Determinism in Electromagnetic Design & Optimization

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Determinism in Electromagnetic Design & Optimization - Part I: Central Force Optimization Richard A. Formato Consulting Engineer & Registered Patent Attorney Of Counsel, Emeritus Cataldo & Fisher, LLC PO Box 1714, Harwich, MA 02645 USA (Email: [email protected])

Abstract—Design and optimization (D&O) problems in applied electromagnetics, in particular antenna D&O, often rely on global search and optimization metaheuristics based on Nature-inspired metaphors. Most are stochastic in nature because the underlying process is, for example the swarming behavior of fish and birds (Particle Swarm Optimization) or the foraging behavior of ants (Ant Colony Optimization). While these algorithms generally work well (enough), their inherent randomness frequently limits utility in solving real-world antenna problems. A better approach is to use an inherently deterministic optimizer or to implement a stochastic algorithm so that it is effectively deterministic. This article discusses in detail Central Force Optimization (CFO) which is a deterministic algorithm analogizing gravitational kinematics. It provides an overview and emphasizes CFO's utility in solving the real-world problems routinely encountered by practicing engineers with examples drawn from antenna optimization. Because CFO is inherently deterministic only a single run is required, which can be a major advantage in addressing electromagnetic problems. Even in cases where the balance between decision space exploration and exploitation is better achieved using a stochastic approach, CFO offers the advantage of being easily hybridized to include pure or pseudo randomness. One method of injecting pseudo randomness is using π fractions which are discussed in the companion paper (Part II).

Index Terms—Central Force Optimization, CFO, π fractions, Pseudo randomness, Global Search and Optimization, Metaheuristic, Antenna Design, Antenna Optimization, Algorithm.

I. INTRODUCTION Central Force Optimization (CFO) was introduced in 2007 [1] because the author was dissatisfied with the stochastic algorithms often used for antenna design or optimization (D&O). The term "design" refers to meeting minimum designer-specified performance criteria, while "optimization" refers to determining the "best" value of a designer-defined fitness function or figure-of-merit. This article focuses on optimization, but its concepts are equally applicable to design problems. Optimization problems fall broadly into two categories: benchmarks and real-world. This discussion addresses the real-world optimization problems that practicing engineers routinely encounter. A benchmark problem comprises a known objective (fitness) function with a known solution (see, for example, [67-69]), whereas a real-world problem requires both (i) formulating from scratch a suitable fitness function and then (ii) determining its unknown solution. Defining a good objective function can be a daunting task invariably compounded by a stochastic optimizer. If the optimizer itself generates random results there simply is no good way of knowing whether one objective function is better than another. A deterministic algorithm like CFO mitigates this problem by always returning the same results for the same setup because there is no inherent randomness. CFO also provides the additional advantage of allowing hybridization by including some measure of pure or pseudo randomness if doing so is beneficial. For example, purely deterministic CFO might be used first to decide which of many candidate objective functions is best followed by a randomized implementation to improve decision space (DS) exploration. Another important advantage of determinism is that different algorithm implementations can be compared quickly and with certainty. Taking CFO as an example,

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investigating whether or not variable "gravitation" improves its performance is easily accomplished: only two runs are required, one with and another without variable gravitation. Stochastic algorithms, on the other hand, do not permit direct comparison because they require many runs to generate statistical performance data. And even then the questions of which objective function is best or which algorithm implementation is best cannot be answered with certainty. This paper is organized as follows: Section II provides an overview of CFO. Section III discusses generally why determinism is important using a typical benchmark and realworld examples. Sections IV and V provide real-world examples that illustrate the difficulties in formulating suitable fitness functions: a broadband HF monopole and a wireless power transfer (WPT) multi-arm helix. Section VI introduces basic CFO theory. Section VII provides CFO pseudocode and discusses methods for creating Initial Probe Distributions (IPD's). Section VIII is the Conclusion. II. CFO OVERVIEW A.

CFO's Gravitational Kinematics Metaphor

CFO analogizes gravitational kinematics, the branch of physics that deals with the motion of masses under the influence of gravity. In physical space the fundamental equations are Newton's Law of Universal Gravitation and his Second Law of Motion. Imagine a group of small "probe" satellites flying through space, say, the Solar System. As these very small objects approach a massive planet, for example, Jupiter, they may become gravitationally trapped in orbit around it. Trapping generally requires a loss of energy, but under certain conditions, specifically "resonant returns," energy is conserved [52-55], and the trapped mass eventually escapes orbit. It seems reasonable to speculate that over time more and more probes would become trapped around the more massive planets than around the less massive ones, that is, the group of satellites eventually "converges" on the planet with the greatest gravitational pull. This is the process metaphorically implemented by CFO. CFO generalizes Newton's laws in 3D physical space to an optimization problem's N d -dimensional decision space [5659]. The most massive object ("largest planet") is the optimum (maximum) value of the objective function (note that unlike most metaheuristics CFO performs maximization, not minimization). Each of CFO's "probes" is assigned a user-defined "mass" computed from the objective function's value at each sample point in the decision space. Newton's laws are generalized to define two equations of motion governing each probe's trajectory through the optimization problem's "landscape" (the objective function's topology over the decision space). As CFO progresses its probes converge of the objective function's maximum value, and the run terminates when some user-specified criteria are met.

B.

Exploitation vs. Exploration in CFO

Because gravitational kinematics is perfectly deterministic, so too is CFO. Every CFO run with the same setup yields precisely the same results, this in stark contrast to stochastic algorithms that invariably yield different results one run to the next. But there is a price to be paid for repeatability: deterministic algorithms frequently do not traverse the decision space as effectively as stochastic ones (§2.1.1 in [60]). Determinism encourages exploitation, converging quickly on a solution using current data, but often at the expense of exploration, seeking still better solutions in other regions of the decision space. While generally it is difficult or impossible to add determinism to a stochastic algorithm because the underlying equations are fundamentally stochastic, CFO has the significant advantage of being readily randomized. This characteristic allows the algorithm designer to strike a balance between DS exploitation and exploration by creating hybridized versions of CFO. One effective approach to hybridization is to include pseudo randomness in CFO's Initial Probe Distribution [61]. In a truly stochastic algorithm each random variable (RV) is computed from a probability distribution and consequently is a priori unknowable. This is fundamentally different from a pseudo random variable (PRV) whose value is precisely known but arbitrarily assigned. A PRV can be specified in advance, for example an arbitrary sequence of numbers like the π fractions [62] discussed in the companion article Part II [70], or it can be calculated in a prescribed manner. The PRV’s "randomness" derives from its being uncorrelated with the decision space topology, not that it is uncertain in the sense of a true RV. Because each PRV is known with absolute precision, either explicitly or by calculation, CFO’s probe trajectories remain deterministic. Including PRVs in CFO may enhance its exploration while preserving determinism so that the algorithm always yields the same results on runs with the same setup while effectively exploring DS. C.

