Abstract
Stochastic ranking (SR) is a popular constraint handling method for constrained evolutionary optimization. A constant probability parameter P f in the classic SR is applied to choose one of constraint violation and fitness as the basis for comparison of a pair of individuals. This paper presents two adaptive SR schemes, referred to as a linear decline SR and a reciprocal decline SR respectively, which are inspired by cooling schedules for simulated annealing. Both adaptive SRs aim to achieve different optimization goals at the various stages of constrained evolutionary optimization by adaptively adjusting the probability P f according to the current generation. Experimental results on 13 benchmarks problems show that both adaptive SRs are more competitive for local search at the terminal stage than the classic SR, and the reciprocal decline SR obtains the best performance in terms of solution accuracy and convergence speed.
Keywords
1 Introduction
Many optimization problems with various types of constraints could be modeled as constrained optimization problems (COPs) in a wide range of real-world applications [1]. Without loss of generality, COPs in the minimization sense can be described as follows:
where f(x) represents the objective to be minimized, and S denotes an n-dimensional decision space bounded by the following parametric constraints:
gj (x) and hj(x) are, separately, the j-th inequality constraint and the (j–l)-th equality constraint.
Unlike unconstraint problems, the degree of constraint violation becomes an additional evaluation criterion for any solution of constrained problems. It can be calculated as follows:
where the tolerance parameter δ is generally extremely small. The algorithms [2–4] or optimization mechanisms [5] should compare or select solutions based on their objective values and violation degrees of constraints [6] for constrained optimization. Consequently, the additional constraint handling techniques about how to deal with the constraints and evaluate individuals play an important role in constrained evolutionary optimization. Although Michalewicz divided the existing constraint handling techniques into five categories, there has been a great deal of interest in the research community in the primary methodologies of penalty function, stochastic ranking and multi-objective optimization methods in recent years. Penalty function methods are easy to implement but hard to tune some effective penalty factors. The multi-objective techniques are employed to solve COPs in diverse optimization algorithms [7, 8]. In multi-objective techniques, the concept of Pareto dominance is a crucial relation and the optimization goal is to find a uniformly spread of non-dominated solutions. The main difficulty of the multi-objective techniques is how to reduce the time of non-dominance checking. The stochastic ranking method with simple preference rules [9, 10] not only has fewer parameters than the penalty function methods but also spends less time checking for non-dominance than the multi-objective techniques.
The classic SR algorithm introduced a random probability to keep a balance between constraint violation and fitness during optimization. The ranking sequence of each population depends on the values of a random number. Apart from the case that both of two compared adjacent individuals are feasible, if the random value is less than or equal to the probability parameter Pf, the comparison between the adjacent individuals is decided by their values of objective function; otherwise, their degrees of constraint violation are utilized to determine the comparison result. Runarsson and Yao [10] proposed an improved SR evolutionary algorithm based on evolution strategy and differential variation operator. The parameter of Pf is usually set as a constant value in both optimization process of the classic SR and improved SR methods. However, there are several stages with different optimization goals in the whole evolutionary process. It is blind to apply an invariable biases search at different optimization stages. The invariable biases search may lead to the fact that the population converges slowly or even converges to infeasible regions at one certain specific stage of the evolutionary process.
In this paper, we analyze the features of biases search at different evolutionary stages. Further, two adaptive SR schemes, respectively referred to as a linear decline SR and a reciprocal decline SR, are presented to enhance the performance at variable evolutionary stages by borrowing the idea of cooling schedules of simulated annealing. This paper is organized as follows. After the introduction, Sect. 2 discusses characteristic of various optimization stages for solving COPs, analyses the roles of the probability parameter Pf during the whole evolutionary process, and presents our proposed methods. Section 3 gives the experimental results and analysis. Finally, Sect. 4 concludes the paper.
2 Theoretical Analysis and Proposed Approach
There are usually some difficulties about the search space when solving COPs. The searching space is constructed by a variety of feasible and infeasible regions. For some particular COPs, there are several disjoined feasible regions in the search space. Sometimes, these feasible zones may be small size, or the proportion of feasible regions in the search space is extremely low. Even the global feasible optima for these particular COPs maybe locate on the boundaries between feasible and infeasible regions. Therefore, some additional techniques is supposed to be adopted to keep diversity of population in the beginning of evolution or to improve the ability of local search at the late stage of evolution since a lot of local feasible optima for COPs make the search difficult. In order to satisfy the above optimization goals of two stages in some degree, the value of Pf in SR can be used to adjust the ratio between infeasible and feasible solutions and determine the comparison preference on constraint violation or fitness.
