 Research
 Open Access
Memetic computing through bioinspired heuristics integration with sequential quadratic programming for nonlinear systems arising in different physical models
 Muhammad Asif Zahoor Raja^{1},
 Adiqa Kausar Kiani^{2},
 Azam Shehzad^{3} and
 Aneela Zameer^{4}Email author
 Received: 26 May 2016
 Accepted: 26 November 2016
 Published: 1 December 2016
Abstract
Background
In this study, bioinspired computing is exploited for solving system of nonlinear equations using variants of genetic algorithms (GAs) as a tool for global search method hybrid with sequential quadratic programming (SQP) for efficient local search. The fitness function is constructed by defining the error function for systems of nonlinear equations in mean square sense. The design parameters of mathematical models are trained by exploiting the competency of GAs and refinement are carried out by viable SQP algorithm.
Results
Twelve versions of the memetic approach GASQP are designed by taking a different set of reproduction routines in the optimization process. Performance of proposed variants is evaluated on six numerical problems comprising of system of nonlinear equations arising in the interval arithmetic benchmark model, kinematics, neurophysiology, combustion and chemical equilibrium. Comparative studies of the proposed results in terms of accuracy, convergence and complexity are performed with the help of statistical performance indices to establish the worth of the schemes.
Conclusions
Accuracy and convergence of the memetic computing GASQP is found better in each case of the simulation study and effectiveness of the scheme is further established through results of statistics based on different performance indices for accuracy and complexity.
Keywords
 Nonlinear systems
 Hybrid computing
 Genetic algorithms
 Sequential quadratic programming
 Nonlinear equations
Background
Introduction
Solving system of nonlinear equations in particular based on large set are generally considered to be the most difficult and a challenging problem for the research community in the field of numerical computation. These systems arise frequently in a spectrum of applied mathematics and engineering applications including trajectory planning, kinematics, combustion theory and neurophysiology etc. (Grosan and Abraham 2008a, b; Morgan 1987). Currently, number of numerical methods have been developed to deal with nonlinear equations effectively, however, one of the simplest, oldest and widely used solvers for these problems is the Newton–Raphson method (NRM) (Ortega and Rheinboldt 1970; Kelley 2003). Similar to most of the numerical methods for solving system of nonlinear equations, the performance of the NRM can be highly sensitive to the initial guess of the problem and generally fail with bad initial parameters. Therefore, normally, any global search methodology is used to determine the initial bias values which are then supplied to the NRM for solving viably the system of nonlinear equations. Besides NRM, many other iterative methods for solving linear and nonlinear equations are reported in the literature with their own strengths, limitations and applicability domain on specific scenarios or environments. For instance, Kelley, Campbell, and Broyden’s classically provide different solvers for these equations (Kelley 1999; Campbell et al. 1996; Darvishi and Barati 2007). Moreover, the Jacobianfree Newton–Krylov method is applied broadly for nonlinear equations arising in many applications in which an effective two sided bicolouring method is used to get the lower triangular half of the sparse Jacobian matrix via automatic differentiation (Broyden 1971; Knoll and Keyes 2004; Saad and van der Vorst 2000). Recently, many researchers, including Jaffari and Gejji, Abbasbandy, Sharma et al., Vahidi et al. have given an updated version of methods to solve the nonlinear system of equation reliably and efficiently (Jafari and Gejji 2006; Abbasbandy 2005; Vahidi et al. 2012; Sharma and Guha 2013; Sharma and Gupta 2013, 2014; Sharma and Arora 2014).
Most of the existing literature available for solving nonlinear system of equation is based on iterative and recursive procedure, and working on these methods is usually dependent on values of initial guess or start point of the algorithms. On the other hand, these systems of equations have been used in modelling of many physical problems arising in a wide spectrum of fields (Morgan 1987; de Soares 2013). Therefore, design of numerical procedures that are accurate, reliable, robust, and efficient, has attracted the research community significantly. The aim of this study is to step further in this domain by exploring and exploiting the strength of soft computing framework (SCF) to determine the solution of systems of nonlinear equation without prior knowledge of biased initial guess or weights. The soft computing techniques based on genetic algorithms and swarming intelligence has been used extensively for different applications such as VanderPol oscillatory systems (Khan et al. 2015), reliable feature selection for Arabic text summarization (AlZahrani et al. 2015), effective navigation of mobile robot in unknown environment(Algabri et al. 2014), robust feature selection and classification (Nekkaa and Boughaci 2015), fuel ignition model in combustion theory (Raja 2014), change detection mechanism in synthetic aperture radar images (Li et al. 2015), optimization of multirate quadrature mirror filter bank (Baicher 2012), integrated process planning and scheduling problems (Li et al. 2014), thin film flow of third grade fluids (Raja et al. 2014), Troesch’s problem (Raja 2014), second order system of boundary value problems (Arqub and AboHammour 2014; AbuArqub et al. 2014), prediction of linear dynamical systems (AboHammour et al. 2013), JefferyHamel Flow in the presence of high magnetic field (Raja and Samar 2014), Painlevé equations (Raja et al. 2015), modeling of electrical conducting solids (Raja et al. 2016), nanofludics problems (Raja et al. 2016), Riccati fractional differential equations (FrDEs) (Raja et al. 2015), real time cross layer optimization (Elias et al. 2012) and BagleyTorvik FrDEs (Raja et al. 2011). These are the motivating factors for the authors to explore in this domain. The objective of this study is to design memetic evolutionary techniques based on effective global search and efficient local search methodologies and then apply the proposed SCF for an accurate, effective and reliable solution of system of nonlinear equations.
The rest of the organization of the paper is as follows. In “Methods” section, proposed design methodology is presented for the solutions of nonlinear system of equation by formulation of fitness functions, stepwise working criteria and its learning mechanism. In “Results and discussion” section, the results of numerical experimentations of proposed schemes are presented for six benchmark problems, including application arising in combustion theory, neurophysiology and Kinetic modelling etc. along the comparison of the results in term of performance operators. In “Comparative studies” section, results of the proposed algorithms are compared using statistical performance indicators for both accuracy and complexity. Concluding remarks as well as future research directions are given in the last section.
Methods
In this section, design methodology is presented for finding the solution of a system of nonlinear equations. Our aim is to provide a platform for optimization of variables for the given system in order to find the accurate and precise solution. Genetic algorithms (GAs) is an optimization tool which can be used effectively for finding the solution of a given system of nonlinear equation without using the initial guess.
Formulation of fitness function
The next step is to optimize the formulated fitness function (3) by using reliable optimization mechanisms such that for \(\varepsilon \to 0\) then \({\mathbf{F}}(t) \to 0\). Eventually the correspondingly values of the vector \(t = (t_{1} ,t_{2} ,t_{3} , \ldots ,t_{n} )\) are the solution set for the given system of nonlinear equation.
Learning methodology
From the last few decades’ mathematicians and researchers made serious efforts to produce quality of initial guesses to find the optimal final solutions. This was really tough, especially when the number of variables involving in the system of nonlinear equations exceeds a specific level. Genetic algorithms (GAs) are basically the modelling of the phenomenon of natural evolution (Miettinen 1999). The effectiveness and systematic operation of GA’s depends upon not only the selection of different constitutional operators, but also on the settings of algorithm, variable parameters and the design constraints.
GAs is considered to be one of the best optimization algorithm as compared to the others because of their multidimensionally operations to produce most feasible solutions. GAs are also effective for the problems for which the construction of fitness function is much complex (Johnson et al. 2014; Kociecki and Adeli 2014). In our daily life, most problems have a very large solution space, which is generally not dealt by the ordinary algorithms while GAs deals these situations efficiently and correctly. GAs is considered to be the most viable and accurate method for finding the numerical approximate solutions of the given system of nonlinear equations by determining the best fit from the extensive range of search space. In the present study, the memetic computing approach is developed based on variants of GAs (Raja et al. 2015) hybrid with sequential quadratic programming (SQP) technique to obtain unknown design variables of nonlinear system given in Eq. (1).
 Step 1:

Initialization Initialize the chromosome with number of elements equal to the number of variable in nonlinear system of equations as:where n represents variables in nonlinear system. Set of chromosome represents an initial population and it is given mathematically as:$$c = (t_{1} ,t_{2} ,t_{3} , \ldots ,t_{n} ),$$here m is the total number of chromosomes in the population P.$$P = (c_{1} ,c_{2} ,c_{3} , \ldots ,c_{m} ),$$
Initial assignments and declarations of GAs program and are set using MATLAB builtin functions of the optimization toolbox based on ‘ga’ and ‘gaoptimset’ routines. The fix parameter settings of all twelve variants of GAs are used such as 200 generations. The choice of these settings is made with care, after a lot of experimentation, and experience of operating optimization solvers.
 Step 2:

Fitness calculation Evaluate the fitness values of each individual or chromosome of the population using a problem specific fitness function as defined in (3).
 Step 3:

Termination criteria Terminate the updating process of the algorithm, if any of the following predefined conditions are fulfilled.Go to step 5 in case of termination conditions are satisfied.

