Construction Example for Algebra System Using Harmony Search Algorithm

The construction example of algebra system is to verify the existence of a complex algebra system, and it is a NP-hard problem. In this paper, to solve this kind of problems, firstly, a mathematical optimization model for construction example of algebra system is established. Secondly, an improved harmony search algorithm based on NGHS algorithm (INGHS) is proposed to find as more solutions as possible for the optimization model; in the proposed INGHS algorithm, to achieve the balance between exploration power and exploitation power in the search process, a global best strategy and parameters dynamic adjustment method are present. Finally, nine construction examples of algebra system are used to evaluate the optimizationmodel and performance of INGHS.The experimental results show that the proposed algorithmhas strong performance for solving complex construction example problems of algebra system.


Introduction
Algebra system is one of the broad parts of mathematics, together with number theory, geometry, and analysis.It has wide applications in astronomy, biology, construction, computer science, and so on.Many scientific researchers focus on its studies.However, with the rapid development of science and technology, more and more complex algebra systems emerge in large numbers, so the algebra system researchers meet a lot of difficult problems.One of them is to construct appropriate examples for the algebra system so as to prove the existence of the algebra system.This is because there is the tremendous computing workload to construct appropriate examples that are satisfied with the operations of the algebra system.In the current study of algebra system, the construction examples of algebra system are often manually achieved.However, for a multielement complex algebra system, the construction examples are very difficult.In order to understand the construction example of algebra system better, we introduce its construction by an example of N(2, 2, 0) algebra system.Above all, we begin this section by introducing a simplified definition of N(2, 2, 0) algebra system [1,2].Definition 1.An N(2, 2, 0) algebra is a system (, * , Δ, 0) where  is nonempty set, 0 is a constant element of , and * and Δ are two binary operations on , obeying the following axioms.For all , ,  ∈ , (F 1 )  * (Δ) =  * ( * ), (F 2 ) (Δ) *  =  * ( * ), (F 3 ) 0 *  = .
For Definition 1, there are two works for researchers to do.(1) Give some examples to prove the existence of this algebra.
Next we consider For this algebra system, researchers hope to construct some examples that satisfy the conditions (F 1 ), (F 2 ), and (F 3 ) and to prove the existence of the algebra.However, the calculated amount of this work is so large that computing cost is typically high.Like Example 2, each variable   or   ( = 1, 2, . . ., ;  = 1, 2, . . ., ) can take  different values ( 1 ,  2 , . . .,   ); the combinatorial number with the repetition for all   and   is  2 2  .Obviously it is a NP-hard problem.It will result in the combination blast when the dimension increases.So the enumeration method can hardly be used to construct an example for the algebra system (, * , Δ, 0).

The Optimization Model of 𝑁(2, 2, 0)
Algebra System For the construction example of an (2, 2, 0) algebra system, we can consider it as an optimized constraint satisfaction problem (OCSP) that satisfies conditions (F 1 ), (F 2 ), and (F 3 ).Then it can be solved by the intelligent optimization algorithm.
To predigest the solving process and promote its practicability, we turn this problem into nonconstrained optimization problem.The optimization model is as follows: where X = ( The functions  1 ,  2 , and  3 are to transform the constraint satisfaction problem to a nonconstraint optimization problem by adding the penalty factors.When  1 ,  2 , and  3 meet conditions (F 1 ), (F 2 ), and (F 3 ), respectively, the objective values of each functions are 0; otherwise, a penalty factor 1 is as the objective value.
For this problem, there are multiple solutions that meet conditions (F 1 ), (F 2 ), and (F 3 ).Therefore, sometimes researchers hope to get as many solutions as possible.To resolve the multisolution problem, we introduce tube table (tube table: TT) to store the obtained solutions and to avoid duplication of solutions.The definite means are as follows: where TT is the tube It is clear that, for this optimization problem, the objective value of optimal solution is min  * = 0.
In order to solve the nonconstrained optimization problem, we propose a new harmony search (HS) algorithm which is a big improvement for classical HS algorithm [1] and NGHS [3] method.We called it improved NGHS (INGHS).

Classical Harmony Search (HS) and NGHS
Classical harmony search (HS) is derivative-free metaheuristic algorithm [4,5].It mimics the improvisation process of music players and attains the ideal harmony by adjusting the HM.HS algorithm has the same feature as genetic algorithm (GA).The HS algorithm is good at identifying the new regions of the searching space in a reasonable time; however, it has difficulties performing a local search for numerical applications [3,[6][7][8].
In the HS algorithm, X represents the harmony and (X) denotes the melody of harmony X. HS uses three rules (memory consideration, pitch adjustments, and randomization) to optimize the harmony memories (HM) composed of HMS harmony vectors.

