Heap-Based Optimizer Algorithm with Chaotic Search for Nonlinear Programming Problem Global Solution

In this paper, a heap-based optimizer algorithm with chaotic search has been presented for the global solution of nonlinear programming problems. Heap-based optimizer (HBO) is a modern human social behavior-influenced algorithm that has been presented as an effective method to solve nonlinear programming problems. One of the difficulties that faces HBO is that it falls into locally optimal solutions and does not reach the global solution. To recompense the disadvantages of such modern algorithm, we integrate a heap-based optimizer with a chaotic search to reach the global optimization for nonlinear programming problems. The proposed algorithm displays the advantages of both modern techniques. The robustness of the proposed algorithm is inspected on a wide scale of different 42 problems including unimodal, multi-modal test problems, and CEC-C06 2019 benchmark problems. The comprehensive results have shown that the proposed algorithm effectively deals with nonlinear programming problems compared with 11 highly cited algorithms in addressing the tasks of optimization. As well as the rapid performance of the proposed algorithm in treating nonlinear programming problems has been proved as the proposed algorithm has taken less time to find the global solution.


Introduction
Optimization is a research field that aims to take decisions that give the best result under given conditions.Many important technological and managerial decisions must be taken by engineers to find maximum benefits and minimum required efforts in any engineering system at several stages [1].Engineers try to see the suitable conditions which give the maximum or minimum values of their objective function using different optimization techniques.Practical life problems are usually nonlinear programming problems.The nonlinear programming problems include nonlinear parts in the objective function or the constraints or both.It plays an essential role in real-world applications, such as engineering design, structural optimization, economics, and many other applications [2].
Researchers found that is no general method used to find the optimal value for a nonlinear programming problem [3].Some traditional optimization techniques appeared to solve some optimization problem types.For example, optimization techniques for solving unconstrained problems can be grouped to direct search techniques and descent techniques.Also, some traditional optimization techniques that are used for solving constrained problems can be assorted into two main methods: direct techniques and indirect techniques.All traditional techniques focus that the derivatives of functions that must be existence, and they find that it is so hard to treat noisy search spaces, discontinuous, and vast multi-modal [2,3].
Chaos means "A condition or place of great disorder or confusion".Chaos theory studies the behavior of systems that follow deterministic laws but appear random and unpredictable.It studies a dynamical system that has a sensitive dependence on its initial conditions.A small change in the initial conditions may cause quite different outcomes.Henon first described chaos theory in 1976 and Lorenz concluded it in 1993 [33].Chaos theory has many applications in mathematics, physics, engineering, economics, etc. [34,35].In mathematics, scientists apply search based on chaos theory to the optimization problems searching for the global optimal solution.The search based on chaos theory is called a chaotic search.In addition, chaos theory could be used to accelerate the optimum seeking operation.It has an advantage more than other optimization techniques that it can efficiently search the space and escape from local optimal solutions [32].Researchers attracted much attention to the use of chaotic search to improve the work of their algorithms.Yousria et al. presented a hybrid algorithm by combining an enhanced genetic algorithm and chaos searching technique for solving bilevel programming problems [36].They used a chaos theory with a genetic algorithm to solve an optimization problem with two levels that helps the algorithm reach the global solution in less time.
In this paper, we improve one modern optimization technique which simulates human beings' social behavior, the heap-based optimizer algorithm.HBO algorithm has been presented as an effective way to treat nonlinear optimization problems.The modification of the HBO algorithm depends on improving the performance of the initialization phase in the HBO algorithm using chaos theory.We have applied chaos theory in the initialization phase of the HBO algorithm.The integration between the HBO algorithm and chaos theory has improved the performance of optimization procedures, has escaped from local optimal traps, and has reached the optimal solution quickly.The effectiveness of our proposed algorithm is tested on 42 various benchmark problems, such as unimodal problems, multi-modal problems, and CEC-C06 2019 benchmark problems.
The organization of this paper is as follows: in Sect.2, the mathematical model of nonlinear optimization problems is introduced.Section 3 shows a brief survey of HBO.In Sect.4, popular chaotic maps are described.Section 5 introduces the proposed algorithm.In Sect.6, we present experimental results.Finally, Sect.7 introduces our conclusion.

