FORM 2
THE PATENTS ACT, 1970
(39 of 1970)
&
THE PATENT RULES, 2003
COMPLETE SPECIFICATION (See Section 10 and Rule 13)
Title of invention:
METHOD AND SYSTEM FOR OPTIMIZATION BASED ON PERISCOPIC, PHEROMONIC AND FRACTAL SEARCH
Applicant
Tata Consultancy Services Limited A company Incorporated in India under the Companies Act, 1956
Having address:
Nirmal Building, 9th floor,
Nariman point, Mumbai 400021,
Maharashtra, India
Preamble to the description
The following specification particularly describes the invention and the manner in which it is to be performed.

TECHNICAL FIELD [001] The disclosure herein generally relates solving optimization problems, and, more particularly, to systems and methods for optimization based on periscopic, pheromonic and fractal search.
BACKGROUND [002] In general, an optimization problem involves a mathematical or algorithmic process of finding the best combination of parameters of a device or an experiment that minimizes or maximizes its output, defined by a target-, cost-, or fitness-function. However, an exhaustive search across possible parameter-values and ranges is impractical. Typical solutions reduce the search space but can settle at a local optimum instead of the global optimum. Metaheuristic algorithms increase the probability of finding the global optimum. They do so by striking a balance between the spread of randomly searching various regions of the search space and refining the solutions in each region.
SUMMARY
[003] Embodiments of the present disclosure present technological improvements as solutions to one or more of the above-mentioned technical problems recognized by the inventors in conventional systems.
[004] In an aspect, there is provided a processor implemented method comprising the steps of: defining, via one or more hardware processors, an M-dimensional search space for an optimal solution to an optimization problem being a minimization problem or a maximization problem, wherein M is based on the optimization problem; employing N entities, N being empirically determined, via the one or more hardware processors, in the defined M-dimensional search space, wherein each of the N entities acts as a caterpillar in search of food being representative of the optimal solution, in the defined M-dimensional search space, and wherein a change in each of the N entities is governed by a periscopic property, a pheromone persistence property and a fractal dimension; randomly initializing, via the one or more hardware processors, positions of the N entities in an N * M

matrix represented as a matrix W; iteratively updating the matrix W, via the one or more hardware processors, until a termination criterion is met, using the steps of: performing, for each of the N entities, the steps of: computing a fitness value associated with each of the N entities in the matrix W by evaluating an objective function associated with the optimization problem; evaluating, using a Katz Fractal Dynamics (FD) method, the fractal dimension of a time series that comprises the computed fitness value associated with each of the N entities, wherein the fractal dimension is indicative of roughness of the defined M-dimensional search space; computing new positions of each of the N entities in the matrix W to obtain a matrix U representative of a new state corresponding to movement of each of the N entities, based on (i) a randomly generated waving probability ri associated with each of the N entities in the matrix W, ri being greater than a pre-defined randomly generated initial waving probability Pw, (ii) position of a neighboring entity therein using the pheromone persistence property and (iii) the evaluated fractal dimension; updating the matrix W using the computed new positions of each of the N entities in the matrix U by comparing the fitness value associated with each of the N entities in the matrix W with the fitness value associated with each of the N entities in the matrix U and replacing the entity wi in the matrix W with the entity ui in the matrix U, if the fitness value of ui is optimal compared to the fitness value of wi; and identifying the entity wi in the matrix W having an optimal fitness value compared to the fitness values of the rest of the N entities in the matrix W, and designating a position of the identified entity wi as an interim optimal solution, wherein the termination criterion is one of (i) completion of a predefined number of iterations and (ii) if the designated interim optimal solution is lower than a tolerance value for a predetermined global minimum for the minimization problem or higher than the tolerance value for a predetermined global maximum for the maximization problem; and designating the interim optimal solution associated with a last iteration as the optimal solution for the optimization problem.
[005] In another aspect, there is provided a system comprising: one or more data storage devices operatively coupled to one or more hardware processors and configured to store instructions configured for execution via the one or more

