Benchmark and Phase Stability Problems
4.6 Benchmark Problems Similar to Phase Stability Problems It is clear from the previous section that phase stability problems have
special characteristics of a few but comparable minima, similar to phase equilibrium problems (Srinivas and Rangaiah, 2006); the local minimum in these problems is some times in a narrow valley. Although there are many benchmark problems in the literature with different characteristics (such as flat objective function and huge number of local minima), none
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of them represents the comparable minima as in phase stability problems.
Motivated from the unique characteristic of phase stability problems, a new benchmark problem with similar characteristics is developed. This and the benchmark problem proposed in Srinivas and Rangaiah (2006) are used for evaluating the four methods. The proposed benchmark problem has only a few comparable minima whereas the test problem in Srinivas and Rangaiah (2006) has huge number of comparable minima. Owing to their simple and mathematical nature, these test functions can easily be used by all researchers in global optimization.
The benchmark problem is developed from the Rosenbrock function as it has a few minima as in phase stability and phase equilibrium problems.
The minima in this function are made comparable by adding a multiplier (αr/N) to the quadratic term:
f(x) =
N−1 i=1
100(xi2− xi+1)2+αr
N(xi − 1)2
(4.12) whereαris a constant and N is the dimension of the problem. Each variable is bounded between−5 and 10 as in the Rosenbrock function (Table 4.1).
Both Rosenbrock function and the modified one do not have local minima up to and including 3 variables but they have several local minima (many of them are constrained minima) beginning from 4 variables. The minima in the modified function are made comparable to the global minimum by decreasing the effect of the quadratic term viaαr, but the global minimum is unaffected (i.e. 0.0 at xi = 1.0 for i = 1, 2, . . . , N). These effects can be seen in Table 4.10 for the 5-variable case; as αr decreases, the objective function becomes slightly flat and the minima become comparable but the global minimum and its location remain unaffected. The comparable minima for the modified Rosenbrock function with 4 to 20 variables are given in Table 4.11 forαr = 1.5 × 10−3. The difference in function value between the nearest local and global minimum is in the range of 1.39×10−3 to 3× 10−4for 4 to 20 variables, which is within the range (10−2to 10−6) of phase stability problems tested.
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120 M. Srinivas and G.P. Rangaiah
Srinivas and Rangaiah (2006) developed a new benchmark problem from the N-dimensional test function reported in Cetin et al. (1993). Unlike Rosenbrock function, this function has huge number(2N) of local minima.
The minima in this function are made comparable by adding a quadratic
Table 4.10 Trend of comparable minima withαrfor the modified Rosenbrock function (5 variables case)
αr 1 2.5 × 10−2 1.5 × 10−2 5× 10−3 1.5 × 10−3 Local minimum 3.93084 1.972 × 10−2 1.183 × 10−2 3.944 × 10−3 1.183 × 10−3
Global minimum 0.0 0.0 0.0 0.0 0.0
Table 4.11 Function values at the comparable minimum for the modi-fied Rosenbrock function withαr= 1.50 × 10−3. The global minimum is 0.0 at xi= 1.0 for i = 1, 2, . . . , N
Number of variables (N)
Function value at the comparable minimum
4 1.394 × 10−3
5 1.183 × 10−3
6 9.968 × 10−4
8 7.498 × 10−4
10 5.999 × 10−4
12 4.999 × 10−4
14 4.2857 × 10−4
16 3.75 × 10−4
18 3.333 × 10−4
20 3× 10−4
Table 4.12 Trend of comparable minima withαnfor the modified N-dimensional test function (4 variables case)
αn 0.0 0.3 0.4304
Local minimum −142.52794 −152.29792 −156.66095 Global minimum −156.66466 −156.66466 −156.66466 Differential Evolution in Chemical Engineering Downloaded from www.worldscientific.com by NANYANG TECHNOLOGICAL UNIVERSITY on 06/03/17. For personal use only.
term multiplied with a constant(αn):
f(x) =
1 2
N
i=1
(xi4− 16xi2+ 5xi) − αn
N i=1
(xi + 2.90353)2, (4.13)
where the search domain is −5 ≤ xi ≤ 5. The global minimum of the modified function for some values of αn (e.g. 0.3, 0.42, 0.4304) is still located at xi = −2.90353 for i = 1, 2, . . . , N. As αn value changes in equation 4.13, the minima become comparable and are given in Table 4.12 for 4 variables case. At αn = 0.4304, the difference in function value between the nearest local and global minimum is around 3.70 × 10−3for 2 to 20 variables (Table 4.13). This modified N-dimensional test function is more difficult than the modified Rosenbrock function since it has huge number of local minima compared to the latter.
