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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,

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])



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 real-

world 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 dN -dimensional decision space [56-

59]. 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 deter-

ministic, 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 beam-

forming 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



functions for broadband HF monopoles [27]; solving

nonlinear circuits [28]; optimizing "smart" antennas [29] and

a high voltage air-break disconnector [30]; improving two-

level 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].

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 real-

world 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 dN -dimensional decision space,

this objective function is defined as ∑=

=dN

i

ii xxxf1

)]sin([)( ,

500500 ≤≤− ix . It is highly multimodal with a single global

maximum of dN×9829.418 at dN

]9687.420[ ([63] @

p.4467). The 2D maximum, which can be visualized, is

9658.837 at. )9687.420,9687.420( , 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.

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.

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 vehicle-

mounted 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



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.

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

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 base-

fed high-frequency (HF) monopole antenna on a perfectly

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, maxG ; input impedance,

ininin XjRZ += ;

and voltage standing wave ratio relative to a reference

impedance 0Z , )( 0ZVSWR (the industry standard is

Ω= 500Z ). Three different objective functions are

considered:

)(

)(min)(min),,(

0

max01

ZVSWR

GZHRf

∆

+=

ε (1)

)(max)(max

)(min),,(

0

02

inin XRZZHRf

⋅−=

ε (2)

)](min)([max)()(max

)(min),,(

00

03XinXinZVSWRRZ

ZHRfin −⋅∆⋅−

=ε (3)

where )( 0ZVSWR∆ is the difference between maximum and

minimum VSWR over the 5-30 MHz HF band.

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 1f 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. 2f is so

pathologically spiky that many algorithms probably cannot

efficiently locate its maximum. Only 3f is well-behaved

with a reasonably sharp maximum not too close to the DS

boundary.

Fig. 3. Base-fed Loaded Monopole Antenna.



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 real-

world engineering problems as opposed to benchmarks.

V. ANOTHER REAL-WORLD ANTENNA PROBLEM:

WIRELESS POWER TRANSFER

Another real-world example involves optimizing a multi-

arm 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.

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 cut-

and-try the following less than obvious fitness function was

determined to be a good one for achieving the objectives:

in

inin R

XRZAGTka

F

+−+−⋅

=

01

1...(4)

Fig. 4. Landscape of Objective Function )50,,(1 HRf .

Note that VSWR does not appear explicitly whereas

components of the antenna input impedance do, and that the

value of 0Z was determined by CFO using Variable Z0

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-of-

Moments modeling engine considered to be the "gold

standard" for wire antenna modeling.



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



assume PEC wires because the radiation efficiency with

annealed Cu conductors is %99≈ε .

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.

Fig. 7. Fitness 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 dN -

dimensional hyperspace Ω for the global maxima of a

function ),...,,( 21 dNxxxf . Stated more precisely,

determine RRfRXXf nn →⊂Ω⊂Ω∈ :,:)(max with

,...,1,|:maxmin

diii NixxxX =≤≤=Ω being the bounded

region of feasible (allowable) solutions (decision space);

)( Xf a pointer to the objective function; and L

Ω∈∪Ω= XXf ,)( the problem's landscape. The ix are

the decision variables for each coordinate number i . Fitness

refers to the value of the objective function )(xfr

at sample

point xr

= ),...,,( 21 dNxxx in Ω . There is no a priori

information about the objective function’s extrema, that is,

s)'(xfr

landscape (topology) is completely unknown.

CFO searches Ω by flying "probes" through the space at

discrete "time" steps (iterations). Each probe’s location is

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.

The position vector at step j is k

N

k

jp

k

p

j exRd

ˆ1

,∑=

=r

, where

the jp

kx,

are the probe's coordinates and ke is the unit vector

along the kx -axis. Indices p , pNp ≤≤1 , and j ,

tNj ≤≤0 are the probe number and iteration number,

respectively, where pN and

tN are the corresponding total

number of probes and total number of iterations (time steps).

Fig. 8. Davg Evolution, 200 MHz Helix.

Fig. 9. 200 MHz 4-Arm Helix.

(axis 0.05m, L=0.148m, D=0.089 m, #12AWG)

Step #

0 20 40 60 80 100

Fit

nes

s

0

2

4

6

8

10

12

14Fitness (200 Mhz Multi-Arm Folded Helix)

Step#

0 20 40 60 80 100

Dav

g

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07Davg (200 MHz Multi-Arm Folded Helix)



Fig. 10. 200 MHz Helix VSWR // 95.67Ω [PEC wires].

