A hybrid cuckoo search algorithm with Nelder Mead method for solving global optimization problems
 Ahmed F. Ali^{1, 2} and
 Mohamed A. Tawhid^{2, 3}Email author
Received: 21 August 2015
Accepted: 28 March 2016
Published: 18 April 2016
Abstract
Cuckoo search algorithm is a promising metaheuristic population based method. It has been applied to solve many real life problems. In this paper, we propose a new cuckoo search algorithm by combining the cuckoo search algorithm with the Nelder–Mead method in order to solve the integer and minimax optimization problems. We call the proposed algorithm by hybrid cuckoo search and Nelder–Mead method (HCSNM). HCSNM starts the search by applying the standard cuckoo search for number of iterations then the best obtained solution is passing to the Nelder–Mead algorithm as an intensification process in order to accelerate the search and overcome the slow convergence of the standard cuckoo search algorithm. The proposed algorithm is balancing between the global exploration of the Cuckoo search algorithm and the deep exploitation of the Nelder–Mead method. We test HCSNM algorithm on seven integer programming problems and ten minimax problems and compare against eight algorithms for solving integer programming problems and seven algorithms for solving minimax problems. The experiments results show the efficiency of the proposed algorithm and its ability to solve integer and minimax optimization problems in reasonable time.
Keywords
Cuckoo search algorithm Nelder–Mead method Integer programming problems minimax problemsBackground
Cuckoo search (CS) is a population based metaheuristic algorithm that was developed by Yang et al. (2007). CS (Garg 2015a, d) and other metaheuristic algorithms such as ant colony optimization (ACO) (Dorigo 1992), artificial bee colony (Garg et al. 2013; Garg 2014; Karaboga and Basturk 2007), particle swarm optimization (PSO) (Garg and Sharma 2013; Kennedy and Eberhart 1995), bacterial foraging (Passino 2002), bat algorithm (Yang 2010a), bee colony optimization (BCO) (Teodorovic and DellOrco 2005), wolf search (Tang et al. 2012), cat swarm (Chu et al. 2006), firefly algorithm (Yang 2010b), fish swarm/school (Li et al. 2002), genetic algorithm (GA) (Garg 2015a), etc., have been applied to solve global optimization problems. These algorithms have been widely used to solve unconstrained and constrained problems and their applications. However, few works have been applied to solve minimax and integer programming problems via these algorithms.
A wide variety of real life problems in logistics, economics, social science, politics, game theory, and engineering can be formulated as integer optimization and minimax problems. The combinatorial problems, like the knapsackcapital budgeting problem, warehouse location problem, traveling salesman problem, decreasing costs and machinery selection problem, network and graph problems, such as maximum flow problems, set covering problems, matching problems, weighted matching problems, spanning trees problems, very large scale integration (LSI) circuits design problems, robot path planning problems, and many scheduling problems can also be solved as integer optimization and minimax problems (see, e.g., Chen et al. 2010; Du and Pardalos 2013; Hoffman and Padberg 1993; Little et al. 1963; Mitra 1973; Nemhauser et al. 1989; Zuhe et al. 1990).
Branch and bound (BB) is one of the most famous exact integer programming algorithm. However, BB suffers from high complexity, since it explores a hundred of nodes in a big tree structure when it solves a large scale problems. Recently, there are some efforts to apply some of swarm intelligence algorithms to solve integer programming problems such as ant colony algorithm (Jovanovic and Tuba 2011, 2013), artificial bee colony algorithm (Bacanin and Tuba 2012; Tuba et al. 2012), particle swarm optimization algorithm (Petalas et al. 2007), cuckoo search algorithm (Tuba et al. 2011) and firefly algorithm (Brown et al. 2007).
The minimax problem, as well as all other problems containing max (or min) operators, is considered to be difficult because max function is not differentiable. So many unconstrained optimization algorithms with the use of derivatives can not be applied to solve the nondifferentiable unconstrained optimization problem directly.
There are several different approaches that have been taken to solve minimax problem. Many researchers have derived algorithms for the solution to minimax problem by solving an equivalent differentiable program with many constraints (see, e.g., Liuzzi et al. 2006; Polak 2012; Polak et al. 2003; Yang 2010b and the references therein), which may not be efficient in computing.
Some swarm intelligence (SI) algorithms have been applied to solve minimax problems such as PSO (Petalas et al. 2007). The main drawback of applying swarm intelligence algorithms for solving minimax and integer programming problems is the slow convergence and the expensive computation time for these algorithms.
Recent studies illustrate that CS is potentially far more efficient than PSO, GAs, and other algorithms. For example, in Yang et al. (2007), the authors showed that CS algorithm could outperform is very promising the existing algorithms such as GA and PSO. Also, CS algorithm has shown good performance both on benchmark unconstrained functions and applications (Gandomi et al. 2013; Yang and Deb 2013). Also, the authors in Singh and Abhay Singh (2014) compared latest metaheuristic algorithms such as Krill Herd algorithm (Gandomi and Alavi 2012), firefly algorithm and CS algorithm and found that CS algorithm is superior for both unimodal and multimodal test function in terms of optimization fitness and time processing.
Moreover, the CS algorithm has a few number of parameters and easy to implement which is not found on other metaheuristics algorithms such as GA and PSO. Due to these advantage of the CS algorithm, many researchers have applied it on their work for various applications such as Garg et al. (2014), Garg (2015b, c, d). The CS algorithm is combined with other methods such as Nelder–Mead method to solve various problems (Chang et al. 2015; Jovanovic et al. 2014).
