Unimodal Problems Research Articles

Multi-modal Battle Royale optimizer

Multimodal optimization poses a challenging problem in the field of optimization as it entails the discovery of multiple local and global optima, unlike unimodal optimization, which seeks a single global solution. In recent years, the significance of addressing multimodal optimization challenges has grown due to the real-world complexity of many problems. While numerous optimization methods are available for unimodal problems, multimodal optimization techniques have garnered increased attention. However, these approaches often grapple with a common issue: the determination of the niching parameter, necessitating prior knowledge of the problem space. This paper introduces a novel multimodal optimization approach that circumvents the need for prior problem space knowledge and avoids the challenge of predefining the niching parameter. Building upon the Battle Royal Optimization (BRO) algorithm, this extended version formulates a multimodal solution by utilizing Coulomb's law to identify suitable neighbors. The incorporation of Coulomb's law serves the dual purpose of identifying potential local and global optima based on fitness values and establishing optimal distances from solution candidates. A comparison study was done between the MBRO and seven well-known multimodal optimization algorithms using 14 benchmark problems from the CEC 2013 and CEC 2015 competitions to see how well it worked. The experimental results underscore MBRO's proficiency in successfully identifying most, if not all, local and global optima, positioning it as a superior solution when compared to its competitors.

Extension of Stochastic Point Location for Multimodal Problems.

Stochastic point location (SPL) involves a learning mechanism (LM) determining an optimal point on the line when the only inputs LM receives are stochastic information about the direction in which LM should move. The complexity of SPL comes from the stochastic responses of the environment, which may lead LM completely astray. SPL is a fundamental problem in optimization and was studied by many researchers during the last two decades, including improvement of its solution and all-pervasive applications. However, all existing SPL studies assume that the whole search space contains only one optimal point. Since a multimodal optimization problem (MMOP) contains multiple optimal solutions, it is significant to develop SPL's multimodal version. This article extends it from a unimodal problem to a multimodal one and proposes a parallel partition search (PPS) solution to address this issue. The heart of the proposed solution involves extracting the feature of the historical sampling information to distinguish the subintervals that contain the optimal points or not. Specifically, it divides the whole search space into multiple subintervals and samples them parallelly, then utilizes the feature of the historical sampling information to adjust the subintervals adaptively and to find the subintervals containing the optimal points. Finally, the optimal points are located within these subintervals according to their respective sampling statistics. The proof of the ε-optimal property for the proposed solution is presented. The numerical testing results demonstrate the power of the scheme.

Redefining teaching-and-learning-process in TLBO and its application in cloud

Teaching-Learning-Based Optimization (TLBO) algorithm was developed to solve single-objective optimization problems. TLBO is inspired by the theory of teaching-learning mechanism. Basic TLBO works better for unimodal problems but poorly for multi-model problems because of its poor exploration. To provide a fair exploration for solving complex optimization problems, we redefined the learning strategy to the basic TLBO. This newly redefined variant is called Multi-Teacher Teaching-Learning Based Optimization (MT-TLBO). The performance of MT-TLBO was tested on the latest optimization benchmark functions, CEC-C06, 2019. It is observed that the performance of MT-TLBO is superb comparatively. After that, it is simulated to solve workflow scheduling problems by minimizing the execution cost of the standard workflow and maximizing the workload on computing resources in a cloud environment. Finally, it is found that MT-TLBO produces minimal execution cost and a fair workload distribution on resources on standard benchmark workflow, comparatively.

Energy efficiency task scheduling for battery level‐aware mobile edge computing in heterogeneous networks

This paper focuses on a mobile edge-computing-enabled heterogeneous network. A battery level-aware task-scheduling framework is proposed to improve the energy efficiency and prolong the operating hours of battery-powered mobile devices. The formulated optimization problem is a typical mixed-integer nonlinear programming problem. To solve this nondeterministic polynomial (NP)-hard problem, a decomposition-based task-scheduling algorithm is proposed. Using an alternating optimization technology, the original problem is divided into three subproblems. In the outer loop, task offloading decisions are yielded using a pruning search algorithm for the task offloading subproblem. In the inner loop, closed-form solutions for computational resource allocation subproblems are derived using the Lagrangian multiplier method. Then, it is proven that the transmitted power-allocation subproblem is a unimodal problem; this subproblem is solved using a gradient-based bisection search algorithm. The simulation results demonstrate that the proposed framework achieves better energy efficiency than other frameworks. Additionally, the impact of the battery level-aware scheme on the operating hours of battery-powered mobile devices is also investigated.