Nonlinear Programming Problems
Nonlinear problems appear if the objective or constraints are nonlinear functions, and we cannot express them as linear functions.The general nonlinear programming problems can be represented as shown [1]: nonlinear programming problem: where g k (x) refers to a group of k nonlinear inequality constraints, f (x) represents an objective function,, h v (x) indicates a group of v nonlinear equality constraints, T , where x is decision variable vector, the values of decision variable are a real values, and the decision variables' total numbers are indicated by b .There are two boundaries; upper boundary and lower boundary of each decision variable x b can be written as x l b , x u b .In solving nonlinear optimization problems, we are looking for a global solution and not stock on a local solution.Definition 1 introduces the difference between the local solution and the global solution [1]. Figure 1 illustrates this definition. (1) ) be a feasible solution to a minimization problem with objective function f (x) .We call x is

Heap-Based Optimizer Algorithm
HBO simulates human beings' social behavior.We can find one type of human interaction in a company, the officials that are given designations that represent the workers' job descriptions.Although the designations are different from one company to another one.They are arranged in a hierarchy called a hierarchy ranked corporation (HRC), as shown in Fig. 2a.
The HRC is effectively based on the HBO algorithm.To know the meaning of HRC, we should understand how to manage the arrangement in the hierarchy of search agents' suitability.This arrangement is done by taking a heap treebased data structure in proceeding with priority queues.Three degrees (3-array) of min-heap appear in Fig. 2b.The behaviors of employees can be arranged in three sorts which play an essential role in the HBO algorithm.The sorts are the subordinates' interaction with their immediate head, coworkers' interaction, and the individuals' self-contribution.HBO's steps have been discussed as follows.

Step 1: Modeling of Hierarchy-Ranked Cooperation
The modeling of HRC with the structure of heap data can be represented by the meaning that the heap is a nonlinear data structure which takes the shape of the tree, as shown in Fig. 3.In Fig. 3, the whole HRC is considered as the population.We treat each official designation as the search agent.The search agent's fitness is considered as the node's key and the population index of the search agent is considered as the node's value.Every parent node's key in a minimum of the heap is either less than or equal to the keys of its children.x i is the population's ith search agent.The search agents are plotted on the fitness landscape by their fitness, and the curve in the objective space represents the landscape of an assumed objective function.A heap is created using each search agent's fitness as a key.For instance, x 4 is the best solution in the population, so is the root of the heap.

Step 2: Mathematical Representation of the Interaction with the Immediate Leader
The regulations in a centralized organizational structure are decreed from the higher levels and subordinates must where u is the current iteration, n is the nth vector's com- ponent, D indicates the parent node, r is a random number (2) ) n = (2r − 1), located in the range [0, 1] , U is the iterations' maximum number, and C represents a user-defined parameter.

Step 3: Mathematical Representation of Colleagues' Interaction
The workers in the same positions are said to be colleagues and they work with each other to do the best in their tasks.To represent the mathematical representation of the interaction between colleagues who are considered as the nodes at the same level in a heap, and the position of each search agent x i can be updated according to randomly chosen colleague Sr as shown in the following: (5) . The self-contribution of a worker can be represented as follows: Step 5: Putting it All Together For balancing exploration and exploitation, it should be determined the probabilities of three equations as their selection probabilities play an important part in this balance.To realize a balance of possibilities, the roulette wheel can be used.The roulette wheel is considered as having three values: p 1 , p 2 , and p 3 where p 1 makes a population changes its position.And p 2 , p 3 are evaluated as shown in the following: Therefore, the mathematical representation of the mechanism used to update a general position-of the HBO algorithm can be written as follows: where b is a random number located in the range (0, 1).