hardware processors to: define an M-dimensional search space for an optimal solution to an optimization problem being a minimization problem or a maximization problem, wherein M is based on the optimization problem; employ N entities, N being empirically determined in the defined M-dimensional search space, wherein each of the N entities acts as a caterpillar in search of food being representative of the optimal solution, in the defined M-dimensional search space, and wherein a change in each of the N entities is governed by a periscopic property, a pheromone persistence property and a fractal dimension; randomly initialize positions of the N entities in an N * M matrix represented as a matrix W; iteratively update the matrix W until a termination criterion is met, the steps of: perform, for each of the N entities, using the steps of: computing a fitness value associated with each of the N entities in the matrix W by evaluating an objective function associated with the optimization problem; evaluating, using a Katz Fractal Dynamics (FD) method, the fractal dimension of a time series that comprises the computed fitness value associated with each of the N entities, wherein the fractal dimension is indicative of roughness of the defined M-dimensional search space; computing new positions of each of the N entities in the matrix W to obtain a matrix U representative of a new state corresponding to movement of each of the N entities, based on (i) a randomly generated waving probability ri associated with each of the N entities in the matrix W, ri being greater than a pre-defined randomly generated initial waving probability Pw, (ii) position of a neighboring entity therein using the pheromone persistence property and (iii) the evaluated fractal dimension; updating the matrix W using the computed new positions of each of the N entities in the matrix U by comparing the fitness value associated with each of the N entities in the matrix W with the fitness value associated with each of the N entities in the matrix U and replacing the entity wi in the matrix W with the entity ui in the matrix U, if the fitness value of ui is optimal compared to the fitness value of wi; and identifying the entity wi in the matrix W having an optimal fitness value compared to the fitness values of the rest of the N entities in the matrix W, and designating a position of the identified entity wi as an interim optimal solution, wherein the termination criterion is one of (i) completion of a predefined number of iterations and (ii) if the

designated interim optimal solution is lower than a tolerance value for a predetermined global minimum for the minimization problem or higher than the tolerance value for a predetermined global maximum for the maximization problem; and designate the interim optimal solution associated with a last iteration as the optimal solution for the optimization problem.
[006] In yet another aspect, there is provided a computer program product comprising a non-transitory computer readable medium having a computer readable program embodied therein, wherein the computer readable program, when executed on a computing device, causes the computing device to: define an M-dimensional search space for an optimal solution to an optimization problem being a minimization problem or a maximization problem, wherein M is based on the optimization problem; employ N entities, N being empirically determined in the defined M-dimensional search space, wherein each of the N entities acts as a caterpillar in search of food being representative of the optimal solution, in the defined M-dimensional search space, and wherein a change in each of the N entities is governed by a periscopic property, a pheromone persistence property and a fractal dimension; randomly initialize positions of the N entities in an N * M matrix represented as a matrix W; iteratively update the matrix W until a termination criterion is met, the steps of: perform, for each of the N entities, using the steps of: computing a fitness value associated with each of the N entities in the matrix W by evaluating an objective function associated with the optimization problem; evaluating, using a Katz Fractal Dynamics (FD) method, the fractal dimension of a time series that comprises the computed fitness value associated with each of the N entities, wherein the fractal dimension is indicative of roughness of the defined M-dimensional search space; computing new positions of each of the N entities in the matrix W to obtain a matrix U representative of a new state corresponding to movement of each of the N entities, based on (i) a randomly generated waving probability ri associated with each of the N entities in the matrix W, ri being greater than a pre-defined randomly generated initial waving probability Pw, (ii) position of a neighboring entity therein using the pheromone persistence property and (iii) the evaluated fractal dimension; updating the matrix W using the computed new

positions of each of the N entities in the matrix U by comparing the fitness value associated with each of the N entities in the matrix W with the fitness value associated with each of the N entities in the matrix U and replacing the entity wi in the matrix W with the entity ui in the matrix U, if the fitness value of ui is optimal compared to the fitness value of wi; and identifying the entity wi in the matrix W having an optimal fitness value compared to the fitness values of the rest of the N entities in the matrix W, and designating a position of the identified entity wi as an interim optimal solution, wherein the termination criterion is one of (i) completion of a predefined number of iterations and (ii) if the designated interim optimal solution is lower than a tolerance value for a predetermined global minimum for the minimization problem or higher than the tolerance value for a predetermined global maximum for the maximization problem; and designate the interim optimal solution associated with a last iteration as the optimal solution for the optimization problem.
[007] In accordance with an embodiment of the present disclosure, the periscopic property is the ability of each of the N entities to move such that an associated vicinity is viewed beyond line of sight in a rest position; and the pheromone persistence property is the ability of each of the N entities to leave a trail of the position thereof.
[008] In accordance with an embodiment of the present disclosure, the one or more hardware processors are configured to compute new positions of each of the N entities in the matrix W as ui = wi + [rand() × Hw] × [wj - (wi × α)], wherein ui represents an entity in the matrix U, wi represents a corresponding entity in the matrix W, Hw represents a waving height of an associated entity, wj represents a neighboring entity in the matrix W, α represents the fractal dimension and wherein rand() is a function configured to generate values uniformly distributed between 0 and 1.
[009] In accordance with an embodiment of the present disclosure, the waving height Hw of each of the N entities is related to an empirically determined fixed length Lw and is represented by Hw = Lw + [rand() + 1] × 0.5.
[010] In accordance with an embodiment of the present disclosure, the optimization problem is the minimization problem with the objective function being