Differential Evolution Algorithm with Hierarchical Fair Competition Model

This paper presents the study of differential evolution algorithm with hierarchical fair competition model (HFC-DE). HFC model is based on the fair competition of societal system found in natural world. In this model, the population is split into hierarchy and the competition is allowed between the hierarchical members. During evolution, the population members are allowed to move within the hierarchy levels. The standard differential evolution algorithm is used for population evolution. Experimentation has carried out to define the parameter for proposed model on test suit having unimodal problems and multi-model problems. After analyzing the results, the two variants of HFC-DE are proposed, named hierarchical fair competition model in differential evolution algorithm with replacement, I,e, HFCDE-R and hierarchical fair competition model in differential evolution algorithm without replacement, I,e, HFCDE-wR. The problem-solving capabilities of both algorithms are checked and the results are compared with differential evolution algorithm (DE) and other variants of DE. The early results are encouraging and motivating.

A new method for global optimization

This paper presents a new method for global optimization. We use exact quadratic regularization for the transformation of the multimodal problems to a problem of a maximum norm vector on a convex set. Quadratic regularization often allows you to convert a multimodal problem into a unimodal problem. For this, we use the shift of the feasible region along the bisector of the positive orthant. We use only local search (primal-dual interior point method) and a dichotomy method for search of a global extremum in the multimodal problems. The comparative numerical experiments have shown that this method is very efficient and promising.

Finite element method coupled with TLBO for shape control optimization of piezoelectric bimorph in COMSOL Multiphysics

Finite element methods (FEMs) are more advantageous for analyzing complex geometry and structures than analytical methods. Local search optimization techniques are suitable for the unimodal problem because final result depends on the starting point. On the other hand, to optimize the parameters of multi-minima/maxima problems, global optimization-based FEM is used. Unfortunately, global optimization solvers are not present in, COMSOL Multiphysics, a versatile tool for solving varieties of problems using FEM. Teaching–learning-based optimization (TLBO) is a global optimization technique and does not require any algorithm-specific parameter. In this paper, FEM is coupled with TLBO algorithms in COMSOL Multiphysics for solving the global optimization problem. The TLBO algorithm is implemented in COMSOL Multiphysics using the JAVA application programming interface and tested with the standard benchmark functions. The solutions of the standard benchmark problem in COMSOL Multiphysics are in close agreement with the results presented in literature. Furthermore, the optimization procedure thus established is used for the optimization of actuator voltage for piezoelectric bimorphs to achieve the desired shapes. The FEM-based TLBO method is compared with two optimization methods present in COMSOL Multiphysics for a shape control problem; (i) method of moving asymptotes (MMA) and (ii) Bound Optimization BY Quadratic Approximation (BOBYQA). The root mean square error shows that the FEM-based TLBO algorithm converges to a global minimum and gives the same result (19.3 nm) at multiple runs, whereas MMA and BOBYQA trapped in local minimum and gave different results for different starting points.

A variant of teaching‐learning‐based optimization and its application for minimizing the cost of Workflow Execution in the Cloud Computing

AbstractTeaching learning‐based optimization (TLBO) was proposed by Rao to solve optimization problems. It is based on the theory of teaching‐learning mechanism. Although it performs well in unimodal problems yet its performance is not good in multimodal problems. To further improve this algorithm's performance and make it suitable for both unimodal problems and multimodal problems, we made some major changes in the theory and the algorithm's operators. The proposed algorithm is able to capture diverse optimal solutions in less number of iterations and is very good for solving multimodal problems. This newly created variant of TLBO is named generalized TLBO (GTLBO). The performance of GTLBO is tested on CEC−06, 2019 benchmark functions and other 15 classical benchmark functions, and it is found that the proposed algorithm is performing better comparatively. Then it is simulated for solving the workflow scheduling problem in CloudSim. Standard scientific workflow applications as Montage, Epigenomics, Sipht, and a sample workflow are used as dataset to test algorithms' performance in cloud environments. Our proposed approach, GTLBO, provides the proper distribution of workloads and offers minimal execution‐cost for the workflow applications. Results reflect the supremacy of the proposed algorithm GTLBO comparatively.