Chaos Theory
Chaos theory is a scientific theory interested in studying deterministic laws which are more sensitive to initial conditions in dynamic systems.Scientists think that these systems have disorders in random states and irregular random states.Chaos is not completely like statistical randomness, according to its effective searching for interesting space and improving optimization's performance of procedures.A chaotic map offers some behavior of chaos theory.The parameters of such maps can be discrete-time or continuous-time parameters.Discrete maps are usually formed as iterated (6) functions.This section presents several renowned chaotic charts which we get in the literature [33][34][35][36][37][38][39][40].The discretetime or continuous-time parameter can be used to parameterize different chaotic maps.Ordinarily, discrete maps take the shape of ordered functions.Figure 4 presents a different chaotic map used to produce chaotic numbers.The chaotic equations used to produce different chaotic numbers are represented by the following: -Sinusoidal Map: -Chebyshev Map: -Singer Map: -Tent Map: -Sine Map: where 0 < a ≤ 4.
-Gauss Map: where and are real numbers.

Chaotic Heap-Based Optimizer Algorithm (CHBO)
CHBO is a modified version of the HBO algorithm.The enhancement of the HBO algorithm's performance is mainly based on the initialization part of our algorithm.We have applied chaos theory to produce initial solutions.Chaos theory develops the behavior of the convergence capability and provides different diversified solutions in the search space.CHBO algorithm has generated the first population of search agents using the logistic chart.As discussed in the previous section, there are different chaotic maps, and we have chosen the chart as it had proved in [41] that the logistic chart had increased the solution quality.In [41], authors applied their algorithm to 12 chaotic charts and compared the results.
From their simulation result, they have proved that using a Logistic map may increase their proposed algorithm and avoid trapping in local optima.In addition, using a logistic map gave the best performance.Therefore, in our paper, we have chosen to use the logistic chart.The detailed description of the initialization phase using the logistic chart in CHBO is displayed in the following:

Generation Using Logistic Map
Step boundaries.The chaos boundary range is determined by Step 2: Produce the Chaotic Number In this step, a chaotic random number z k is produced using a logistic chaotic map using Eq.(19).

Step 3: Generating the Initial Chaotic Variables Values
The variance range of optimization valuable a i , b i is used to generate the initial chaotic variable values using the following equation: which leads A basic deterministic system can generate complicated behavior without the requirement for a random sequence, as shown by logistic maps.The dynamics of a biological population are described by an easy-to-understand polynomial equation on which the system is based.Figure 5 shows a chaotic variable produced using logistic map.The red points appear the variable values.The lower and upper boundaries are The initialization steps have repeated for a specified number until generating the initial chaotic population.After that, (23)  the algorithm has been completed using the HBO algorithm as explained in Sect.3.

Updating Solutions using HBO Steps
Step 4: Hierarchy-Ranked Modeling of the Initial Chaotic Population The modeling of HRC with the structure of heap data takes the shape of a tree, as shown in Fig. 2.

Step 5: Mathematical Representation of the Interaction with the Immediate Leader
The regulations in a centralized organization produced in step 4 structure are decreed from the higher levels and subordinates must follow the instructions of their leaders.The mathematical representation of the interaction with an immediate leader can be represented by Eqs. 2, 3, and 4.

Step 6: Mathematical Representation of Colleagues' Interaction
As explained in Sect.3, the workers produced in the same positions are said to be colleagues and they work with each other to do the best in their tasks.To identify the workers in the population produced in step 4. The mathematical representation of the interaction between colleagues who are considered as the nodes at the same level in a heap, and the position of each search agent x i can be updated according to Eq. 5.

St e p 7 : Re p re s e n t a t i o n o f a n E mp l oye e ' s Self-contribution
The self-contribution of a worker can be represented according to Eq. 6.

Step 8: Putting it All Together
This step is to balance exploration and exploitation.To realize a balance of possibilities, the roulette wheel can be used.The roulette wheel is considered as having three values: p 1 , p 2 , andp 3 according to Eqs. 7, 8, and 9.Then, the updated positions of the HBO algorithm are according to Eq. 10.

