Evolutionary Programming: an Efficient Alternative to Genetic
Algorithms for Electromagnetic Optimization Problems
Kumar Chellapilla*,
Dept. of ECE, U.C San Diego, La Jolla, CA 92093-4007, USA
Ahmad Hoorfar,
Dept. of ECE, Villanova University, Villanova, PA 19085, USA
1. Introduction
Evolutionary algorithms, such as genetic algorithms (GAs) [1], evolutionary programming (EP) [2], and evolutionary strategies (ES), have recently received much attention for global optimization of electromagnetic problems [3-5,7]. These evolutionary
algorithms are heuristic population-based search procedures that incorporate random variation and selection. Of the three paradigms GAs are well known to the electromagnetic
community. Even though several successful applications have been reported, recent
research has identified some inefficiencies in GA performance [6]. This degradation in
efficiency is apparent in applications with highly epistatic objective functions, i.e., where
the parameters being optimized are highly correlated. On the other hand, EP and ES are
more robust to epistatic objective functions and coordinate rotations. EP has been shown
to be more efficient than GA on many function optimization problems [2].
The dynamics of GA are explained through the building block hypothesis and the
schema theorem [1], which are not fully accepted in the evolutionary computation literature. On the other hand, the convergence theory for EP is well established and EP has been
proven to asymptotically converge to the global optimum with probability one, under elitist selection [2]. Further, as will be demonstrated, EP is well suited for optimizing continuous, discrete, and mixed parameter optimization problems. Binary GAs require the
parameters to be coded as bits. The selection of the crossover and mutation probabilities is
quite arbitrary and they are not adapted during evolution. The selection of the initial values for the strategy parameters for EP and ES are well defined and efficient adaptive and
self-adaptive techniques exist for adapting these parameters during evolution.
In this work, the capabilities of EP will be demonstrated and contrasted with those
obtained using GAs on three challenging electromagnetic optimization problems, namely,
the design of optimally thinned linear arrays, aperiodic arrays, and Yagi-Uda antennas. A
more complicated problem on the gain optimization of a multilayered microstrip Yagi
array will be discussed in a separate paper in this symposium [8].
2. Evolutionary Programming
Thinning an array means turning off some elements in a uniformly spaced or periodic array (with an inter-element spacing of one-half wavelength) to create a desired
amplitude density across the aperture. For a 2N-element symmetric array, thinning has 2 N
possible combinations which is substantially large for large N. The problem of thinning an
2N element symmetric array consists of finding an optimal amplitude weight vector a =
{an, n = 1, 2, ... ,N}, consisting of ones and zeros (a vector of bits, or a bit vector) such that
the relative side lobe level, rsllmax, is minimized. Typical antenna designs desire an rsllmax
of -20 dB or lower. At the same time it is desirable to have arrays with few elements
turned on, resulting in lower power requirements and cheaper cost. Evolutionary programming is briefly presented for the design of optimally thinned linear arrays. For a fully
detailed description the reader is referred to [7]. The objective function to be minimized
consists of the sum of rsllmax and a credit term for the number of elements turned off. The
credit term is set to zero when rsll max > -20 dB and is equal to −(the number of elements
turned off) for rsll max < -20 dB. EP is used to optimally thin linear arrays using this objective function as follows:
1. Initialization: A set of initial trial solutions {Λi = (ai,Si), i = 1, 2, ... , µ} is generated
by randomly selecting the bits with uniform probability of selecting ones and zeros.
The ai’s represent the amplitude weight vectors and the Si’s represent the associated
floating point strategy parameters. These initial trial solutions constitute the parent
population at generation, k = 0. All the strategy parameters, i.e., the Si’s, are initialized
to max(N/10, 4). The fitness of each initial solution, Λi, is computed.
2. Mutation: Each parent Λi produces one offspring, Λµ+i as follows:
a) First, the offspring’s strategy parameter, Sµ+i, is generated by perturbing the corresponding parent’s strategy parameter, Si, with a gaussian random variable with
mean zero and standard deviation equal to 0.5. Mathematically,
S µ + i = S i + 0.5N ( 0, 1 )
(1)
3.
4.
5.
6.
b) Using the offspring’s strategy parameter as the mean a poisson random number, γ,
is generated.
c) γ bits in aµ are selected at random and flipped to obtain aµ+ι.
Fitness evaluation: The fitnesses of the µ offspring are evaluated. The parents and the
offspring constitute the population.
Tournament: Each member in the population, Λi, is compared with 10 opponents that
are selected at random from the population. Fore each comparison, if the rsllmax of Λi
is less than or equal to that of it’s opponent, it receives a win.
Selection: The µ members that have the highest number of wins are selected to be the
parents for the next generation. The generation number is incremented by 1.
Steps 2 through 5 are repeated Ngen number of times.
Linear symmetric arrays with 100 and 200 elements were optimally thinned. A
population size µ = 100 was used and evolution lasted Ngen = 200 generations.
For continuous parameter optimization problems, let x represent the n-dimensional vector of parameters to be optimized in light of a fitness function F(x). In this case,
EP is implemented as follows:
1. Initialization: A set of initial trial solutions {Λi = (xi,Si), i = 1, 2, ... , µ} is generated
by randomly selecting the parameters in each vector xi with uniform probability over
the search space. The Si’s, once again, represent the associated floating point strategy
parameters which in this case are vectors representing the standard deviations along
each of the dimensions. These initial trial solutions constitute the parent population at
generation, k = 0. All the strategy parameters, i.e., the Si’s, are initialized to (xi,max −
xi,min)/6. The fitness of each initial solution, Λi, is the associated objective function
value F(xi).