CFO Applications and Extensions

CFO has been used to solve a wide range of practical problems, representative examples being training neural networks [2]; optimizing microstrip patch antennas [3,4]; designing broadband absorbers [5]; designing drinking water networks [6]; optimizing E-shaped laptop patch antennas [7,8]; designing tri-band slotted bowties [9]; analyzing financial systems [10]; assessing power grid reliability [11]; developing iris recognition software [12]; designing multilayer electromagnetic absorbers [13]; designing finite impulse response filters [14]; optimizing adaptive beamforming antennas [15, 16]; designing UAV 3D flight paths [17]; designing reconfigurable [18] and tri-band handheld [19] RFID antennas; optimizing bandpass filters [20]; optimizing electromagnetic absorbers [21]; synthesizing antenna arrays [22,23]; increasing antenna impedance bandwidth [24,25,26]; investigating different objective

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functions for broadband HF monopoles [27]; solving nonlinear circuits [28]; optimizing "smart" antennas [29] and a high voltage air-break disconnector [30]; improving twolevel clustering for multi-view data analysis [31]; and optimizing transmitter locations for indoor optical wireless networks [32]. Proofs of convergence for CFO and extensions have been developed [33,34]. The algorithm has been implemented on a GPU using various neighborhood topologies [35,36]. Several CFO extensions and hybridizations have been developed, for example, adaptive DS and variable initial probes [37]; neural network ensembles using copula function theory [38]; real-time adaptation [39]; variable population size [40]; multistart CFO [41]; clustering-simplex hybridization [42]; CFO with differential evolution operator mutation [43]; adaptation based on stability analysis [44]; and hybridization with back-propagation simplex search [45]. CFO is included in several surveys of established optimization metaheuristics [46-51].

Topology aside, testing against the Schwefel is straightforward because the problem is fully specified, both the functional form and its known solution. The only question is how well a particular algorithm performs against it. A stochastic algorithm might be run tens, hundreds, possibly thousands of times to determine how close it comes to the 30D Schwefel's known maximum of 12,569.487. Does knowing that an algorithm works well against this benchmark help when it comes to solving a problem where the objective function itself is unknown? Maybe, but probably not.

III. WHY DETERMINISM MATTERS A distinction has been drawn between benchmark and real-world problems because real-world problems usually require a two-step solution whereas benchmark problems require only one. Optimization algorithms almost always are evaluated against benchmark suites for the very reason that the solutions are known in advance. An algorithm's effectiveness is measured by the quality of its solutions (how close it comes to the known answers) and by the required computational effort (usually the number of objective function evaluations). At no point is the fitness function itself an unknown, but this almost always is the case in realworld problems. The user must define a suitable objective function before any optimization can be done, and this added requirement makes solving the problem much harder because not all objective functions are well-behaved or for that matter yield good results even if they are. This idea is illustrated with some examples. A.

A Typical Benchmark: Schwefel's Problem 2.26

Schwefel's Problem 2.26 is a well-known benchmark used in many test suites. In an N d -dimensional decision space, Nd

this objective function is defined as f ( x) = ∑ [ xi sin( xi )] , i =1

− 500 ≤ xi ≤ 500 . It is highly multimodal with a single global

maximum of 418.9829 × N d at [ 420.9687 ]N d ([63] @ p.4467). The 2D maximum, which can be visualized, is 837.9658 at ( 420 .9687 ,420 .9687 ) , but it is the 30D maximum of 12,569.487 that most often is used as a benchmark. The 2D Schwefel's landscape, Fig. 1, is extremely multimodal which shows why this function is an especially challenging benchmark. .

Fig. 1. 2D Landscape of Schwefel's Problem 2.26.

When the very form of the objective function is unknown the entire optimization process must be repeated for every candidate function. This is not a fundamental limitation, of course, but in many real-world cases as a practical matter it turns out to be one, and it cannot be overcome because as a practical matter computation times simply become too long. Evaluating the Schwefel is straightforward and quick because it requires only built-in functions (sine, square root, absolute value), but real problems usually are quite different. B.

A Real-World Problem: Vehicle-Mounted Antenna

Real-world problems often require an external "modeling engine" that frequently is far more computation intensive than using built-in functions. In antenna D&O, for example, Finite Element and Method of Moments codes are popular with their attendant long run times. This limitation renders statistical evaluations completely impractical. As an example, the run time for the 7,303-segment vehiclemounted antenna model shown in Fig. 2 using the Numerical Electromagnetics Code (NEC4/MP, [66]) is 369 seconds on a 2.30GHz 8-core Core i7-3610QM machine running a multithreaded version of NEC4 under 64-bit Windows 7. An optimization run requiring, say, 1,000 evaluations would take more than 102 hours, well over 4 days! And this must be repeated for each candidate fitness function! Knowing that an algorithm is effective against Schwefel 2.26 tells the practitioner absolutely nothing about how to formulate useful fitness functions and whether or not the algorithm will work well against those functions, whatever form they may take. Stochastic algorithms make answering

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this and related questions very difficult and consequently are not well-suited to solving the real-world problems routinely encountered by practicing engineers. IV. A REAL-WORLD PROBLEM: MONOPOLE ANTENNA Because most real-world problems do not come with predefined objective functions, the first mandatory step is formulating one. How can the engineer know that a particular mathematical form is suitable to achieve the desired goals? How can the practitioner differentiate one function from another? These questions cannot be answered using a stochastic optimizer because every run yields a different result.