The aim of this work is to propose a new hybrid cuckoo search algorithm with a Nelder–Mead method in order to overcome the slow convergence of the standard cuckoo search. The Nelder–Mead method accelerates the search of the proposed algorithm and increases the convergence of the proposed algorithm. The proposed algorithm is called hybrid cuckoo search with Nelder–Mead (HCSNM). In HCSNM algorithm, we combine the cuckoo search with a Nelder Mead method in order to accelerate the search and avoid running the algorithm with more iterations without any improvements.
The main difference between our proposed algorithm and the other hybrid Cuckoo search and Nelder–Mead algorithms is the way of applying the Nelder–Mead method. The authors in Chang et al. (2015), Jovanovic et al. (2014) have invoked the Nelder–Mead method in the cuckoo search algorithm instead of the levy Flight operator. The drawback of this idea is the computation time because the calling for NM method at each iteration in the Cuckoo search algorithm. However in our proposed algorithm we run the standard CS algorithm for some iterations then we pass the best found solution to the Nelder–Mead method to start from good Solution which help the NM method to get the global minimum of the functions in reasonable time.
Also, we test the HCSNM algorithm on seven integer programming and ten minimax benchmark problems. The experimental results show that the proposed HCSNM is a promising algorithm and can obtain the optimal or near optimal solution for most of the tested function in reasonable time.
The outline of the paper is as follows. “Definition of the problems and an overview of the applied algorithms” section presents the definitions of the integer programming and the minimax problems and gives an overview of the Nelder–Mead method. “Overview of cuckoo search algorithm” section summarizes the main concepts of cuckoo search algorithm (CS). “The proposed HCSNM algorithm” section describes the main structure of the proposed HCSNM algorithm. “Numerical experiments” section gives the experimental results and details of implementation in solving integer programming and minimax problems. Finally, we end with some conclusions and future work in “Conclusion and future work” section.
Definition of the problems and an overview of the applied algorithms
In this section, we present the definitions of the integer programming and the minimax problems as follows.
The integer programming problem definition
Minimax problem definition
Nelder Mead method
The Nelder–Mead algorithm (NM) is one of the most popular derivativefree nonlinear optimization algorithms. Nelder and Mead (1965) proposed NM algorithm. It starts with \(n+1\) vertices (points) \(x_1,x_2,\ldots ,x_{n+1}\). The vertices are evaluated, ordered and relabeled in order to assign the best point and the worst point. In minimization optimization problems, the \(x_1\) is considered as the best vertex or point if it has the minimum value of the objective function, while the worst point \(x_{n+1}\) with the maximum value of the objective function. At each iteration, new points are computed, along with their function values, to form a new simplex. Four scalar parameters must be specified to define a complete NM algorithm: coefficients of reflection \(\rho\), expansion \(\chi\), contraction \(\tau\), and shrinkage \(\phi\) where \(\rho > 0\), \(\chi > 1\), \(0< \tau < 1\), and \(0< \phi < 1\). The main steps of the NM algorithm are presented as shown below in Algorithm 1. The vertices are ordered according to their fitness functions. The reflection process starts by computing the reflected point \(x_r = {\bar{x}}+ \rho ({\bar{x}} x_{(n+1)})\), where \({\bar{x}}\) is the average of all points except the worst. If the reflected point \(x_r\) is lower than the nth point \(f(x_{n})\) and greater than the best point \(f(x_1)\), then the reflected point is accepted and the iteration is terminated. If the reflected point is better than the best point, then the algorithm starts the expansion process by calculating the expanded point \(x_e = {\bar{x}} + \chi (x_r  {\bar{x}})\). If \(x_e\) is better than the reflected point nth, the expanded point is accepted. Otherwise the reflected point is accepted and the iteration will be terminated. If the reflected point \(x_r\) is greater than the nth point \(x_n\) the algorithm starts a contraction process by applying an outside \(x_{oc}\) or inside contraction \(x_{ic}\) depending on the comparison between the values of the reflected point \(x_r\) and the nth point \(x_n\). If the contracted point \(x_{oc}\) or \(x_{ic}\) is greater than the reflected point \(x_r\), the shrink process is starting. In the shrink process, the points are evaluated and the new vertices of simplex at the next iteration will be \(x^{\prime }_2,\ldots ,x^{\prime }_{n+1}\), where \(x^{\prime} = x_1 + \phi (x_i  x_1) , i = 2,\ldots , n + 1\).
Overview of cuckoo search algorithm
In the following subsection, we summarize the main concepts and structure of the cuckoo search algorithm.
Main concepts
Cuckoo search algorithm is a population based metaheuristic algorithm inspired from the reproduction strategy of the cuckoo birds (Yang and Deb 2009). The cuckoo birds lay their eggs in a communal nests and they may remove other eggs to increase the probability of hatching their own eggs (Payne and Karen Klitz 2005). This method of laying the eggs in other nests is called obligate brood parasitism. Some host birds can discover the eggs are not their own and throw these eggs away or abandon their nest and build a new nest in a new place. Some kind of cuckoo birds can mimic the color and the pattern of the eggs of a few host bird in order to reduce the probability of discovering the intruding eggs. Since the cuckoo eggs are hatching earlier than the host bird eggs, the cuckoos laid their eggs in a nest where the host bird just laid its own eggs. Once the eggs are hatching, the cuckoo chick’s starts to propel the host eggs out the of the nest in order to increase its share of food provided by its host bird.
L\(\acute{e}\)vy flights
Recent studies show that the behavior of many animals when searching for foods have the typical characteristics of L\(\acute{e}\)vy Flights, see, e.g., Brown et al. (2007), Pavlyukevich (2007) and Reynolds and Frye (2007). L\(\acute{e}\)vy flight (Brown et al. 2007) is a random walk in which the steplengths are distributed according to a heavytailed probability distribution. After a large number of steps, the distance from the origin of the random walk tends to a stable distribution.
Cuckoo search characteristic