DETL-E and DETL-G are further tested for the modified Rosenbrock and the modified N-dimensional problems with αr = 1.5 × 10−3 and αn = 0.4304, and the results are compared to those of DE and TS. The optimal parameter values used for DE, DETL-E and DETL-G are the same as those for moderate functions and difficult functions (Table 4.2) respec-tively for modified Rosenbrock function and modified N-dimensional test function except for NP = 30, Genmax = 100 N and Scmax = 12 N. For TS, the parameter values used are the same as those for difficult functions (Table 4.2).
The performance results (SR and NFE) averaged over successful trials out of 100, are given in Tables 4.14 and 4.15 for the modified Rosenbrock and the modified N-dimensional test functions respectively. SR of DE, DETL-E and DETL-G is comparable, and is better than that of TS for both the functions. SR of the methods tried either improves or is unaffected as the number of variables increases for the modified Rosenbrock function. This is because the number of minima in this function is only a few although the minima are comparable as in phase stability problems. On the other hand, SR of all the methods decreases with the number of variables for the modified N-dimensional test function (Table 4.15) due to the increase in the number of minima with the number of variables according to 2N. SR of DETL-G is high (93%) for the 2 variable N-dimensional test function compared to DE and DETL-E. This could be due to the implementation of TS concept in the generation step itself, which in turn is able to explore the
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122 M. Srinivas and G.P. Rangaiah
Table 4.13 Function values at the comparable minimum for the modified N-di-mensional test function withαn= 0.4304
Number of Function value at the Function value at the global minimum variables (N) comparable minimum (xi = −2.90353 for i = 1, 2, . . . , N)
2 −78.32862 −78.33231
4 −156.66095 −156.66466
5 −195.82712 −195.83082
6 −234.99328 −234.99699
8 −313.32562 −313.32932
10 −391.65795 −391.66165
12 −469.99028 −469.99398
14 −548.32261 −548.32631
16 −626.65494 −626.65865
18 −704.98727 −704.99098
20 −783.31960 −783.32331
Table 4.14 Results for the modified Rosenbrock function withαr = 1.5 × 10−3
DE TS DETL-E DETL-G
Dimension SR NFE SR NFE SR NFE SR NFE
4 82 10,832 76 4749 86 9988 85 10,631
5 90 17,235 76 7111 94 14,459 81 15,438
10 96 47,530 81 25,838 99 33,528 98 41,484
15 97 86,995 81 64,360 98 60,425 98 76,104
20 100 138,865 78 98,849 96 99,522 100 125,085
global minimum region successfully. Compared to DE, NFE of DETL-E and DETL-G is respectively 22% and 9% less for the modified Rosen-brock function, and 47% and 49% less for the modified N-dimensional test function.