Delay Robustness of PID Control of Second-Order Systems: Pseudoconcavity, Exact Delay Margin, and Performance Tradeoff

In this paper we study delay robustness of PID controllers in stabilizing systems containing uncertain delays. We consider second-order systems and seek analytical characterization and exact computation of the PID delay margin, where by PID delay margin we mean the maximal range of delay values within which the system can be robustly stabilized by a PID controller. Our primary contribution is threefold. First, we show that under a nonrestrictive assumption, the delay margin achieved by PID control coincides with that by PD controllers. Second, we show that the PID delay margin can be computed efficiently by solving a pseudo-concave unimodal problem, that is, a univariate optimization problem that admits a unique maximum and hence is a convex optimization problem in one variable. As such, from a computational standpoint, the PID delay margin problem can be considered resolved. Finally, we demonstrate analytically the trade-off between achieving delay margin and tracking performance, showing that for several canonical performance criteria, integral control reduces the delay margin. These results not only ensure that the PID delay margin problem be readily solvable, b

A memetic animal migration optimizer for multimodal optimization

Unimodal optimization algorithms can find only one global optimum solution, while multimodal ones have the ability to detect all/most existing local/global optima in the problem space. Many practical scientific and engineering optimization problems have multiple optima to be located. There are a considerable number of optimization approaches in the literature to address the unimodal problems. Although multimodal optimization methods have not been studied as much as the unimodal ones, they have attracted an enormous amount of attention recently. However, most of them suffer from a common niching parameter problem. The main difficulty faced by existing approaches is determining the proper niching radius. Determining the appropriate radius of the niche requires prior knowledge of the problem space. This paper proposes a novel multimodal optimization scheme that does not face the dilemma of having prior knowledge of the problem space as it does not require the niching parameter to be determined in advance. This scheme is the extended version of the unimodal animal migration optimization (AMO) algorithm that has the capability of taking advantage of finding multiple solutions. Like other multimodal optimization approaches, the proposed MAMO requires specific modifications to make it possible to locate multiple optima. The local neighborhood policy is modified to adapt the multimodal search by utilizing Coulomb's law. Also, Coulomb's law is also applied to decide the movement direction of the individuals. Hence, instead of moving an individual toward the two randomly chosen individuals, it moves toward the near and good enough two neighborhoods. Additionally, a further local search step is performed to improve the exploitation. To investigate the performance of the MAMO, the comparisons are conducted with five existing multi-modal optimization algorithms on nine benchmarks of the CEC 2013 competition. The experimental results reveal that the MAMO performs success in locating all or most of the local/global optima and outperforms other compared methods. Note that the source codes of the proposed MAMO algorithm are publicly available at www.taymaz.dev .

Проверка численной эффективности методов глобальной оптимизации

In this paper, new difficult test problems are proposed to test the numerical efficiency of global optimization methods. These are problems of unconstrained optimization with unknown solutions. The proposed test problems are inseparable and have arbitrary dimensions. The author also proposes to include the test functions by J. Nie in the list of test functions for numerical verification of the effectiveness of methods. These functions are also inseparable functions of arbitrary dimensions with unknown solutions. The proposed test problems have many local extrema. Testing the effectiveness of global optimization methods for such functions is simplified. If the method allows improving the found solutions to test problems, then it will be more effective. The existing global optimization methods are compared with the exact quadratic regularization method developed by the author. This method is compared with known software packages that implement modern methods of global optimization. These packages include several methods. The best of them use convex relaxation of the problem to obtain estimates of solutions with subsequent use of local optimization programs. But even such powerful packages have difficulties in solving the considered test problems. Some test problems, for example, with the Rana or Egg Holder function, have been solved by different methods for over 20 years. During this time, no method has allowed obtaining results that are obtained by the method of exact quadratic regularization. For almost all complex test problems with unknown solutions, this method yielded better solutions. Sometimes the advantage of this method was significant, as is the case with the Rana test function. The essence of the exact quadratic regularization method is to transform any global optimization problem to maximize the square of the Euclidean norm of a vector on a convex set. This problem is computationally much simpler. Often, with such a transformation, the multimodal problem becomes unimodal, which is easy to solve. Keywords: test problems, global optimization, unimodal problems, multimodal problems, numerical methods.

Optimal correction of the absolute value equations

In this paper, we study the optimum correction of the absolute value equations through making minimal changes in the coefficient matrix and the right hand side vector and using spectral norm. This problem can be formulated as a non-differentiable, non-convex and unconstrained fractional quadratic programming problem. The regularized least squares is applied for stabilizing the solution of the fractional problem. The regularized problem is reduced to a unimodal single variable minimization problem and to solve it a bisection algorithm is proposed. The main difficulty of the algorithm is a complicated constraint optimization problem, for which two novel methods are suggested. We also present optimality conditions and bounds for the norm of the optimal solutions. Numerical experiments are given to demonstrate the effectiveness of suggested methods.