At a time, cuckoo randomly chooses a nest to lay an egg.

The best nests with high quality of eggs (solutions) will carry over to the next generations.

The number of available host nests is fixed. The probability of discovering an intruding egg by the host bird is \(p_a \in [0,1]\). If the host bird discovers the intruding egg, it throws the intruding egg away the nest or abandons the nest and starts to build a new nest elsewhere.
Cuckoo search algorithm
We present in details the main steps of the Cuckoo search algorithm as shown in Algorithm 2.

Step 1 The standard cuckoo search algorithm starts with the initial values of population size n, probability \(p_a \in [0,1]\), maximum number of iterations \(Max_{itr}\) and the initial iteration counter t (Lines 1–2).

Step 2 The initial population n is randomly generated and each solution \(x_i\) in the population is evaluated by calculating its fitness function \(f(x_i)\) (Lines 3–6).

Step 3 The following steps are repeated until the termination criterion is satisfied.

Step 3.1 A new solution is randomly generated using a L\({\acute{e}}\)vy flight as follows.where \(\oplus\) denotes entrywise multiplication, \(\alpha\) is the step size, and L\({\acute{e}}\)vy \((\lambda )\) is the L\({\acute{e}}\)vy distribution (Lines 8–9).$$x_i^{t+1}=x_i^{t}+\alpha \oplus L\acute{e}vy(\lambda ),$$(6)

Step 3.2 If its objective function is better than the objective function of the selected random solution, then the new solution is replaced with a random selected solution (Lines 10–13).

Step 3.3 A fraction \((1p_a)\) of the solutions is randomly selected, abandoned and replaced by new solutions generated via using local random walks as follows.where \(x_j^t\) and \(x_k^t\) are two different solutions randomly selected and \(\gamma\) is a random number (Lines 14–15).$$x_i^{t+1}=x_i^{t}+\gamma \left( x_j^tx_k^t\right) ,$$(7)

Step 3.4 The solutions are ranked according to their objective values, then the best solution is assigned. The iteration counter increases (Lines 16–18).

Step 4 The operation is repeated until the termination criteria are satisfied (Line 19).


Step 6 Produce the best found solution so far (Line 20).
The proposed HCSNM algorithm
The steps of the proposed HCSNM algorithm are the same steps of the standard CS algorithm till line 19 in Algorithm 2 then we apply the NM method in Algorithm 1 as an intensification process in order to refine the best obtained solution from the previous stage in the standard CS algorithm.
Numerical experiments
In order to investigate the efficiency of the HCSNM, we present the general performance of it with different benchmark functions and compare the results of the proposed algorithm against variant of particle swarm optimization algorithms. We program HCSNM via MATLAB and take the results of the comparative algorithms from their original papers. In the following subsections, we report the parameter setting of the proposed algorithm with more details and the properties of the applied test functions. Also we present the performance analysis of the proposed algorithm with the comparative results between it and the other algorithms.
Parameter setting
In Table 1, we summarize the parameters of the HCSNM algorithm with their assigned values.

Population size n The experimental tests show that the best population size is \(n=20\), we applied the proposed algorithm with different population size in order to test the efficiency of the selected population size number. Figure 1 shows that the best population size is \(n = 20,\) while increasing this number to \(n = 25\) will increase the function evaluation without a big improvement in the function values.

A fraction of worse nests \(p_a\) In order to increase the diversification ability of the proposed algorithm, the worst solutions are discarded and the new solutions are randomly generated to replace the worst solutions. The number of the discarded solutions depends on the value of a fraction of worse nests \(p_a\). The common \(p_a\) value is 0.25.

