Hybrid Simulated Annealing and Nelder-Mead Algorithm for Solving Large-Scale Global Optimization Problems

International Journal of Research in Computer Science
eISSN 2249-8265 Volume 4 Issue 3 (2014) pp. 1-11
www.ijorcs.org, A Unit of White Globe Publications
doi: 10.7815/ijorcs.43.2014.084
HYBRID SIMULATED ANNEALING AND
NELDER-MEAD ALGORITHM FOR SOLVING
LARGE-SCALE GLOBAL OPTIMIZATION
PROBLEMS
Ahmed Fouad Ali
Suez Canal University, Dept. of Computer Science, Faculty of Computers and Information, Ismailia, EGYPT
ahmed_fouad@ci.suez.edu.eg
Abstract: This paper presents a new algorithm for
solving large scale global optimization problems
based on hybridization of simulated annealing and
Nelder-Mead algorithm. The new algorithm is called
simulated Nelder-Mead algorithm with random
variables updating (SNMRVU). SNMRVU starts with
an initial solution, which is generated randomly and
then the solution is divided into partitions. The
neighborhood zone is generated, random number of
partitions are selected and variables updating process
is starting in order to generate a trail neighbor
solutions. This process helps the SNMRVU algorithm
to explore the region around a current iterate
solution. The Nelder- Mead algorithm is used in the
final stage in order to improve the best solution found
so far and accelerates the convergence in the final
stage. The performance of the SNMRVU algorithm is
evaluated using 27 scalable benchmark functions and
compared with four algorithms. The results show that
the SNMRVU algorithm is promising and produces
high quality solutions with low computational costs.
Keywords: Global optimization, Large-scale
optimization, Nelder-Mead algorithm, Simulated
annealing.
I. INTRODUCTION
Simulated annealing (SA) applied to optimization
problems emerge from the work of S. Kirkpatrick et al.
[18] and V. Cerny [3]. In these pioneering works, SA
has been applied to graph partitioning and VLSI
design. In the 1980s, SA had a major impact on the
field of heuristic search for its simplicity and
efficiency in solving combinatorial optimization
problems. Then, it has been extended to deal with
continuous optimization problems [25] and has been
successfully applied to solve a variety of applications
like scheduling problems that include project
scheduling [1, 4], parallel machines [5, 17, 21, 34].
However, implementing SA on the large scale
optimization problems is still very limited in
comparison with some other meta-heuristics like
genetic algorithm, differential evolution, particle
swarm, etc. The main powerful feature of SA is the
ability of escaping from being trapped in local minima
by accepting uphill moves through a probabilistic
procedure especially in the earlier stages of the search.
In this paper, we produce a new hybrid simulated
annealing and Nelder-Mead algorithm for solving
large scale global optimization problems. The
proposed algorithm is called simulated Nelder-Mead
algorithm with random variables updating
(SNMRVU). The goal of the SNMRVU algorithm is
construct an efficient hybrid algorithm to obtain
optimal or near optimal solutions of a given objective
functions with different properties and large number
of variables. In order to search the neighborhood of
the current solution with large variables, we need to
reduce the dimensionality. The proposed SNMRVU
algorithm searches neighborhood zones of smaller
number of variables at each iteration instead of
searching neighborhood zones of all the n variables.
Many promising methods have been proposed to solve
the mentioned problem in Equation 1, for example,
genetic algorithms [14, 24], particle swarm
optimization [23, 28], ant colony optimization [31],
tabu search [9, 10], differential evolution [2, 6] scatter
search [15, 20], and variable neighborhood search [11,
27]. Although the efficiency of these methods, when
applied to lower and middle dimensional problems,
e.g., D < 100, many of them suffer from the curse of
dimensionality when applied to high dimensional
problems.
The quality of any meta-heuristics method is its
capability of performing wide exploration and deep
exploitations process, these two processes have been
invoked in many meta-heuristics works through
different strategies, see for instance [8, 12, 135].