In order to see the effect of comparable minima when they are in large number, DETL-E and DETL-G are evaluated forαn = 0, 0.3 and 0.42 for the modified N-dimensional test function, and the results are presented in Table 4.16. Atαn = 0, SR of DETL-E and DETL-G is 100%, and decreases asαnincreases. NFE of both DETL-E and DETL-G increases as the minima become comparable (i.e.αnchanges from 0 to 0.4304). These results clearly
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Table 4.15 Results for the modified N-dimensional test function withαn= 0.4304
DE TS DETL-E DETL-G
Dimension SR NFE SR NFE SR NFE SR NFE
2 70 4503 21 1483 70 2000 93 2301
4 85 11463 5 2779 78 4868 81 4167
5 73 13198 3 5524 79 5998 73 5640
10 44 21843 0 — 42 11905 35 11714
15 19 33908 0 — 20 20316 5 20283
20 2 47541 0 — 3 33854 4 28941
Table 4.16 Effect ofαnvalue on the performance of DETL-E and DETL-G for modified N-dimensional test function
SR and NFE (in brackets) of SR and NFE (in brackets) of DETL-E for the function withαn= DETL-G for the function with αn=
Dimension 0.0 0.3 0.42 0.4304 0.0 0.3 0.42 0.4304
2 100 100 98 70 100 100 100 93
(1689) (1665) (1736) (2000) (1804) (1872) (2027) (2301)
4 100 100 98 78 100 100 97 81
(3402) (3559) (4022) (4868) (3287) (3421) (3912) (4167)
5 100 100 95 79 100 100 98 73
(4053) (4324) (4943) (5998) (3934) (4226) (4762) (5640)
10 100 100 89 42 100 100 83 35
(7106) (7764) (9304) (11905) (7180) (7817) (9330) (11714)
15 100 100 68 20 100 98 67 5
(9772) (11301) (14694) (20316) (10245) (11570) (15023) (20283)
20 100 97 46 3 99 97 41 4
(12431) (15211) (21737) (33854) (13249) (15755) (22154) (28941)
show the challenging nature of the comparable minima to a global optimiza-tion algorithm, particularly when the dimension of the problem is high.
Overall, the performance of DETL-E and DETL-G is better than that of DE and TS in terms of NFE and SR for the examples tested. In terms of CPU time, TS performs better for small variable problems (up to 5 variables) whereas DE performs better for 10 to 20 variable problems compared to
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124 M. Srinivas and G.P. Rangaiah
all other methods. The relative performance of DETL-E and DETL-G is comparable; this is consistent with the comparable performance of DE and MDE (Srinivas and Rangaiah, 2007b) since the former is similar to DETL-E whereas the latter is similar to DETL-G except for the inclusion of the tabu list and checks. In the present study, DETL-E and DETL-G are evaluated and compared with DE and TS for continuous problems only. They need to be tested for non-differentiable and constrained problems.
4.7 Conclusions
This study describes and evaluates two methods, namely, DETL-E and DETL-G with tabu list and check in the evaluation and generation step of DE respectively. Initially, the methods are applied to two sets of benchmark problems, namely, moderate and difficult functions, which involve 2 to 20 variables and a few to thousands of local minima. The reliability of DETL-E and DETL-G is found to be comparable to that of DE, and is better than that of TS for both moderate and difficult functions. NFE of DETL-E and DETL-G is around 37% and 35% less, and 34% and 24%
less compared to DE, respectively for moderate and difficult functions. The methods are then tested for challenging phase stability problems which include several components. Both DETL-E and DETL-G located the global minimum successfully for these problems with almost 100% reliability similar to DE, and about 60% less NFE compared to DE. Overall, the performance of DETL-E and DETL-G is found to be better than that of DE and TS. A new benchmark problem (modified Rosenbrock function) with characteristics similar to phase stability problems is proposed. This facilitates the development and testing of global optimization algorithms for phase stability type of problems.
Nomenclature
A Amplification factor CR Crossover constant gE Excess Gibbs free energy G Gibbs free energy
¯G Partial molar Gibbs free energy Genmax Maximum number of generations
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hn Length of the hyper rectangle H Tangent plane distance function Itermax Maximum number of iterations
nc Number of components
N Number of decision variables in the problem
NP Population size
NPinit Initial population size
Nt and Np Number of tabu and promising points Nneigh Number of neighbors in each iteration
P System pressure
R Universal gas constant
Scmax Maximum number of successive generations without improvement in the best function value
t Tangent plane distance function tls Tabu list size
tr Tabu radius
T System temperature
vi,J+1 Mutant individual i for generation J+ 1 xi,J Target individual i in generation J x Vector of continuous variables
xli and xiu Lower and upper bounds on decision variable, xi
Greek letters
αr Constant in the modified Rosenbrock function αn Constant in the modified N-dimensional test function βi A new decision variable introduced in place of xi
γi Activity coefficient of component i φi Fugacity coefficient of component i
ε Radius
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Chapter 5