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

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

B. Equations of Motion

In metaphorical CFO-space each of the pN probes is

accelerated by the gravitational pull of the surrounding

"masses." At step 1−j probe p experiences an acceleration

given by the acceleration equation

βα

p

j

k

j

p

j

k

j

N

pkk

p

j

k

j

p

j

k

j

p

j

RR

RRMMMMUGa

p

11

11

1

11111

)()()(

−−

−−

≠=

−−−−−

−

−×−⋅−= ∑ rr

rr

r (5)

which is the first of CFO’s two equations of motion.

),...,,( 1,1,

2

1,

11

−−−

− = jp

N

jpjpp

j dxxxfM is the fitness at probe p ’s

location at time step 1−j . Every other probe at that step

(iteration) has associated with it the fitness

p

k

j NppkM ,...,1,1,...,1,1 +−=−. G is CFO’s gravitational

constant, and )(⋅U is the Unit Step function,

≥

=otherwise

zzU

,0

0,1)(

.

The acceleration p

ja 1−

r causes probe p to move from

position p

jR 1−

r at step 1−j to position

p

jRr

at step j

according to the trajectory equation

1,2

1 2

11 ≥∆+= −− jtaRR p

j

p

j

p

j

rrr (6)

which is CFO’s second equation of motion. t∆ is the "time

interval" between steps during which the acceleration is

constant.

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.

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



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. 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 CFO-

space in stark contrast "mass" is a user-defined positive-

definite function of the objective function’s fitness (not

necessarily the fitness itself).

In the acceleration equation (5) above mass by definition isα

)()( 1111

p

j

k

j

p

j

k

jCFO MMMMUMASS −−−− −⋅−= , 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 CFO-

space 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. 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 min

ii xx < or max

ii xx > , 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 10 ≤≤∆< reprep FF

of the distance between the probe’s starting coordinate and

the corresponding boundary coordinate.

repF 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 repF∆ , 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 repF 's

value is uncorrelated with the landscape thereby adding a

measure of pseudo randomness.

VII. CFO PSEUDOCODE & IPD SPECIFICATION

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

d

p

N

N probes per dimension uniformly distributed on lines

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 1x and 2x axes. The probe

lines intersect at a point )( minmaxmin XXXDrrrr

−+= γ where

i

N

i

i exXd

ˆ1

min

min ∑=

=r

and i

N

i

i exXd

ˆ1

max

max ∑=

=r

are the diagonal’s

endpoint vectors. Parameter 10 ≤≤ γ 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


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 of 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

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

can be done. Stochastic algorithms 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

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

parameters 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 r

real world problems and has been improved and extended by

many researchers. A bibliography of CFO

included that may assist practitioners and researchers who

are new to the algorithm.



random IPD or they could be used to pseudo

Fig. 12. Typical 2D Probe Line IPD.

using the variable probe

Of course many adaptations are

possible, for example, adding other loops to vary, say, the

f 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

randomness.

Central Force Optimization has become an established

metaheuristic for global search and optimization. It is

problems that require the

of a suitable objective function before any D&O

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 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

s whose values are precisely known but

uncorrelated with the optimization problem's landscape.

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 range of

real world problems and has been improved and extended by

bibliography of CFO-related work is

assist practitioners and researchers who

Fig. 13. Sample 2D 12-probes/probe

( γ values: 0.0, 0.4, 0.9 top to bottom)


probes/probe line IPD’s.

, 0.4, 0.9 top to bottom)


Fig. 14. Typical 3D 6-probes/probe line IPD’s

( γ values: 0.0, 0.5, 0.8 top to bottom)



probes/probe line IPD’s.

: 0.0, 0.5, 0.8 top to bottom)

Fig. 15. CFO Pseudocode with Variable Probe Line IPDs.

REFERENCES

[1] Formato, R.A. (2007). Central force optimization: a

new metaheuristic with applications in applied

electromagnetics. Progress In Electromagnetics Research

PIER 77: 425-491 (2007).

[2] Green, R., Wang, L., Alam, M. (2012). Training

neural networks using Central Force Optimizati

Particle Swarm Optimization: Insights and comparisons.

Expert Systems with Applications

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

[4] Mahmoud, KR (2011) Central force optimization:

Nelder-Mead hybrid algorithm for rectangul

antenna design. Electromagnetics

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

26:101-113, 2010. doi: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.

Management: 26(8), 2347-2363.

[7] Montaser, A.M., Mahmoud, K.R., Elmikati, H.A.