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2 Ahmed Fouad Ali
SNMRVU algorithm invoked these two processes
and combined them together in order to improve its
performance through three strategies. The first
strategy is a variable partitioning strategy, which
allows SNMRVU algorithm to intensify the search
process at each iteration. The second strategy is an
effective neighborhood structure, which uses the
neighborhood area to generate trail solutions. The last
strategy is final intensification, which uses the Nelder-
Mead algorithm as a local search algorithm in order to
improve the best solutions found so far. The
performance of the SNMRVU algorithm is tested
using 27 functions, 10 of them are classical function
[16] and they are reported in Table 3, the reminder 17
are hard functions [22], which are reported in Table
10. SNMRVU is compared with four algorithms as
shown in Section IV, Section V. The numerical results
show that SNMRVU is a promising algorithm and
faster than other algorithms, and it gives high quality
solutions. The paper is organized as follows. In
Section II, we define the global optimization problems
and give an overview of the Nelder-Mead algorithms.
Section III describes the proposed SNMRVU
algorithm. Sections IV and V discuss the performance
of the proposed algorithm and report the comparative
experimental results on the benchmark functions.
Section VI summarizes the conclusion.
II. PROBLEM DEFINITION AND OVERVIEW OF
THE NELDER-MEAD ALGORITHM
In the following subsections, we define the global
optimization problems and present an overview of the
Nelder-Mead algorithms and the necessary
mechanisms in order to understand the proposed
algorithm.
A. Global optimization problems definition
Meta-heuristics have received more and more
popularity in the past two decades. Their efficiency
and effectiveness to solve large and complex problems
has attracted many researchers to apply their
techniques in many applications. One of these
applications is solving the global optimization
problems, this problem can express as follows:
Where f(x) is a nonlinear function, x = (x1, ..., xn) is
a vector of continuous and bounded variables, x, l, u ϵ
Rn
.
B. The Nelder-Mead algorithm
The Nelder-Mead algorithm [29] is one of the most
popular derivative free nonlinear optimization
algorithm. The Nelder-Mead algorithm starts with n +
1 point (vertices) as x1, x2, ..., xn+1. The algorithm
evaluates, order and re-label the vertices. 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
Nelder-Mead algorithm; coefficients of reflection ρ,
expansion χ, contraction γ , and shrinkage σ. These
parameters are chosen to satisfy ρ > 0, χ > 1, 0 < γ < 1
and 0 < σ < 1.
III. THE PROPOSED SNMRVU ALGORITHM
The proposed SNMRVU starts with an initial
solution generated randomly, which consists of n
variables. The solution is divided into η partitions,
each partition contains υ variables (a limited number
of dummy variables may be added to the last partition
if the number of variables n is not a multiple of υ). At
a fixed temperature, random partition(s) is/are
selected, and trail solutions are generated by updating
random numbers of variables in the selected partition.
The neighbor solution with the best objective function
value is always accepted. Otherwise, the neighbor is
selected with a given probability that depends on the
current temperature and the amount of degradation ∆E
of the objective value. ∆E represents the difference in
the objective value between the current solution and
generated neighboring solution. At a particular level of
temperature, many trails solutions are explored until
the equilibrium state is reached, which is a given
number of iterations executed at each temperature T in
our in SNMRVU algorithm. The temperature is
gradually decreased according to a cooling schedule.
The scenario is repeated until T reached to Tmin.
SNMRVU uses the Nelder-Mead algorithm as a local
search algorithm in order to refine the best solution
found so far.
A. Variable partitioning and trail solution generating
An iterate solution in SNMRVU is divided into η
partitions. Each partition contains υ variables, i.e., η =
n/υ. The partitioning mechanism with υ = 5 is shown
in Figure 1. The dotted rectangular in Figure 1 shows
the selected partition with its variables. Trail solutions
are generating by updating all the variables in the
selected partition(s). The number of generated trail
solutions is μ, the number of the selected partition(s)
at each iteration in the algorithm inner loop (Markove
chain) is w, where w is a random number, w ϵ [1, 3].
Once the equilibrium state is reached (a given number
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Hybrid Simulated Annealing and Nelder-Mead Algorithm for Solving Large-Scale Global Optimization Problems 3
of iterations equal to μ), the temperature is gradually
decreased according to a cooling schedule, and the
operation of generating new trail solutions is repeated
until stopping criteria satisfied, e.g., T ≤ Tmin.
Figure 1: Variable Partitioning
B. The SNMRVU algorithm
SNMRVU algorithm starts with an initial solution
x0
generated randomly. At each iteration the solution
is divided into η partitions. The neighborhood zone is
generated in order to generate a trail neighborhood
solutions. The generated neighbor solution that
improves the objective function is always selected.
