Constrained Global Optimization Problems Research Articles

Integrated Local Search Technique With Reptile Search Algorithm for Solving Large‐Scale Bound Constrained Global Optimization Problems

Integrated Local Search Technique With Reptile Search Algorithm for Solving Large‐Scale Bound Constrained Global Optimization Problems

A new filled function method for solving constrained global optimization problems

ABSTRACT Filled function methods have been considered as effective algorithms for solving global optimization problems. However, their effectiveness is greatly affected by the selection of parameters, the noncontinuous or non-differentiable properties of the constructed filled function. In addition, many of the constructed filled functions are only for unconstrained optimization problems, and they are unable to solve constrained optimization problems. In this paper, a new filled function is constructed for solving constrained global optimization problems. The new filled function has only one parameter which needs to be adjusted, and, when the objective functions and constrained functions are all continuously differentiable functions, the constructed filled function is also a continuously differentiable function. Then, the classical local optimization methods can be used to find a better minimum of the proposed filled function and a few parameter adjustments are needed. At last, a new filled function algorithm for constrained global optimization is developed based on the proposed filled function. The new algorithm is applied to several test examples. The results of the numerical experiments show that the new filled function algorithm is effective and efficient.

A Gaussian random walk salp swarm algorithm for optimal dynamic charging of electric vehicles

Salp swarm algorithm (SSA) is one of the recently developed meta-heuristic optimization algorithms. Since SSA outperforms other swarm-based algorithms, it has recently been employed in various applications, including feature selection, neural network training and renewable energy systems. In this paper, an improved salp swarm algorithm based on a Gaussian random walk is proposed, which enhances the algorithm’s performance particularly for multidimensional constrained global optimization problems. The integration of a Gaussian random walk into the algorithm balances between its exploration and exploitation capabilities. Furthermore, the proposed algorithm introduces a new re-dispersion strategy in the case of stagnation at local optimum points, which considerably enhances exploration. The performance of the proposed algorithm is evaluated using a set of twenty-three benchmark test functions and is compared to the performance of prevalent metaheuristic algorithms. Statistical analysis is performed using Wilcoxon signed-rank test, and the results reveal considerable improvement over the competing algorithms. Then, 21 real-world optimization problems are used to further evaluate the efficacy of the proposed algorithm. The winners of the CEC2020 Competition on Real-World Single Objective Constrained Optimization, SASS, sCMAgES, EnMODE, and COLSHADE algorithms, are used as four comparable algorithms in the real-world optimization problems. The convergence curves and simulations provide very competitive performance compared to the comparative algorithms. The proposed algorithm is used to address one of the most challenging real-world constrained problems in power system applications, namely, determining the optimal charging schedule for electric vehicles at charging stations. The results reveal that the proposed algorithm outperforms other existing algorithms in terms of increasing the charging revenues and achieving maximum power grid stability.

AOBLMOA: A Hybrid Biomimetic Optimization Algorithm for Numerical Optimization and Engineering Design Problems.

The Mayfly Optimization Algorithm (MOA), as a new biomimetic metaheuristic algorithm with superior algorithm framework and optimization methods, plays a remarkable role in solving optimization problems. However, there are still shortcomings of convergence speed and local optimization in this algorithm. This paper proposes a metaheuristic algorithm for continuous and constrained global optimization problems, which combines the MOA, the Aquila Optimizer (AO), and the opposition-based learning (OBL) strategy, called AOBLMOA, to overcome the shortcomings of the MOA. The proposed algorithm first fuses the high soar with vertical stoop method and the low flight with slow descent attack method in the AO into the position movement process of the male mayfly population in the MOA. Then, it incorporates the contour flight with short glide attack and the walk and grab prey methods in the AO into the positional movement of female mayfly populations in the MOA. Finally, it replaces the gene mutation behavior of offspring mayfly populations in the MOA with the OBL strategy. To verify the optimization ability of the new algorithm, we conduct three sets of experiments. In the first experiment, we apply AOBLMOA to 19 benchmark functions to test whether it is the optimal strategy among multiple combined strategies. In the second experiment, we test AOBLMOA by using 30 CEC2017 numerical optimization problems and compare it with state-of-the-art metaheuristic algorithms. In the third experiment, 10 CEC2020 real-world constrained optimization problems are used to demonstrate the applicability of AOBLMOA to engineering design problems. The experimental results show that the proposed AOBLMOA is effective and superior and is feasible in numerical optimization problems and engineering design problems.