The classic SR algorithm is sensitive to the parameter value of Pf which is normally set in an interval [0, 1]. The most recommended value of Pf is set as a constant value of 0.45 in the classic SR. And it would not be changed in the whole evolutionary process. The comparison rules in SR is described as follows: two adjacent feasible individuals are surely compared based on their values of objective function; if there are less than two feasible individuals, the comparison preference between the adjacent two individuals depends on the value of random number. If the random value is less than or equal to Pf, the fitness is chosen as a basis for comparison of the adjacent individuals; otherwise, they are compared according to the violation degree. It means the parameter Pf in the above rules is used to control how much proportion of individuals in the population to be compared based on the objective function values, no matter they are feasible or infeasible.
When the value of Pf is set large, some solutions with the better values of objective function, which may be feasible or infeasible, have a larger chance to lie at the top of the ranking sequence. When Pf becomes low, the randomness degree of ranking would decrease and the top of the sequence would include more feasible solutions and a few of infeasible solutions with a small value of constraint violation. It is worth noting that, the selection of parent individuals which are used to generate new offspring candidates and the generation of the next population, are closely related to the individuals at the top of the ranking sequence. Consequently, it is not necessary to keep the value of Pf constant at various evolutionary stages with different optimization goals.
A relatively large value of Pf in the early evolution is beneficial to keep high degree of randomness and to maintain population diversity. It results in that many solutions with good values of objective function, regardless of feasible or infeasible individuals, would be ranked at the top of the ranking sequence and would be retained to the next generation with a high probability. The individuals at the top of the ranking sequence play a great role in improving the capability of exploration to search more extensive areas. As the SR algorithm evolves, the decrease of Pf could reduce the search randomness and could more strictly compare the individuals according to their degrees of constraint violation. It leads to that many individuals with the smaller values of constraint violation would be ranked in the top order. This means that infeasible individuals with small degrees of constraint violation and feasible individuals would be sorted at the top of the ranking sequence and would be allowed to survive to the next generation with a high probability. In other words, the individuals at the top are focusing on searching feasible regions and a part of infeasible regions which is close to the feasible regions. It is beneficial to search the global optima located on the feasible boundaries and to escape from the local optima. As a result, the main goal of improving the ability of local search could be achieved.
According to the above analysis, the parameter Pf is expected to be set as a relatively high initial value to make searching more stochastic and to explore as wide regions as possible at the beginning of optimization. As the SR algorithm evolves, Pf should be gradually decreased. With a small value of Pf, more individuals are ranked in order of constraint violation value with a high probability. It helps the algorithm enhance the performance of local search. In simulated annealing, the temperature cools slowly so that the system finally converges to a state of minimum energy steadily. The standard cooling schedule is normally described as Tt = T0*αt where T0 is the initial temperature of the system, the value of cooling parameter α is usually set between 0 and 1, and t denotes the cooling time for the system. As the cooling time goes on, the temperature exhibits an exponential decline. Inspired by cooling schedules of simulated annealing, two schemes to decrease the value of parameter Pf adaptively in the SR algorithm are designed in this paper. The first scheme reduces the value of Pf at a linear rate, in which Pf changes dynamically in accordance with the ratio between the current generation and the maximum generation maxGen in the following form:
Taking into consideration the initial population at the current generation t = 0, parameter Pf is equal to the initial value Pf 0. When an modified SR algorithm with the adaptive linear decline parameter, referred to as SR-LD, reaches one half of the maximum generation, the value of Pf would decline to one half of the initial value Pf 0. The value would become much smaller as the current generation increases to the maximum generation. The details of the adaptive SR scheme in one generation of the algorithm SR-LD are presented in the pseudocode of Algorithm 1.
The experimental results on 13 benchmark problems show that the classic SR algorithm find lots of feasible solutions, respectively, within less than 100 generations for problems g01, g02, g04, g06, g08, g09, g10, and g12, and after more than 300 generations for problems g03, g05, g11 and g13. This is because the feasible regions are extremely small and the estimated ratios between the feasible regions and the whole search space roughly equal to 0 for problems g03, g05, g11 and g13. It suggests that the optimization should be concentrated on improving the ability of local search when many feasible solutions had been found during the evolutionary process. If the declining speed is too fast, the likelihood of converging in local optima is very large and there exists a high risk of not taking full advantage of valuable infeasible solutions. As the value of Pf decreases quickly, infeasible solutions are sorted at the end with a high probability and survive to the next generation of population with an extremely small probability.
The second scheme is aiming to adjust the balance between convergence speed and ability of local search. As shown in the experimental results of the classic SR algorithm, the appropriate declining speed of Pf is supposed to be slower and smoother after several hundred generations than that in the linear decline SR. Therefore, the second scheme is designed with a reciprocal decline rate. The adaptive decline value of Pf in the modified SR algorithm with a reciprocal decline parameter, referred as SR-RD, is calculated as follows:
The only difference between SR-LD and SR-RD is that the value of Pf in SR-RD is updated in the form of reciprocal decline.