Otherwise the solution is accepted with a given
probability e-∆E/T, where T is the current
temperature, and ∆E is the amount of the degradation
of the objective function. The scenario is repeated
until the equilibrium state is reached. In SNMRVU
algorithm, the equilibrium state is a given number of
iterations executed at each temperature, this number is
equals to μ, μ is a user predefined number. Once the
equilibrium state is reached, the temperature is
decreased gradually according to a cooling schedule.
This process is repeated until the stopping criteria
satisfied, which is in our algorithm T ≤ Tmin. In order
to refine the best solution, SNMRVU uses the Nelder-
Mead algorithm as a local search algorithm in the final
stage. The structure of the SNMRVU with the formal
detailed description is given in Algorithm 1, all
variables in Algorithm 1 and its values are reported in
Table 1, 2.
Algorithm 1 SNMRVU Algorithm
1. Choose an initial solution x0
.
2. Set initial values for Tmax, Tmin, β, μ, υ , z0.
3. Set z= z0, T= Tmax, x=x0
.
4. Repeat
5. k:=0.
6. Repeat
7. Partition the solution x into η partitions, where
η=n/ υ.
8. Generate neighborhood trail solutions y1
,...yn
around x.
9. Set x' equal to the best trail solution from
y1
,...yn
.
10. ∆E =f(x')-f(x).
11. If ∆E ≤ 0 Then
12. x=x'.
13. z=τzmax, τ>1
14. Else
15. z=αzmin, α<1
16. If rand( )< e-∆E/T
Then
17. x=x'.
18. EndIf
19. EndIf
20. k=k+1.
21. Until k ≤ μ
22. T=T-β.
23. Until T≤ T min
24. Apply Nelder-Mead algorithm at Nelite solutions.
Table 1: Parameter used in SNMRVU Algorithm
Parameters Definitions
n No. of variables
υ Partition size
η No. of total partitions
μ No of maximum trail solutions
ζ Id no, of the selected partition
w No of the selected updated partition(s)
z Radius of neighborhood
z0 Initial radius of neighborhood
zmax Maximum setting of z
zmin Minimum setting of z
Tmax Initial temperature
Tmin Final temperature
∆E The different objective value between the
current solution and the generated solution
β Temperature reduction value
Nmax No. of maximum iteration in the Nelder-
Mead algorithm
Nelite No. of best solutions for intensification
Table 2: Parameter Setting
Parameter Value
υ 5
η n/ υ
μ 3
zmax (u-l)/2
zmin (u-l)/50
z0 (zmax+ zmin)/2
Tmax n
β 1
Nmax 5n
Nelite 1
IV. NUMERICAL EXPERIMENTAL
This section presents the performance of our
proposed SNMRVU algorithm and the comparison
results between it and other four benchmark
algorithms. SNMRVU uses a selected two sets of
benchmark functions with different properties. The
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4 Ahmed Fouad Ali
first set of functions contains of 10 classical functions
which listed in Table 3, the second set of functions
contains of 17 hard functions from special session on
real parameter optimization of IEEE congress on
evolutionary computations, CEC2005 [22], the results
of the benchmark algorithms and our SNMRVU
algorithm are averaged over 30 runs. The numerical
results of all tested functions are reported in Tables 4,
5, 6, 7, 9, and 10.
A. Performance analysis
In the following subsection, we present the
efficiency of the applied strategies in our proposed
SNMRVU algorithm.