Maximum number of iterations \(Max_{itr}\) The main termination criterion in standard cuckoo search algorithm is the number of iterations. In the proposed algorithm, we run the standard CS algorithm 3d iterations, then the best found solution is passed to the NM method. The effect of the maximum number of iteration is shown in Table 2. Table 2 shows that function values of six random selected functions (three integer functions and three minmax function). The results in Table 2 shows that there is no big different in the function value after applying 3d and 4d iterations which indicates that the number of iteration 3d is the best selection in term of function evaluation

Number of best solution for NM method \(N_{elite}\) In the final stage of the algorithm, the best obtained solution from the cuckoo search is refined by the NM method. The number of the refined solutions \(N_{elite}\) is set to 1.
Parameter setting
Parameters  Definitions  Values 

n  Population size  20 
\(p_a\)  A fraction of worse nests  0.25 
\(Max_{itr}\)  Maximum number of iterations  3d 
\(N_{elite}\)  No. of best solution for final intensification  1 
The effect of maximum number of iteration before applying Nelder–Mead method
Function  d  2d  3d  4d 

\(FI _1\)  117.60  18.26  2.46  2.04 
\(FI _2\)  2379.15  350.54  179.85  175.14 
\(FI _7\)  870.11  1.014  0.0095  0.0042 
\(FM _3\)  454.79  −39.14  −41.92  −41.93 
\(FM _6\)  15.73  6.15  1.19  1.15 
\(FM _{10}\)  459.25  1.05  0.114  0.114 
Integer programming optimization test problems
Integer programming optimization testproblems
Test problem  Problem definition 

Problem 1 (Rudolph 1994)  \(FI _{1}(x)=\Vert x\Vert _1=x_1+\cdots +x_n\) 
Problem 2 (Rudolph 1994)  \(FI _{2}(x)=x^Tx= \left[ \begin{array}{ccc} x_1&\ldots&x_n\end{array}\right] \left[ \begin{array}{c} x_1\\ \vdots \\ x_n\end{array}\right]\) 
Problem 3 (GlankwahmdeeL et al. 1979)  \(FI _{3}(x)= \left[ \begin{array}{ccccc} 15&\quad 27&\quad 36&\quad 18&\quad 12 \end{array}\right] x +x^T \left[ \begin{array}{ccccc} 35&{}\quad 20&{}\quad 10&{}\quad 32&{}\quad 10\\ 20&{}\quad 40&{}\quad 6&{}\quad 31&{}\quad 32\\ 10&{}\quad 6&{}\quad 11&{}\quad 6&{}\quad 10\\ 32&{}\quad 31&{}\quad 6&{}\quad 38&{}\quad 20\\ 10&{}\quad 32&{}\quad 10&{}\quad 20&{}\quad 31\\ \end{array}\right] x\) 
Problem 4 (GlankwahmdeeL et al. 1979)  \(FI _{4}(x)=(9x_1^2+2x_2^211)^2+(3x_1+4x_2^27)^2\) 
Problem 5 (GlankwahmdeeL et al. 1979)  \(FI _{5}(x)=(x_1+10x_2)^2+5(x_3x_4)^2+(x_22x_3)^4+10(x_1x_4)^4\) 
Problem 6 (Rao 1994)  \(FI _{6}(x)=2x_1^2+3x_2^2+4x_1x_26x_13x_2\) 
Problem 7 (GlankwahmdeeL et al. 1979)  \(FI _{7}(x)=3803.84138.08x_1232.92x_2+123.08x_1^2+203.64x_2^2 + 182.25x_1x_2\) 
The properties of the Integer programming test functions
Function  Dimension (d)  Bound  Optimal 

\(FI _1\)  5  [−100 100]  0 
\(FI _2\)  5  [−100 100]  0 
\(FI _3\)  5  [−100 100]  −737 
\(FI _4\)  2  [−100 100]  0 
\(FI _5\)  4  [−100 100]  0 
\(FI _6\)  2  [−100 100]  −6 
\(FI _7\)  2  [−100 100]  −3833.12 
The efficiency of the proposed HCSNM algorithm with integer programming problems
The efficiency of invoking the Nelder–Mead method in the final stage of SSSO algorithm for \(FI_1  FI_{7}\) integer programming problems
Function  Standard CS  NM method  HCSNM 

\(FI _1\)  11,880.15  1988.35  638.3 
\(FI _2\)  7176.23  678.15  232.64 
\(FI _3\)  6400.25  819.45  1668.1 
\(FI _4\)  4920.35  266.14  174.04 
\(FI _5\)  7540.38  872.46  884.48 
\(FI _6\)  4875.35  254.15  155.89 
\(FI _7\)  3660.45  245.47  210.3 
The general performance of the HCSNM algorithm with integer programming problems
HCSNM and other algorithms

RWMPSOg RWMPSOg is random walk memetic particle swarm optimization (with global variant), which combines the particle swarm optimization with random walk (as direction exploitation).