|G - C|, wherein G represents ground truth, C represents corrected gaze coordinates associated with raw gaze coordinates received from an eye tracker and |G - C | is
the Euclidean distance between G and C.
[011] In accordance with an embodiment of the present disclosure, the randomly generated waving probability n has a value in between 0 and 1.
[012] It is to be understood that both the foregoing general description and the following detailed description are exemplary and explanatory only and are not restrictive of the invention, as claimed.
BRIEF DESCRIPTION OF THE DRAWINGS
[013] The accompanying drawings, which are incorporated in and constitute a part of this disclosure, illustrate exemplary embodiments and, together with the description, serve to explain the disclosed principles:
[014] FIG.1 illustrates an exemplary block diagram of a system for optimization based on periscopic, pheromonic and fractal search, in accordance with some embodiments of the present disclosure.
[015] FIG.2A and FIG.2B illustrate an exemplary flow diagram of a computer implemented method for optimization based on periscopic, pheromonic and fractal search, in accordance with some embodiments of the present disclosure.
[016] FIG.3 illustrates Area Under the Curve (AUC) metric to analyze performance of the method in accordance with some embodiments of the present disclosure.
[017] FIG.4 illustrates a systematic error in eye tracker data, as known in the art.
[018] FIG. 5 illustrates corrected gaze coordinates for the actual eye tracker data, with the transformation matrices computed in accordance with some embodiments of the present disclosure, when training and testing data are the same.
[019] FIG.6 illustrates corrected gaze coordinates for the actual eye tracker data, with the transformation matrices computed.in accordance with some

embodiments of the present disclosure, when training and testing data are not the same.
[020] FIG.7 and FIG.8 illustrate root mean square (RMSE) values between gaze coordinates and ground truth locations corresponding to FIG.5 and FIG.6 respectively, in accordance with some embodiments of the present disclosure.
DETAILED DESCRIPTION OF EMBODIMENTS [021] Exemplary embodiments are described with reference to the accompanying drawings. In the figures, the left-most digit(s) of a reference number identifies the figure in which the reference number first appears. Wherever convenient, the same reference numbers are used throughout the drawings to refer to the same or like parts. While examples and features of disclosed principles are described herein, modifications, adaptations, and other implementations are possible without departing from the scope of the disclosed embodiments. It is intended that the following detailed description be considered as exemplary only, with the true scope being indicated by the following claims.
[022] Applicant has addressed the problem of increasing the probability of finding an optimal solution for an optimization problem in an efficient and quick manner by providing a metaheuristic approach based on the food search behavior of caterpillars. Caterpillars are typically voracious feeders. The movement or the dispersal of caterpillars is governed by the characteristics of natural surfaces. Along with this, the pheromone of caterpillars guide their peers in the food search task. The nature of the surfaces may be assessed using fractal analysis as the surface area and distances are a function of the scale at which the measurements are made. Another important movement behavior of caterpillars is the waving of their body. They lift their bodies up to a certain height and wave themselves in order to look for food in their vicinity. This nature along with the fractal analysis and the usage of pheromone is used by the Applicant in the metaheuristic approach of the present disclosure.
[023] Referring now to the drawings, and more particularly to FIG. 1 through FIG.8, where similar reference characters denote corresponding features