2. Mutation: Each parent Λi produces one offspring, Λµ+i as follows:
a) First, the offspring’s strategy parameter, Sµ+i, is generated by perturbing the corresponding parent’s strategy, Si, using
S µ + i = S i exp ( τ’N ( 0, 1 ) + τN j ( 0, 1 ) )
(2)
where τ’ = 1/sqrt(2sqrt(n)) and τ = 1/sqrt(2n).
b) Using the offspring’s strategy parameter the offspring parameter vector xµ+i is
obtained using
x µ + i = x i + S µ + i N ( 0, 1 )
(3)
3. Fitness evaluation: The fitnesses of the µ offspring are evaluated as F(xi).
Steps 4 through 6 are the same as for the discrete parameter optimization case. EP is used
to optimize the currents and inter element spacings of a 21 element symmetric aperiodic
array for minimum relative side lobe level with the constraint that the radiation pattern
have atleast a 10 degree bandwidth. In another optimization problem, the lengths and
spacings of a six element Yagi-Uda antenna are optimized to achieve maximum directivity. In both these cases, the population size is set to 50 and Ngens is 200 generations. For
this problem the Yagi antenna is numerically modeled using the method of moments.
3. Results
Figure 1. depicts the radiation pattern of the optimally thinned 100 element symmetric array, that had 80 elements on with an rsll max of -20.00 dB. The solution obtained
for the 200 element array had 124 elements turned on and an rsllmax of -20.21 dB. The best
solution in the GA literature had 145 elements turned on [4], which has 16.94 percent
more number of elements. Figure 2. gives the radiation pattern of the optimized 21 element aperiodic array with an rsll max of -27.88 dB with a beamwidth of 10 degrees. The
corresponding currents and spacings are given in table 1. The optimized six element YagiAntenna with a directivity of 13.70 dB is given in table 2. For comparison, the best solution obtained using GAs from [5] is also presented.
4. Conclusion
Evolutionary programming is proposed as an efficient alternative to genetic algorithms for electromagnetic global optimization problems and its efficiency is demonstrated through three antenna array examples.
5. References
[1] D. E. Goldberg, Genetic algorithms in search, optimization and machine learning, Reading,
MA: Addison Wesley, 1989.
[2] D. B. Fogel, Evolutionary Computation: Toward a New Philosophy of Machine Intelligence,
Piscataway, NJ: IEEE Press, 1995.
[3] D. S. Weile. and E. Michielssen, “Genetic Algorithm Optimization Applied to Electromagnetics: A Review,” IEEE Trans. on Antennas and Propagation, vol. 45, no. 3, March 1997.
[4] R. L. Haupt, “Thinned Arrays Using Genetic Algorithms,” IEEE Trans. on Antennas and Propagation, vol. 42, no. 7, July 1994.
[5] E. A. Jones, W. T. Joines, “Design of Yagi-Uda Antennas Using Genetic algorithms,” IEEE
Trans. on Antennas and Propagation, vol. 45, no. 9, September 1997.
[6] R. Solomon, “Reevaluating Genetic Algorithm Performance under Coordinate Rotation of
Benchmark Functions; A survey of some theoretical and practical aspects of genetic algorithms,”
BioSystems, 39 (3): 263-278, 1996, Elvier Science.
[7] K. Chellapilla, A. Hoorfar, and S.S.Rao, “Optimization of Thinned Phased Arrays using Evolutionary Programming,” EP98: Proc. of the Seventh Intl. Conf. on Evolutionary Programming,
Springer-Verlag: New York.
[8] A. Hoorfar and K. Chellapilla, “Gain Optimization of a Multi-Layer Printed Dipole Array
Using Evolutionary Programming,” submitted to the IEEE AP-S Intl. Symposium, Atlanta, GA,
June 1998.
11111111111111111111111010111111001011100111001101
0
0
−5
−5
−10
−10
Relative gain in dB
−15
−15
−20
−25
−20
−25
−30
−35
−30
−40
−35
−40
−45
0
20
40
60
80
100
Angle (in degrees)
120
140
160
−50
0
180
Figure 1. The radiation pattern of the optimally thinned 100 element antenna array
with 80 elements turned on. This solution
has the elements numbered 24, 26, 33, 34,
36, 40, 41, 45, 46, and 49 turned off and has
an rsllmax of -20.0094 dB.
20
40
60
80
100
Angle in degrees
120
140
160
180
Figure 2. The radiation pattern of the optimized 21 element symmetric aperiodic
antenna designed using EP with an rsllmax
of -27.88 dB and a bandwidth of 10.013
degrees.
Table 1: Inter-element spacings (in terms of λ) and relative currents for the optimized 21
element aperiodic antenna with a beamwidth of 10.013 degrees.
Element Number
Parameter
1
2
3
4
5
6
7
8
9
10
11
Spacings (λ)
-
0.60
0.40
0.40
0.42
0.41
0.41
0.60
0.40
0.60
0.40
Currents
0.78
1.00
0.99
0.64
0.59
0.80
0.95
0.91
0.99
0.65
0.11
Table 2: Optimized six element Yagi-Uda antenna for maximum directivity,
radius of wires, r = 0.003369λ
EP optimized for Directivity
Element
GA optimized for Directivity
Length (λ)
Spacing (λ)
Length (λ)
Spacing (λ)
1
0.493
−
0.504
−
2
0.574
0.191
0.601
0.101
3
0.443
0.250
0.442
0.321
4
0.427
0.330
0.438
0.274
5
0.418
0.363
0.420
0.428
6
0.435
0.367
0.422
0.435
Directivity (dB)
13.701
11.462