RWMPSOl RWMPSOl is random walk memetic particle swarm optimization (with local variant), which combines the particle swarm optimization with random walk (as direction exploitation).

PSOg PSOg is standard particle swarm optimization with global variant without local search method.

PSOl PSOl is standard particle swarm optimization with local variant without local search method.
Comparison between RWMPSOg, RWMPSOl, PSOg, PSOl and HCSNM for integer programming problems
Experimental results (min, max, mean, standard deviation and rate of success) of function evaluation for \(FI_1  FI_{7}\) test problems
Function  Algorithm  Min  Max  Mean  SD  Suc 

\(FI _1\)  RWMPSOg  17,160  74,699  27,176.3  8657  50 
RWMPSOl  24,870  35,265  30,923.9  2405  50  
PSOg  14,000  261,100  29,435.3  42,039  34  
PSOl  27,400  35,800  31,252  1818  50  
HCSNM  626  650  638.3  4.34  50  
\(FI _2\)  RWMPSOg  252  912  578.5  136.5  50 
RWMPSOl  369  1931  773.9  285.5  50  
PSOg  400  1000  606.4  119  50  
PSOl  450  1470  830.2  206  50  
HCSNM  208  238  232.64  4.28  50  
\(FI _3\)  RWMPSOg  361  41,593  6490.6  6913  50 
RWMPSOl  5003  15,833  9292.6  2444  50  
PSOg  2150  187,000  12,681  35,067  50  
PSOl  4650  22,650  11,320  3803  50  
HCSNM  1614  1701  1668.1  43.2  50  
\(FI _4\)  RWMPSOg  76  468  215  97.9  50 
RWMPSOl  73  620  218.7  115.3  50  
PSOg  100  620  369.6  113.2  50  
PSOl  120  920  390  134.6  50  
HCSNM  163  191  174.04  6.21  50  
\(FI _5\)  RWMPSOg  687  2439  1521.8  360.7  50 
RWMPSOl  675  3863  2102.9  689.5  50  
PSOg  680  3440  1499  513.1  43  
PSOl  800  3880  2472.4  637.5  50  
HCSNM  769  1045  884.48  56.24  50  
\(FI _6\)  RWMPSOg  40  238  110.9  48.6  50 
RWMPSOl  40  235  112  48.7  50  
PSOg  80  350  204.8  62  50  
PSOl  70  520  256  107.5  50  
HCSNM  139  175  155.89  5.16  50  
\(FI _7\)  RWMPSOg  72  620  242.7  132.2  50 
RWMPSOl  70  573  248.9  134.4  50  
PSOg  100  660  421.2  130.4  50  
PSOl  100  820  466  165  50  
HCSNM  119  243  210.3  6.39  50 
HCSNM and other metaheuristics and swarm intelligence algorithms for integer programming problems
HCSNM and other metaheuristics algorithms for \(FI_1  FI_{7}\) integer programming problems
Function  GA  PSO  FF  GWO  HCSNM 

\(FI _1\)  
Avg  1640.23  20,000  1617.13  860.45  613.48 
SD  425.18  0.00  114.77  43.66  21.18 
\(FI _2\)  
Avg  1140.15  17,540.17  834.15  880.25  799.23 
SD  345.25  1054.56  146.85  61.58  41.48 
\(FI _3\)  
Avg  4120.25  20,000  1225.17  4940.56  764.15 
SD  650.21  0.00  128.39  246.89  37.96 
\(FI _4\)  
Avg  1020.35  16,240.36  476.16  2840.45  205.48 
SD  452.56  1484.96  31.29  152.35  39.61 
\(FI _5\)  
Avg  1140.75  13,120.45  1315.53  1620.65  792.56 
SD  245.78  1711.83  113.01  111.66  53.32 
\(FI _6\)  
Avg  1040.45  1340.14  345.71  3660.25  294.53 
SD  115.48  265.21  35.52  431.25  33.90 
\(FI _7\)  
Avg  1060.75  1220.46  675.48  1120.15  222.35 
SD  154.89  177.19  36.36  167.54  33.55 
HCSNM and the branch and bound method
We apply further investigation to verify of the powerful of the proposed algorithm with the integer programming problems, by comparing the HCSNM algorithm against the branch and bound (BB) method (Borchers and Mitchell 1991, 1994; Lawler and Wood 1966; Manquinho et al. 1997).
Comparison between the BB method and HCSNM for integer programming problems
Experimental results (mean, standard deviation and rate of success) of function evaluation between BB and HCSNM for \(FI_1  FI_{7}\) test problems
Function  Algorithm  Mean  SD  Suc 