electrically conducting (PEC) ground plane. The antenna is shown in Fig. 3 (see [27] for details). This problem is 2D with decision variables R and H (loading resistor value and height above the ground plane, respectively). The optimization objectives are maximum gain and as flat as possible an impedance bandwidth from 5 to 30 MHz. The relevant performance parameters are radiation efficiency, ε ; maximum gain, Gmax ; input impedance, Z in = Rin + j X in ; and voltage standing wave ratio relative to a reference impedance Z 0 , VSWR ( Z 0 ) (the industry standard is Three different objective functions are Z 0 = 50 Ω ). considered: f1 ( R , H , Z 0 ) = f 2 ( R, H , Z 0 ) =

f 3 ( R, H , Z 0 ) =

min (ε ) + min (Gmax ) ∆VSWR ( Z 0 )

(1)

min (ε ) Z 0 − max ( Rin ) ⋅ max ( X in )

(2)

min (ε ) Z 0 − max ( Rin ) ⋅ ∆VSWR ( Z 0 ) ⋅ [max ( Xin) − min ( Xin)]

(3)

where ∆VSWR ( Z 0 ) is the difference between maximum and minimum VSWR over the 5-30 MHz HF band.

Fig. 2. NEC Model of Vehicle-mounted Antenna.

At first blush it appears that each of these functions should achieve the desired objectives, but closer examination of the visualized functions shows this is not at all the case. Landscapes are shown in Figs. 4-6, and a cursory inspection reveals potentially serious problems. Function f1 has a diffuse maximum close to the DS boundary. Its width may make locating the actual peak difficult, and its location near the boundary may inhibit effective exploration. f 2 is so pathologically spiky that many algorithms probably cannot efficiently locate its maximum. Only f 3 is well-behaved with a reasonably sharp maximum not too close to the DS boundary.

There simply is no way of knowing why the next run's results are different from the previous one's. Is it because the objective function is different? Or is it because of the algorithm's inherent randomness? Trying to answer these questions requires statistics generated over many runs, and even then the answer may not be correct because it is probabilistic. Deterministic optimizers like CFO avoid this conundrum entirely. If the objective function is changed, for example by adding a term or modifying one, then that change alone accounts for different results in subsequent CFO runs made with the same setup. Only deterministic optimizers allow different objective functions to be compared quickly and without ambiguity. A simple real-world problem illustrates how difficult defining a suitable objective can be and why using a deterministic optimizer like CFO can make a big difference. The example is optimizing a simple resistively loaded basefed high-frequency (HF) monopole antenna on a perfectly

Fig. 3. Base-fed Loaded Monopole Antenna.

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How can the antenna engineer decide which of these three ostensibly reasonable objective functions is best? In the 2D case the answer is simple: plot the landscape, look for the maximum. But 2D problems are rarely if ever the case and in any event do not require multi-dimensional search and optimization algorithms. What happens if the loaded monopole includes reactive loading in addition to resistance? Adding an inductor or capacitor increases the problem's dimensionality from 2D to 4D, the new decision variables being the value and location of the reactive element. Unlike the 2D problem the 4D problem cannot be visualized. The only way to decide which of many candidate objective functions is best is to make optimization runs with each one, a process that mandates the use of a deterministic optimizer like CFO in order to avoid the plethora of runs required by a stochastic optimizer. This example highlights the efficacy of algorithms like CFO when it comes to dealing with realworld engineering problems as opposed to benchmarks.

technology [64] (VZ0) instead of being specified in advance. The other parameters are the free space wavenumber, k ; the largest dimension fully enclosing the helix, a ; and NEC's free-space Average Gain Test, AGT (a measure of the antenna model's fidelity). The Numerical Electromagnetics Code (NEC) [65], is a widely used external Method-ofMoments modeling engine considered to be the "gold standard" for wire antenna modeling.

V. ANOTHER REAL-WORLD ANTENNA PROBLEM: WIRELESS POWER TRANSFER Another real-world example involves optimizing a multiarm helical antenna for wireless power transfer. This is a 4D problem with decision variables (i) helix length, (ii) diameter, (iii) number of turns, and (iv) number of arms. For the practitioner the first question is, How to combine these parameters? Choosing a suitable fitness function requires investigating a variety of forms, which as discussed above is best done with a deterministic optimizer.

Fig. 5. Landscape of Objective Function f 2 ( R, H ,50) .

The WPT optimization objectives are minimum VSWR , maximum power gain, and minimum helix volume at 200 MHz. CFO was used as the optimizer, and after much cutand-try the following less than obvious fitness function was determined to be a good one for achieving the objectives:

F =

1 ka ⋅ AGT − 1 + Z 0 − Rin +

X in

...(4)

Rin

Fig. 4. Landscape of Objective Function f1 ( R, H ,50) .

Note that VSWR does not appear explicitly whereas components of the antenna input impedance do, and that the value of Z 0 was determined by CFO using Variable Z0

Fig. 6. Landscape of Objective Function f 3 ( R, H ,50) .

Evolution of CFO's fitness is plotted in Fig. 7. The monotonic increase is typical of CFO's behavior even without elitism (preserving the best solution step to step) which was not included here. Davg, plotted in Fig. 8, is the average distance in DS between the best probe and all other probes. A value of zero means that all probes have coalesced on a single point. It is quite interesting that oscillatory behavior in Davg apparently correlates with local trapping and perhaps can be used to signal that condition [54,55] and improve DS exploration. The optimized 200 MHz WPT helix is shown in Fig. 9. It has four upper and lower 1.537-turn arms wound with #12 AWG wire. Its dimensions are 0.148m long by 0.089m diameter with the arms connected by 0.01m jumpers. The VZ0 optimized impedance is Z0 = 95.67Ω. VSWR//Z0 is plotted in Fig. 10 (standing wave ratio relative to Z0), and the corresponding input impedance appears in Fig. 11. The 3D radiation pattern is shown in Fig. 12. Note that these data

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specified by its position vector computed from two equations of motion that analogize their real-world counterparts for material objects moving through physical space under the influence of gravity without energy dissipation.