The Classical Harmony Search (HS) Algorithm.
The steps in the procedure of classical harmony search algorithm are as follows.
Step 1 (initialize the harmony memory).The harmony memory (HM) consists of HMS harmony vectors.Each harmony vector is generated from a uniform distribution in the feasible space, as where  and HMS represent the number of decision variables and the size of harmony memory, respectively, and  denotes a uniform distribution random number between 0 and 1, as Step 2 (improvise a new harmony via three rules).There are three rules that can be used to improvise a new harmony vector X new .
(a) Memory Consideration.A decision variable value of the harmony vector will be adopted by choosing from the harmony memory with probability HMCR.
(b) Pitch Adjustment.Get a component randomly from an adjacent value of one decision variable of a harmony vector with probability PAR.
(c) Random Generation.Generate a component randomly in the feasible region with probability 1-HMCR.
The improvisation procedure of a new harmony vector works as Algorithm 2.
A trial harmony vector X new is generated in Step 2. Next, in Step 3, it will be decided whether to survive.
Step 3 (select operator).Get the worst harmony vector X worst from the HM (see Algorithm 1).
Step 4 (check stopping criterion).If the stopping criterion (maximum function evaluation times: maxFEs) is satisfied, computation is terminated.Otherwise, Steps 2 and 3 are repeated.

The NGHS Algorithm.
In the novel global harmony search (NGHS) algorithm [9], three significant parameters, harmony memory considering rate (HMCR), fret width (FW), and pitch adjusting rate (PAR), are excluded from NGHS and a random selection rate (  ) is included in the NGHS.In Step 3, NGHS works as Algorithm 3, where  best  and  worst  , respectively, represent indexes of the best harmony and the worst harmony in HM. rand(0, 1) is a uniformly generated random number in [0, 1], and it should set parameter   = 2/ for the 0-1 knapsack problem and set parameter   = 0.01 ∼ 0.1 for continuous optimal problem.

The Proposed HS Method.
In this section, we proposed a new harmony search algorithm which is improved based on classical HS algorithm and NGHS method, so we call the proposed algorithm INGHS.
Since the HS algorithm origination, it has been applied to many practical optimization problems.However, for large scale optimization problems, classical HS has slow convergence and low precision, which is due to the fact that because a new decision variable value in harmony memory (HM) can If fitness (X new ) is better than fitness (X worst ) then X worst = X new ; (X worst is the worst harmony vector in HM).EndIf Algorithm 1 Algorithm 2: The improvisation procedure of new harmony vector by HS.

EndIf EndFor
Algorithm 3: The NGHS modifies improvisation step.be generated only by using pitch adjustment and randomization strategies during the search procedure, the memory consideration rule is only used to adjust the existing decision variable values according to the harmony memory (HM).Thus HS can maintain a strong performance of exploration, but not a good performance of exploitation, and, in the later stage of search, it is characterized by slow convergent.Therefore, for solving the large scale optimization problem, the key is to balance the global exploration performance and the local exploitation ability.
Because the construction example of the algebra is a large scale high-dimensional optimization problem, to achieve the most satisfactory optimization performance by applying the HS algorithm to a given problem, we adopt four optimization strategies and dynamical parameter control method to balance the global exploration power and the local exploitation ability.
It is of very importance between the convergence and the diversity in order to improve the efficiency of the search.In the classical HS algorithm, a new harmony is generated in Step 2. After the selecting operation in Step 3, the population variance may increase or decrease.With a high population variance, the diversity and exploration power will increase, and in the same time the convergence and the exploitation power will decrease accordingly.Conversely, with a low population variance, the convergence and the exploitation power will increase [8]; the diversity and the exploration power will decrease.So it is significant how to keep the balance between the convergence and the diversity.Classical HS algorithm loses its ability easily at the later evolution process [3], because of improvising a new harmony from HM with a high HMCR and local adjusting with PAR.And HM diversity decreases gradually from the early iteration to the last.However, in HS algorithm, a low HMCR employed will increase the probability (1-HMCR) of random select in search space; the exploration power will improve, but the local search ability and the exploitation accuracy cannot be improved by the single pitch adjusting strategy.
To overcome the inherent weaknesses of HS, in this section, we present INGHS method to construct example for algebra.The INGHS algorithm works as Algorithm 4 and Figure 1.In Algorithm 4, to resolve the discrete optimization problems, the function Round() is used to round each element of  to the nearest integer.
(1) Novel Global (NG) Best Strategy.In NGHS algorithm, the novel global best strategy is very effective to explore the global best solution.So, in the proposed method, the novel global best strategy is adopted (see Algorithm 5), where the best and worst, respectively, represent indexes of the best harmony and the worst harmony in HM. ) , (see [8]).It can be seen that the parameter HMCR gradually increased from HMCR min to HMCR max linearly and the parameter NGP increased with low velocity in the early stage and it increased sharply in the final stage.That is because, in the beginning, in order to explore the global optimal solution, the harmony consideration rules and NG strategies are carried out with a smaller probability, and, in later stage, INGHS methods begin to focus on the local exploitation that needs a high probability to employ the NG strategy and harmony consideration rules.The benefits of doing so can get more opportunities to reinforce the global exploration by strengthening disturbance in the early stage and can acquire high precision solution by carrying local intensification search in the later stage.For the same reason, FW is decreased gradually in order to reduce perturbation step size step by step, and the variation of PAR from 0.55 to 0.3 is to reduce the probability of pitch adjustment.