Population size in Particle Swarm Optimization

Abstract Particle Swarm Optimization (PSO) is among the most universally applied population-based metaheuristic optimization algorithms. PSO has been successfully used in various scientific fields, ranging from humanities, engineering, chemistry, medicine, to advanced physics. Since its introduction in 1995, the method has been widely investigated, which led to the development of hundreds of PSO versions and numerous theoretical and empirical findings on their convergence and parameterization. However, so far there is no detailed study on the proper choice of PSO swarm size, although it is widely known that population size crucially affects the performance of metaheuristics. In most applications, authors follow the initial suggestion from 1995 and restrict the population size to 20–50 particles. In this study, we relate the performance of eight PSO variants to swarm sizes that range from 3 up to 1000 particles. Tests are performed on sixty 10- to 100-dimensional scalable benchmarks and twenty-two 1- to 216-dimensional real-world problems. Although results do differ for the specific PSO variants, for the majority of considered PSO algorithms the best performance is obtained with swarms composed of 70–500 particles, indicating that the classical choice is often too small. Larger swarms frequently improve efficiency of the method for more difficult problems and practical applications. For unimodal problems slightly lower swarm sizes are recommended for the majority of PSO variants, but some would still perform best with hundreds of particles.

Adversarial Uni- and Multi-modal Stream Networks for Multimodal Image Registration.

Deformable image registration between Computed Tomography (CT) images and Magnetic Resonance (MR) imaging is essential for many image-guided therapies. In this paper, we propose a novel translation-based unsupervised deformable image registration method. Distinct from other translation-based methods that attempt to convert the multimodal problem (e.g., CT-to-MR) into a unimodal problem (e.g., MR-to-MR) via image-to-image translation, our method leverages the deformation fields estimated from both: (i) the translated MR image and (ii) the original CT image in a dual-stream fashion, and automatically learns how to fuse them to achieve better registration performance. The multimodal registration network can be effectively trained by computationally efficient similarity metrics without any ground-truth deformation. Our method has been evaluated on two clinical datasets and demonstrates promising results compared to state-of-the-art traditional and learning-based methods.

Оптимизация в конечномерном евклидовом пространстве

In this paper, optimization models in Euclidean space are divided into four complexity classes. Ef-fective algorithms have been developed to solve the problems of the first two classes of complexity. These are the primal-dual interior-point methods. Discrete and combinatorial optimization problems of the third complexity class are recommended to be converted to the fourth complexity class with continuous change of variables. Effective algorithms have not been developed for problems of the third and fourth complexity classes, with the exception of a narrow class of problems that are unimodal. The general optimization problem is formulated as a minimum (maximum) objective function in the presence of constraints. The complexity of the problem depends on the structure of the objective function and its feasible region. If the functions that determine the optimization model are quadratic or polynomial, then semidefinite programming can be used to obtain estimates of so-lutions in such problems. Effective methods have been developed for semidefinite optimization problems. Sometimes it’s enough to develop an algorithm without building a mathematical model. We see such an example when sorting an array of numbers. Effective algorithms have been devel-oped to solve this problem. In the work for sorting problems, an optimization model is constructed, and it coincides with the model of the assignment problem. It follows from this that the sorting problem is unimodal. Effective algorithms have not been developed to solve multimodal problems. The paper proposes a simple and effective algorithm for the optimal allocation of resources in mul-tiprocessor systems. This problem is multimodal. In the general case, for solving multimodal prob-lems, a method of exact quadratic regularization is proposed. This method has proven its compara-tive effectiveness in solving many test problems of various dimensions. Keywords: Euclidean space, optimization, unimodal problems, multimodal problems, complexity classes, numerical methods.

A Novel Bat Algorithm based on Cross Boundary Learning and Uniform Explosion Strategy

Population-based algorithms have been used in many real-world problems. Bat algorithm (BA) is one of the states of the art of these approaches. Because of the super bat, on the one hand, BA can converge quickly; on the other hand, it is easy to fall into local optimum. Therefore, for typical BA algorithms, the ability of exploration and exploitation is not strong enough and it is hard to find a precise result. In this paper, we propose a novel bat algorithm based on cross boundary learning (CBL) and uniform explosion strategy (UES), namely BABLUE in short, to avoid the above contradiction and achieve both fast convergence and high quality. Different from previous opposition-based learning, the proposed CBL can expand the search area of population and then maintain the ability of global exploration in the process of fast convergence. In order to enhance the ability of local exploitation of the proposed algorithm, we propose UES, which can achieve almost the same search precise as that of firework explosion algorithm but consume less computation resource. BABLUE is tested with numerous experiments on unimodal, multimodal, one-dimensional, high-dimensional and discrete problems, and then compared with other typical intelligent optimization algorithms. The results show that the proposed algorithm outperforms other algorithms.