assume PEC wires because the radiation efficiency with annealed Cu conductors is ε ≈ 99% . The optimized helix is electrically small (maximum dimension less than 0.1 wavelength at 200 MHz). Its VSWR is nearly perfect at 1.013:1. And the power gain is excellent at 1.88 dBi. The optimization objectives have been fully met because a suitable fitness function was employed whose form was determined by comparing candidate functions using deterministic CFO. Accomplishing this with a stochastic optimizer might require hundreds of runs for each candidate fitness function, which is clearly prohibitive for most real antenna D&O problems.

r

the xkp, j are the probe's coordinates and along the

p, j k k

eˆ , where

eˆk is the unit vector

xk -axis. Indices p , 1 ≤ p ≤ N p , and

j,

0 ≤ j ≤ Nt are the probe number and iteration number, respectively, where N p and Nt are the corresponding total number of probes and total number of iterations (time steps). Davg (200 MHz Multi-Arm Folded Helix)

0.07

12

0.06

10

0.05

8

Davg

Fitness

∑x k =1

Fitness (200 Mhz Multi-Arm Folded Helix)

14

Nd

The position vector at step j is R jp =

6

0.04 0.03

4 0.02

2 0.01

0 0

20

40

60

80

100 0.00

Step #

0

20

40

60

80

Step#

Fig. 7. Fitness Evolution, 200 MHz Helix.

Fig. 8. Davg Evolution, 200 MHz Helix.

VI. BASIC CFO THEORY Basic CFO theory is presented in this section. Proofs of convergence and extensions developed by other researchers and are not discussed.

A.

Problem Statement

Central Force Optimization searches a bounded N d dimensional hyperspace Ω for the global maxima of a function f ( x1 , x2 ,..., x N d ) . Stated more precisely, determine max{ f ( X ) : X ∈ Ω ⊂ R n }, f : Ω ⊂ R n → R with Ω := { X | ximin ≤ xi ≤ ximax , i = 1,..., N d } being the bounded region of feasible (allowable) solutions (decision space); f ( X ) a pointer to the objective function; and L

= Ω ∪ f ( X ) , X ∈ Ω the problem's landscape. The xi are the decision variables for each coordinate number i . Fitness r refers to the value of the objective function f (x ) at sample

r

point x = ( x1 , x2 ,..., xN d ) in Ω .

There is no a priori

information about the objective function’s extrema, that is, r f (x )' s landscape (topology) is completely unknown. CFO searches Ω by flying "probes" through the space at discrete "time" steps (iterations). Each probe’s location is

Fig. 9. 200 MHz 4-Arm Helix. (axis 0.05m, L=0.148m, D=0.089 m, #12AWG)

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r a jp−1 = G

Np

∑ U (M

k j −1

−M

p j −1

) ⋅ (M

k j −1

−M

k =1 k≠p

r r ( R kj−1 − R jp−1 ) (5) ) × r r β R kj−1 − R jp−1

α p j −1

which is the first of CFO’s two equations of motion. M jp−1 = f ( x1p, j −1 , x2p, j −1 ,..., xNpd, j −1 ) is the fitness at probe p ’s location at time step j − 1 . Every other probe at that step (iteration) has associated with it the fitness M kj−1 , k = 1,..., p − 1, p + 1,..., N p . G is CFO’s gravitational constant,

and

U (⋅)

is

the

Unit

Step

function,

1, z ≥ 0 . U ( z) =   0 , otherwise  

r

The acceleration a jp−1 causes probe p to move from Fig. 10. 200 MHz Helix VSWR // 95.67Ω [PEC wires].

rp

r

position R j −1 at step j − 1 to position R jp at step j according to the trajectory equation

r r 1r R jp = R jp−1 + a jp−1 ∆t 2 , j ≥ 1 2

(6)

which is CFO’s second equation of motion. ∆t is the "time interval" between steps during which the acceleration is constant.

Fig. 11. 200 MHz Helix Input Impedance [PEC wires].

In the original CFO paper the counterpart of equation (6) above, that is, eq. (5) in [1], includes a velocity term that intentionally is excluded here. That term was set to zero in [1] as a matter of convenience because in the case of rectilinear motion it simply is an additive constant. But what eventually became clear is that including the velocity term is a mistake. It never should have been included in equation (5) in [1] in the first place because generally a probe's motion is not rectilinear. Trajectories instead are curvilinear which results in the acceleration and velocity vectors being in different directions. As an example, in the case of purely circular motion in real space the velocity vector is tangent to the circle while the acceleration vector is along the radius to the circle's center. The acceleration and velocity vectors actually are mutually perpendicular. This limiting case illustrates why, as a general proposition, the velocity term in Newton's real-space kinematic equations should not be included in the metaphorical CFO-space analogues - doing so erroneously changes the direction of a probe's acceleration.

Fig. 12. 200 MHz Helix Radiation Pattern [PEC wires].

B.

Equations of Motion In metaphorical CFO-space each of the N p probes is

accelerated by the gravitational pull of the surrounding "masses." At step j − 1 probe p experiences an acceleration given by the acceleration equation

While CFO’s terminology mimics that of gravitational kinematics, it is important to remember that CFO, like all metaheuristics, is a metaphor, not a precise model of the physical system inspiring it. CFO's terminology has no significance beyond reflecting its kinematic roots, as, for example, does the factor ½ in equation (6). The gravitational constant, G , and time increment, ∆t , have direct analogues in Newton’s equations in physical space, but the exponents α and β do not. In keeping with CFO's metaphorical nature, these exponents are included to provide the algorithm

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designer with added flexibility, say, by changing how gravity varies with distance or with mass or both if doing so creates a more effective algorithm metaphorical CFO-space. C.

