Handling constraints using multiobjective optimization concepts

In this paper, we propose a new constraint‐handling technique for evolutionary algorithms which we call inverted‐shrinkable PAES (IS‐PAES). This approach combines the use of multiobjective optimization concepts with a mechanism that focuses the search effort onto specific areas of the feasible region by shrinking the constrained search space. IS‐PAES also uses an adaptive grid to store the solutions found, but has a more efficient memory‐management scheme than its ancestor (the Pareto archived evolution strategy for multiobjective optimization). The proposed approach is validated using several examples taken from the standard evolutionary and engineering optimization literature. Comparisons are provided with respect to the stochastic ranking method (one of the most competitive constraint‐handling approaches used with evolutionary algorithms currently available) and with respect to other four multiobjective‐based constraint‐handling techniques. Copyright© 2004 John Wiley & Sons, Ltd.     

A constrained optimization algorithm based on the simplex search method

In this article, a robust method is presented for handling constraints with the Nelder and Mead simplex search method, which is a direct search algorithm for multidimensional unconstrained optimization. The proposed method is free from the limitations of previous attempts that demand the initial simplex to be feasible or a projection of infeasible points to the nonlinear constraint boundaries. The method is tested on several benchmark problems and the results are compared with various evolutionary algorithms available in the literature. The proposed method is found to be competitive with respect to the existing algorithms in terms of effectiveness and efficiency.     

Constrained multi-objective optimization algorithm with an ensemble of constraint handling methods

Different constraint handling techniques have been used with multi-objective evolutionary algorithms (MOEA) to solve constrained multi-objective optimization problems. It is impossible for a single constraint handling technique to outperform all other constraint handling techniques always on every problem irrespective of the exhaustiveness of the parameter tuning. To overcome this selection problem, an ensemble of constraint handling methods (ECHM) is used to tackle constrained multi-objective optimization problems. The ECHM is integrated with a multi-objective differential evolution (MODE) algorithm. The performance is compared between the ECHM and the same single constraint handling methods using the same MODE (using codes available from http://www3.ntu.edu.sg/home/EPNSugan/index.htm). The results show that ECHM overall outperforms the single constraint handling methods.     

基于空间收缩的并行演化算法

提出了一种基于空间收缩的求解MINLP问题的新算法。算法应用了快速有效的不完全演化搜索较优解的分布信息，通过分布信息定位最优解的可能分布，再由精英个体信息决定下次搜索空间。仿真结果表明该算法在搜索效率、应用范围、解的精确性和鲁棒性上都优于其他现存演化算法。     

Cultural algorithms,an alternative heuristic to solve the job shop scheduling problem

In this work, an approach for solving the job shop scheduling problem using a cultural algorithm is proposed. Cultural algorithms are evolutionary computation methods that extract domain knowledge during the evolutionary process. Additional to this extracted knowledge, the proposed approach also uses domain knowledge given a priori (based on specific domain knowledge available for the job shop scheduling problem). The proposed approach is compared with respect to a Greedy Randomized Adaptive Search Procedure (GRASP), a Parallel GRASP, a Genetic Algorithm, a Hybrid Genetic Algorithm, and a deterministic method called shifting bottleneck. The cultural algorithm proposed in this article is able to produce competitive results with respect to the two approaches previously indicated at a significantly lower computational cost than at least one of them and without using any sort of parallel processing.     

MULTIOBJECTIVE DESIGN OPTIMIZATION BY AN EVOLUTIONARY ALGORITHM

This paper presents an evolutionary algorithm for generic multiobjective design optimization problems. The algorithm is based on nondominance of solutions in the objective and constraint space and uses effective mating strategies to improve solutions that are weak in either. Since the methodology is based on nondominance, scaling and aggregation affecting conventional penalty function methods for constraint handling does not arise. The algorithm incorporates intelligent partner selection for cooperative mating. The diversification strategy is based on niching which results in a wide spread of solutions in the parametric space. Results of the algorithm for the design examples clearly illustrate the efficiency of the algorithm in solving multidisciplinary design optimization problems.     