3 Experimental Results
13 benchmark problems [10] were used to evaluate the performance of two proposed adaptive SR algorithms and the classic SR algorithm. The properties of these test instances were reported in [8, 10]. All experiments for SR, SR-LD and SR-RD were carried out in the same experimental conditions. The population size is set as λ = 200 and the parent population size µ=30 in these algorithms. The probability parameter Pf in SR is set as a constant value of 0.45, which is also the value of Pf 0 in SR-LD and SR-RD. The tolerance parameter δ = 0.0001, and the other parameters in these algorithms are set as the same as those in [10]. In our experiments, every algorithm performed 30 times independently for each among 13 instances. The statistical results of the best objective value, median, mean, standard deviation are recorded, respectively, in Table 1 for g01-g06 and Table 2 for g07-g13. The “optimal” means the objective value of the best known solution.
For problems g01, g03, g04, g05, g06, g08, g11 and g12, the three algorithms always find the optimal solutions in each run. For problem g09, the optimal solutions are found by all algorithms, but the SR-LD and SR-RD are performed better than the classic SR. The results achieved by SR-RD are consistently with the higher precision. The estimated feasible region proportion of problem g02 is more than 90%, which means most of the search regions are feasible [8, 10]. SR-LD and SR-RD obtain extremely approximate optimal results and they discover the better “best” result than SR. While SR finds the better “worst” result than SR-LD, SR-RD outperforms SR in terms of the criteria of “best”, “median”, “mean” and “worst”. Most of the search regions are infeasible and the estimated feasible region ratio is very small for problem g07, g10 and g13. All the three algorithms find extremely approximate optimal results for the above three problems. SR-LD performs better in terms of the “mean” results as well as the “best” results and worse than “worst” results than SR for problems g07 and g10. SR-RD achieves the best performance in terms of all criteria among three algorithms. Especially, SR-RD can consistently find the optimal solution in all 30 runs while SR-LD and SR have the similar performance and sometimes get trapped into a local optimal solution of 0.438851219909 for problem g13.
Figure 1 plots the convergence curves of average objective errors in logarithmic scalar in 30 independent runs of each algorithm for problems g02, g07, g09 and g10, respectively. These problems are more difficult to be solved than the others. Most of the search regions for problem g02 are feasible and one half of the search regions for problem g09 is also feasible. Conversely, the estimated feasible region ratio in the whole search space for problem g07 and g10 nearly equal to 0 which means most of their search regions are infeasible.
As shown in Fig. 1(a) and (b), the convergence curves of SR and SR-LD are nearly the same for problem g02 and g07. After about 200 generations, the two algorithms converge to an approximate optimal solution and fail to improve the solution towards higher accuracy. However, SR-RD continues to converge towards the optimal solution. It obtains the solutions with a much higher accuracy after about 200 generations than SR and SR-LD. It is evident from Fig. 1(c) that the performance of SR-RD is the best among the three algorithms for problem g09 while SR-LD performs better than the classic SR. For problem g10, the convergence speeds of the three algorithms are similar during the first 200 generations while the SR-LD and SR-RD perform better than the SR in terms of convergence speed as well as solution accuracy at the terminal stage of evolution, as shown in Fig. 1(d). In other words, two adaptive SR algorithms could explore much more extensive areas at the beginning stage of evolution and could enhance local search capacity to improve solution quality at the terminal stage.
4 Conclusion
On the basis of the classic SR algorithm, this paper proposes an adaptive linear decline SR and an adaptive reciprocal decline SR to enhance the performance at various evolutionary stages by decreasing the value of parameter Pf. They are able to maintain population diversity at the early stage of evolution and improve local search capacity and solution quality at the later stage. Experimental results on 13 benchmark problems show that SR-LD and SR-RD are capable of converging to highly accurate solutions and jumping out of local optima by adaptively reducing the value of Pf when solving COPs. Meanwhile, SR-RD is the most competitive among all the three algorithms. Further research focuses on extending the adaptive SR schemes to solve constraint multi-objective optimization problems.
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Acknowledgments
This work was supported in part by the National Natural Science Foundation of China (Nos. 61203310 and 61503087), the Natural Science Foundation of Guangdong Province, China (No. 2015A030313204), the China Scholarship Council (CSC) (Nos. 201406155076 and 201408440193), the Pearl River Science & Technology Nova Program of Guangzhou (No. 2014J2200052), and the Fundamental Research Funds for the Central Universities, SCUT (No. 2013ZZ0048).
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Wu, Y., Ying, W., Wu, B., Peng, D. (2017). Adaptive Stochastic Ranking Schemes for Constrained Evolutionary Optimization. In: Xhafa, F., Patnaik, S., Yu, Z. (eds) Recent Developments in Intelligent Systems and Interactive Applications. IISA 2016. Advances in Intelligent Systems and Computing, vol 541. Springer, Cham. https://doi.org/10.1007/978-3-319-49568-2_14
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