(2012). Integration of an optimized E

into a laptop structure for bluetooth and notched

standards using optimization techniques.

(Applied Computational Electromagnetics Soc

p. 784 (2012).

For startγγ = to

stopγ by γ∆

(a.1) Compute initial probe distribution.

(a.2) Compute initial fitness matrix.

(a.3) Assign initial probe accelerations.

(a.4) Set initial repF .

For 0=j to tN (or earlier termination criterion)

(b) Compute probe position vectors

p

p

j NpR ≤≤1,r

[eq.(6)].

(c) Retrieve errant probes ( p≤1

If ˆˆ minmin

ii

p

jii

p

j xeRxeR +=⋅∴≤⋅rr

If ˆˆ maxmax

ii

p

jii

p

j xeRxeR +=⋅∴≥⋅rr

(d) Compute fitness matrix for current probe

(e) Compute accelerations using current probe distribution

and fitnesses [eq. (5)].

(f) Assign repF .


Fig. 15. CFO Pseudocode with Variable Probe Line IPDs.

EFERENCES

Central force optimization: a

new metaheuristic with applications in applied

Progress In Electromagnetics Research,

Green, R., Wang, L., Alam, M. (2012). Training

neural networks using Central Force Optimization and

Particle Swarm Optimization: Insights and comparisons.

Applications. 39(1), 555-563 (2012).

Qubati, G., Dib, N. (2010). Microstrip patch antenna

optimization using modified central force optimization.

Progress In Electromagnetics Research B, 21, 281-298.

Mahmoud, KR (2011) Central force optimization:

Mead hybrid algorithm for rectangular microstrip

Electromagnetics, 31(8):578-592. doi:

Asi M, Dib NI (2010) Design of multilayer

microwave broadband absorbers using central force

Progress In Electromagnetics Research B,

10.2528/PIERB10090103

A, Ramos (2012). HM Detection of

leakage freshwater and friction factor calibration in drinking

networks using central force optimization. Water Resources

Montaser, A.M., Mahmoud, K.R., Elmikati, H.A.

(2012). Integration of an optimized E-shaped patch antenna

into a laptop structure for bluetooth and notched-UWB

standards using optimization techniques. ACES Journal

(Applied Computational Electromagnetics Society). 27(10),

(a.1) Compute initial probe distribution.

(a.2) Compute initial fitness matrix.

(a.3) Assign initial probe accelerations.

ermination criterion)

(b) Compute probe position vectors

pNp ≤ ):

)ˆ(min

1 ii

p

jrep xeRF −⋅+ −

r.

)ˆ( 1

max

i

p

jirep eRxF ⋅−+ −

r.

(d) Compute fitness matrix for current probe distribution.

(e) Compute accelerations using current probe distribution



[8] Montaser, A.M., Mahmoud, K.R., Abdel-Rahman,

A.B., Elmikati, H.A. (2013) "Design bluetooth and notched-

UWB 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 finite-

impulse 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 beam-

forming applications. PIERS Proceedings, Tapei, March 25-

28, 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. Magneto-

Thermal Modeling and Optimization of a High Voltage Air-

Break Disconnector. Sixteenth Biennial IEEE Conference on

Electromagnetic Field Computation (CEFC2014), Annecy,

France. 25-28 May 2014, pp.1-1.



[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/s11107-

015-0574-3

[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. 17-

20. 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 gravitationally-

inspired 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.


[55] Formato, R.A. (2010). Central Force Optimization

and NEOs – First Cousins? Journal of Multiple

& Soft Computing. 16:547–565 (2010).

[56] Formato, R. A. (2009). Central force optimization: A

new deterministic gradient-like optimization metaheuristic.

OPSEARCH 46(1):25–51 (2009).

[57] Formato, R.A. (2009). Central force optimisation: a

new gradient-like metaheuristic for multidimensional search

and optimisation. Int. J. Bio-Inspired Computation

217-238 (2009).

[58] Formato, R.A. (2008). Central Force Optimization: A

New Nature Inspired Computational Framework for

Multidimensional Search and Optimization

Computational Intelligence (SCI) 129

Springer-Verlag Berlin Heidelberg.

[59] Formato, R.A. (2010). Parameter-

Global Search with Simplified Central Force Optimization.

D.-S. Huang et al. (Eds.): ICIC 2010, LNCS (

in Computer Science) 6215:309–318 (2010). Springer

Verlag Berlin Heidelberg.