Guided Hybrid Modified Simulated Annealing Algorithm for Solving Constrained Global Optimization Problems

In this paper, a hybrid gradient simulated annealing algorithm is guided to solve the constrained optimization problem. In trying to solve constrained optimization problems using deterministic, stochastic optimization methods or hybridization between them, penalty function methods are the most popular approach due to their simplicity and ease of implementation. There are many approaches to handling the existence of the constraints in the constrained problem. The simulated-annealing algorithm (SA) is one of the most successful meta-heuristic strategies. On the other hand, the gradient method is the most inexpensive method among the deterministic methods. In previous literature, the hybrid gradient simulated annealing algorithm (GLMSA) has demonstrated efficiency and effectiveness to solve unconstrained optimization problems. In this paper, therefore, the GLMSA algorithm is generalized to solve the constrained optimization problems. Hence, a new approach penalty function is proposed to handle the existence of the constraints. The proposed approach penalty function is used to guide the hybrid gradient simulated annealing algorithm (GLMSA) to obtain a new algorithm (GHMSA) that finds the constrained optimization problem. The performance of the proposed algorithm is tested on several benchmark optimization test problems and some well-known engineering design problems with varying dimensions. Comprehensive comparisons against other methods in the literature are also presented. The results indicate that the proposed method is promising and competitive. The comparison results between the GHMSA and the other four state-Meta-heuristic algorithms indicate that the proposed GHMSA algorithm is competitive with, and in some cases superior to, other existing algorithms in terms of the quality, efficiency, convergence rate, and robustness of the final result.

Memory based hybrid crow search algorithm for solving numerical and constrained global optimization problems

Memory based hybrid crow search algorithm for solving numerical and constrained global optimization problems

Parametric study on mass minimization of radial network arch pedestrian bridges

This article deals with the optimization of radial network arch bridges for pedestrian traffic. This is a challenging optimization problem that involves determining certain topological parameters, as well as shape configurations and all sizing parameters of structural members, in order to seek minimum mass. An optimal bridge scheme was sought tuning a large set of design parameters of diverse character: the type of hanger arrangement, the number of hangers, their inclination and spread angles, the alteration of these angles, the arch rise, etc. Mathematically, the optimization of the bridge scheme is a mixed-integer constrained global optimization problem solved employing a stochastic evolutionary algorithm. Plane light-deck bridges of 60 m span were optimized using the proposed optimization technique. Decisive design parameters and their promising ranges were revealed. In addition, the influence of some simplified effects was shown: using constant spread and inclination angles, limiting the diameters of the hangers to a given value, etc. The results show that the available design recommendations for automotive and railway network arch bridges are not fully applicable to similarly designed pedestrian bridges.

A Modified Bat Algorithm for Solving Large-Scale Bound Constrained Global Optimization Problems

In the last two decades, the field of global optimization has become very active, and, in this regard, many deterministic and stochastic algorithms were developed for solving various optimization problems. Among them, swarm intelligence (SI) is a stochastic algorithm that is more flexible and robust and has had the ability to find an optimum solution for high-dimensional optimization and search problems. SI-based algorithms are mainly inspired by the social behavior of fish schooling or bird flocking. Among the SI-based algorithms, Bat algorithm (BA) is one of the recently developed evolutionary algorithms. It employs an echolocation behavior of microbats by varying pulse rates of emission and loudness to perform their search process. In this paper, a modified Bat algorithm (MBA) is developed. The main focus of the MBA is to further enhance the exploration and exploitation search abilities of the original Bat algorithm. The performance of the modified Bat algorithm (MBA) is examined over the benchmark functions designed for evolutionary algorithms competition in the special session of 2005 IEEE Congress on Evolutionary Computation. The used benchmark functions include the unimodal, multimodal, and hybrid benchmark functions with high dimensionality. Furthermore, the impact analysis with respect to different values of temperatures is conducted by executing the proposed algorithm twenty-five times independently by using each benchmark function with different random seeds.

On the Performance of Differential Evolution Variants in Constrained Structural Optimization

Constrained optimization is a highly important field of engineering as most real-world optimization problems are associated with one or several constraints. Such problems are often challenging to solve due to their complexity and high nonlinearity. Differential evolution (DE) is arguably one of the most versatile and stable population-based search algorithms that exhibits robustness to multi-modal problems and has shown to be very efficient when solving constrained global optimization problems. In this paper we investigate the performance of several DE variants existing in the literature such as the traditional DE, the composite DE (CoDE), the adaptive DE with optional external archive (JADE) and the self-adaptive DE (jDE and SaDE), for handling constrained structural optimization problems. The performance of each DE variant is quantified by using three well-known benchmark structures in 2D and 3D. It is shown that JADE, which updates control parameters in an adaptive way, truly exhibits superior performance and outperforms the other DE variants in all the cases examined.