\(FI _1\)  BB  1167.83  659.8  30 
HCSNM  638.26  4.41  30  
\(FI _2\)  BB  139.7  102.6  30 
HCSNM  230.86  4.68  30  
\(FI _3\)  BB  4185.5  32.8  30 
HCSNM  1670.5  39.90  30  
\(FI _4\)  BB  316.9  125.4  30 
HCSNM  173.73  5.57  30  
\(FI _5\)  BB  2754  1030.1  30 
HCSNM  898.3  66.54  30  
\(FI _6\)  BB  211  15  30 
HCSNM  150.63  3.10  30  
\(FI _7\)  BB  358.6  14.7  30 
HCSNM  211.1  5.20  30 
Minimax optimization test problems
Minimax optimization test problems
Test problem  Problem defination 

Problem 1 (Yang 2010b)  \(FM _{1}(x)=\text {max}\;{f_i(x)},\; i=1,2,3,\) 
\(f_1(x)=x_1^2+x_2^4,\)  
\(f_2(x)=(2x1)^2+(2x_2)^2,\)  
\(f_3(x)=2exp(x_1+x_2)\)  
Problem 2 (Yang 2010b)  \(FM _{2}(x)=\text {max}\;{f_i(x)}, \;i=1,2,3,\) 
\(f_1(x)=x_1^4+x_2^2\)  
\(f_2(x)=(2x1)^2+(2x_2)^2,\)  
\(f_3(x)=2exp(x_1+x_2)\)  
Problem 3 (Yang 2010B)  \(FM _{3}(x)=x_1^2+x_2^2+2x_3^2+x_4^25x_15x_221x_3+7x_4,\) 
\(g_2(x)=x_1^2x_2^2x_3^3x_4^2x_1+x_2x_3+x_4+8,\)  
\(g_3(x)=x_1^22x_2^2x_3^22x_4+x_1+x_4+10,\)  
\(g_4(x)=x_1^2x_2^2x_3^22x_1+x_2+x_4+5\)  
Problem 4 (Yang 2010B)  \(FM _{4}(x)=\text {max}\;{f_i(x)}\;i=1,\ldots ,5\) 
\(f_{1}(x)=(x_110)^2+5(x_212)^2+x_3^4+3(x_411)^2 +10x_5^6+7x_6^2+x_7^44x_6x_710x_68x_7,\)  
\(f_2(x)=f_1(x)+10(2x_1^2+3x_2^4+x_3+4x_4^2+5x_5127),\)  
\(f_3(x)=f_1(x)+10(7x_1+3x_2+10x_3^2+x_4x_5282),\)  
\(f_4(x)=f_1(x)+10(23x_1+x_2^2+6x_6^28x_7196),\)  
\(f_5(x)=f_1(x)+10(4x_1^2+x_2^23x_1x_2+2x_3^2+5x_611x_7\)  
Problem 5 (Schwefel 1995)  \(FM _{5}(x)=\text {max}\;{f_i(x)},\;i=1,2,\) 
\(f_1(x)=x_1+2x_27\),  
\(f_2(x)=2x_1+x_25\)  
Problem 6 (Schwefel 1995)  \(FM _{6}(x)=\text {max}\;{f_i(x)},\) 
\(f_i(x)=x_i,\;i= 1,\ldots ,10\)  
Problem 7 (Lukšan and Vlcek 2000)  \(FM _{7}(x)= \text {max}\;{f_i(x)},\;i=1,2,\) 
\(f_1(x)=(x_1\sqrt{(x_1^2+x_2^2)}cos\sqrt{x_1^2+x_2^2})^2+0.005(x_1^2+x_2^2)^2,\)  
\(f_2(x)=(x_2\sqrt{(x_1^2+x_2^2)}sin\sqrt{x_1^2+x_2^2})^2+0.005(x_1^2+x_2^2)^2\)  
Problem 8 (Lukšan and Vlcek 2000)  \(FM _{8}(x)= \text {max}\;{f_i(x)},\;i=1,\ldots ,4,\) 
\(f_1(x)=(x_1(x_4+1)^4)^2+(x_2(x_1(x_4+1)^4)^4)^2 +2x_3^2+x_4^25(x_1(x_4+1)^4)5(x_2(x1(x_4+1)^4)^4)21x_3+7x_4,\)  
\(f_2(x)=f_1(x)+10[(x_1(x_4+1)^4)^2+(x_2(x_1(x_4+1)^4)^4)^2 +x_3^2+x_4^2+(x_1(x_4+1)^4)(x_2(x_1(x_4+1)^4)^4)+x_3x_48],\)  
\(f_3(x)=f_1(x)+10[(x_1(x_4+1)^4)^2+2(x_2(x_1 (x_4+1)^4)^4)^2+x_3^2+2x_4^2(x_1(x_4+1)^4)x_410]\)  
\(f_4(x)=f_1(x)+10[(x_1(x_4+1)^4)^2+(x_2(x_1 (x_4+1)^4)^4)^2+x_3^2+2(x_1(x_4+1)^4)(x_2(x_1(x_4+1)^4)^4)x_45]\)  
Problem 9 (Lukšan and Vlcek 2000)  \(FM _{9}(x)= \text {max}\;{f_i(x)},\;i=1,\ldots ,5,\) 
\(f_1(x)=(x_110)^2+5(x_212)^2+x_3^4+3(x_411)^2+10x_5^6+7x_6^2+x_7^44x_6x_710x_68x_7\),  
\(f_2(x)=2x_1^22x_3^4x_34x_4^25x_5+127\),  
\(f_3(x)=7x_13x_210x_3^2x_4+x_5+282\),  
\(f_4(x)=23x_1x_2^26x_6^2+8x_7+196\),  
\(f_5(x)=4x_1^2x_2^2+3x_1x_22x_3^25x_6+11x_7\)  
Problem 10 (Lukšan and Vlcek 2000)  \(FM _{10}(x)=\text {max}\;{f_i(x)}, \;i=1,\ldots ,21,\) 
\(f_i(x)=x_1exp(x_3t_i)+x_2exp(x_4t_i)\frac{1}{1+t_i},\)  
\(t_i=0.5+\frac{i1}{20}\) 
Minimax test functions properties
Function  Dimension (d)  Desired error goal 