Two efficient algorithms for constructing almost even approximations of the Pareto front in multi-objective optimization problems

In this article, two algorithms are proposed for constructing almost even approximations of the Pareto front of multi-objective optimization problems. The first algorithm is a hybrid of the ε-constraint and Pascoletti–Serafini scalarization methods for solving bi-objective problems. The second is a modification of the successive Pareto optimization (SPO) algorithm for solving three-objective problems. In these algorithms, the MATLAB fmincon solver is used to solve single-objective optimization problems, which returns a local optimal solution. Some metrics are considered to evaluate the quality of approximations obtained by the suggested algorithms on six test problems, and their results are compared with other algorithms (normal constraint, weighted constraint, SPO, differential evolution, multi-objective evolutionary algorithm/decomposition–differential evolution, non-dominated sorting genetic algorithm-II and S-metric selection evolutionary multi-objective algorithm). Experimental results show that the proposed algorithms provide almost even approximations of the whole Pareto front, and better quality of approximation and CPU time compared with established algorithms.     

A generic framework for handling constraints with agent-based optimization algorithms and application to aerodynamic design

A generic constraint handling framework for use with any swarm-based optimization algorithm is presented. For swarm optimizers to solve constrained optimization problems effectively modifications have to be made to the optimizers to handle the constraints, however, these constraint handling frameworks are often not universally applicable to all swarm algorithms. A constraint handling framework is therefore presented in this paper that is compatible with any swarm optimizer, such that a user can wrap it around a chosen swarm algorithm and perform constrained optimization. The method, called separation-sub-swarm, works by dividing the population based on the feasibility of individual agents. This allows all feasible agents to move by existing swarm optimizer algorithms, hence promoting good performance and convergence characteristics of individual swarm algorithms. The framework is tested on a suite of analytical test function and a number of engineering benchmark problems, and compared to other generic constraint handling frameworks using four different swarm optimizers; particle swarm, gravitational search, a hybrid algorithm and differential evolution. It is shown that the new framework produces superior results compared to the established frameworks for all four swarm algorithms tested. Finally, the framework is applied to an aerodynamic shape optimization design problem where a shock-free solution is obtained.     

THE ROLE OF EVOLUTIONARY AND ADAPTIVE SEARCH DURING WHOLE SYSTEM,CONSTRAINED AND DETAILED DESIGN OPTIMIZATION

The paper introduces various strategies which incorporate evolutionary and adaptive search techniques. These strategies incorporate genetic algorithms (GA) and ant colony models combined within co-operating frameworks that provide a capability for decision support and optimization during whole system design and constraint satisfaction/ constrained optimization during the engineering design process. The objective during whole system design is to determine an optimum initial configuration for large engineering systems. Strategies for the efficient integration of evolutionary techniques with detailed design are also introduced. Each of these areas presents specific problems to the evolutionary/adaptive search processes and the overall objective here is to identify the main areas of difficulty and provide solutions that will lead to successful integration. The paper illustrates the flexibility and utility of the various techniques when applied across the various stages of the design process, i.e. from providing decision support during the high-risk stages of preliminary design to the identification of definitive optimal solutions during the more deterministic stages of detailed design.     

Hessian approximation algorithms for hybrid optimization methods

This article introduces Hessian approximation algorithms to estimate the search direction of the quasi-Newton methods for solving optimization problems of continuous parameters. The proposed algorithms are quite different from other well-known quasi-Newton methods, such as symmetric rank-one, Davidon–Fletcher–Powell, and Broyden–Fletcher–Goldfarb–Shanno, in that the Hessian matrix is not calculated from the gradient information, rather directly from the function values. The proposed algorithms are designed for a class of hybrid algorithms that combine evolutionary search with the gradient-based methods of quasi-Newton type. The function values calculated for the evolutionary search are used for estimation of the Hessian matrix (or its inverse) as well as the gradient vector. Since the estimation process of the Hessian matrix is independent of that of the gradient vector, more reliable Hessian estimation with a small population is possible compared with the previous methods based upon the classical quasi-Newton methods. Numerical experiments show that the proposed algorithms are very competitive with state-of-the-art evolutionary algorithms for continuous optimization problems.     

A multi-objective metamodel-assisted memetic algorithm with strength-based local refinement

Metamodel-assisted evolutionary algorithms are low-cost optimization methods for CPU-demanding problems. Memetic algorithms combine global and local search methods, aiming at improving the quality of promising solutions. This article proposes a metamodel-assisted memetic algorithm which combines and extends the capabilities of the aforementioned techniques. Herein, metamodels undertake a dual role: they perform a low-cost pre-evaluation of population members during the global search and the gradient-based refinement of promising solutions. This reduces significantly the number of calls to the evaluation tool and overcomes the need for computing the objective function gradients. In multi-objective problems, the selection of individuals for refinement is based on domination and distance criteria. During refinement, a scalar strength function is maximized and this proves to be beneficial in constrained optimization. The proposed metamodel-assisted memetic algorithm employs principles of Lamarckian learning and is demonstrated on mathematical and engineering applications.     