Fitness ε values are less than or equal to 10^{−35}.

Generations, i.e., 400 times step increment are made in the program execution.

Limited values of any of the functions, tolerance (TolFun), constraints tolerance (TolCon) and stall generation limit (StallGenLimit) is achieved.

 Step 4:

Reproduction Create the next generation of each variant of GAs based on a different set of combination for the reproduction mechanism using selection, crossover, and mutation routines as listed in Table 1. Go to step 2
 Step 5:

Hybridization Global best individual of GAs variants is given to a local search method based on SQP algorithm, i.e., the memetic computing approach of learning, for further refinements in the results. SQP algorithm is implemented by invoking ‘fmincon’ function of MATLAB optimization toolbox for constraint problems as per following procedure:
 (a)
Initialization Initial weights or start point of SQP algorithm is the global best chromosomes of GAs variants. The bounds, declarations and initial parameters are given in ‘optimset’ function such as number of iteration 500.
 (b)
Fitness calculation Calculate the fitness value using the Eq. (3).
 (c)Termination criteria Terminate the cyclic process of updating in variables if any of the following predefine conditions fulfilled.Go to the step 6 in case of termination conditions satisfied.

Total number of Iterations/cycles are executed.

Limited values for any TolFun, maximum function evaluations (MaxFunEvals), Xtolerance (TolX), and TolCon are achieved as given in ‘optimset’ function.

 (d)
Updating of variables: Updating of weights is made on each step increment as per SQP procedure and continues from step 5(b).
 (a)
 Step 6:

Storage Store the values of the weights, fitness, generations, MaxFunEvals and time taken for this run in case of all twelve hybrid schemes based on GASQP.
 Step 7:

Statistical analysis Repeat the procedure for a sufficient large number of times from step 1 to step 6 to generate large data set for reliable and effective analysis of the performance of the algorithms.
Proposed algorithms for system of nonlinear equations
Memetic algorithms  Global search operators for gas  Local search method  

Selection  Crossover  Mutations  
GASQP1  Stochastic uniform  Heuristic  Adaptive feasible  SQP 
GASQP2  Stochastic uniform  Heuristic  Gaussian  SQP 
GASQP3  Stochastic uniform  Arithmetic  Adaptive feasible  SQP 
GASQP4  Stochastic uniform  Arithmetic  Gaussian  SQP 
GASQP5  Reminder  Heuristic  Adaptive feasible  SQP 
GASQP6  Reminder  Heuristic  Gaussian  SQP 
GASQP7  Reminder  Arithmetic  Adaptive feasible  SQP 
GASQP8  Reminder  Arithmetic  Gaussian  SQP 
GASQP9  Roulette  Heuristic  Adaptive feasible  SQP 
GASQP10  Roulette  Heuristic  Gaussian  SQP 
GASQP11  Roulette  Arithmetic  Adaptive feasible  SQP 
GASQP12  Roulette  Arithmetic  Gaussian  SQP 
Results and discussion
In this section, results of proposed schemes for solving system of nonlinear equations are presented. Six different problems are taken for the study and proposed methods base on twelve variants of memetic computing using GAs and SQP algorithms are applied for these equations. Six different models of system of nonlinear equation are taken for numerical experimentation to check the effectiveness of the proposed design schemes. These models with governing mathematical relations are described in six problems.
Problem 1: generic nonlinear system of equations
Comparison of trained parameters along with their fitness for GA and GASQP algorithms in case of problem 1
Method  Proposed solutions  ɛ  

t _{1}  t _{2}  t _{3}  t _{4}  
GA1  −0.999404287  0.999479968  −0.027480501  −1.999042387  1.8262E−08 
GA2  0.979068695  −0.981748424  0.163179035  1.966420060  2.1943E−05 
GA3  0.868284707  0.877541519  0.392426227  −1.784981013  7.0007E−04 
GA4  −0.987183026  −0.987817444  0.124553341  −1.978604103  8.6898E−06 
GA5  −0.990822818  −0.991856983  −0.106387112  1.985169394  4.1780E−06 
GA6  0.999913716  −0.999917219  −0.010038843  1.999856079  3.8862E−10 
GA7  0.966617064  0.970313641  0.198841379  1.946148641  5.2871E−05 
GA8  −0.950190958  0.965065042  −0.277206136  1.925861774  2.0243E−04 
GA9  0.973598309  −0.976106840  0.178440757  1.956961516  3.3514E−05 
GA10  0.999854603  0.999934568  0.017078799  −1.999804148  5.4235E−09 
GA11  −0.812150872  0.832453398  0.475826424  1.699196146  1.2769E−03 
GA12  0.980516502  −0.984157549  −0.162937302  −1.969458167  2.1478E−05 
GASQP1  0.999999973  −0.999999974  0.000178418  −1.999999955  3.8017E−17 
GASQP2  −0.999999973  −0.999999974  0.000178410  −1.999999955  3.8011E−17 
GASQP3  0.999999973  0.999999974  0.000178411  1.999999955  3.8012E−17 
GASQP4  −0.999999973  0.999999974  0.000178416  1.999999955  3.8015E−17 
GASQP5  −0.999999973  0.999999974  −0.000178417  1.999999955  3.8016E−17 
GASQP6  0.999999973  −0.999999974  −0.000178410  1.999999955  3.8011E−17 
GASQP7  −0.999999973  0.999999974  −0.000178410  1.999999955  3.8011E−17 
GASQP8  0.999999975  −0.999999977  −6.283010586  1.999999959  3.3944E−17 
GASQP9  0.999999973  0.999999974  0.000178410  1.999999955  3.8012E−17 
GASQP10  −0.999999973  −0.999999974  −0.000178408  1.999999955  3.8010E−17 
GASQP11  −0.999999973  0.999999974  0.000178430  1.999999955  3.8025E−17 
GASQP12  0.999999973  0.999999974  −0.000178417  −1.999999955  3.8016E−17 
Comparison of the performance on the basis of absolute values of constitutional equations of problem 1
Method  Absolute values  Method  Absolute values  