Table 3: Classical Functions
f Function name Bounds Global
minimum
F1 Branin Function [-5,10] F(x*
)= 0.397887
F2 Booth Function [-10,10] F(x*
)= 0
F3 Matyas Function [-5,10] F(x*
)= 0
F4 Rosenbrock
Function
[-5,10] F(x*
)= 0
F5 Zakhrof Function [-5,10] F(x*
)= 0
F6 Trid Function [-n2
,n2
] at n=6,
F(x*
)= -50
F7 SumSqure function [-5,10] F(x*
)= 0
F8 Sphere Function [-100,100] F(x*
)= 0
F9 Staircased
Rosenbrock
[-5,10] F(x*
)= 0
F10 Staircased LogAbs
Function
[-5,10] F(x*
)= 0
1. The efficiency of variables partitioning:
In order to check the efficiency of variable
partitioning, we compare our SNMRVU algorithm
with the basic SA algorithm using the same
termination criteria and the same SA parameters, the
results are shown in Figure 2. The dotted line
represents the results of the basic SA (without
partitioning mechanism), the solid line represents the
result SNMRVU algorithm with the variables
partitioning process. f2 is selected with dimensions
100, 500 and 1000 by plotting the number of iterations
versus the function values. Figure 2 shows that the
function values of SNMRVU are rapidly decreases as
the number of iteration increases rather than the
performance of the basic SA. We can conclude from
Figure 2 that exploring the neighborhood of all
variables at the same time as the basic SA do, can
badly effects the progress of the search. However,
exploring the neighborhood gradually through
partition with small number of variables as applied in
the SNMRVU algorithm can give better results.
2. The performance of the Nelder-Mead algorithm
SNMRVU uses the Nelder-Mead algorithm as an
intensification process in order to accelerate the
convergence in the final stage instead of let the
algorithm run for several iterations without significant
improvement of the function values. Figure 3 shows
the performance of the Nelder-Mead algorithm on two
functions f3, h2 by plotting the number of iterations
versus the function values. The dotted line in Figure 3
represents the behavior of the final intensification
process.
B. SNMRVU and other algorithms.
We compared our SNMRVU algorithm with four
benchmark algorithm. The results of all algorithms are
taken from its original papers.
1. SSR (Scatter search with randomized subset
generation).
In [16] two scatter search methods were proposed,
they called SSR, SSC, and some computational testes
were applied on classical direct search methods [7, 19,
30, 32, 33, 36, 37], and joined them with scatter search
method. In SSR method, a subset is created in two
steps. First, the subset size is decided, and second
selecting the solutions from the reference set randomly
until the required subset size is obtained. SSR uses
biased randomized method in order to select the subset
size. The different size probabilities are adjusted
dynamically based on search history.
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Figure 2: Basic SA vs. SNMRVU.
Figure 3: The Performance of the Nelder-Mead Algorithm.
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6 Ahmed Fouad Ali
2. SSC (Scatter search with clustering subset
generation).
SSC method uses k clustering algorithm [26] in
order to create subsets by dividing the reference set
points into k clusters. The cluster are then ranked by
the quality of the points, which they contain, and the
cluster are selected as subset sequentially.
C. Comparison between SNMRVU, SSR and SSC on
functions with 16-512 dimensions.
The proposed SNMRVU algorithm was tested and
compared with SSR, SSC algorithm on 10 benchmark
functions, which reported in Table 3. The dimensions
of functions are 16, 32, 64, 128, 256, and 512,
respectively, the results of the three algorithms are
reported in Tables 4, 5, 6. SNMRVU was run 30 times
with different initial solutions, the average of function
evaluation values of each test function is reported. In
the case where at least one run fails to the global
solution, to produce a converged point, the ratio of
successful run is recorded in parentheses. The
experimental results for f5 in Table 6 at 256, 512
dimensional and f6 in Table 5 at 128 dimensional were
not reported in its original paper [16]. The best mean
values of the four methods are marked in bold face.
SNMRVU algorithm can obtain better function values
if we apply one termination criterion, which is the
number of evaluation function values is ≤ 50, 000.
Table 4: Function Evaluations of SSR, SSC and SNMRVU with 16, 32 Dimensions.