consistently throughout the figures, there are shown preferred embodiments and these embodiments are described in the context of the following exemplary system and/or method.
[024] FIG.1 illustrates an exemplary block diagram of a system 100 for optimization based on periscopic, pheromonic and fractal search, in accordance with some embodiments of the present disclosure. In an embodiment, the system 100 includes one or more processors 104, communication interface device(s) or input/output (I/O) interface(s) 106, and one or more data storage devices or memory 102 operatively coupled to the one or more processors 104. The one or more processors 104 that are hardware processors can be implemented as one or more microprocessors, microcomputers, microcontrollers, digital signal processors, central processing units, state machines, graphics controllers, logic circuitries, and/or any devices that manipulate signals based on operational instructions. Among other capabilities, the processor(s) are configured to fetch and execute computer-readable instructions stored in the memory. In the context of the present disclosure, the expressions ‘processors’ and ‘hardware processors’ may be used interchangeably. In an embodiment, the system 100 can be implemented in a variety of computing systems, such as laptop computers, notebooks, hand-held devices, workstations, mainframe computers, servers, a network cloud and the like.
[025] I/O interface(s) 106 can include a variety of software and hardware interfaces, for example, a web interface, a graphical user interface, and the like and can facilitate multiple communications within a wide variety of networks N/W and protocol types, including wired networks, for example, LAN, cable, etc., and wireless networks, such as WLAN, cellular, or satellite. In an embodiment, the I/O interface(s) can include one or more ports for connecting a number of devices to one another or to another server.
[026] The memory 102 may include any computer-readable medium known in the art including, for example, volatile memory, such as static random access memory (SRAM) and dynamic random access memory (DRAM), and/or non-volatile memory, such as read only memory (ROM), erasable programmable ROM, flash memories, hard disks, optical disks, and magnetic tapes. In an

embodiment, one or more modules (not shown) of the system 100 can be stored in the memory 102.
[027] FIG.2A and FIG.2B illustrate an exemplary flow diagram of a computer implemented method 300 for optimization based on periscopic, pheromonic and fractal search, in accordance with some embodiments of the present disclosure. In an embodiment, the system 100 includes one or more data storage devices or memory 102 operatively coupled to the one or more processors 104 and is configured to store instructions configured for execution of steps of the method 200 by the one or more processors 104. The steps of the method 200 will now be explained in detail with reference to the components of the system 100 of FIG.1. Although process steps, method steps, techniques or the like may be described in a sequential order, such processes, methods and techniques may be configured to work in alternate orders. In other words, any sequence or order of steps that may be described does not necessarily indicate a requirement that the steps be performed in that order. The steps of processes described herein may be performed in any order practical. Further, some steps may be performed simultaneously.
[028] Accordingly, in an embodiment of the present disclosure, the one or more processors 104, are configured to define, at step 202, an M-dimensional search space for an optimal solution to an optimization problem being a minimization problem or a maximization problem, wherein M is based on the optimization problem. For instance, in an exemplary use case explained later in the description, to remove systematic error from eye tracker data, gaze coordinates returned by the eye tracker are analysed. The value of M in this use case is 2 (corresponding to the horizontal and vertical screen coordinates).
[029] In an embodiment of the present disclosure, the one or more processors 104, are configured to employ, at step 204, N entities in the search space defined at step 202. In accordance with the present disclosure, the N entities are M-dimensional variables. Each of the N entities acts as a caterpillar in search of food being representative of the optimal solution, in the defined M-dimensional search space, and wherein a change in each of the N entities is governed by a periscopic

property, a pheromone persistence property and a fractal dimension. In accordance with the present disclosure, the periscopic property is the ability of each of the N entities to move such that an associated vicinity is viewed beyond line of sight in a rest position; and the pheromone persistence property is the ability of each of the N entities to leave a trail of the position thereof.
[030] In an embodiment of the present disclosure, the one or more processors 104, are configured to randomly initialize, at step 206, positions of the N entities in an N * M matrix represented as a matrix W. The matrix W may be considered to represent an initial state of each of the N entities.

Every entity (caterpillar) w1,* is a candidate to obtain an optimal solution and number of decision variables (dimensions) is represented by M. For brevity,w1,* is denoted as wi
[031] Further, at step 208, the matrix W is iteratively updated, until a termination criterion is met. The iterative updation is performed for each of the N entities and includes steps 208a through 208e described hereinafter. At step 208a, a fitness value associated with each of the N entities is computed by evaluating an objective function associated with the optimization problem, wherein the objective function is a function that is desired to be maximized or minimized. At step 208b, the fractal dimension of a time series that comprises the computed fitness value associated with each of the N entities is evaluated using a Katz Fractal Dynamics (FD) method. In accordance with the present disclosure, the fractal dimension is indicative of roughness of the defined M-dimensional search space.
[032] In accordance with the present disclosure, at step 208c, new positions of each of the N entities is computed in the matrix W to obtain a matrix U representative of a new state corresponding to movement of each of the N entities in the search space. The new positions are computed based on (i) a randomly