The Construction
Example for (2, 2, 0) Algebra System with the Proposed HS Method.For an N(2, 2, 0) algebra system, there are multiple solutions that meet the conditions.Sometimes we need to obtain multiple solutions.So we adopt the proposed HS algorithm and tube table technology to resolve the multisolution problem.Its flow chart is as shown in Figure 2 and in Pseudocode 1.

Effect of Parameters HMCR and NGP on INGHS Performance.
In this section, we determine the effect of parameters HMCR max , HMCR min , NGP max , and NGP min on the performance of the proposed algorithm INGHS.The experiments are investigated for algebra system (, * , Δ, 0), where  = {, , , , , , }.
(2) Investigation for  max and  min (initialize NGP max = 0.5, NGP min = 0.15).Let HMCR min = 0.2 to 0.7 and let HMCR max = 0.5 to 1, respectively.For each pair of parameters (HMCR min , HMCR max ), the proposed algorithm INGHS is performed || 4 (7 4 = 2401) times and then we record the number of solutions obtained.The number of solutions obtained is shown in Figure 4.
It can be seen from Figure 3 that most solutions can be obtained when NGP min = 0.2 and NGP max = 0.5.From Figure 4, we find that most solutions are found when HMCR min = 0.6 and HMCR max = 0.9.
From the above, the proposed algorithm has better performance when it is set as HMCR min = 0.6, HMCR max = 0.9, NGP min = 0.2, and NGP max = 0.5.

Computational Experiments and Results
In this section, we have tested the proposed algorithm over a set of N(2, 2, 0) algebra system construction examples problems.
In order to deal with the optimization problems easily, we replace the elements , , , , , , , ℎ, , and  with 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10, respectively.So, for each , the upper limit of variables about optimization problems is set as || and the lower limit is set as 1.
When  = 2 and  = 3, there are 3 and 16 solutions for the N(2, 2, 0) algebra system, respectively.So we set max looping times (ML) = 3 and 16, respectively.When  > 3 and  < 9, we do not know how many solutions are there for the algebra system.So we set ML =  4 .However, with the increasing of , there are much more solutions, and computer's running time is very long.So we set ML = 8000 and 5000 when  = 9 and  = 10.

The Experiment Results and Analysis.
In order to evaluate the performance of the proposed algorithm INGHS, we compared its success rate of finding the solutions with HS, HSTL, and NGHS algorithms in the same conditions.The results are shown in Table 1.
The first solution on  = 2 is as follows: The second solution on  = 2 is as follows: The third solution on  = 2 is as follows: The first solution on  = 3 is as follows: From Table 1, it can be seen obviously that the success rate of all instances of the proposed method is higher than HS, HSTL, and NGHS.
For instance,  = 6, 7, 8, 9, and 10, and Figures 5, 6, 7, 8, 9, and 10 show the convergence curves of three algorithms (HS, NGHS, and INGHS).It is evident from Figures 5-10 that the proposed algorithm is better than HS and NGHS for all instances.The convergence curve of the proposed algorithm can maintain falling until it finds the best solution.When  > 6, the convergence curve of INGHS method falls faster than HS, HSTL, and NGHS algorithms, and it keeps declining actively until the best solution is found.

Conclusions
This paper has investigated nine construction example problems of algebra system and converted the construction example problems into optimization problems.A novel harmony search algorithm (INGHS) is proposed to solve the problems.Global best strategy and dynamic parameters adjustment are employed in INGHS.The experimental results on 9 instances of algebra system demonstrate that the proposed algorithm is more effective than HS, HSTL, and NGHS algorithms.Further research will investigate the INGHS algorithm to solve the combinatorial optimization problems and some practical optimization problems.

Figure 1 :
Figure 1: The flow chart of the proposed algorithm.

( 2 )
Parameters Dynamically Changed.To balance the exploration and exploitation power of the INGHS algorithm efficiently, HMCR, PAR, FW, and NGP parameters are dynamically adapted to a suitable range with the increase of generations.Equation (11) shows the dynamic change of HMCR, PAR, and NGP, respectively, HMCR = HMCR min + (HMCR max − HMCR min ) × (   max

Figure 2 :
Figure 2: The flow chart for construction example of algebra system based on INGHS algorithm.

Figure 5 :
Figure 5: The convergence curve and the box plot when  = 5.

Figure 6 :Figure 7 :
Figure 6: The convergence curve and the box plot when  = 6.

Figure 8 :
Figure 8: The convergence curve and the box plot when  = 8.

table .
Set TT as empty For  = 1 to Max Loop times (ML)Run the proposed HS algorithm to get a best solution X new ; If the objective function value (X new ) == 0 Put X new into tube table TT.
(F 1 ), (F 2 ), and (F 3 ) Perform INGHS algorithm to find an optimal solution X new X new X new

Table 1 :
The success rate comparison of the proposed method with HS and NGHS algorithm.