Step 9: Termination Criteria
The algorithm is terminated when either the maximum number of generations has been produced, or the maximal number of evaluations is achieved.
The flowchart of CHBO can be represented in Fig. 6.As shown the algorithm starts by initializing the population using the logistic chart in the problem boundary, defining the size of the population, decision variables' numbers, and iteration numbers.The decision variables' number depends on the problem.
The population size and iteration numbers and iteration numbers affect the performance of the algorithms, and they can be chosen to give the best results.In the proposed algorithm, the population size is 40 and the number of iterations is about 1250.These values give the best results for test problems.After defining the variables, HBO equations can   be calculated for specified iteration numbers unless we get the optimal solution of the different benchmark functions.After generating the initial population using the logistic chart, the value of vectors Ƴ, λ have been calculated by Eqs.
(3), and (4).Then, the value of x i has updated by Eq. ( 10) and the fitness function has calculated.Finally, the solution has been checked and the value of the best result has been returned.

Experimental Results
For checking the proposed algorithm's performance for global optimization, CHBO is coded in MATLAB R2013a.Ink.The simulations have been executed on an Intel core (TM) i3-1005G1 (CPU) 1.2 GHz 1.19 GHz processor.The capability of the proposed algorithm CHBO is clarified in checking 42 wide scale optimization test functions problems including unimodal, multi-model [32], and CEC-C06 2019 [42].The descriptions of unimodal variable-dimension, multi-modal variable-dimension benchmark problems are shown in Tables 1, 2.
The results obtained with the proposed CHBO technique are compared with other optimization algorithms, including the sine cosine algorithm (SCA) [43], Moth-Flame Optimization (MFO) [44], Gravitational Search Algorithm (GSA) [45], Cuckoo Search algorithm (CS) [46], particle swarm optimization (PSO) [47], Multi-Verse Optimizer (MVO) [48], and heap-based optimizer (HBO) [32].The parameter setting of all compared algorithms and the proposed algorithm is proposed in Table 3.The parameter settings of all compared algorithms are from their original papers.In the proposed algorithm, the population size is 40 and

Evaluating the Performance of CHBO Compared with Other Algorithms
The  median result, the mean result, and the standard deviation have been used for comparison.

Results for Unimodal Problems, Multi-modal Test Problems
The results obtained by the proposed CHBO technique have been compared with SCA [43], MFO [44], GSA [45], CS [46], PSO [47], MVO [48], and HBO [32].Table 4 shows results for unimodal variable-dimension test functions from F 1 to F 15 of the CHBO and other algorithms.The best result of the compared algorithms is shown in bold.
As shown in Table 4, we reach the global optimal solution for eight functions, and we reach the best solution for 12 functions from 15 problems compared to other compared algorithms.The result shows the effectiveness of our proposed algorithm CHBO to reach the optimal solution for unimodal variable benchmark functions.Table 5 shows results for multi-modal variable benchmark functions from F 16 to F 32 for the proposed algorithm CHBO and other algorithms.The best result is shown in bold.The results of the proposed algorithm CHBO and other well-known algorithms have been ordered according to the best-obtained results of all compared algorithms in Tables 6,  7. The algorithm that has achieved order 1 is the algorithm that has reached the best result and the algorithm that has achieved order 2 is the algorithm that has reached the second-best result and so on.Table 7 shows statistics of ordering how many every algorithm has obtained every rank.
Figure 7 presents the ranking comparison between the well-known algorithms.This arrangement has proved that we achieve first order more than other algorithms to reach the optimal solution.Figure 8 presents the percentages of rank for famous solved problems used by the proposed algorithm (CHBO) and other compared algorithms.Our proposed algorithm reaches first rank for about 68.75% of test problems and for about 21.88% as a second rank of solved problems, as shown in Fig. 8. Figure 9 offers the resulted ranks compared with CHBO, and HBO algorithms for Fn1 to Fn24.The simulation results proved that the proposed algorithm (CHBO) effectively deals with nonlinear programming problems and have proved the effectiveness and robustness of the proposed algorithm.
The standard deviation is a very important problem in practical production areas.The standard deviation should be as small as possible.From the results, the standard deviation is smaller than other algorithms which indicates that CHBO possesses excellent robustness.