generated waving probability ri associated with each of the N entities in the matrix W (ii) position of a neighboring entity therein using the pheromone persistence property and (iii) the evaluated fractal dimension. In accordance with the present disclosure, if the randomly generated waving probability ri is greater than a pre-defined randomly generated initial waving probability Pw, it is taken as an instance of the entity (caterpillar) waving its body in a periscopic manner. In an embodiment, the value of the randomly generated waving probability ri has a value between 0 and 1.
[033] In an embodiment, the step of computing new positions of each of the N entities in the matrix W may be represented by ui = wi + [rand() × Hw] × [wj - (wi × α)]
→ (2) wherein ui represents an entity in the matrix U, wi represents a corresponding entity
in the matrix W, Hw represents a waving height of an associated entity, wj represents
a neighboring entity in the matrix W and α represents the fractal dimension. When
an entity (caterpillar) i waves its body, it gets an idea of the nearby vicinity and
follows the path of the entity (caterpillar) j due to its pheromone (j is selected
randomly). Initially, in a first iteration, α may be taken as 0.5 as the fitness values
are not present.
[034] The Katz Fractal Dynamics (FD) may be represented as

→(3) wherein d is the Euclidean distance between the first point in the time series
(computed fitness value associated with each of the N entities) and the point lying
at the farthest distance; and L is the total length of the time series. Let a be the
average distance between the successive points in the time series. Then, the number
of steps is given by n = La.
[035] In an embodiment, the waving height Hw of each of the N entities is related to an empirically determined fixed length Lw and is represented by Hw=Lw + [rand() + 1] × 0.5,

→(4) wherein rand() is a function configured to generate values uniformly distributed
between 0 and 1. The factor [rand() + 1] × 0.5 ensures that the entity (caterpillar)
lifts at least 50% of its total length during a waving phase. Hence, the mean waving
height is 75% of the caterpillar’s total length. If the worm does not wave, i.e. ri ≤
Pw, then the position of the entity (caterpillar) is reset randomly. After computing
new positions of each of the N entities in the matrix W, fitness value f(.) is
obtained.
[036] In accordance with the present disclosure, at step 208d, the matrix
W is updated using the computed new positions of each of the N entities in the
matrix U by comparing the fitness value associated with each of the N entities in
the matrix W with the fitness value associated with each of the N entities in the
matrix U and replacing the entity wi in the matrix W with the entity ui in the matrix
U, if the fitness value of ui is optimal compared to the fitness value of wi. Updating
the state of the matrix W may be represented as given below.
wi = { ui if f(ui)is better/optimal than f(wi)
Wi otherwise
→ (5) [037] In accordance with the present disclosure, at step 208e, the entity wi
in the matrix W having an optimal fitness value compared to the fitness values of
the rest of the N entities in the matrix W is identified and its position is designated
as an interim optimal solution.
[038] In an embodiment of the present disclosure, the termination criterion
may be completion of a predefined number of iterations. In another embodiment,
the termination criterion may be based on a tolerance value. For instance, the
iteratively performed steps may be terminated if the designated interim optimal
solution is lower than a tolerance value for a predetermined global minimum for the
minimization problem or higher than the tolerance value for a predetermined global
maximum for the maximization problem. Once the iterations are terminated, at step
210, the interim optimal solution associated with a last iteration is designated as the
optimal solution for the optimization problem.

[039] The method 200 of the present disclosure, as described above, may
be represented as shown below.
Algorithm 1:
1 Randomly initialize the positions of N entities (caterpillars) in the search space
2 W =Initialize the state for each worm
3 while termination condition is not met do

4 Compute α for the fitness value (refer equation 3)
5 for i=1 to N entities (caterpillars) do

6 Choose one entity randomly, denoted by si
7 Define entity’s waving probability ri
8 if ri≥Pw then

9 Evaluate Hw according to equation (4)
10 ui = wi + [rand() × Hw] × [wj - (wi × α)]

11 end
12 else
13 ui = rand() × α
14 end
15 end
16 Update W from U based on equation (5)
17 end
STUDY AND OBSERVATIONS:
[040] 26 standard test functions publicly available in literature were selected for evaluating the method of the present disclosure. Details of the functions with the parameters used are presented in Table 1 below.
Table 1: Details of the test functions used with Function ids being represented by the column header F-id and Dimensions represented by the column header Dim and D in the formulae.