16 dimensions 32 dimensions
F SSR SSC SNMRVU SSR SSC SNMRVU
F1 406.6 374.2 456.4 779.7 783.2 1086.2
F2 806.4 737.2 218.3 1732.4 1436.9 393.6
F3 1109.8 902.1 217.3 2422.8 1834.6 327.1
F4 (0.9) 12500.4 5134 (0.9) 21397.7 13213.4
F5 6827.8 4030.5 439.4 32626.4 16167.1 888.3
F6 4357.8 3447.5 643.1 (0.8) (0.3) 2076
F7 298.3 344.7 312.6 622.5 647.5 734.5
F8 227.2 323.8 133.4 462.5 463.2 261.5
F9 31978.3 13575.7 2645.3 26315.3 22706.1 5112.3
F10 441.8 445.8 395.4 893.7 896.3 698.7
64 dimensions 128dimensions
F1 1636.6 1626.4 1996.1 3448.8 3443.7 4130.2
F2 3154.8 3148.5 842.8 6737.8 7516.3 1545.4
F3 3854.2 3827.1 777.4 8575 9687.9 1674
F4 39193.2 (0.2) 39212.3 (0.0) (0.0) (0.0)
F5 (0.0) (0.1) 1654.2 (0.0) (0.0) 4125.3
F6 (0.0) (0.0) 7407.3 - - 34440.3
F7 1369 1348.4 1996.4 2840.7 2808.8 4135.2
F8 958 983.9 517.3 1979.4 2111.7 1029.1
F9 37090.5 (0.0) (0.2) (0.0) (0.0) (0.0)
F10 1858.8 1861 1559.7 3880.2 3899.9 2128.9
256 dimensions 512 dimensions
F1 7173.4 7184.6 8273.7 15206.9 15221.8 1734.8
F2 16293.8 16915.3 3081.3 36840 36635.4 7179.2
F3 24207 24903.1 3338.1 (0.0) (0.0) 6656.8
F4 (0.0) (0.0) (0.0) (0.0) (0.0) (0.0)
F5 - - 9113.1 - - 18312.5
F6 (0.0) (0.0) (0.0) (0.0) (0.0) (0.0)
F7 6057.1 6105.4 7934.3 12866.2 13021.4 23215.7
F8 4236.8 4258.6 2310.3 8890.4 8890.2 4614.1
F9 (0.0) (0.0) (0.0) (0.0) (0.0) (0.0)
F10 8070.3 8212.7 6811.2 17180.3 16947.2 12593.3
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Table 7 shows the mean number of function values.
For function f1, we obtained the exact global minimum
value for all given dimensions with function
evaluation values ≤ 50, 000. Also we obtained the
exact global minimum values for functions f6, h2 till n
= 128, we can reach to the global minimum of these
functions for n > 128 by increasing the number of
function evaluation values, for example for function f6
our SNMRVU algorithm can obtains the value -
2828800, which is the global minimum of it within a
cost of 97400 at dimension n = 256. For the other
functions, we obtained very close results to their
global minimum. We can conclude that, the SNMRVU
algorithm is robust and can obtains better function
values with lower computational cost than the other
two algorithms values with lower computational cost
than the other two algorithms.
Table 7: Mean Number of Function Values with 16- 512
Dimensions.
Mean number of function values
F 16 32 64
F1 0 0 0
F2 2.15e-11 5.33e-12 2.7e-11
F3 1.73e-13 2.6e-08 1.31e-12
F4 2.34e-09 6.86e-10 3.17e-08
F5 2.1e-10 0 1.34e-08
F6 0 1.77e-08 0
F7 1.44e-09 0 7.88e-08
F8 0 1.75e-03 0
F9 5.3e-04 1.44e-09 (0.0)
F10 4.12e-09 2.45e-07
F 128 256 512
F1 0 0 0
F2 7.14e-11 9.06e-10 3.34e-09
F3 5.57e-11 6.56e-11 2.12e-09
F4 (0.0) (0.0) (0.0)
F5 6.01e-07 1.54e-05 4.32e-05
F6 0 (0.0) (0.0)
F7 3.85e-07 2.11e-06 9.85e-06
F8 4.9e-07 1.42e-14 2.81e-14
F9 (0.0) (0.0) (0.0)
F10 4.3e-07 3.12e-04 1.12e-04
V. HARD BENCHMARK TEST FUNCTIONS
In this section, we evaluate the SNMRVU
algorithm with another type of functions with various
properties provided by CEC2005 special session [23].
These functions are based on classical benchmark
functions after applying some modifications. These
modifications make these functions harder to solve
and resistance to simple search. We applied our
SNMRVU algorithm on 17 functions as reported
Table 8. The functions in Table 8 are shifted, rotated,
expanded and combined variants of classical functions
such as sphere's, Rastrigin's, Rosenbrock's, Ackley's
function. In the next subsection, we study the
performance of SNMRVU on some hard functions,
before comparing the results of SNMRVU algorithm
with the results of two algorithms.
A. The performance of SNMRVU algorithm with the
hard functions.