\(f_{1} ({\mathbf{t}})\)  \(f_{2} ({\mathbf{t}})\)  \(f_{3} ({\mathbf{t}})\)  \(f_{4} ({\mathbf{t}})\)  \(f_{1} ({\mathbf{t}})\)  \(f_{2} ({\mathbf{t}})\)  \(f_{3} ({\mathbf{t}})\)  \(f_{4} ({\mathbf{t}})\)  
GA1  1.81E−04  2.85E−05  3.00E−05  1.96E−04  GASQP1  7.48E−09  2.68E−09  4.22E−09  8.44E−09 
GA2  6.14E−03  8.59E−04  8.89E−04  6.96E−03  GASQP2  7.48E−09  2.68E−09  4.22E−09  8.43E−09 
GA3  3.19E−02  8.07E−03  1.57E−02  3.83E−02  GASQP3  7.48E−09  2.68E−09  4.22E−09  8.43E−09 
GA4  3.48E−03  6.64E−05  2.22E−03  4.21E−03  GASQP4  7.48E−09  2.68E−09  4.22E−09  8.43E−09 
GA5  2.74E−03  6.52E−04  6.89E−04  2.88E−03  GASQP5  7.48E−09  2.68E−09  4.22E−09  8.43E−09 
GA6  2.16E−05  6.80E−06  1.46E−05  2.88E−05  GASQP6  7.48E−09  2.68E−09  4.22E−09  8.43E−09 
GA7  1.02E−02  3.92E−03  3.01E−03  9.15E−03  GASQP7  7.48E−09  2.68E−09  4.22E−09  8.43E−09 
GA8  1.84E−02  5.89E−03  1.00E−02  1.83E−02  GASQP8  5.42E−09  3.58E−10  3.06E−09  9.85E−09 
GA9  7.89E−03  1.73E−03  3.00E−03  7.74E−03  GASQP9  7.48E−09  2.68E−09  4.22E−09  8.43E−09 
GA10  8.50E−05  7.18E−05  7.49E−05  6.08E−05  GASQP10  7.48E−09  2.68E−09  4.22E−09  8.43E−09 
GA11  4.72E−02  5.69E−03  1.38E−02  5.16E−02  GASQP11  7.48E−09  2.68E−09  4.22E−09  8.44E−09 
GA 12  6.53E−03  3.53E−04  6.19E−04  6.54E−03  GASQP12  7.48E−09  2.68E−09  4.22E−09  8.43E−09 
It is seen from Table 2 that the values of fitness ε for variants of GAs are around 10^{−04} to 10^{−10} while these values for the hybrid approach GASQP variants are around 10^{−17}. Moreover, it is seen from Table 3 that the values of, \(f_{i} ({\mathbf{t}})\), i = 1–4, for GAs and GASQP algorithms lie 10^{−02} to 10^{−05}, and 10^{−09}, respectively. There is no noticeable difference between the performances of the memetic computing approaches; however, the best results are obtained with GASQP8 algorithm. It is observed that generally highly accurate results are determined by memetic computing techniques than variants of GAs.
Problem 2: interval arithmetic benchmark model
Comparison of trained parameters along with their fitness for GA and GASQP algorithms in case of problem 2
Method  Absolute values  

\(f_{1} ({\mathbf{t}})\)  \(f_{2} ({\mathbf{t}})\)  \(f_{3} ({\mathbf{t}})\)  \(f_{4} ({\mathbf{t}})\)  \(f_{5} ({\mathbf{t}})\)  \(f_{6} ({\mathbf{t}})\)  \(f_{7} ({\mathbf{t}})\)  \(f_{8} ({\mathbf{t}})\)  \(f_{9} ({\mathbf{t}})\)  \(f_{10} ({\mathbf{t}})\)  
GA1  1.95E−04  8.59E−05  8.04E−05  6.73E−05  2.14E−06  1.99E−04  2.03E−04  5.08E−05  2.38E−05  1.28E−04 
GA2  1.11E−04  6.11E−05  1.45E−04  1.64E−05  1.59E−04  3.00E−04  9.51E−05  1.51E−04  1.06E−04  3.37E−04 
GA3  7.70E−03  2.72E−03  7.92E−03  9.06E−03  8.95E−03  8.62E−04  3.63E−03  7.65E−03  6.87E−03  2.55E−03 
GA4  1.16E−03  8.85E−04  7.90E−04  5.93E−04  2.00E−03  8.81E−04  2.61E−04  4.13E−04  8.22E−05  1.26E−03 
GA5  6.24E−05  1.51E−06  1.10E−04  1.18E−04  3.31E−05  8.62E−06  1.83E−05  3.80E−05  9.69E−05  3.09E−05 
GA6  7.95E−05  1.03E−04  6.85E−05  1.35E−04  4.90E−05  2.66E−04  2.10E−04  2.02E−04  2.03E−04  8.51E−05 
GA7  1.15E−03  1.79E−03  1.38E−03  2.72E−04  2.27E−03  3.74E−03  2.23E−03  7.52E−04  6.01E−04  2.69E−03 
GA8  1.33E−04  4.64E−04  1.22E−04  8.29E−05  9.25E−04  5.37E−05  2.72E−05  3.39E−04  7.71E−04  7.23E−04 
GA9  4.50E−04  3.23E−05  3.72E−04  3.20E−04  8.02E−05  6.25E−04  1.68E−04  2.28E−04  5.79E−05  3.10E−04 
GA10  4.43E−04  1.85E−05  4.26E−04  7.06E−05  4.04E−05  2.16E−04  1.50E−04  2.12E−04  5.02E−04  5.50E−04 
GA11  1.13E−02  1.90E−03  2.95E−03  1.00E−02  2.57E−03  6.58E−03  4.03E−03  7.04E−03  4.98E−03  7.41E−03 
GA12  1.09E−03  7.57E−04  1.41E−03  1.54E−04  1.86E−03  1.98E−03  1.13E−03  4.83E−04  1.59E−04  4.41E−03 
GASQP1  3.51E−17  9.63E−17  8.33E−17  1.69E−17  2.21E−17  1.09E−16  5.20E−18  3.08E−17  1.52E−16  2.47E−17 
GASQP2  1.46E−16  1.65E−17  2.69E−17  1.78E−17  1.44E−16  8.24E−17  5.07E−17  3.69E−17  4.24E−17  2.58E−17 
GASQP3  9.06E−17  7.33E−17  7.98E−17  8.24E−18  1.32E−16  2.60E−18  5.64E−17  9.02E−17  4.36E−17  2.63E−17 
GASQP4  2.39E−17  1.65E−17  8.76E−17  8.67E−18  3.73E−17  3.17E−17  6.29E−17  6.11E−17  1.25E−16  7.98E−17 
GASQP5  8.89E−17  3.73E−17  2.69E−17  6.42E−17  3.51E−17  2.65E−17  5.29E−17  6.33E−17  9.89E−17  2.92E−17 
GASQP6  3.34E−17  7.16E−17  8.07E−17  3.77E−17  2.34E−17  8.24E−17  1.16E−16  9.11E−18  6.90E−17  3.01E−17 
GASQP7  3.43E−17  1.65E−17  2.60E−17  3.73E−17  2.17E−17  5.46E−17  6.07E−17  4.77E−18  4.23E−17  8.58E−17 
GASQP8  3.47E−17  1.69E−17  7.98E−17  6.42E−17  2.13E−17  4.34E−19  1.15E−16  2.17E−17  9.82E−17  2.57E−17 
GASQP9  3.47E−17  1.65E−17  3.21E−17  4.68E−17  9.11E−17  2.60E−18  1.17E−16  6.07E−18  4.23E−17  8.03E−17 
GASQP10  2.21E−17  1.69E−17  1.36E−16  6.64E−17  3.25E−17  8.67E−19  5.25E−17  5.07E−17  6.79E−17  2.96E−17 
GASQP11  2.13E−17  9.45E−17  2.78E−17  4.60E−17  8.98E−17  2.60E−17  6.07E−17  1.86E−17  6.87E−17  8.65E−17 
GASQP12  1.32E−16  1.47E−17  2.08E−17  4.38E−17  3.38E−17  2.95E−17  1.73E−18  9.41E−17  1.20E−17  3.01E−17 
Problem 3: chemical equilibrium applications
Comparison of trained parameters along with their fitness for GA and GASQP algorithms in case of problem 3
Method  Proposed solutions  ɛ  