A reinforcement learning-based communication topology in particle swarm optimization

Recently, a multitude of researchers have considered the fully connected topology (Gbest) as a default communication topology in particle swarm optimization (PSO). Despite many earlier studies of this issue indicating that the Gbest might favor unimodal problems, the topology with fewer connections, e.g., Lbest, might perform better on multimodal problems. It seems that different topologies make PSO a problem-related algorithm, while in this paper a problem-free PSO which integrates a reinforcement learning method has been proposed, referred to as QLPSO. In the new proposed algorithm, each particle acts as an agent independently, selecting the optimal topology under the control of Q-learning (QL) during each iteration. Two variants of QLPSO consider the different dimensions of the communication topology, respectively. In order to investigate the performance of QLPSO, experiments on 28 CEC 2013 benchmark functions are carried out when comparing with static and dynamic topologies. The reported computational results show that the proposed QLPSO is more superior compared with several state-of-the-art methods.

Multi-modal forest optimization algorithm

Multi-modal optimization algorithms are one of the most challenging issues in the field of optimization. Most real-world problems have more than one solution; therefore, the potential role of multi-modal optimization algorithms is rather significant. Multi-modal problems consider several global and local optima. Therefore, during the search process, most of the points should be detected by the algorithm. The forest optimization algorithm has been recently introduced as a new evolutionary algorithm with the capability of solving unimodal problems. This paper presents the multi-modal forest optimization algorithm (MMFOA), which is constructed by applying a clustering technique, based on niching methods, to the unimodal forest optimization algorithm. The MMFOA operates by dividing the population of the forest into subpopulations to locate existing local and global optima. Subpopulations are generated by the Basic Sequential Algorithmic Scheme with a radius neighborhood. As population size is self-adaptive in MMFOA, population size can be increased in functions with too many local and global optima. The proposed algorithm is evaluated by a set of multi-modal benchmark functions. The experiment results show that not only is the population size low, but also that the convergence speed is high, and that the algorithm is efficient in solving multi-modal problems.

KGSA: A Gravitational Search Algorithm for Multimodal Optimization based on K-Means Niching Technique and a Novel Elitism Strategy

AbstractGravitational Search Algorithm (GSA) is a metaheuristic for solving unimodal problems. In this paper, a K-means based GSA (KGSA) for multimodal optimization is proposed. This algorithm incorporates K-means and a new elitism strategy called “loop in loop” into the GSA. First in KGSA, the members of the initial population are clustered by K-means. Afterwards, new population is created and divided in different niches (or clusters) to expand the search space. The “loop in loop” technique guides the members of each niche to the optimum direction according to their clusters. This means that lighter members move faster towards the optimum direction of each cluster than the heavier members. For evaluations, KGSA is benchmarked on well-known functions and is compared with some of the state-of-the-art algorithms. Experiments show that KGSA provides better results than the other algorithms in finding local and global optima of constrained and unconstrained multimodal functions.

Multi-Leader Comprehensive Learning Particle Swarm Optimization with Adaptive Mutation for Economic Load Dispatch Problems

Particle swarm optimization (PSO) is one of the most popular, nature inspired optimization algorithms. The canonical PSO is easy to implement and converges fast, however, it suffers from premature convergence. The comprehensive learning particle swarm optimization (CLPSO) can achieve high exploration while it converges relatively slowly on unimodal problems. To enhance the exploitation of CLPSO without significantly impairing its exploration, a multi-leader (ML) strategy is combined with CLPSO. In ML strategy, a group of top ranked particles act as the leaders to guide the motion of the whole swarm. Each particle is randomly assigned with an individual leader and the leader is refreshed dynamically during the optimization process. To activate the stagnated particles, an adaptive mutation (AM) strategy is introduced. Combining the ML and the AM strategies with CLPSO simultaneously, the resultant algorithm is referred to as multi-leader comprehensive learning particle swarm optimization with adaptive mutation (ML-CLPSO-AM). To evaluate the performance of ML-CLPSO-AM, the CEC2017 test suite was employed. The test results indicate that ML-CLPSO-AM performs better than ten popular PSO variants and six other types of representative evolutionary algorithms and meta-heuristics. To validate the effectiveness of ML-CLPSO-AM in real-life applications, ML-CLPSO-AM was applied to economic load dispatch (ELD) problems.