Figure 4 represents general performance of
SNMRVU algorithm on three functions h1, h2, h6, with
different properties by plotting the number of
iterations versus the function values. Figure 4 shows
that the function values are rapidly decreases as the
number of iterations increases, and the performance of
SNMRVU algorithm is promising and can obtain good
solutions with some of hard functions.
We compared our SNMRVU algorithm with the
following methods:
Table 8: Benchmark Hard functions
H Function name Bounds Global
minimum
H1 Shifted Sphere [-100,100] -450
H2 Shifted Schwefel's 1.2 [-100,100] -450
H3 Shifted rotated high
conditioned elliptic
[-100,100] -450
H4 Shifted Schwefel's 1.2
with noise in fitness
[-100,100] -450
H5 Schwefel's 2.6 with
global optimum on
bounds
[-100,100] -310
H6 Shifted Rosenbrock's [-100,100] 390
H7 Shifted rotated
Griewank's without
bounds]
[0,600] -180
H8 Shifted rotated
Ackley's with global
optimum
[-32,32] -140
H9 Shifted Rastrigin's [-5,5] -330
H10 Shifted rotated
Rastrigin's
[-5,5] -330
H11 Shifted rotated
Weierstrass
[-0.5,0.5] 90
H12 Schwefel's 2.13 [-100,100] -460
H13 Expanded extended
Griewank's +
Rosenbrock's
[-3,1] -130
H14 Expanded rotated
extended Sca_e's
[-100,100] -300
H15 Hybrid composition 1 [-5,5] 120
H16 Rotated hybrid comp. [-5,5] 120
H17 Rotated hybrid comp.
Fn 1 with noise in
fitness
[-5,5] 120
1. DECC-G( Deferential Evolution with cooperative
co-evolution with a group-based problem) [37].
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8 Ahmed Fouad Ali
DECC-G designed to increase the chance of
optimizing interacting variable together by changing
grouping structure dramatically. It splits an objective
vector into smaller subcomponents and evolves each
component with an evolutionary algorithm EA. After
each recycle the DECC-G applied a weight to each
subcomponent and evolves the weight vector with a
certain optimizer. DECC-G uses one termination
criterion, which is the number of evaluation number is
2.5 e6
, 5 e6
, for n = 500, 1000, respectively.
2. AMALGAM-SO (A multi-algorithm genetically
adaptive method for single objective optimization)
[35].
AMALGAM-SO is a hybrid evolutionary search
algorithm. It considers the covariance matrix
adaptation (CMA), genetic algorithm (GA),
evolutionary strategy, deferential evolution(DE),
parental-centric recombination operator (PCX), and
particle swarm optimizer (PSO) for population
evolution. It applies a self-adaptive learning strategy to
favor individual algorithm that demonstrate the
highest reproductive success during the search. The
termination criterion of AMALGAM-SO method is to
reach the global minimum of some functions within
the tolerance limit 10-6
and 10-6
for the others.
Figure 4 : The Performance of SNMRVU Algorithm With the Hard Functions.
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B. Comparison between AMALGAM-SO and
SNMRVU.
We compare the SNMRVU algorithm with
AMALGAM-SO algorithm. The two algorithms are
tested on 17 benchmark functions which listed in
Table 8. The average means and standard deviations
are listed in Tables 9. These results for 50 dimensions.
The results between parentheses represent the average
of the final objective function values, when none of
the optimization runs reached to the recorded
tolerance limit in Tol column in Table 9. The reported
results in Table 9 show that the SNMRVU algorithm
is outperforms the other algorithm for functions h1, h2,
h3,h6, h7, h9, h10, h12, h13. The AMALGAM-SO results
for functions h10, h12 are not reported in its original
paper. The two algorithms are unable to locate
solutions within the tolerance limit for function h8, h13,
h14, h16, h17.
Table 9: Functions 1-17 in Dimension n = 50: Mean
Number of Function Evaluations.