[60] Luke, S. (2015). Essentials of Metaheuristics

edition, online Ver. 2.2, October 2015.

http://cs.gmu.edu/!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.

Computational Information Systems, 7(12):4463

(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 Method of Moments, Part I: User's Manual.

Lawrence Livermore Nat. Lab., Livermore, CA, Rep. LLNL

SM-490875, Jul. 2011.

[66] Athan Papadimitriou, NEC4/MP, Ver. 1.5, 05

2012 (no longer available). Email: [email protected].

Additional info at http://fornectoo.freeforums.org/nec2

a-refactored-multithreaded-nec2-engine-for

[67] Comparing Continuous Optimizers: COCO

http://coco.gforge.inria.fr/

[68] Finck, S., Hansen, N., Ros, R. and

Parameter Black-Box Optimization Benchmarking

Presentation of the Noiseless Functions.

2009/20, compiled November 17, 2015. Online at

http://coco.gforge.inria.fr/doku.php?id=downloads



Central Force Optimization

Journal of Multiple-Valued Logic

Formato, R. A. (2009). Central force optimization: A

like optimization metaheuristic.

Formato, R.A. (2009). Central force optimisation: a

like metaheuristic for multidimensional search

Inspired Computation. 1(4):

Central Force Optimization: A

New Nature Inspired Computational Framework for

Multidimensional Search and Optimization, Studies in

129:221-238 (2008).

-Free Deterministic

Global Search with Simplified Central Force Optimization.

. (Eds.): ICIC 2010, LNCS (Lecture Notes

318 (2010). Springer-

Essentials of Metaheuristics. 2nd

sean/book/metaheuristics/

Formato, R.A. (2013). Pseudorandomness in Central

British Journal of Mathematics &

Formato, R.A. Pi Fractions for Generating Uniformly

Distributed Sampling Points in Global Search and

Wang, J. (2011). Particle Swarm Optimization with

Adaptive Parameter Control and Opposition. Journal of

, 7(12):4463-4470

Antenna Device Design System and

. Patent No. US8776002B2, Jul. 8,

Burke, G. J. (2011). Numerical Electromagnetics

4.2 Method of Moments, Part I: User's Manual.

Lawrence Livermore Nat. Lab., Livermore, CA, Rep. LLNL-

C4/MP, Ver. 1.5, 05-25-

2012 (no longer available). Email: [email protected].

Additional info at http://fornectoo.freeforums.org/nec2-mp-

for-4nec2-t371.htm

Comparing Continuous Optimizers: COCO

, R. and Auger, A. Real-

Box Optimization Benchmarking 2010:

. Working Paper

Online at


[69] Finck, S., Hansen, N., Ros

Parameter Black-Box Optimization Benchmarking 2010:

Presentation of the Noisy Functions. Working Paper

2009/21, compiled November 17, 2015


[70] Formato, R. A. Determinism in Electromagnetic

Design & Optimization - Part II: BBP

Generating Uniformly Distributed Sampling Points in Global

Search and Optimization Algorithms

Online at www.e-fermat.org.

Richard A. Formato

a Consulting Engineer a

Patent Attorney.

from Suffolk University Law School,

PhD and MS degrees from the

University of Connecticut, and MSEE

and BS (Physics) degrees from

Worcester Polytechnic Institute.

early 1990s, he began applying genetic algori

design and developed YGO3 freewa

Optimizer). His interest in optimization algorithms led to the

development of Central Force Optimization

Dynamic Threshold Optimization (DTO), and Very Simple

Optimization (VSO). Variable Z0

invented in a true "aha" moment when he realized that an

antenna's feed system impedance should be treated as

optimization variable, not as a fixed parameter.


Ros, R. and Auger, A. Real-

Box Optimization Benchmarking 2010:

Presentation of the Noisy Functions. Working Paper

November 17, 2015. Online at


Determinism in Electromagnetic

Part II: BBP-Derived π fractions for

Generating Uniformly Distributed Sampling Points in Global

Algorithms.

Richard A. Formato (IEEE Life SM) is

a Consulting Engineer and 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

orcester Polytechnic Institute. In the

early 1990s, he began applying genetic algorithms to antenna

and developed YGO3 freeware (Yagi Genetic

His interest in optimization algorithms led to the

f Central Force Optimization (CFO) ,

Dynamic Threshold Optimization (DTO), and Very Simple

0 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

variable, not as a fixed parameter.

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Determinism in Electromagnetic Design Optimization - Part ......process is, for example the swarming behavior of fish and birds (Particle Swarm Optimization) or the foraging ... One