Feasibility-Guided Constraint-Handling Techniques for Engineering Optimization Problems

The particle swarm optimization (PSO) algorithm is an established nature-inspired population-based meta-heuristic that replicates the synchronizing movements of birds and fish. PSO is essentially an unconstrained algorithm and requires constraint handling techniques (CHTs) to solve constrained optimization problems (COPs). For this purpose, we integrate two CHTs, the superiority of feasibility (SF) and the violation constraint-handling (VCH), with a PSO. These CHTs distinguish feasible solutions from infeasible ones. Moreover, in SF, the selection of infeasible solutions is based on their degree of constraint violations, whereas in VCH, the number of constraint violations by an infeasible solution is of more importance. Therefore, a PSO is adapted for constrained optimization, yielding two constrained variants, denoted SF-PSO and VCH-PSO. Both SF-PSO and VCH-PSO are evaluated with respect to five engineering problems: the Himmelblau’s nonlinear optimization, the welded beam design, the spring design, the pressure vessel design, and the three-bar truss design. The simulation results show that both algorithms are consistent in terms of their solutions to these problems, including their different available versions. Comparison of the SF-PSO and the VCH-PSO with other existing algorithms on the tested problems shows that the proposed algorithms have lower computational cost in terms of the number of function evaluations used. We also report our disagreement with some unjust comparisons made by other researchers regarding the tested problems and their different variants.     

Real-parameter quantum evolutionary algorithm for economic load dispatch

A novel real-parameter optimisation algorithm called the 'real-parameter quantum evolutionary algorithm' is presented. The algorithm pieces together the ideas from evolutionary algorithms (EA) and quantum computing to provide a robust optimisation technique that can be utilised to optimise highly constrained non-linear real-parameter functions. Quantum bits have immense representational power due to their being in superposition of all the basic states at the same time. New quantum operators designed in this work enable the search to effectively handle the twin objectives of exploitation and exploration. This enables the search to be pursued with small population sizes, thereby speeding up the search process and also ensuring that there is no problem of premature convergence that often plagues pure EA implementations. The power of the proposed algorithm is demonstrated by solving the economic load dispatch (ELD) in power systems. ELD is to find the optimal loadings on the generators so as to achieve minimum operating cost while satisfying various system and unit-level constraints. The proposed method has been applied to standard load dispatch problems reported in the literature including the IEEE 30 bus system, IEEE 57 bus system and a 110-generator problem, and its performance has been compared with the results obtained by other methods. The results adequately demonstrate the enhanced search power of the proposed algorithm in terms of obtaining better solutions and provide motivation for its application to other real-parameter optimisation problems in power systems.     

A genetic algorithm for solving the economic lot scheduling problem in flow shops

In this study, we propose a hybrid genetic algorithm (HGA) to solve the economic lot scheduling problem in flow shops. The proposed HGA utilizes a so-called Proc PLM heuristic that tests feasibility for the candidate solutions obtained in the evolutionary process of genetic algorithm. When a candidate solution is infeasible, we propose to use a binary search heuristic to ‘fix’ the candidate solution so as to obtain a feasible solution with the minimal objective value. To evaluate the performance of the proposed HGA, we randomly generate a total of 2100 instances from seven levels of utilization rate ranged from 0.45 to 0.80. We solve each of those 2100 instances by the proposed HGA and the other solution approaches in the literature. Our experiments show that the proposed HGA outperforms traditional methods for solving the economic lot scheduling problem in flow shops.     

Mathematical modelling of container transfers and storage locations at seaport terminals

This paper models the seaport system with the objective of determining the optimal storage strategy and container-handling schedule. It presents an iterative search algorithm that integrates a container-transfer model with a container-location model in a cyclic fashion to determine both optimal locations and corresponding handling schedule. A genetic algorithm (GA), a tabu search (TS) and a tabu search/genetic algorithm hybrid are used to solve the problem. The implementation of these models and algorithms are capable of handling the very large problems that arise in container terminal operations. Different resource levels are analysed and a comparison with current practise at an Australian port is done.     