VII. CFO PSEUDOCODE & IPD SPECIFICATION

Mass in CFO-Space

The concept of mass in CFO-space is very important, and it is quite different than it is in real space. In the Universe mass is an inherent, immutable property of matter. In CFOspace in stark contrast "mass" is a user-defined positivedefinite function of the objective function’s fitness (not necessarily the fitness itself). In the acceleration equation (5) above mass by definition is

MASSCFO = U ( M kj−1 − M jp−1 ) ⋅ ( M kj−1 − M jp−1 )α , that is, the difference in fitness values raised to the α power multiplied by the Unit Step. Of course, the algorithm designer is free to choose any functional form that works well. In this case the Unit Step is included to preclude negative mass because without it CFO's mass could be negative depending on which fitness is greater. Real mass in the physical Universe is always positive and consequently the force of gravity always attractive. But, depending on how it is defined, mass in metaphorical CFOspace can be positive or negative, which can lead to some very undesirable effects. Negative mass creates a repulsive gravitational force that flies probes away from maxima instead of toward them, thereby defeating the very purpose of the algorithm. While any definition of mass is permissible in CFO-space, only positive-definite ones allow the algorithm to perform as it should. D.

value is uncorrelated with the landscape thereby adding a measure of pseudo randomness.

Errant Probes in CFO-Space

Another important consideration is the possibility that a probe’s computed acceleration may be so large that it flies outside Ω . If any coordinate in the trajectory update equation (6) satisfies xi < ximin or xi > ximax , then probe is errant because it enters a region of unfeasible solutions (ones that are not valid for the problem at hand). Errant probes arise in many algorithms, and the question is what to do with them. While there is a multitude of approaches, a simple empirically determined one is suggested here. On a coordinate-by-coordinate basis, probes flying outside DS are placed a fraction 0 < ∆Frep ≤ Frep ≤ 1 of the distance between the probe’s starting coordinate and the corresponding boundary coordinate.

Frep is the variable repositioning factor discussed in more detail in [57-59]. Its value can be assigned by calculation, for example by incrementing it by a fixed amount ∆ Frep , or by specification, for example, using π fractions [62]. Not only does this scheme guaranty that all probes remain within Ω , it probably improves CFO's exploration because Frep 's

Central Force Optimization is simple and easily implemented. It comprises only the two equations of motion, (5) and (6) above, and may be implemented with the following steps: (1) (i) Compute (or assign) the initial probe distribution (IPD); (ii) Compute corresponding objective function fitnesses; and (iii) Assign initial probe accelerations (usually zero). (2) Probe-by-probe successively compute a new position in Ω at the current time step using the previously computed (or assigned) accelerations. (3) Verify that each probe remains inside Ω and make corrections as required. (4) Update the objective function fitnesses at each new probe position. (5) Compute accelerations for the next time step using the new probe distribution. (6) Loop over time steps. There are many ways to generate an IPD. The details are entirely up to the algorithm designer, and some IPDs will be better than others. It also is likely that different IPDs will work better with different classes of objective function landscapes. The "probe line" IPD [37] that seems to work well across a wide range of objective functions is described here. Fig. 12 is a schematic representation of a typical 2D IPD using probe lines. It comprises an orthogonal array of Np probes per dimension uniformly distributed on lines Nd

parallel to the coordinate axes that intersect at a point along Ω’s principal diagonal. The figure shows nine probes on each of two lines parallel to the x1 and x2 axes. The probe r r r r lines intersect at a point D = X min + γ ( X max − X min ) where Nd Nd r r X min = ∑ ximin eˆi and X max = ∑ ximax eˆi are the diagonal’s i =1

i =1

endpoint vectors. Parameter 0 ≤ γ ≤ 1 determines where along Ω 's diagonal the probe line array is placed. Typical uniform 2D and 3D IPD's using variable probe lines (different γ values) appear in Figs. 13 and 14. It is not necessary that each probe line contain the same number of probes, and under some conditions different numbers may be more useful. As examples, equal probe spacing might be desired in a decision space with unequal boundaries or possibly to avoid overlapping probes. While the probe line IPD is completely deterministic, a random IPD could be used instead. π fractions, for example, could be used to create a

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pseudo random IPD or they could be used to pseudo randomly place probes on probe lines.

Fig. 12. Typical 2D Probe Line IPD.

Representative CFO pseudocode using the variable probe line IPD appears in Fig. 15. Of course many adaptations are possible, for example, adding other loops to vary, say, the number off probes, the gravitational constant, or the CFO exponents. Other adaptations could include changing parameters or shrinking DS during a run in response to metrics such as rate of change or spread of fitnesses or perhaps probe convergence as measured by Davg. The possibilities are endless, and the efficacy of a particular CFO implementation is easily investigated because CFO is deterministic even with pseudo randomness. VIII. CONCLUSION Central Force Optimization has become an established metaheuristic for global search and optimization. It is particularly useful for real problems that require the definition of a suitable objective function before any D&O can be done. Stochastic algorithms thms make it difficult if not impossible to assess the utility of different fitness functions in real problems because each run yields different results, and there is no way of attributing the differences to the objective function being used or to the algo algorithm's inherent randomness. CFO eliminates this problem because it is completely deterministic, always yielding precisely the same result for the same run setup. CFO retains this characteristic even when it is implemented pseudo randomly using parameterss whose values are precisely known but uncorrelated with the optimization problem's landscape. Adding pseudo randomness improves CFO's ability to explore a decision space, and some methods to that end have been discussed. CFO has been applied to a wide rrange of real world problems and has been improved and extended by many researchers. A bibliography of CFO CFO-related work is included that may assist practitioners and researchers who are new to the algorithm.

Fig. 13. Sample 2D 12-probes/probe probes/probe line IPD’s. ( γ values: 0.0,, 0.4, 0.9 top to bottom)

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For (a.1) (a.2) (a.3) (a.4)

γ = γ start

to

γ stop

by

∆γ

Compute initial probe distribution. Compute initial fitness matrix. Assign initial probe accelerations. Set initial Frep .

For j = 0 to N t (or earlier termination ermination criterion) (b) Compute probe position vectors r R jp , 1 ≤ p ≤ N p [eq.(6)].

(c) Retrieve errant probes (1 ≤ p ≤ N p ): r r r If R jp ⋅ eˆi ≤ ximin ∴ R jp ⋅ eˆi = ximin + Frep ( R jp−1 ⋅ eˆi − ximin ) . r r r If R jp ⋅ eˆi ≥ ximax ∴ R jp ⋅ eˆi = ximax + Frep ( ximax − R jp−1 ⋅ eˆi ) . (d) Compute fitness matrix for current probe distribution. (e) Compute accelerations using current probe distribution and fitnesses [eq. (5)]. (f) Assign Frep . Fig. 15. CFO Pseudocode with Variable Probe Line IPDs.