t _{1}  t _{2}  t _{3}  t _{4}  t _{5}  
GA1  0.03228963  2.85389240  0.21989919  −0.85284380  0.03635484  6.6071E−05 
GA2  0.03754135  2.38952055  0.23779156  0.85132675  0.03623233  9.0098E−05 
GA3  0.06308618  1.18624138  0.32683618  0.84488974  0.03568812  2.6746E−04 
GA4  0.05434276  1.46521255  0.29761121  −0.84773318  0.03592740  1.9541E−04 
GA5  0.02704748  3.51739016  0.19943025  0.85423864  0.03648805  4.5340E−05 
GA6  0.02302472  4.24026955  0.18171406  0.85266824  0.03634051  3.8101E−05 
GA7  0.06294020  1.18814861  0.32687402  −0.84344415  0.03555521  2.6983E−04 
GA8  0.06329114  1.14885683  −0.33415631  −0.84616131  0.03575514  2.8793E−04 
GA9  0.03477842  2.60523708  0.22955001  0.85209604  0.03630524  7.6619E−05 
GA10  0.03333412  2.61711165  0.23398167  0.85554477  0.03670055  1.2481E−04 
GA11  0.06151041  1.23009392  −0.32203695  −0.84478930  0.03567868  2.5537E−04 
GA12  0.05962671  1.14482903  0.33821936  −0.83854375  0.03516478  3.9369E−04 
GASQP1  0.00275613  39.25753931  −0.06137573  0.85972527  0.03698507  6.6525E−16 
GASQP2  0.00311281  34.61266395  0.06502805  0.85937893  0.03695189  1.4845E−15 
GASQP3  0.00275605  39.25869183  −0.06137484  0.85972528  0.03698507  4.2918E−16 
GASQP4  0.00275616  39.25721209  −0.06137599  0.85972527  0.03698506  4.0331E−17 
GASQP5  0.00275615  39.25724045  −0.06137596  0.85972527  0.03698506  3.6564E−16 
GASQP6  0.00275614  39.25739493  −0.06137584  0.85972527  0.03698507  4.1288E−16 
GASQP7  0.00311297  34.61091320  0.06502969  0.85937892  0.03695189  2.6794E−16 
GASQP8  0.00311298  34.61075588  0.06502984  0.85937892  0.03695189  1.5478E−16 
GASQP9  0.00311294  34.61123963  0.06502938  0.85937892  0.03695189  1.3468E−15 
GASQP10  0.00275615  39.25730254  −0.06137592  0.85972527  0.03698506  4.0902E−16 
GASQP11  0.00275613  39.25760857  −0.06137568  0.85972527  0.03698507  6.9394E−16 
GASQP12  0.00311298  34.61075277  0.06502984  0.85937892  0.03695189  6.8128E−16 
Comparison of the performance on the basis of absolute values of constitutional equations of problem 3
Method  Absolute values  Method  Absolute values  

\(f_{1} ({\mathbf{t}})\)  \(f_{2} ({\mathbf{t}})\)  \(f_{3} ({\mathbf{t}})\)  \(f_{4} ({\mathbf{t}})\)  \(f_{5} ({\mathbf{t}})\)  \(f_{1} ({\mathbf{t}})\)  \(f_{2} ({\mathbf{t}})\)  \(f_{3} ({\mathbf{t}})\)  \(f_{4} ({\mathbf{t}})\)  \(f_{5} ({\mathbf{t}})\)  
GA1  1.5E−02  8.7E−03  4.3E−03  4.1E−04  5.2E−04  GASQP1  4.3E−08  9.2E−09  3.2E−08  3.5E−10  2.0E−08 
GA2  1.9E−02  9.9E−03  2.6E−03  2.9E−04  1.5E−03  GASQP2  2.2E−07  1.2E−07  5.6E−08  3.1E−10  2.7E−09 
GA3  3.1E−02  1.7E−02  9.5E−03  1.9E−04  5.3E−04  GASQP3  1.4E−07  8.2E−08  4.5E−08  2.7E−10  9.7E−10 
GA4  2.6E−02  1.6E−02  6.7E−03  1.6E−04  1.9E−04  GASQP4  7.0E−09  1.0E−09  1.2E−08  9.2E−10  2.9E−10 
GA5  1.3E−02  7.2E−03  3.7E−03  2.8E−05  5.9E−05  GASQP5  1.2E−08  2.4E−08  2.6E−08  1.8E−10  2.0E−08 
GA6  1.2E−02  4.5E−03  2.5E−03  5.9E−04  5.3E−03  GASQP6  2.8E−08  1.5E−08  2.7E−08  3.4E−10  1.8E−08 
GA7  3.1E−02  1.6E−02  1.1E−02  5.5E−04  3.0E−03  GASQP7  3.1E−08  1.3E−08  1.3E−08  3.2E−10  3.1E−09 
GA8  2.9E−02  2.1E−02  1.3E−02  1.7E−03  1.4E−03  GASQP8  4.8E−09  2.6E−08  3.8E−09  1.7E−10  8.6E−09 
GA9  1.6E−02  9.4E−03  4.9E−03  1.2E−06  6.0E−04  GASQP9  7.1E−08  1.6E−08  3.5E−08  2.9E−10  1.4E−08 
GA10  1.0E−02  1.5E−02  1.5E−02  4.0E−03  6.9E−03  GASQP10  2.4E−08  2.6E−08  2.0E−08  4.6E−10  2.1E−08 
GA11  3.0E−02  1.7E−02  9.4E−03  1.6E−04  2.0E−03  GASQP11  4.8E−08  4.7E−10  3.0E−08  3.4E−10  1.6E−08 
GA 12  2.2E−02  2.4E−02  2.5E−02  3.1E−04  1.6E−02  GASQP12  2.5E−08  3.1E−08  3.3E−08  3.0E−10  2.7E−08 
Problem 4: neurophysiology applications
Comparison of trained parameters along with their fitness for GA and GASQP algorithms in case of problem 4
Method  Proposed solutions  ɛ  

t _{1}  t _{2}  t _{3}  t _{4}  t _{5}  t _{6}  
GA1  0.46763361  −0.33451289  0.88392240  −0.94239114  −0.00000019  0.00000009  3.2086E−15 
GA2  0.99936492  −0.62752141  −0.03562736  0.77859916  −0.00000002  0.00000126  2.2147E−13 
GA3  0.24762472  0.95525282  0.96885624  −0.29579072  −0.00000035  0.00000019  5.1515E−14 
GA4  −0.05955973  −0.91317817  −0.99822579  0.40756671  0.00000306  −0.00000217  6.6386E−12 
GA5  −0.91497150  −0.59493980  −0.40351842  −0.80377028  0.00000008  −0.00000018  2.9425E−15 
GA6  0.00637146  0.92806866  0.99997967  −0.37240907  0.00000066  0.00000061  1.4258E−13 
GA7  0.70302906  −0.48804791  −0.71116111  0.87281683  0.00000002  −0.00000003  1.9908E−16 
GA8  −0.39538307  0.98389132  0.91850661  −0.17866286  0.00004137  −0.00000002  5.2853E−10 
GA9  −0.88160354  0.49496188  −0.47199049  −0.86891466  0.00000031  −0.00000009  1.7079E−14 
GA10  0.60385584  −0.97776199  0.79709341  −0.20971714  0.00000222  0.00000171  7.0492E−13 
GA11  −0.81473843  0.27084100  0.57982768  0.96262427  −0.00000213  0.00000511  3.8003E−12 
GA12  0.70613629  −0.93203731  −0.70808850  −0.36240691  0.00008539  −0.00002124  1.1376E−09 
GASQP1  0.28971734  0.00668370  0.95711225  0.99997766  0.00000000  0.00000000  8.4176E−24 
GASQP2  0.64464707  −0.70847890  −0.76448032  0.70573200  0.00000000  0.00000000  1.4960E−23 
GASQP3  0.21239967  0.58108779  −0.97718288  0.81384088  0.00000000  0.00000000  5.3650E−24 
GASQP4  −0.99315627  0.58140420  0.11679307  −0.81361487  0.00000000  0.00000000  1.0099E−23 
GASQP5  0.91699104  0.89690900  0.39890780  0.44221515  0.00000000  0.00000000  1.8177E−22 
GASQP6  −0.22521963  0.57489425  0.97430802  0.81822772  0.00000000  0.00000000  4.3799E−23 
GASQP7  −0.97134152  0.04316905  −0.23768813  0.99906778  0.00000000  0.00000000  3.1433E−23 
GASQP8  −0.57465370  −0.99701922  −0.81839668  −0.07715361  0.00000000  0.00000000  5.2944E−23 
GASQP9  0.81550584  −0.34778348  0.57874884  0.93757488  0.00000000  0.00000000  3.1435E−23 
GASQP10  −0.97410591  0.98078519  −0.22609217  0.19509079  0.00000000  0.00000000  1.5465E−23 
GASQP11  −0.60502940  −0.52408103  −0.79620313  0.85166840  0.00000000  0.00000000  1.0865E−22 
GASQP12  0.96895420  −0.90489448  −0.24724029  −0.42563598  0.00000000  0.00000000  1.4094E−23 
Comparison of the performance on the basis of absolute values of constitutional equations of problem 4
Method  Absolute values  