Generating $\alpha $-dense curves in non-convex sets to solve a class of non-smooth constrained global optimization

This paper deals with the dimensionality reduction approach to study multi-dimensional constrained global optimization problems where the objective function is non-differentiable over a general compact set $D$ of $\mathbb{R}^{n}$ and H\{o}lderian. The fundamental principle is to provide explicitly a parametric representation $x_{i}=\ell _{i}(t),1\leq i\leq n$ of $\alpha $-dense curve $\ell_{\alpha }$ in the compact $D$, for $t$ in an interval $\mathbb{I}$ of $\mathbb{R}$, which allows to convert the initial problem to a one dimensional H\{o}lder unconstrained one. Thus, we can solve the problem by using an efficient algorithm available in the case of functions depending on a single variable. A relation between the parameter $\alpha $ of the curve $\ell _{\alpha }$ and the accuracy of attaining the optimal solution is given. Some concrete $\alpha $ dense curves in a non-convex feasible region $D$ are constructed. The numerical results show that the proposed approach is efficient.

Locating critical failure surface using meta-heuristic approaches: a comparative assessment

Slope stability analysis is performed to access the slope failures and to design safe and optimal slopes with regard to safety and reliability and for saving economic loss. The prime objective of this study is to find endangered areas and observe their safety factors and failure mechanism under which any structured pattern of slope tends to fail soon. An important aspect is to achieve this by locating the critical failure surface associated with a minimum factor of safety. Limit equilibrium analysis is one of the most established procedures which computes the factor of safety under all kinematic and geometric constraints. Locating the critical failure surface is very cumbersome and becomes a difficult constrained global optimization problem due to the presence of discontinuous function and multiple local minima points. The classical techniques are also unable to converge to a valid solution due to high computation complexity. In the proposed work, an analysis has been performed using meta-heuristic approaches due to their fast and elegant efficiency. A comparative study has been carried out to ensure the applicability and reliability of the methodology to evaluate the minimum factor of safety. The effectiveness of the proposed methods is validated by solving three different benchmark problems from the literature, where the implementation results confirm the higher stability analysis and acquire a more efficient and acceptable result over relevant existing methods.

A new auxiliary function approach for inequality constrained global optimization problems

In this study, we deal with the nonlinear constrained global optimization problems. First, we introduce a new smooth exact penalty function for constrained optimization problems. We combine the exact penalty function with the auxiliary function in regard to constrained global optimization. We present a new auxiliary function approach and the adapted algorithm for solving non-linear inequality constrained global optimization problems. Finally, we illustrate the efficiency of the algorithm on some numerical examples.

Penalty functions and two-step selection procedure based DIRECT-type algorithm for constrained global optimization

Applied optimization problems often include constraints. Although the well-known derivative-free global-search DIRECT algorithm performs well solving box-constrained global optimization problems, it does not naturally address constraints. In this article, we develop a new algorithm DIRECT-GLce for general constrained global optimization problems incorporating two-step selection procedure and penalty function approach in our recent DIRECT-GL algorithm. The proposed algorithm effectively explores hyper-rectangles with infeasible centers which are close to boundaries of feasibility and may cover feasible regions. An extensive experimental investigation revealed the potential of the proposed approach compared with other existing DIRECT-type algorithms for constrained global optimization problems, including important engineering problems.

GPGPU based Multi-hive ABC Algorithm for Constrained Global Optimization Problems

INTRODUCTION: The artificial bee colony (ABC) algorithm is a nature-inspired technique used for solving different optimization problems. This paper presents a multi-hive ABC algorithm for solving constrained benchmark functions of CEC2006. The CEC2006 data set contains the global benchmark functions with different design variables, number and type of constraints. OBJECTIVES: The objective of the proposed work is to design and apply the GPGPU based multi-hive ABC algorithm to solve constrained optimization problems. METHODS: The proposed approach is a multi-population coarse-grained system implemented using General Purpose Graphics Processing Unit (GPGPU). The performance of the proposed approach is compared with the serial ABC algorithm for eleven benchmark functions and results in the literature. The multi-hive ABC algorithm has multiple hives, each running separate ABC algorithm on different cores of GPGPU. RESULTS: The proposed approach provides global best solutions in significantly reduced time for all benchmark functions. The speed-up obtained is approximately 7X to 9X. The GPGPU device utilization is approximately 57% to 91%. CONCLUSION: The GPGPU based multi-hive ABC algorithm is a found good with respect to best results, speed up factor and GPU utilization to solve constrained optimization problems.