REFERENCES

probes/probe line IPD’s IPD’s. Fig. 14. Typical 3D 6-probes/probe ( γ values:: 0.0, 0.5, 0.8 top to bottom)

[1] Formato, R.A. (2007). Central force optimization: a new metaheuristic with applications in applied electromagnetics. Progress In Electromagnetics Research, Research PIER 77: 425-491 (2007). [2] Green, R., Wang, L., Alam, M. (2012). Training neural networks using Central Force Optimization Optimizati and Particle Swarm Optimization: Insights and comparisons. Expert Systems with Applications. Applications 39(1), 555-563 (2012). doi:10.1016/j.eswa.2011.07.046 [3] Qubati, G., Dib, N. (2010). Microstrip patch antenna optimization using modified central force optimization. Progress In Electromagnetics Research B, B 21, 281-298. [4] Mahmoud, KR (2011) Central force optimization: Nelder-Mead Mead hybrid algorithm for rectangular rectangul microstrip antenna design. Electromagnetics, Electromagnetics 31(8):578-592. doi: 10.1080/02726343.2011.621110. [5] Asi M, Dib NI (2010) Design of multilayer microwave broadband absorbers using central force optimization. Progress In Electromagnetics Research B, B 26:101-113, 2010. doi:10.2528/PIERB10090103 10.2528/PIERB10090103 [6] Ali Haghighi A, Ramos (2012). HM Detection of leakage freshwater and friction factor calibration in drinking networks using central force optimization. Water Resources Management: 26(8), 2347-2363. [7] Montaser, A.M., Mahmoud, K.R., Elmikati, H.A. (2012). Integration of an optimized E-shaped E patch antenna into a laptop structure for bluetooth and notched-UWB notched standards using optimization techniques. ACES Journal (Applied Computational Electromagnetics Society). Soc 27(10), p. 784 (2012).

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[8] Montaser, A.M., Mahmoud, K.R., Abdel-Rahman, A.B., Elmikati, H.A. (2013) "Design bluetooth and notchedUWB e-shape antenna using optimization techniques," Progress In Electromagnetics Research B, 47:279-295, 2013. doi:10.2528/PIERB12101407 [9] Montaser, A.M., Mahmoud, K.R., Elmikati, H.A. (2012). Tri-band slotted bow-tie antenna design for RFID reader using hybrid CFO-NM algorithm, Paper presented at 29th National Radio Science Conference (NRSC), 10-12 April 2012, Cairo, 119-126. doi: 10.1109/NRSC.2012.6208515. [10] Yilmaz, A.E., Weber, G-W. (2011). Why you should consider nature-inspired optimization methods in financial mathematics. In Nonlinear and Complex Dynamics: Applications in Physical, Biological, and Financial Systems. Machado, Baleanu, and Luo, Springer-Verlag New York (pp.241-255). [11] Green, R.C., Wang, L., Alam, M. (2012). Intelligent state space pruning with local search for power system reliability evaluation. Paper presented at 3rd IEEE PES International Conference and Exhibition on Innovative Smart Grid Technologies (ISGT Europe), 14-17 Oct. 2012 , Berlin, pp. 1-8. doi: 10.1109/ISGTEurope.2012.6465775 [12] Shaikh, N.F., Doye, D.D. (2014). A novel iris recognition system based on central force optimization. International Journal of Tomography and Simulation. 27(3) [13] Gonzalez, J.E., Amaya, I., Correa, R., (2013). Design of an optimal multi-layered electromagnetic absorber through central force optimization. Paper presented at PIERS, Stockholm, Sweden, 12-15 Aug. 2013, Proceedings p. 1082. [14] A.khaleel, A. A. J. (2009). Linear phase finiteimpulse response filter design using central force optimization algorithm. M.Sc. Elec. Eng. Thesis, online at https://theses.ju.edu.jo/Original_Abstract/JUF0706056/JUF0 706056.pdf [15] Ahmed, E., Mahmoud, K.R., Hamad, S., Fayed, Z.T. (2013) CFO parallel implementation on GPU for adaptive beam-forming applications. International Journal of Computer Applications. 70(12):10-16, May 2013. doi: 10.5120/12013-8001 [16] Amed, E., Mahmoud, K.R., Hamad, S., Fayed, Z.T. (2013). Real time parallel PSO and CFO for adaptive beamforming applications. PIERS Proceedings, Tapei, March 2528, 2013, p. 816. [17] Chen, Y., Yu, J., Mei, Y., Wang, Y., Su, X. (2015). Modified central force optimization (MCFO) algorithm for 3D UAV path planning. Neurocomputing. Online 26 July 2015,http://www.sciencedirect.com/science/article/pii/S0925 231215010267. doi: 10.1016/j.neucom.2015.07.044 [18] Montaser, A.M. (2014). Reconfigurable antenna for RFID reader and notched UWB applications using DTO algorithm. Progress in Electromagnetics Research C, Vol. 47, 65-74, 2014