Results for CEC-C06 2019 Benchmark Problems
The results obtained by the proposed algorithm CHBO have been compared with other renowned algorithms; Learner Performance-based Behavior Algorithm (LPB) [32], Dragonfly algorithm (DA) [49], and Particle Swarm Optimization algorithm (PSO) [49].CHBO has run 30 times with about 40 search agents and 1000 iterations and other compared algorithms have run 30 times with 80 search agents over 500 iterations.The CHBO parameters have been chosen to give the best results and execute the proposed algorithm with the same number of objective function evaluations of the compared algorithms.Table 8 provides the results of CHBO and other compared algorithms.Figure 10 shows the ordering of CHBO and other algorithms.From the results, the proposed algorithm has reached the best result for eight functions that proved the effectiveness of the CHBO algorithm.The standard deviation is smaller than other algorithms which indicates that CHBO has good robustness.

Effect of Using Chaos Theory in Enhancing the Performance of HBO Algorithm
The integration between chaos theory and the HBO algorithm has made the quality of the solution more rapid.Chaos theory has provided different diversified solutions in the search space, as shown in Fig. 4. Also, it has improved the behavior of the convergence capability of HBO. the 1 benchmark function.By comparing the two figures, the CHBO can reach the optimal solutions in a smaller number of iterations than HBO which means that CHBO is faster than HBO, and using chaos theory in initialization helps our algorithm to get the optimal solution in a small number of iterations and faster.Also, CHBO has started searching for the solution from a very close value to the optimal solution compared to HBO which helps the search to reach the optimal solution faster.Figures 12,13 show the comparison between CHBO and HBO according to the results obtained.From the figure, the proposed algorithm has reached first rank in 22 functions and the heap-based optimizer algorithm has reached first rank in four functions.This has proved that the integration between chaos theory and heap-based optimizer helps to get the optimal solution in a small number of iterations.

Convergence Speed
In this section, computation time analysis has been provided for measuring the convergence speed behavior of the CHBO algorithm compared to other algorithms.The iteration number cannot be used as a reference where every algorithm performs a different amount of work in its inner loops and have different population sizes.The number of fitness evaluations (FEs) was applied as a measuring tool of computational time.The number of function evaluations is equal to the sum of the number of the main population with the number of new individuals multiplied by the number of iterations.Table 9 provides the FEs of CHBO and other algorithms for CEC-C06 2019.The least FEs of all compared algorithms is in bold.Figure 14 illustrates the results.
As indicated from the results, the proposed algorithm has the smallest number of FEs for five problems and has the same number of FEs of DA and PSO algorithms for the other

Fig. 1
Fig. 1 Global minimum and local minimum

149 Page 6 of 21 Fig. 4 3 Page 7
Fig. 4 Different chaotic maps used to produce chaotic numbers

1 :
Identify Chaotic Search Boundary Range Chaos boundary range is decision variables range a k , b k , k = 1, 2, … , N where a k , b k are lower and upper(20) a k = −5, b k = 15.Then = 10 and x * k = 5.

Fig. 5
Fig. 5 Chaotic variable produced using a Logistic map

Fig. 8
Fig.8The percentages of Rank for unimodal problems and multimodal test problems by CHBO

Fig. 12 32 International
Fig. 12 Ranks of CHBO and HBO algorithms for F 1 to F 32

Fig. 13
Fig.13 The objective function values for F 1 to F 32 using CHBO and HBO algorithms

Table 2
Details of multi-modal benchmark problems

Table 3
The parameter settings of all compared algorithms

Table 4
Results of unimodal benchmark problems obtained by CHBO and other optimization algorithms

Table 5
The results of multi-model variable benchmark problems obtained by CHBO and optimization algorithms

Note that high Rank show the best result achieved by the different algorithms. Rank F 1 Rank F 2 Rank F 3
proposed algorithm CHBO has been tested on known 42 functions, including unimodal, multi-modal test problems, and CEC-C06 2019 benchmark problems.In this section, the performance of CHBO has been compared with other renowned algorithms.Statistical measurements like the Ranks of CHBO and other algorithms for F 1 to F 9 .Ranks of CHBO and other algorithms for F 10 to F 24

Table 9
FEs of CHBO and other algorithms for IEEE CEC-C06 2019 benchmark test Bold values indicate the best results of all compared algorithms