\(f_{1} ({\mathbf{t}})\)  \(f_{2} ({\mathbf{t}})\)  \(f_{3} ({\mathbf{t}})\)  \(f_{4} ({\mathbf{t}})\)  \(f_{5} ({\mathbf{t}})\)  \(f_{6} ({\mathbf{t}})\)  
GA1  2.99E−09  5.85E−08  6.21E−08  2.27E−08  9.66E−08  4.62E−08 
GA2  4.46E−07  2.28E−07  7.66E−07  3.32E−07  4.80E−07  3.88E−07 
GA3  4.12E−07  9.72E−08  3.08E−07  1.61E−07  6.44E−08  7.22E−08 
GA4  2.08E−06  5.00E−06  2.69E−06  1.65E−06  1.48E−07  7.48E−07 
GA5  4.00E−08  3.44E−08  1.02E−07  2.48E−08  5.67E−08  2.34E−08 
GA6  5.64E−08  5.66E−08  7.50E−07  4.91E−07  8.33E−08  1.97E−07 
GA7  2.10E−08  1.22E−08  1.24E−08  8.73E−09  1.57E−08  1.13E−08 
GA8  1.78E−05  3.74E−05  3.49E−05  2.57E−06  1.38E−05  5.94E−06 
GA9  1.77E−07  4.37E−08  7.95E−10  2.26E−07  9.64E−08  9.50E−08 
GA10  2.15E−07  2.14E−07  1.48E−06  1.11E−06  7.78E−07  3.02E−07 
GA11  1.14E−06  3.37E−07  4.02E−06  1.25E−06  1.86E−06  4.58E−07 
GA12  1.78E−05  3.23E−05  4.00E−05  4.73E−05  3.28E−05  2.35E−05 
GASQP1  5.72E−12  4.20E−12  3.08E−13  8.00E−15  8.73E−14  2.64E−14 
GASQP2  3.67E−11  3.63E−11  7.92E−13  9.68E−13  1.21E−12  1.08E−12 
GASQP3  1.17E−12  1.33E−11  7.56E−12  4.28E−12  7.02E−12  6.43E−12 
GASQP4  7.27E−12  1.91E−12  1.64E−12  2.48E−13  9.48E−13  6.52E−13 
GASQP5  4.56E−13  3.24E−11  2.31E−12  4.38E−12  2.00E−12  3.16E−12 
GASQP6  6.32E−13  1.44E−11  6.68E−12  1.32E−12  2.29E−12  2.02E−12 
GASQP7  5.52E−12  8.59E−12  1.05E−13  8.91E−12  5.52E−13  2.18E−12 
GASQP8  4.19E−12  1.64E−11  4.45E−12  1.27E−12  2.56E−12  1.80E−12 
GASQP9  4.80E−12  1.25E−11  2.00E−12  1.55E−12  1.71E−12  6.34E−13 
GASQP10  8.82E−12  3.30E−12  1.07E−12  1.58E−12  2.31E−13  7.05E−13 
GASQP11  6.73E−12  1.04E−11  1.22E−11  6.17E−12  8.42E−12  1.55E−11 
GASQP12  6.51E−12  3.04E−13  2.06E−12  3.33E−12  7.03E−13  5.12E−12 
Problem 5: combustion theory applications
Comparison of the performance on the basis of absolute values of constitutional equations of problem 5
Method  Absolute values  

\(f_{1} ({\mathbf{t}})\)  \(f_{2} ({\mathbf{t}})\)  \(f_{3} ({\mathbf{t}})\)  \(f_{4} ({\mathbf{t}})\)  \(f_{5} ({\mathbf{t}})\)  \(f_{6} ({\mathbf{t}})\)  \(f_{7} ({\mathbf{t}})\)  \(f_{8} ({\mathbf{t}})\)  \(f_{9} ({\mathbf{t}})\)  \(f_{10} ({\mathbf{t}})\)  
GA1  2.34E−12  4.11E−12  1.33E−12  4.96E−13  3.16E−09  1.34E−09  4.15E−09  8.55E−12  1.38E−09  1.63E−15 
GA2  2.22E−13  2.55E−12  5.55E−13  1.33E−13  4.83E−09  5.90E−10  1.98E−09  2.14E−12  3.85E−09  6.43E−16 
GA3  3.21E−12  4.13E−12  3.70E−12  1.50E−12  6.53E−09  1.78E−10  9.56E−10  4.33E−12  1.93E−09  2.33E−15 
GA4  3.40E−12  1.54E−11  3.21E−12  1.53E−12  3.60E−09  4.19E−10  2.46E−09  2.81E−11  1.15E−08  2.98E−15 
GA5  1.43E−12  2.36E−13  6.44E−12  1.58E−12  1.67E−09  1.46E−10  1.00E−09  3.36E−12  5.68E−10  1.07E−15 
GA6  9.49E−14  1.24E−13  6.97E−14  9.58E−15  4.29E−10  3.29E−09  6.22E−10  4.71E−13  2.49E−09  2.95E−17 
GA7  2.63E−12  2.38E−12  3.20E−12  2.40E−12  6.10E−09  8.68E−10  1.75E−09  5.29E−13  8.11E−09  8.00E−16 
GA8  2.18E−12  3.48E−12  7.63E−12  2.96E−12  7.82E−09  3.49E−10  1.38E−09  1.43E−11  3.93E−10  8.13E−16 
GA9  8.43E−13  2.10E−12  9.20E−13  7.86E−12  2.71E−09  2.04E−11  2.43E−10  7.98E−12  7.34E−09  1.09E−15 
GA10  3.26E−12  5.36E−12  7.68E−12  4.39E−13  8.81E−10  3.05E−09  1.23E−09  1.81E−11  5.57E−10  5.88E−17 
GA11  2.34E−12  4.11E−12  1.33E−12  4.96E−13  3.16E−09  1.34E−09  4.15E−09  8.55E−12  1.38E−09  1.63E−15 
GA12  2.22E−13  2.55E−12  5.55E−13  1.33E−13  4.83E−09  5.90E−10  1.98E−09  2.14E−12  3.85E−09  6.43E−16 
GASQP1  3.21E−12  4.13E−12  3.70E−12  1.50E−12  6.53E−09  1.78E−10  9.56E−10  4.33E−12  1.93E−09  2.33E−15 
GASQP2  3.40E−12  1.54E−11  3.21E−12  1.53E−12  3.60E−09  4.19E−10  2.46E−09  2.81E−11  1.15E−08  2.98E−15 
GASQP3  1.43E−12  2.36E−13  6.44E−12  1.58E−12  1.67E−09  1.46E−10  1.00E−09  3.36E−12  5.68E−10  1.07E−15 
GASQP4  9.49E−14  1.24E−13  6.97E−14  9.58E−15  4.29E−10  3.29E−09  6.22E−10  4.71E−13  2.49E−09  2.95E−17 
GASQP5  2.63E−12  2.38E−12  3.20E−12  2.40E−12  6.10E−09  8.68E−10  1.75E−09  5.29E−13  8.11E−09  8.00E−16 
GASQP6  2.18E−12  3.48E−12  7.63E−12  2.96E−12  7.82E−09  3.49E−10  1.38E−09  1.43E−11  3.93E−10  8.13E−16 
GASQP7  8.43E−13  2.10E−12  9.20E−13  7.86E−12  2.71E−09  2.04E−11  2.43E−10  7.98E−12  7.34E−09  1.09E−15 
GASQP8  3.26E−12  5.36E−12  7.68E−12  4.39E−13  8.81E−10  3.05E−09  1.23E−09  1.81E−11  5.57E−10  5.88E−17 
GASQP9  2.34E−12  4.11E−12  1.33E−12  4.96E−13  3.16E−09  1.34E−09  4.15E−09  8.55E−12  1.38E−09  1.63E−15 
GASQP10  2.22E−13  2.55E−12  5.55E−13  1.33E−13  4.83E−09  5.90E−10  1.98E−09  2.14E−12  3.85E−09  6.43E−16 
GASQP11  3.21E−12  4.13E−12  3.70E−12  1.50E−12  6.53E−09  1.78E−10  9.56E−10  4.33E−12  1.93E−09  2.33E−15 
GASQP12  3.40E−12  1.54E−11  3.21E−12  1.53E−12  3.60E−09  4.19E−10  2.46E−09  2.81E−11  1.15E−08  2.98E−15 
Problem 6: economics modelling application
Comparison of trained parameters along with their fitness for GA and GASQP algorithms in case of problem 6
Method  Proposed solutions  ɛ  