Derived heuristics-based consistent optimization of material flow in a gold processing plant

Material flow in a chemical processing plant often follows complicated control laws and involves plant capacity constraints. Importantly, the process involves discrete scenarios which when modelled in a programming format involves if–then–else statements. Therefore, a formulation of an optimization problem of such processes becomes complicated with nonlinear and non-differentiable objective and constraint functions. In handling such problems using classical point-based approaches, users often have to resort to modifications and indirect ways of representing the problem to suit the restrictions associated with classical methods. In a particular gold processing plant optimization problem, these facts are demonstrated by showing results from MATLAB®'s well-known fmincon routine. Thereafter, a customized evolutionary optimization procedure which is capable of handling all complexities offered by the problem is developed. Although the evolutionary approach produced results with comparatively less variance over multiple runs, the performance has been enhanced by introducing derived heuristics associated with the problem. In this article, the development and usage of derived heuristics in a practical problem are presented and their importance in a quick convergence of the overall algorithm is demonstrated.     

An adaptive penalty scheme for genetic algorithms in structural optimization

A parameter‐less adaptive penalty scheme for genetic algorithms applied to constrained optimization problems is proposed. Using feedback from the evolutionary process the procedure automatically defines a penalty parameter for each constraint. The user is thus relieved from the burden of having to determine sensitive parameter(s) when dealing with every new constrained optimization problem. The procedure is shown to be effective and robust when applied to test problems from the evolutionary computation literature as well as several optimization problems from the structural engineering literature. Copyright © 2003 John Wiley & Sons, Ltd.     

Visual data association for real‐time video tracking using genetic and estimation of distribution algorithms

In this article, an efficient and novel approach for video data association is developed. This new method is formulated as a search across the hypotheses space defined by the possible association among tracks and detections, carried out for each frame of a video sequence. The full data association problem in visual tracking is formulated as a combinatorial hypotheses search with a heuristic evaluation function taking into account structural and specific information such as distance, shape, color, etc. To guarantee real‐time performance, a time limit is set for the search process explore alternative solutions. This time limit defines the upper bound of the number of evaluations depending on search algorithm efficiency. Estimation distribution algorithms are proposed as an efficient evolutionary computation technique to search in this hypothesis space. Finally, an exhaustive comparison of the performance of alternative algorithms is carried out considering complex representative situations in real video sets. © 2009 Wiley Periodicals, Inc. Int J Imaging Syst Technol, 19, 208–220, 2009     

An artificial bee colony algorithm for the economic lot scheduling problem

In this study, we present an artificial bee colony (ABC) algorithm for the economic lot scheduling problem modelled through the extended basic period (EBP) approach. We allow both power-of-two (PoT) and non-power-of-two multipliers in the solution representation. We develop mutation strategies to generate neighbouring food sources for the ABC algorithm and these strategies are also used to develop two different variable neighbourhood search algorithms to further enhance the solution quality. Our algorithm maintains both feasible and infeasible solutions in the population through the use of some sophisticated constraint handling methods. Experimental results show that the proposed algorithm succeeds to find the all the best-known EBP solutions for the high utilisation 10-item benchmark problems and improves the best known solutions for two of the six low utilisation 10-item benchmark problems. In addition, we develop a new problem instance with 50 items and run it at different utilisation levels ranging from 50 to 99% to see the effectiveness of the proposed algorithm on large instances. We show that the proposed ABC algorithm with mixed solution representation outperforms the ABC that is restricted only to PoT multipliers at almost all utilisation levels of the large instance.     

A new multi-objective particle swarm optimizer using empirical movement and diversified search strategies

Most real-world optimization problems involve the optimization task of more than a single objective function and, therefore, require a great amount of computational effort as the solution procedure is designed to anchor multiple compromised optimal solutions. Abundant multi-objective evolutionary algorithms (MOEAs) for multi-objective optimization have appeared in the literature over the past two decades. In this article, a new proposal by means of particle swarm optimization is addressed for solving multi-objective optimization problems. The proposed algorithm is constructed based on the concept of Pareto dominance, taking both the diversified search and empirical movement strategies into account. The proposed particle swarm MOEA with these two strategies is thus dubbed the empirical-movement diversified-search multi-objective particle swarm optimizer (EMDS-MOPSO). Its performance is assessed in terms of a suite of standard benchmark functions taken from the literature and compared to other four state-of-the-art MOEAs. The computational results demonstrate that the proposed algorithm shows great promise in solving multi-objective optimization problems.