[19] Montaser, A.M., Mahmoud, K.R., Abdel-Rahman, A.B., Elmikati, H.A. (2013). Design of a compact tri-band antenna for RFID handheld applications using optimization techniques. Paper presented at 30th National Radio Science Conference (NRSC), Cairo, Egypt, 16-18 April, 2013. doi: 10.1109/NRSC.2013.6587905. [20] Abdel-Rahman, A.B., Montaser, A.M., Elmikati, H.A., (2012). Design of a novel bandpass filter with microstrip resonator loaded capacitors using CFO-HC algorithm. Paper presented at Middle East Conference on Antennas and Propagation, Cairo, Egypt, 29-31 Dec. 2012. doi:10.1109/MECAP.2012.6618188 [21] Amaya, I., Correa, R., (2013). Optimal design of multilayer EMAs for frequencies between 0.85 GHz and 5.4 GHz. Revista de Ingenieria (Universidad de los Andes). 38, 2013. [22] Formato RA. Improved CFO Algorithm for Antenna Optimization. Progress In Electromagnetics Research B. 2010; 19: 405-425. doi:10.2528/PIERB09112309. [23] Qubati GM, Dib NI. Microstrip Patch Antenna Optimization using Modified Central Force Optimization. Progress in Electromagnetics Research B. 2010;21:281-298. http://www.jpier.org/pierb/pier.php?paper=10050511 [24] Formato RA. New Techniques for Increasing Antenna Bandwidth with Impedance Loading. Progress In Electromagnetics Research B. 2011;29:269-288. doi:10.2528/PIERB11021904. [25] Formato, RA. Improving Bandwidth of Yagi-Uda Arrays, Wireless Engineering and Technology, vol. 3, January 2012, pp. 18-24. http://www.SciRP.org/journal/wet (doi:10.4236/wet.2012.31003). [26] Variable Z0 Antenna Technology: A New Approach for IoT Wireless, Intl. J. Microwave & Wireless Tech. 2014. doi:10.1017/S1759078714000634 [27] Issues in Antenna Optimization – A Monopole Case Study,” Applied Computational Electromagnetics Society J., 28(11):1122. Nov. 2013. [28] Roa, O., Amaya, I., Ramirez, F., Correa, R. (2012). Solution of nonlinear circuits with the Central Force Optimization algorithm. Paper presented at 2011 IEEE 4th Colombian Workshop on Circuits and Systems (CWCAS), Barranquilla, Colombia, 1-2 Nov. 2012. doi: 10.1109/CWCAS.2012.6404079 [29] Montaser, A.M., Mahmoud, K.R., AElmikati, H.A., Abdel-Rahman, A.B., (2012). Circular, hexagonal and octagonal array geometries for smart antenna systems using hybrid CFO-HC algorithm.Journal of Engineering Sciences, Assiut University, 40(6), 1715-1732 (2012). online at http://www.aun.edu.eg/journal_files/90_J_3128.pdf [30] da Luz, M.V.F., Leite, J.V., Coelho, L. MagnetoThermal Modeling and Optimization of a High Voltage AirBreak Disconnector. Sixteenth Biennial IEEE Conference on Electromagnetic Field Computation (CEFC2014), Annecy, France. 25-28 May 2014, pp.1-1.

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[31] Sahu1, N.K., Sharma, S., Agrawal, J. (2014). Improved Two-Level Clustering Technique for Multi-View Data Using Central Force Optimization Algorithm. Engineering Universe for Scientific Research and Management (International Journal), 6(5), May 2014, Paper ID: 2014/EUSRM/5/2014/8290 [32] Eltokhey, M.W., Mahmoud, K.R., Obayya, S.S.A. (2015). Optimized diffusion spots' locations for simultaneous improvement in SNR and delay spread. Photonic Network Communications. Online 15 Oct. 2015. doi: 10.1007/s11107015-0574-3 [33] Ding D, Luo X, Chen J, Wang X, Du P, Guo Y. A Convergence Proof and Parameter Analysis of Central Force Optimization Algorithm. Journal of Convergence Information Technology (JCIT). 2011;6(10):16-23. [34]. Ding D, Qi D, Luo X, Chen J, Wang X, Du P. Convergence analysis and performance of an extended central force optimization algorithm. Applied Mathematics and Computation. 2012;219(4):2246–2259 http://dx.doi.org/10.1016/j.amc.2012.08.071. [35] Green RC, Wang L, Alam M, Formato RA. Central Force Optimization on a GPU: A Case Study in High Performance Metaheuristics using Multiple Topologies. The Journal of Supercomputing. 2012;62(1):378-398 doi:10.1007/s11227-011-0725-y. [36] Green, R.C., Wang, L., Alam, M., Formato, R.A. (2011). Central force optimization on a GPU: A case study in high performance metaheuristics using multiple topologies. Paper presented at 2011 IEEE Congress on Evolutionary Computation (CEC), New Orleans, LA, 5-8 June 2011. doi: 10.1109/CEC.2011.5949667. [37] Formato, R.A., Central Force Optimization with Variable Initial Probes and Adaptive Decision Space, Applied Mathematics and Computation., vol. 217, no. (21), pp. 8866–8872, 2011. doi:10.1016/j.amc.2011.03.151. [38] Chao, M., Xin, S. Z., Min, L. S. (2014) Neural network ensembles based on copula methods and distributed multiobjective central force optimization algorithm. Engineering Applications of Artificial Intelligence. 32:203– 212. doi: 10.1016/j.engappai.2014.02.009. [39] Qian, W-y., Zhang, T-t., (2012). Adaptive Central Force Optimization Algorithm. http://epub.cnki.net/ [40] Jie, L. Adaptive central force optimization with variable population size. (2014). Paper presented at Tenth International Conference on Computational Intelligence and Security (CIS), 15-16 Nov. 2014, Kunming, China, pp. 1720. doi:10.1109/CIS.2014.46. [41] Liu, Y., Tian, P. (2014). A multi-start central force optimization for global optimization. In Intelligent Systems Reference Library, Vol. 62: Innovative Computational Intelligence: A Rough Guide to 134 Clever Algorithms, Xing, B., Gao, W-J., Springer. doi: 10.1016/j.asoc.2014.10.031