AMALGAM-SO SNMRVU
H Tol Mean Std Mean Std
H1 1e-6 7907 209 4653 9.12
H2 1e-6 3.41e4 681 14496 792.13
H3 1e-6 1.29e5 18.3 29732 195.14
H4 1e-6 3.48e5 1.67e5 (3.116e3) (495.19)
H5 1e-6 4.92e5 2.68e4 (1.127e4) (1532.4)
H6 1e-2 2.11e5 1.2e5 39546.24 3245.1
H7 1e-2 9596 463 7214 1.15
H8 1e-2 (21.13) (0.03) (19.99) (8.4e-06)
H9 1e-2 4.76e5 4.09e4 10342.11 95.2
H10 1e-2 - - (198.371) (114.5)
H11 1e-2 2.65e5 1.16e5 (14.23) (3.425)
H12 1e-2 - - 24132.21 1854.4
H13 1e-2 (3.62) (1.08) (1.945) (2.241)
H14 1e-2 (19.69) (0.92) (19.5) (0.45)
H15 1e-2 4.93e5 2.81e4 (9.24) (8.14)
H16 1e-2 (12.9) (3.73) (22.63) (66.4)
H17 1e-2 (65.63) (59.58) (74.5) (61.44)
C. Comparison between DECC-G and SNMRVU.
In order to test the performance of SNMRVU
algorithm with dimensions ≥ 500, SNMRVU is
compared with DECC-G algorithm for 500, 1000
dimensions. The average means of objective function
values and function evaluations are recorded in Table
10. The results in Table 10 shows that the SNMRVU
performed significantly better on functions h1, h5, h6,
h8, h9 and with cheaper cost than the other algorithm.
Table 10 : Mean Number of Function Values and Function
Evaluation with 500, 1000 Dimensions.
Function Values
H n DECC-G SNMRVU
H1 500 3.71e-13 0
1000 6.84e-13 0
H3 500 3.06e8 9.84e8
1000 8.11e8 3.45e9
H5 500 1.15e5 1.09e5
1000 2.20e5 2.14e5
H6 500 1.56e3 4.15e5
1000 2.22e3 8.15e2
H8 500 2.16e1 2.0e1
1000 2.16e1 2.0e1
H9 500 4.5e2 1.20e2
1000 6.32e2 2.11e2
H10 500 5.33e3 1.12e4
1000 9.73e3 2.21e4
H13 500 2.09e2 1.15e3
1000 3.56e2 3.86e3
Function Evaluations
H n DECC-G SNMRVU
H1 500 2.5e6 29455
1000 5.00e6 56324
H3 500 2.5e6 69345
1000 5.00e6 169521
H5 500 2.5e6 65301
1000 5.00e6 132547
H6 500 2.5e6 134520
1000 5.00e6 233548
H8 500 2.5e6 63124
1000 5.00e6 139587
H9 500 2.5e6 63548
1000 5.00e6 91247
H10 500 2.5e6 74501
1000 5.00e6 166487
H13 500 2.5e6 121458
1000 5.00e6 211485
VI. CONCLUSIONS AND FUTURE WORK
In this paper, we proposed a SNMRVU algorithm
in order to solve large scale global optimization
problems. SNMRVU uses three strategies, the first
strategy is variable partitioning process, which help
our algorithm to achieve high performance with high
dimensional problems, the second strategy is the
effective neighborhood structure strategy, which help
the SNMRVU algorithm to explore region around a
current iterate solution by updating variables in the
random selected partitions and the last strategy is the
final intensification by applying the Nelder-Mead
algorithm in the final stage. Invoking these three
strategies together in SNMRVU algorithm represents
the main difference between it and the other related
www.ijorcs.org

10 Ahmed Fouad Ali
methods existing in the literature. SNMRVU
algorithm is evaluated on a set of 27 benchmark
functions with various properties, 10 classical
functions and 17 hard functions. The obtained results
give evidence that the SNMRVU algorithm is
promising algorithm for solving large scale
optimization problems, and in most cases its present
significantly better results and cheaper than the other
algorithms. In future, we intend to develop new
operators to apply our SNMRVU algorithm in higher
dimensional problems till 10,000 dimension.
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How to cite
Ahmed Fouad Ali, “Hybrid Simulated Annealing and Nelder-Mead Algorithm for Solving Large-Scale Global
Optimization Problems”. International Journal of Research in Computer Science, 4 (3): pp. 1-11, May 2014. doi:
10.7815/ijorcs.43.2014.084
www.ijorcs.org

Hybrid Simulated Annealing and Nelder-Mead Algorithm for Solving Large-Scale Global Optimization Problems

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