t _{1}  t _{2}  t _{3}  t _{4}  t _{5}  
GA1  −0.05232134  2.55174864  −1.79439654  −1.70502885  −0.58646497  1.2304E−09 
GA2  −0.05235315  2.55165541  −1.79424191  −1.70503153  −0.58650707  4.9779E−10 
GA3  −0.05237792  2.54633426  −1.79043794  −1.70193969  −0.58882572  3.6107E−06 
GA4  1.04961877  −1.49514638  0.03145951  −0.58636170  −1.70890458  1.7101E−06 
GA5  1.05237831  −1.49657666  0.03069550  −0.58650558  −1.70496102  2.3976E−10 
GA6  −0.05239351  2.55146647  −1.79407554  −1.70496963  −0.58658463  3.0835E−09 
GA7  1.05155963  −1.49559735  0.03101468  −0.58635525  −1.70711513  3.3197E−07 
GA8  1.04986636  −1.49478395  0.03130816  −0.58633776  −1.70956928  1.6279E−06 
GA9  −0.34925861  −0.27768182  −0.21422258  −0.15877634  −6.29819537  1.1820E−09 
GA10  −0.05251523  2.55075517  −1.79319677  −1.70475423  −0.58687119  9.4592E−08 
GA11  1.05132700  −1.49484237  0.03124121  −0.58622944  −1.70773132  9.0707E−07 
GA12  1.05401533  −1.49771615  0.03017421  −0.58658770  −1.70156701  8.8953E−07 
GASQP1  −0.34928445  −0.27770053  −0.21423688  −0.15877814  −6.29809606  4.9304E−33 
GASQP2  −0.34928445  −0.27770053  −0.21423688  −0.15877814  −6.29809606  9.8608E−33 
GASQP3  −0.05235137  2.55164291  −1.79427573  −1.70501581  −0.58650482  2.4652E−33 
GASQP4  −0.34928445  −0.27770053  −0.21423688  −0.15877814  −6.29809606  9.8608E−33 
GASQP5  1.34928445  1.74898460  2.19982701  −6.29809606  −0.15877814  9.8608E−33 
GASQP6  −0.05235137  2.55164291  −1.79427573  −1.70501581  −0.58650482  9.8608E−33 
GASQP7  −0.05235137  2.55164291  −1.79427573  −1.70501581  −0.58650482  9.8608E−33 
GASQP8  −0.34928445  −0.27770053  −0.21423688  −0.15877814  −6.29809606  2.4652E−33 
GASQP9  1.34928445  1.74898460  2.19982701  −6.29809606  −0.15877814  4.9304E−33 
GASQP10  −0.05235137  2.55164291  −1.79427573  −1.70501581  −0.58650482  9.8608E−33 
GASQP11  −0.34928445  −0.27770053  −0.21423688  −0.15877814  −6.29809606  2.4652E−33 
GASQP12  −0.34928445  −0.27770053  −0.21423688  −0.15877814  −6.29809606  2.4652E−33 
Comparison of the performance on the basis of absolute values of constitutional equations of problem 6
Method  Absolute values  Method  Absolute values  

\(f_{1} ({\mathbf{t}})\)  \(f_{2} ({\mathbf{t}})\)  \(f_{3} ({\mathbf{t}})\)  \(f_{4} ({\mathbf{t}})\)  \(f_{5} ({\mathbf{t}})\)  \(f_{1} ({\mathbf{t}})\)  \(f_{2} ({\mathbf{t}})\)  \(f_{3} ({\mathbf{t}})\)  \(f_{4} ({\mathbf{t}})\)  \(f_{5} ({\mathbf{t}})\)  
GA1  1.9E−06  3.0E−05  2.3E−05  3.3E−05  6.0E−05  GASQP1  1.1E−16  0.0E+00  0.0E+00  0.0E+00  1.1E−16 
GA2  2.9E−05  1.2E−05  3.2E−05  1.8E−05  1.3E−05  GASQP2  2.2E−16  0.0E+00  0.0E+00  0.0E+00  1.1E−16 
GA3  1.6E−03  4.2E−04  2.8E−03  1.8E−03  2.1E−03  GASQP3  0.0E+00  2.2E−16  4.4E−16  2.2E−16  0.0E+00 
GA4  4.3E−04  4.8E−05  4.4E−04  2.0E−03  2.0E−03  GASQP4  0.0E+00  0.0E+00  0.0E+00  2.2E−16  0.0E+00 
GA5  8.4E−06  7.0E−06  1.5E−06  1.1E−05  3.1E−05  GASQP5  0.0E+00  2.2E−16  0.0E+00  0.0E+00  0.0E+00 
GA6  2.8E−05  1.8E−05  4.4E−05  2.2E−05  1.1E−04  GASQP6  0.0E+00  2.2E−16  0.0E+00  0.0E+00  0.0E+00 
GA7  6.2E−04  1.1E−04  4.2E−04  3.6E−04  9.8E−04  GASQP7  0.0E+00  2.2E−16  0.0E+00  0.0E+00  0.0E+00 
GA8  5.3E−05  5.6E−04  9.0E−04  1.2E−03  2.4E−03  GASQP8  1.1E−16  0.0E+00  0.0E+00  0.0E+00  0.0E+00 
GA9  6.1E−05  6.4E−06  1.4E−05  4.4E−05  4.4E−06  GASQP9  0.0E+00  0.0E+00  0.0E+00  1.1E−16  1.1E−16 
GA10  2.9E−04  2.6E−04  2.7E−04  1.6E−04  4.7E−04  GASQP10  2.2E−16  0.0E+00  0.0E+00  0.0E+00  0.0E+00 
GA11  1.5E−03  5.4E−04  1.8E−04  8.4E−04  1.1E−03  GASQP11  0.0E+00  0.0E+00  0.0E+00  1.1E−16  0.0E+00 
GA 12  1.1E−04  3.4E−04  5.5E−04  6.9E−04  1.9E−03  GASQP12  0.0E+00  0.0E+00  0.0E+00  0.0E+00  1.1E−16 
Comparative studies
In this section, comparative studies based on the results of statistical analysis are presented for variants of GA and GASQP for all the six systems of nonlinear equations. These analyses are used to draw reliable and constructive inferences on the performance of designed algorithms.
Statistical performance indicators
Statistical performance indicator base on mean and standard deviation (STD) are used to analyze the performance for each variant of hybrid technique GASQP for 100 independent runs of the algorithm to solve all six nonlinear equations. Results based on the data generated from these simulations are used to draw a constructive and effective inference for the performance of each algorithm.
Comparison of results on the basis of statistical performance indices for truncated 75% independent runs with better fitness
Methods  Problem1  Problem2  Problem3  Problem4  Problem5  Problem6  