[42] Liu, J., Wang Y., An improved central force optimization algorithm for multimodal optimization, Journal of Applied Mathematics, vol. 2014, Article ID 895629, 12 pages, 2014. doi:10.1155/2014/895629 [43] Zhang, T-t., Lu, J., Qian, W-y., Central force optimization algorithm based on differential evolution operator mutation (2012). Journal of Bohai University (Natural Science Edition). 2012(3). [44] Qian, W., Wang, B., Feng, Z. (2015). Adaptive central force optimization algorithm based on the stability analysis. Mathematical Problems in Engineering. Vol. 2015, Art. ID 914789, 10 pages. doi: http://dx.doi.org/10.1155/2015/914789. [45] Meng, C., Gong, J., Liu, S., Sun, Z., Convergence analysis and application of the central force optimization algorithm, Science China Information Sciences, 58, May 2015. doi: 10.1007/s11432-015-5301-2 [46] Yang, L., Qian, W., Zhang, Q. (2011). Central Force Optimization. Journal of Bohai University (Natural Science Edition), 2011(3). [47] Singh, S., Kaur, J., Sinha, R.S., (2014) A Comprehensive Survey on Various Evolutionary Algorithms on GPU. International Conference on Communication, Computing & Systems (ICCCS–2014) , 20-21 Feb 2014, Chennai, India, pp. 83-88. [48] Margarita Spichakova, M. (2013). An approach to the inference of finite state machines based on a gravitationallyinspired search algorithm. Proceedings of the Estonian Academy of Sciences, 2013, 62(1), 39–46. doi: 10.3176/proc.2013.1.05. www.eap.ee/proceedings [49] Fister, I. Jr., Yang, X-S., Fister, I., Brest, J., Fister, D., (2013) A Brief Review of Nature-Inspired Algorithms for Optimization. Elektrotehniski Vestnik 80(3):1–7, 2013 (English Ed.). [50] Can, U., Alatas, B. (2015). Physics Based Metaheuristic Algorithms for Global Optimization. American Journal of Information Science and Computer Engineering (1)3,94-106 (2015). http://www.aiscience.org/journal/ajisce [51] Sinha, R.S., Singh, S., (2014). Optimization Techniques on GPU: A Survey. International Multi Track Conference on Science, Engineering & Technical Innovations (IMTC14), 3-4 June 2014, Punjab, India. pp. 566-568. [52] Valsecchi, G.B., Milani, A., Gronchi, G.F., Chesley, S.R. (2003). Resonant returns to close approaches: analytical theory. Astronomy & Astrophysics. 408(3):1179-1196. [53] Schweickart, R, Chapman, C., Durda, D., Hut, P., Bottke, B., Nesvorny, D. (2006) Threat characterization: trajectory dynamics. http://arxiv.org/abs/physics/0608155. [54] Formato, R.A. (2013). Are Near Earth Objects the Key to Optimization Theory? British Journal of Mathematics & Computer Science. 3(3): 341-351, 2013.

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[55] Formato, R.A. (2010). Central Force Optimization and NEOs – First Cousins? Journal of Multiple Multiple-Valued Logic & Soft Computing. 16:547–565 (2010). [56] Formato, R. A. (2009). Central force optimization: A new deterministic gradient-like like optimization metaheuristic. OPSEARCH 46(1):25–51 (2009). [57] Formato, R.A. (2009). Central force optimisation: a new gradient-like like metaheuristic for multidimensional search and optimisation. Int. J. Bio-Inspired Inspired Computation Computation. 1(4): 217-238 (2009). [58] Formato, R.A. (2008). Central Force Optimization: A New Nature Inspired Computational Framework for Multidimensional Search and Optimization Optimization, Studies in Computational Intelligence (SCI) 129 129:221-238 (2008). Springer-Verlag Berlin Heidelberg. [59] Formato, R.A. (2010). Parameter--Free Deterministic Global Search with Simplified Central Force Optimization. D.-S. Huang et al.. (Eds.): ICIC 2010, LNCS ((Lecture Notes in Computer Science) 6215:309–318 318 (2010). Springer SpringerVerlag Berlin Heidelberg. [60] Luke, S. (2015). Essentials of Metaheuristics Metaheuristics. 2nd edition, online Ver. 2.2, October 2015. http://cs.gmu.edu/!sean/book/metaheuristics/ sean/book/metaheuristics/ [61] Formato, R.A. (2013). Pseudorandomness in Central Force Optimization. British Journal of Mathematics & Computer Science, 3(3):241-264, 2013. [62] Formato, R.A. Pi Fractions for Generating Uniformly Distributed Sampling Points in Global Search and Optimization Algorithms. January 2014 http://arxiv.org/abs/1401.3038. [63] Wang, J. (2011). Particle Swarm Optimization with Adaptive Parameter Control and Opposition. Journal of Computational Information Systems,, 7(12):4463 7(12):4463-4470 (2011). http://www.jofcis.com. [64] Variable Z0 Antenna Device Design System and Method, Formato, U.S.. Patent No. US8776002B2, Jul. 8, 2014. [65] Burke, G. J. (2011). Numerical Electromagnetics Code - NEC-4.2 4.2 Method of Moments, Part I: User's Manual. Lawrence Livermore Nat. Lab., Livermore, CA, Rep. LLNL LLNLSM-490875, Jul. 2011. [66] Athan Papadimitriou, NEC4/MP, C4/MP, Ver. 1.5, 05 05-252012 (no longer available). Email: [email protected]. Additional info at http://fornectoo.freeforums.org/nec2 http://fornectoo.freeforums.org/nec2-mpa-refactored-multithreaded-nec2-engine-for for-4nec2-t371.htm [67] Comparing Continuous Optimizers: COCO http://coco.gforge.inria.fr/ [68] Finck, S., Hansen, N., Ros,, R. and Auger, A. RealParameter Black-Box Box Optimization Benchmarking 2010: Presentation of the Noiseless Functions.. Working Paper 2009/20, compiled November 17, 2015. Online at http://coco.gforge.inria.fr/doku.php?id=downloads

[69] Finck, S., Hansen, N., Ros, Ros R. and Auger, A. RealParameter Black-Box Box Optimization Benchmarking 2010: Presentation of the Noisy Functions. Working Paper 2009/21, compiled November 17, 2015. 2015 Online at http://coco.gforge.inria.fr/doku.php?id=downloads [70] Formato, R. A. Determinism in Electromagnetic Design & Optimization - Part II: BBP-Derived BBP π fractions for Generating Uniformly Distributed Sampling Points in Global Search and Optimization Algorithms. Algorithms Online at www.e-fermat.org.

Richard A. Formato (IEEE Life SM) is a Consulting Engineer and a Registered Patent Attorney. He received his JD from Suffolk University Law School, PhD and MS degrees from the University of Connecticut, and MSEE and BS (Physics) degrees from Worcester orcester Polytechnic Institute. In the early 1990s, he began applying genetic algorithms algori to antenna design and developed YGO3 freeware freewa (Yagi Genetic Optimizer). His interest in optimization algorithms led to the development off Central Force Optimization (CFO) , Dynamic Threshold Optimization (DTO), and Very Simple Optimization (VSO). Variable Z0 antenna technology was invented in a true "aha" moment when he realized that an antenna's feed system impedance should be treated as a true optimization variable, not as a fixed parameter.

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