\(FM _1\)  2  1.95222245 
\(FM _2\)  2  2 
\(FM _3\)  4  −40.1 
\(FM _4\)  7  247 
\(FM _5\)  2  \(10^{4}\) 
\(FM _6\)  10  \(10^{4}\) 
\(FM _7\)  2  \(10^{4}\) 
\(FM _8\)  4  −40.1 
\(FM _9\)  7  680 
\(FM _{10}\)  4  0.1 
The efficiency of the proposed HCSNM algorithm with minimax problems
The efficiency of invoking the Nelder–Mead method in the final stage of HCSNM for \(FM_1  FM_{10}\) minimax problems
Function  Standard CS  NM method  HCSNM 

\(FM _1\)  5375.25  1280.35  705.62 
\(FM _2\)  6150.34  1286.47  624.24 
\(FM _3\)  3745.14  1437.24  906.28 
\(FM _4\)  11,455.17  19,147.15  3162.92 
\(FM _5\)  5845.14  1373.15  670.22 
\(FM _6\)  7895.14  18,245.48  4442.76 
\(FM _7\)  11,915.24  1936.12  1103.86 
\(FM _8\)  20,000  2852.15  2629.36 
\(FM _9\)  14,754.14  19,556.14  2724.78 
\(FM _{10}\)  6765.24  1815.26  977.56 
HCSNM and other algorithms

HPS2 (Santo and Fernandes 2011) HPS2 is heuristic pattern search algorithm, which is applied for solving bound constrained minimax problems by combining the Hook and Jeeves (HJ) pattern and exploratory moves with a randomly generated approximate descent direction.

UPSOm (Parsopoulos and Vrahatis 2005) UPSOm is unified particle swarm Optimization algorithm, which combines the global and local variants of the standard PSO and incorporates a stochastic parameter to imitate mutation in evolutionary algorithms.

RWMPSOg (Petalas et al. 2007). RWMPSOg is random walk memetic particle swarm optimization (with global variant), which combines the particle swarm optimization with random walk (as direction exploitation).
Comparison between HPS2, UPSOm, RWMPSOg and HCSNM for minimax problems
Evaluation function for the minimax problems \(FM _{1}FM _{10}\)
Algorithm  Problem  Avg  SD  %Suc 

HPS2  \(FM _1\)  1848.7  2619.4  99 
\(FM _2\)  635.8  114.3  94  
\(FM _3\)  141.2  28.4  37  
\(FM _4\)  8948.4  5365.4  7  
\(FM _5\)  772.0  60.8  100  
\(FM _6\)  1809.1  2750.3  94  
\(FM _7\)  4114.7  1150.2  100  
\(FM _8\)  –  –  –  
\(FM _9\)  283.0  123.9  64  
\(FM _{10}\)  324.1  173.1  100  
UPSOm  \(FM _1\)  1993.8  853.7  100 
\(FM _2\)  1775.6  241.9  100  
\(FM _3\)  1670.4  530.6  100  
\(FM _4\)  12,801.5  5072.1  100  
\(FM _5\)  1701.6  184.9  100  
\(FM _6\)  18,294.5  2389.4  100  
\(FM _7\)  3435.5  1487.6  100  
\(FM _8\)  6618.50  2597.54  100  
\(FM _9\)  2128.5  597.4  100  
\(FM _{10}\)  3332.5  1775.4  100  
RWMPSOg  \(FM _1\)  2415.3  1244.2  100 
\(FM _2\)  –  –  –  
\(FM _3\)  3991.3  2545.2  100  
\(FM _4\)  7021.3  1241.4  100  
\(FM _5\)  2947.8  257.0  100  
\(FM _6\)  18,520.1  776.9  100  
\(FM _7\)  1308.8  505.5  100  
\(FM _8\)  –  –  –  
\(FM _9\)  –  –  –  
\(FM _{10}\)  4404.0  3308.9  100  
HCSNM  \(FM _1\)  705.62  14.721  100 
\(FM _2\)  624.24  20.83  100  
\(FM _3\)  906.28  98.24  100  
\(FM _4\)  3162.92  218.29  90  
\(FM _5\)  670.22  11.07  100  
\(FM _6\)  4442.76  87.159  95  
\(FM _7\)  1103.86  125.36  95  
\(FM _8\)  2629.336  84.80  75  
\(FM _9\)  2724.78  227.24  95  
\(FM _{10}\)  977.56  176.82  100 
HCSNM and other metaheuristics and swarm intelligence algorithms for minmax problems
HCSNM and other metaheuristics algorithms for \(FM_1  FM_{10}\) minmax problems
Function  GA  PSO  FF  GWO  HCSNM 