Mean  STD  Mean  STD  Mean  STD  Mean  STD  Mean  STD  Mean  STD  
GA1  3.6E−03  1.6E−03  1.5E−07  9.4E−08  8.5E−04  3.6E−04  5.5E−10  1.2E−09  5.3E−05  3.2E−05  5.6E−05  1.5E−04 
GA2  2.6E−03  1.7E−03  1.9E−07  1.2E−07  5.5E−03  6.1E−03  2.2E−08  7.5E−08  2.8E−05  1.6E−05  2.4E−05  4.0E−05 
GA3  5.4E−03  1.3E−03  7.3E−04  6.1E−04  1.9E−03  1.0E−03  5.7E−07  1.6E−06  1.0E−03  9.0E−04  2.0E−03  3.4E−03 
GA4  5.2E−03  1.7E−03  5.3E−06  3.3E−06  5.5E−03  5.3E−03  3.3E−07  5.6E−07  1.2E−04  6.7E−05  8.6E−04  8.1E−04 
GA5  3.3E−03  1.7E−03  1.2E−07  8.1E−08  8.2E−04  3.9E−04  9.3E−11  2.4E−10  5.4E−05  2.8E−05  4.1E−05  1.2E−04 
GA6  2.8E−03  1.8E−03  1.6E−07  9.6E−08  5.6E−03  6.4E−03  1.6E−09  6.2E−09  3.0E−05  1.6E−05  1.6E−05  2.4E−05 
GA7  5.4E−03  1.5E−03  5.0E−04  4.6E−04  1.7E−03  7.7E−04  3.1E−07  5.8E−07  6.5E−04  5.1E−04  9.5E−04  1.8E−03 
GA8  5.2E−03  1.6E−03  4.9E−06  3.1E−06  7.4E−03  6.8E−03  1.5E−06  3.0E−06  1.1E−04  6.0E−05  1.1E−03  1.2E−03 
GA9  3.5E−03  1.9E−03  7.0E−07  4.1E−07  1.0E−03  4.7E−04  1.1E−09  2.4E−09  8.1E−05  4.5E−05  5.5E−05  8.3E−05 
GA10  3.1E−03  1.9E−03  1.1E−06  5.5E−07  5.7E−03  7.2E−03  1.3E−07  3.8E−07  5.4E−05  3.1E−05  3.6E−04  1.0E−03 
GA11  5.6E−03  1.2E−03  1.4E−03  9.7E−04  2.1E−03  1.0E−03  8.1E−07  1.5E−06  1.1E−03  9.5E−04  5.9E−03  1.4E−02 
GA12  5.3E−03  1.7E−03  2.9E−05  1.8E−05  7.5E−03  6.5E−03  1.2E−06  2.0E−06  2.1E−04  1.3E−04  2.7E−03  4.1E−03 
GASQP1  6.9E−17  1.7E−17  1.8E−32  7.0E−33  3.1E−08  5.7E−08  3.9E−22  1.0E−22  1.2E−15  1.3E−15  9.7E−32  6.2E−32 
GASQP2  7.0E−17  1.8E−17  1.8E−32  6.8E−33  4.1E−08  6.7E−08  4.9E−22  2.9E−22  7.4E−16  5.9E−16  9.5E−32  6.6E−32 
GASQP3  7.0E−17  1.7E−17  1.6E−32  5.8E−33  6.2E−09  1.2E−08  4.0E−22  8.8E−23  2.4E−16  3.2E−16  1.1E−31  8.9E−32 
GASQP4  6.5E−17  1.8E−17  1.6E−32  7.2E−33  8.4E−09  1.3E−08  3.4E−22  1.3E−22  2.0E−16  2.5E−16  1.0E−31  6.5E−32 
GASQP5  6.6E−17  1.5E−17  2.0E−32  8.6E−33  4.1E−08  7.0E−08  4.0E−22  6.8E−23  6.7E−16  6.6E−16  1.1E−31  7.5E−32 
GASQP6  6.9E−17  1.7E−17  1.7E−32  8.0E−33  4.2E−08  7.0E−08  8.0E−22  1.3E−21  1.0E−15  1.1E−15  9.2E−32  5.8E−32 
GASQP7  6.7E−17  1.8E−17  1.4E−32  5.4E−33  1.2E−08  3.0E−08  4.0E−22  8.2E−23  2.5E−16  3.6E−16  8.0E−32  5.8E−32 
GASQP8  6.9E−17  1.8E−17  1.6E−32  6.0E−33  3.5E−08  6.4E−08  3.6E−22  1.2E−22  2.7E−16  4.3E−16  9.1E−32  6.6E−32 
GASQP9  6.9E−17  1.6E−17  1.7E−32  7.4E−33  3.4E−08  6.2E−08  3.9E−22  1.2E−22  8.4E−16  1.0E−15  7.8E−32  5.1E−32 
GASQP10  6.7E−17  1.7E−17  1.8E−32  7.5E−33  4.9E−08  7.2E−08  1.3E−21  2.2E−21  5.4E−16  4.4E−16  8.2E−32  5.7E−32 
GASQP11  6.8E−17  1.5E−17  1.4E−32  5.6E−33  8.8E−09  1.3E−08  3.9E−22  8.8E−23  4.7E−16  6.5E−16  8.5E−32  6.5E−32 
GASQP12  6.7E−17  1.9E−17  1.4E−32  6.0E−33  4.4E−08  6.9E−08  3.6E−22  1.4E−22  3.0E−16  4.2E−16  8.7E−32  7.1E−32 
Results of proposed variants of GASQP are compared with simulated annealing (SA), patternsearch (PS), Nelder–Mead (NM) and Levenberg–Marquardt (LM) algorithm. Results of SA, PS, NM and LM methods are determined using builtin functions in Matlab environment for all six problems of nonlinear system of equations. The mean fitness values based on 20 runs of the SA, PS, NM and LM approaches lies in the range 10^{−02} to 10^{−04}, 10^{−02} to 10^{−05}, 10^{−03} to 10^{−17} and 10^{−02} to 10^{−03}, respectively. It is clearly inferred that results of proposed GASQP algorithm is much superior from all four algorithm based on SA, PS, NM and LM methods in case of each problem.
Complexity analyses bases on values of MET, MGen and MFE indices
Index  Methods  Problem1  Problem2  Problem3  Problem4  Problem5  Problem6  

Mean  STD  Mean  STD  Mean  STD  Mean  STD  Mean  STD  Mean  STD  
MET  GASQP1  40.38  8.17  14.78  1.78  29.62  7.20  16.02  10.40  11.95  4.36  15.06  2.88 
GASQP2  37.56  9.12  13.93  2.01  28.05  6.62  15.47  10.40  11.57  5.08  13.91  3.41  
GASQP3  38.85  6.90  14.64  1.78  30.29  7.34  19.16  10.91  11.90  3.75  14.55  3.28  
GASQP4  36.41  8.30  13.39  1.40  28.78  7.20  21.22  11.01  10.82  3.25  14.01  3.34  
GASQP5  39.46  7.29  14.51  1.47  29.47  7.96  17.43  11.33  11.95  2.78  14.93  3.41  
GASQP6  38.76  8.04  13.69  1.39  29.53  7.82  13.49  9.75  11.42  3.98  14.18  3.35  
GASQP7  37.73  8.70  14.15  1.24  29.91  7.40  19.66  11.45  11.65  3.57  14.05  4.05  
GASQP8  37.83  6.20  13.65  1.37  29.01  7.55  20.86  10.75  10.81  2.33  13.56  3.91  
GASQP9  38.22  8.21  14.40  1.30  29.91  6.22  18.27  10.80  11.71  1.93  14.61  3.68  
GASQP10  37.19  7.97  13.58  1.43  27.72  7.11  15.66  10.26  10.72  1.46  14.24  2.90  
GASQP11  38.78  7.20  14.24  1.50  30.17  6.69  19.79  10.52  11.53  1.39  13.92  4.20  
GASQP12  37.36  8.11  13.41  1.10  27.44  7.77  20.20  10.01  10.52  1.77  13.86  4.10  
MGen  GASQP1  700.00  0.00  700.00  0.00  700.00  0.00  524.91  226.38  678.62  87.58  690.43  67.33 
GASQP2  688.02  55.52  700.00  0.00  700.00  0.00  545.94  216.72  675.43  91.49  681.12  92.96  
GASQP3  700.00  0.00  700.00  0.00  700.00  0.00  569.84  205.48  692.09  46.80  685.70  81.72  
GASQP4  689.25  60.24  700.00  0.00  700.00  0.00  600.45  196.65  684.31  77.29  685.67  81.89  
GASQP5  700.00  0.00  700.00  0.00  700.00  0.00  531.36  225.23  687.77  61.83  685.60  82.30  
GASQP6  698.12  14.82  700.00  0.00  700.00  0.00  520.02  219.69  692.68  46.07  681.01  93.51  
GASQP7  692.11  47.15  700.00  0.00  700.00  0.00  573.73  207.36  690.56  57.97  662.04  129.39  
GASQP8  696.99  21.62  700.00  0.00  700.00  0.00  601.21  199.05  690.94  53.13  666.38  123.16  
GASQP9  691.12  55.98  700.00  0.00  700.00  0.00  572.49  203.69  691.35  51.01  671.41  113.74  
GASQP10  689.80  56.05  700.00  0.00  700.00  0.00  562.31  204.86  680.02  81.30  690.43  67.33  
GASQP11  693.74  32.13  700.00  0.00  700.00  0.00  584.84  198.30  694.01  44.06  657.35  136.31  
GASQP12  693.01  46.41  700.00  0.00  695.42  45.80  616.24  181.66  683.73  74.37  671.48  113.46  
MFE  GASQP1  39,020.3  4583.9  19,219.8  735.9  28,524.5  5397.6  524.9  226.4  16,724.1  4258.6  17,196.8  2364.9 
GASQP2  38,348.1  6811.0  19,294.1  664.4  27,864.8  5441.3  545.9  216.7  17,137.5  4800.8  16,761.1  2743.6  
GASQP3  39,021.7  4804.0  19,289.2  665.3  29,239.5  5831.6  569.8  205.5  16,904.5  3431.6  16,822.9  2647.3  
GASQP4  37,477.2  6766.0  19,196.5  661.6  28,629.6  5892.8  600.5  196.7  16,610.1  2962.0  17,050.3  2696.2  
GASQP5  39,290.2  5371.4  19,156.5  653.5  28,136.2  6134.7  531.4  225.2  16,757.3  2599.1  16,970.1  2803.3  
GASQP6  39,251.0  5897.7  19,160.5  827.9  28,799.1  5729.5  520.0  219.7  17,034.3  3742.2  16,995.2  2846.3  
GASQP7  37,769.2  6040.8  19,094.9  730.1  28,841.3  6029.7  573.7  207.4  16,724.7  3446.3  16,334.1  3528.9  
GASQP8  38,841.6  4595.0  19,286.5  690.5  28,691.3  5951.9  601.2  199.1  16,677.9  2319.0  16,618.2  3435.4  
GASQP9  38,365.0  6084.0  19,193.7  663.1  28,931.8  5356.7  572.5  203.7  16,625.9  1585.1  16,800.0  3224.3  
GASQP10  38,162.1  6281.8  19,169.2  808.3  27,466.8  5276.7  562.3  204.9  16,413.2  1426.7  17,200.7  2398.4  
GASQP11  39,193.2  5735.7  19,145.4  700.8  29,093.3  5505.6  584.8  198.3  16,541.6  924.1  16,258.2  3736.0  
GASQP12  38,164.5  6395.4  19,270.1  751.2  27,210.9  6193.7  616.2  181.7  16,354.8  1774.4  16,844.8  3260.3 
Comparative studies bases on values of global performance indices
Methods  MFit  TMFit  GMET  GMGens  GMFEs  

