Convergence Rate Of Evolutionary Algorithms Research Articles

Average convergence rate of evolutionary algorithms in continuous optimization

The average convergence rate (ACR) measures how fast the approximation error of an evolutionary algorithm converges to zero per generation. It is defined as the geometric average of the reduction rate of the approximation error over consecutive generations. This paper makes a theoretical analysis of the ACR in continuous optimization. The obtained results are summarized as follows. According to the limit property, the ACR is classified into two categories: (1) linear ACR whose limit inferior value is larger than a positive and (2) sublinear ACR whose value converges to zero. Then, it is proven that the ACR is linear for evolutionary programming using positive landscape-adaptive mutation, but sublinear for that using landscape-invariant or zero landscape-adaptive mutation. The relationship between the ACR and the decision space dimension is also classified into two categories: (1) polynomial ACR whose value is larger than the reciprocal of a polynomial function of the dimension for any generation, and (2) exponential ACR whose value is less than the reciprocal of an exponential function of the dimension for an exponential long period. It is proven that for easy problems such as linear functions, the ACR of the (1 + 1) adaptive random univariate search is polynomial. But for hard functions such as the deceptive function, the ACR of both the (1 + 1) adaptive random univariate search and evolutionary programming is exponential.

Use of Domain Knowledge to Increase the Convergence Rate of Evolutionary Algorithms for Optimizing the Cost and Resilience of Water Distribution Systems

AbstractEvolutionary algorithms (EAs) have been used extensively for the optimization of water distribution systems (WDSs) over the last two decades. However, computational efficiency can be a problem, especially when EAs are applied to complex problems that have multiple competing objectives. In order to address this issue, there has been a move toward developing EAs that identify near-optimal solutions within acceptable computational budgets, rather than necessarily identifying globally optimal solutions. This paper contributes to this work by developing and testing a method for identifying high-quality initial populations for multiobjective EAs (MOEAs) for WDS design problems aimed at minimizing cost and maximizing network resilience. This is achieved by considering the relationship between pipe size and distance to the source(s) of water, as well as the relationship between flow velocities and network resilience. The benefit of using the proposed approach compared with randomly generating initial popu...

Average Convergence Rate of Evolutionary Algorithms

In evolutionary optimization, it is important to understand how fast evolutionary algorithms converge to the optimum per generation, or their convergence rate. This paper proposes a new measure of the convergence rate, called average convergence rate. It is a normalised geometric mean of the reduction ratio of the fitness difference per generation. The calculation of the average convergence rate is very simple and it is applicable for most evolutionary algorithms on both continuous and discrete optimization. A theoretical study of the average convergence rate is conducted for discrete optimization. Lower bounds on the average convergence rate are derived. The limit of the average convergence rate is analysed and then the asymptotic average convergence rate is proposed.

Adaptive double chain quantum genetic algorithm for constrained optimization problems

Optimization problems are often highly constrained and evolutionary algorithms (EAs) are effective methods to tackle this kind of problems. To further improve search efficiency and convergence rate of EAs, this paper presents an adaptive double chain quantum genetic algorithm (ADCQGA) for solving constrained optimization problems. ADCQGA makes use of double-individuals to represent solutions that are classified as feasible and infeasible solutions. Fitness (or evaluation) functions are defined for both types of solutions. Based on the fitness function, three types of step evolution (SE) are defined and utilized for judging evolutionary individuals. An adaptive rotation is proposed and used to facilitate updating individuals in different solutions. To further improve the search capability and convergence rate, ADCQGA utilizes an adaptive evolution process (AEP), adaptive mutation and replacement techniques. ADCQGA was first tested on a widely used benchmark function to illustrate the relationship between initial parameter values and the convergence rate/search capability. Then the proposed ADCQGA is successfully applied to solve other twelve benchmark functions and five well-known constrained engineering design problems. Multi-aircraft cooperative target allocation problem is a typical constrained optimization problem and requires efficient methods to tackle. Finally, ADCQGA is successfully applied to solving the target allocation problem.

Optimal design and operation of pumping stations using NLP-GA

This paper addresses the optimal design and operation of an irrigation pumping station system using hybrid non-linear programming and a genetic algorithm (NLP-GA), and evaluates the algorithm in a practical problem. Results of the NLP-GA are compared with existing optimisation approaches to solve the same problem. The analytical approaches considered are the Lagrange multiplier method, a genetic algorithm and the honey-bee mating optimisation algorithm. The Lagrange multiplier method, genetic algorithm, honey-bee mating optimisation and the NLP-GA hybrid are used to simultaneously optimise the minimum annualised investment cost of the pumping station and its annual operating cost. The solution includes selection of pump type, capacity, number of units and scheduling of pump operation. The hybrid algorithm takes advantage of the high speed of NLP as well as the intelligent searching of evolutionary algorithms to overcome the shortcomings of individual NLP and genetic algorithm methods such as trapping of local optima, reporting only local or near-global optimal solutions and the low convergence rate of evolutionary algorithms in this type of problem. The results highlight the advantages in design, effective operation and ease of the NLP-GA method for solving complex problems of the type considered here. Although the NLP-GA converges rapidly, the results are promising and compare well with those of the Lagrange multiplier method, the genetic algorithm and honey-bee mating optimisation.

A mixed strategy of combining evolutionary algorithms with multigrid methods

Multigrid methods have been proven to be an efficient approach in accelerating the convergence rate of numerical algorithms for solving partial differential equations. This paper investigates whether multigrid methods are helpful to accelerate the convergence rate of evolutionary algorithms for solving global optimization problems. A novel multigrid evolutionary algorithm is proposed and its convergence is proven. The algorithm is tested on a set of 13 well-known benchmark functions. Experiment results demonstrate that multigrid methods can accelerate the convergence rate of evolutionary algorithms and improve their performance.