\(FM _1\)  
Avg  1080.45  3535.46  1125.61  2940.2  275.45 
SD  83.11  491.66  189.56  490.22  6.40 
\(FM _2\)  
Avg  1120.15  20,000  785.17  3740.14  260.53 
SD  65.14  0.00  31.94  712.19  21.60 
\(FM _3\)  
Avg  1270.65  2920.15  695.54  1120.25  262.15 
SD  95.26  269.48  50.03  417.04  15.68 
\(FM _4\)  
Avg  2220.45  9155.35  1788.26  4940.35  1704.28 
SD  488.45  649.12  118.09  313.60  36.63 
\(FM _5\)  
Avg  1040.84  5680.17  582.52  3520.45  265.54 
SD  55.89  937.44  86.77  946.36  12.01 
\(FM _6\)  
Avg  20,000  20,000  13,692.13  2080.35  1658.23 
SD  0.00  0.00  900.12  938.33  201.92 
\(FM _7\)  
Avg  1120.25  5643.65  2685.25  1020.45  177.23 
SD  65.89  4.3.22  610.07  219.90  12.72 
\(FM _8\)  
Avg  1280.35  20,000  7659.45  1620.46  1555.47 
SD  78.23  0.00  583.21  281.25  59.97 
\(FM _9\)  
Avg  20,000  6220.25  8147.45  3760.54  2732.15 
SD  0.00  727.44  1026.22  246.52  66.84 
\(FM _{10}\)  
Avg  1080.65  6680.19  748.17  1630.4  489.17 
SD  68.15  509.34  98.59  37.36  27.29 
The results in Table 13 shows that the proposed HCSNM algorithm is outperform the other metheuristics and swarm intelligence algorithm
HCSNM and SQP method
Another test for our proposed algorithm, we compare the HCSNM with another known method which is called sequential quadratic programming method (SQP) (Boggs and Tolle 1995; Fletcher 2013; Gill et al. 1981; Wilson et al. 1963).
Experimental results (mean, standard deviation and rate of success) of function evaluation between SQP and HCSNM for \(FM_1  FM_{10}\) test problems
Function  Algorithm  Mean  SD  Suc 

\(FM _1\)  SQP  4044.5  8116.6  24 
HCSNM  704  11.84  30  
\(FM _2\)  SQP  8035.7  9939.9  18 
HCSNM  727.53  22.07  30  
\(FM _3\)  SQP  135.5  21.1  30 
HCSNM  913.43  92.11  30  
\(FM _4\)  SQP  20,000  0.0  0.0 
HCSNM  3112.46  211.47  27  
\(FM _5\)  SQP  140.6  38.5  30 
HCSNM  669.23  12.42  30  
\(FM _6\)  SQP  611.6  200.6  30 
HCSNM  4451.9  89.87  26  
\(FM _7\)  SQP  15,684.0  7302.0  10 
HCSNM  1025.46  8.55  24  
\(FM _8\)  SQP  20,000  0.0  0.0 
HCSNM  2629.93  91.58  22  
\(FM _9\)  SQP  20,000  0.0  0.0 
HCSNM  2720.4  222.77  24  
\(FM _{10}\)  SQP  4886.5  8488.4  22 
HCSNM  978.13  183.49  30 
Conclusion and future work
In this paper, a new hybrid cuckoo search algorithm with NM method is proposed in order to solve integer programming and minimax problems. The proposed algorithm is called hybrid cuckoo search and Nelder–Mead algorithm (HCSNM). The NM algorithm helps the proposed algorithm to overcome the slow convergence of the standard by refining the best obtained solution from the cuckoo search instead of keeping the algorithm running with more iterations without any improvements (or slow improvements) in the results. In order to verify the robustness and the effectiveness of the proposed algorithm, HCSNM has been applied on seven integer programming and ten minimax problems. The experimental results show that the proposed algorithm is a promising algorithm and has a powerful ability to solve integer programming and minimax problems faster than other algorithms in most cases.

Apply the proposed algorithms on solving constrained optimization and engineering problems.

Modify our proposed algorithm to solve other combinatorial problems, large scale integer programming and minimax problems.
Declarations
Authors' contributions
This work was carried out in collaboration among the authors. AFA is a postdoctoral fellow for the MAT. Both authors read and approved the final manuscript
Acknowledgements
The research of the MAT is supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC). The postdoctoral fellowship of the AFA is supported by NSERC.
Competing interests
Both authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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