrated response mechanism’s performance and the current evolutionary stage, which can adjust the preference for diversity and convergence during evolution. The comprehensive experimental results on 24 test functions show that HIADRI-AN is superior compared to some state-of-the-art dynamic multimodal algorithms.

Multimodal multiobjective optimization problems, characterized by multiple solutions mapping to identical objective vectors, are ubiquitous in real-world applications. Despite their prevalence, most existing multimodal multiobjective evolutionary algorithms (MMOEAs) predominantly focus on identifying global Pareto sets, often overlooking the equally significant local Pareto sets. While some algorithms attempt to address local Pareto sets, their performance in the objective space remains suboptimal. The inherent challenge lies in the fact that a single strategy cannot effectively tackle problems with and without local Pareto fronts. This study proposes a novel approach that first detects the presence of local Pareto fronts using a neural network, thereby enabling adaptive adjustments to the algorithm’s selection strategy and search scope. Based on this detection mechanism, we design a microscale searching multimodal multiobjective evolutionary algorithm (MMOEA_MS). Through extensive experiments on twenty-two benchmark problems, MMOEA_MS demonstrates superior performance in identifying local Pareto fronts and outperforms existing algorithms in the objective space. This study highlights the effectiveness of MMOEA_MS in solving multimodal multiobjective optimization problems with diverse Pareto front characteristics, thereby addressing key limitations of current methodologies.

Some problems exist in classical optimization algorithms to solve multi-modal optimization problems and other complex systems. A Dynamic Niches-based Improved Hybrid Breeding Optimization (DNIHBO) algorithm is proposed to address the multi-modal optimization problem in the paper. By dynamically adjusting the niche scale, it effectively addresses the issue of niche parameter sensitivity. The structure of the algorithm includes three distinct groups: maintainer, restorer, and sterile lines for updating operations. However, the maintainer individuals often stagnate, leading to the risk of the local optima. To overcome this, neighborhood search and elite mutation strategies are incorporated, enhancing the balance between exploration and exploitation. To further improve individual utilization within niches, a niche restart strategy is introduced, ensuring sustained population diversity. The efficacy of DNIHBO is validated through simulations on 16 multi-modal test functions, followed by comparative analyses with various multi-modal optimization algorithms. The results convincingly demonstrate that DNIHBO not only effectively locates multiple global optima but also consistently outperforms other algorithms on test functions. These findings underscore the superiority of DNIHBO as a high-performing solution for multi-modal optimization.

Multimodal multiobjective optimization problems (MMOPs) have attracted wide attention in recent years. This kind of problem is very challenging since they need to locate different Pareto-optimal solution sets (PSs) that correspond to the same Pareto front. To resolve it, this article proposes a novel multimodal multiobjective differential evolution (DE) algorithm with species conservation, which develops a new way of locating different PSs. Specifically, the proposed algorithm adopts species conservation to determine different PSs in known areas, while it uses a variant of DE as the basic optimizer to explore new areas. There are three operators in species conservation: 1) species division; 2) seed determination; and 3) seed conservation. Species division mainly partitions the joint population of parents and children into various species in the decision space for retaining different PSs. Seed determination selects superior solutions from each species as seeds that need to be kept in the next generation. Seed conservation is to ensure that all species seeds are retained in the new generation by substituting no promising solutions with them, thereby guarantee not missing some known areas that may contain different PSs. Besides, the DE variant is utilized to produce diverse solutions to find new areas in the decision space where PSs may exist. The comparative experiments with ten state-of-the-art algorithms have been performed on the CEC 2019 MMOPs test set and two real-world problems. The experimental results have verified that the proposed algorithm has a competitive performance for MMOPs.

There are some natural similarities among fitness landscapes of different modals involved in a multimodal optimization problem (MMOP). However, existing multimodal evolutionary algorithms (MMEAs) tend to handle these modals separately, which limits their performance to a great extent. As the first attempt, this study proposes a knowledge transfer-based MMEA by mining and exploiting modal similarities. To this end, a center-aligned normalization strategy (CANS) is first designed to map all the subpopulations corresponding to identified modals into a unified 0-1 space, and then a distribution similarity-based transfer strategy (DSTS) is proposed to guide knowledge transfer among subpopulations. DSTS explicitly quantifies the similarity between two subpopulations and probabilistically transfers elite individuals in the subpopulation of the largest similarity to the target subpopulation. By integrating CANS and DSTS into a well-known MMEA, i.e., NEA2, the final knowledge transfer-based NEA named KTNEA is developed. Experimental results on CEC'2013 niching benchmark suite indicate that KTNEA performs competitively in comparison with state-of-the-art MMEAs.

Using swarm intelligence to obtain multiple optima in a single run simultaneously proved efficient for solving multimodal optimization problems (MOPs). However, the existing studies fail to resolve the contradiction between the required solution accuracy and the number of solutions. In this paper, an improved brain storm optimization (BSO) algorithm based on knowledge learning (KLBSO) is proposed as a solution to the problem. The properties of the improved algorithm and the domain knowledge of the problem are combined during the search process. Two factors need to be taken into account to solve a MOP: the accuracy and the diversity of the solution set. In the proposed algorithm, there are two learning approaches. Firstly, improving the learning method by replacing the perturbation operator of the random solution with the inter-solution learning of the worst solutions, improves the optimization ability of the algorithm. Secondly, by analyzing the MOPs, adding an archive set guarantees the solution’s diversity. To assess the efficiency of KLBSO, eight benchmark functions with various sizes and complexities were used. Comparing the results of KLBSO with those of state-of-the-art methods which are brain storm optimization algorithm (BSO), brain storm optimization algorithm in objective space (BSOOS), two kinds of pigeon-inspired optimization algorithms (PIO, PIOr), the comparison results show that the KLBSO is able to solve the contradiction between required solution accuracy and the number of solutions, and improves the outcomes where BSO is ranked first followed by the test algorithms.