Values  STD  Values  STD  Values  STD  Values  STD  Values  STD  
GA1  5.59E−03  9.65E−03  7.55E−04  1.42E−03  3.18  0.20  199.86  0.34  5058.20  2380.43 
GA2  4.89E−03  6.52E−03  1.37E−03  2.29E−03  2.31  0.19  199.95  0.13  5058.28  2380.22 
GA3  8.31E−03  1.29E−02  1.85E−03  1.91E−03  2.91  0.18  200.00  0.00  5058.33  2380.09 
GA4  8.75E−03  1.39E−02  1.94E−03  2.65E−03  2.13  0.18  199.62  0.65  5054.67  2379.08 
GA5  4.38E−03  6.83E−03  6.95E−04  1.30E−03  3.16  0.17  199.82  0.44  5058.16  2380.52 
GA6  4.38E−03  5.55E−03  1.41E−03  2.34E−03  2.37  0.18  199.97  0.08  5058.30  2380.17 
GA7  8.18E−03  1.33E−02  1.53E−03  1.97E−03  2.98  0.17  199.98  0.04  5058.32  2380.13 
GA8  8.81E−03  1.36E−02  2.31E−03  3.21E−03  2.19  0.18  199.51  1.19  5057.85  2381.28 
GA9  6.36E−03  1.11E−02  7.73E−04  1.39E−03  3.11  0.18  200.00  0.00  5058.33  2380.09 
GA10  7.99E−03  1.30E−02  1.54E−03  2.37E−03  2.32  0.18  200.00  0.00  5058.33  2380.09 
GA11  1.36E−02  2.44E−02  2.68E−03  2.46E−03  2.93  0.19  200.00  0.00  5058.33  2380.09 
GA12  1.23E−02  2.13E−02  2.62E−03  3.16E−03  2.14  0.19  199.38  0.84  5045.19  2374.33 
GASQP1  3.08E−03  6.02E−03  5.11E−09  1.25E−08  18.12  11.06  465.80  69.13  15,143.54  11,238.70 
GASQP2  2.04E−03  3.48E−03  6.92E−09  1.69E−08  17.77  10.38  465.14  59.07  14,933.64  10,933.65 
GASQP3  1.15E−03  2.54E−03  1.04E−09  2.54E−09  18.65  10.68  474.61  51.65  15,249.61  11,335.91 
GASQP4  2.20E−03  5.11E−03  1.40E−09  3.44E−09  18.65  10.03  476.99  37.35  14,872.66  10,776.58 
GASQP5  2.01E−03  3.46E−03  6.83E−09  1.67E−08  18.14  10.82  467.63  66.56  15,082.10  11,289.61 
GASQP6  1.78E−03  2.83E−03  7.01E−09  1.72E−08  17.80  11.24  465.34  71.46  15,235.05  11,342.57 
GASQP7  1.72E−03  3.37E−03  1.96E−09  4.79E−09  18.21  10.41  469.76  49.04  14,831.32  10,948.08 
GASQP8  2.20E−03  4.21E−03  5.82E−09  1.42E−08  18.76  10.55  476.41  37.61  15,061.59  11,228.61 
GASQP9  3.86E−03  7.58E−03  5.66E−09  1.39E−08  18.08  10.52  471.06  49.41  15,023.15  11,113.62 
GASQP10  3.24E−03  6.05E−03  8.23E−09  2.02E−08  17.53  10.32  470.43  53.49  14,770.72  10,847.38 
GASQP11  2.29E−03  5.08E−03  1.46E−09  3.59E−09  18.48  10.82  471.66  45.47  15,077.75  11,445.12 
GASQP12  2.53E−03  4.86E−03  7.25E−09  1.78E−08  18.33  10.23  477.27  31.12  14,698.36  10,850.71 
Conclusions and future research directions

Design of stochastic computational intelligence algorithms based on memetic computing using variants of GAs hybrid with SQP algorithm are developed in this study for solving systems of nonlinear equations arising in arithmetic benchmark, chemical equilibrium, neurophysiology, combustion theory, and economics models and proposed results established the accuracy, reliability and effectiveness.

Accuracy and convergence of the memetic computing GASQP are found better than that of GAs in case of simulation studies performed for all six problems.

Validation and verification of the performance of memetic algorithms are evaluated on the basis of statistical indicators in terms of the mean and standard deviations; which are calculated for 100 independent runs for all six nonlinear systems and small values of these indices show that proposed algorithms are consistent.

The correctness of the proposed schemes are examined further based on values of global performance indices, i.e., MFit, GMET, GMGens and GMFEs, and results show that all given schemes provide viable but the performance of GASQP4 is relatively better from the rest.

Computational complexity analyses of the proposed design scheme GASQP in terms of MET, MGens and MFEs values show that there is no prominent variation found in complexity operators. However, the complexity of the memetic computing approaches is on the higher side as compared with GAs but this factor can be overshadowed due to the superior performance of hybrid algorithms from the rest.

Beside the provision of reliable and viable solutions of nonlinear systems of equations other valuable advantages of the proposed schemes are simplicity of the approach, easily understandable methodologies, implementation ease, readily extendable to different applications and availability of the solutions without prior known initial bias guess.

Modern optimization solvers may play their significant role to enhance accuracy and convergence in solving nonlinear system of equations. Few recently introduced such schemes include fractional particle swarm optimization (PSO) algorithm, fractional Darwinian PSO, chaotic PSO, genetic programming, differential evolution, chaos optimization algorithms and gravitational search algorithms etc.

One should explore to extend the application of these variants of memetic algorithms to solve stiff nonlinear differential equation, differential–algebraic systems and integral equations by transforming into a system of nonlinear equations with the help of discretization process.
Declarations
Authors’ contributions
MAZR, AKK, AS, and AZ contributed towards the main idea, structure, organization of the article. The simulation works in Mathematica and Matlab software’s is carried out by MAZR and AS. MAZR, AKK, AS, and AZ contributed equally in the writeup of the paper and combining examples of their fields, while necessary reviews been carried out by AZ. All authors read and approved the final manuscript.
Acknowledgements
None.
Competing interests
The authors declare that they have no competing interests.
Availability of data and materials
The data and material is presented in the manuscript.
Consent for publication
All the authors agreed for publication.
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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