Master-Slave TLBO Algorithm for Constrained Global Optimization Problems

Master-Slave TLBO Algorithm for Constrained Global Optimization Problems

A BBO-based algorithm for slope stability analysis by locating critical failure surface

Determination of the critical failure surface is performed in stability evaluation process for road cut slope, embankments, dam, excavations, retaining walls and many more. Finding the critical failure surface in a rock or soil slope is very cumbersome and becomes a difficult constrained global optimization problem. Due to existence of discontinuous function and strong multiple local minima points, researchers are facing difficulties to employ trial-and-error methods in a large search space. Thus, classical optimization techniques fail to converge to a valid solution. In this study a stochastic method called biogeography-based optimization algorithm was adopted for analyzing the factor of safety. Based on the finding from the implementation and quantitative evaluation, it was found that the proposed method for locating critical failure surface in homogeneous soil slope acquires more efficient results over other implemented methods such as grid search and genetic algorithm. The validation and effectiveness of the proposed method are investigated by solving two benchmark case studies from the literature, while the simulation design for slip surfaces is carried out using ‘Rocscience slide’ software tool for comparing the results.

Parameter Study on Weight Minimization of Network Arch Bridges

The article concerns optimization of network arch bridges. This is challenging optimization problem involving even for conventional scheme of network arch bridge the identification of some topological parameters as well as shape configurations and all sizing parameters of structural members, seeking the minimum weight. Optimal bridge scheme is sought tuning a large set of design parameters of diverse character: the type of hanger arrangement, the number of hangers, their inclination angles and placement distances, the arch shape and rise, etc. Mathematically, the optimization of the bridge scheme is a mixed-integer constrained global optimization problem solved employing stochastic evolutionary algorithm. Plane heavy/moderate/and light-deck bridges of 18, 30, 42 and 54 m spans were optimized using proposed optimization technique. The decisive design parameters and their promising ranges were revealed. Also, the influence of some simplifications is shown: changing the arch shape from elliptical to circular, placing the hangers at equal distances, etc.

On a smoothed penalty-based algorithm for global optimization

This paper presents a coercive smoothed penalty framework for nonsmooth and nonconvex constrained global optimization problems. The properties of the smoothed penalty function are derived. Convergence to an $$\varepsilon $$ -global minimizer is proved. At each iteration k, the framework requires the $$\varepsilon ^{(k)}$$ -global minimizer of a subproblem, where $$\varepsilon ^{(k)} \rightarrow \varepsilon $$ . We show that the subproblem may be solved by well-known stochastic metaheuristics, as well as by the artificial fish swarm (AFS) algorithm. In the limit, the AFS algorithm convergence to an $$\varepsilon ^{(k)}$$ -global minimum of the real-valued smoothed penalty function is guaranteed with probability one, using the limiting behavior of Markov chains. In this context, we show that the transition probability of the Markov chain produced by the AFS algorithm, when generating a population where the best fitness is in the $$\varepsilon ^{(k)}$$ -neighborhood of the global minimum, is one when this property holds in the current population, and is strictly bounded from zero when the property does not hold. Preliminary numerical experiments show that the presented penalty algorithm based on the coercive smoothed penalty gives very competitive results when compared with other penalty-based methods.

Theoretical and Practical Convergence of a Self-Adaptive Penalty Algorithm for Constrained Global Optimization

This paper proposes a self-adaptive penalty function and presents a penalty-based algorithm for solving nonsmooth and nonconvex constrained optimization problems. We prove that the general constrained optimization problem is equivalent to a bound constrained problem in the sense that they have the same global solutions. The global minimizer of the penalty function subject to a set of bound constraints may be obtained by a population-based meta-heuristic. Further, a hybrid self-adaptive penalty firefly algorithm, with a local intensification search, is designed, and its convergence analysis is established. The numerical experiments and a comparison with other penalty-based approaches show the effectiveness of the new self-adaptive penalty algorithm in solving constrained global optimization problems.

A shifted hyperbolic augmented Lagrangian-based artificial fish two-swarm algorithm with guaranteed convergence for constrained global optimization

This article presents a shifted hyperbolic penalty function and proposes an augmented Lagrangian-based algorithm for non-convex constrained global optimization problems. Convergence to an -global minimizer is proved. At each iteration k, the algorithm requires the -global minimization of a bound constrained optimization subproblem, where . The subproblems are solved by a stochastic population-based metaheuristic that relies on the artificial fish swarm paradigm and a two-swarm strategy. To enhance the speed of convergence, the algorithm invokes the Nelder–Mead local search with a dynamically defined probability. Numerical experiments with benchmark functions and engineering design problems are presented. The results show that the proposed shifted hyperbolic augmented Lagrangian compares favorably with other deterministic and stochastic penalty-based methods.