F¬id
f1 Name Formula Lower bound
-4.5 Upper bound
4.5 Dim
2

Beale (1.5 -x1+ x1x2)2 + (2.25 -x1+ x1x22)2 + (2.625-X1+X1x23)2

f2
f3 f4
f5 f6
f7 f8
f9 f10 f11
f12
f13 f14 f15 f16 f17 f18 Easom

Matyas

Bohachvesky 1

Booth

Michalewicz 2

Schaffer

Six Hump Camel Back

Bohachvesky 2

Bohachvesky 3

Shubert

Colville

Michalewicz 5

Zakharov

Michalewicz 10

Step

Sphere

Sum Squares

-100 100 2
1 2
2 2
2 2
2 2 2
4
5 10 10 30 30 30
-10 10

-100 100

-10 10

0 n

-100 100

-5 5

-100 100

-100 100

-10 10

-10 10

0 n

-5 10

0 n

-100 100

-100 100

-10 10

f19 f20 f21 f22 f23 f24 f25 f26 Quartic

Schwefel 2.22

Schwefel 1.2

Rosenbrock

Dixon-Price

Rastringin

Griewank

Ackley

-1.28 1.28 30 30 30 30 30 30 30 30
-10 10

-100 100

-30 30

-10 10

-5.12 5.12

-600 600

-32 32

[041] The method 200 of the present disclosure was compared with the conventional methods for performance characteristics using two performance analysis metrics. The first metric relates to the solution obtained by the optimization method. The obtained solution is expected to be equal to or very close to the global minimum for each of the test functions. Table 2 below presents the average (Standard Deviation SD) best solution obtained by each of the optimization methods by executing 30 times (obtain statistically significant results) for each of the 26 test functions in Table 1.
Some of the column headers of Table 2 are modified for brevity as below. Function ids are represented by the header F-id.
Crow represents the crow search algorithm by Alireza Askarzadeh in Computers & Structures, vol. 169, pp. 1–12, 2016.

Bat represents the bat algorithm by Xin-She Yang and Amir Hossein Gandomi in
Engineering Computations, vol. 29, no. 5, pp. 464–483, 2012.
Whale represents the whale optimization algorithm by Seyedali Mirjalili and
Andrew Lewis in Advances in engineering software, vol. 95, pp. 51–67, 2016.
Butterfly represents the butterfly optimization algorithm by Sankalap Arora and
Satvir Singh in Soft Computing, vol. 23, no. 3, pp. 715–734, 2019.
Grey Wolf represents the Grey wolf optimizer by Seyedali Mirjalili, Seyed
Mohammad Mirjalili, and Andrew Lewis in Advances in engineering software, vol.
69, pp. 46–61, 2014.
DE represents Differential evolution by Swagatam Das and Ponnuthurai
Nagaratnam Suganthan in IEEE transactions on evolutionary computation, vol. 15,
no. 1, pp. 4–31,
2011.
Table 2: Mean (SD) of solutions obtained over 30 runs. SD>mean may indicate
many/far off outliers.

F-id
f1
f2
f3 f4 f5 f6 Global min Method 200 Crow Bat Whale Butterfl y Grey Wolf DE

0 0 1.60E-06 1.51E-06 1.44E-28 3.31E-28 0.64
1.51E+0
0 0.05 1.90E-01 0.26 2.50E-01 0.03 1.40E-01 0
0.00E+0
0

-1 0 -1 3.38E-05 -1 0 -0.09 2.60E-01 -1 5.53E-08 -1.80E-01
0.55 -1 7.08E-08 -1
0.00E+0
0

0 0 2.72E-41 1.35E-40 8.01E-30 1.19E-29 0.09 0.1 0
0.00E+0
0 1.64E-18 5.30E-19 0 0.00E+00 1.31E-131
5.24E-131

0 0 0 0 0 0 196.12
4.08E+0
2 0 0 0.12 0.19 0 0 0 0

0 0 0.13 0.2 0.08 0.18 3.77
4.67E+0
0 0.26 0.26 0.38 0.2 0.3 0.25 0.22 0.26

-1.8 0 -1.80E+00 1.70E-05 -1.8 9.03E-16 -1.27 0.29 -1.8 6.76E-08 -Inf NaN -1.77 0.15 -1.8 9.03E-16

0.00E+0 2.10E- 0.00E+0
f7 0 0 0 01 0.02 0.02 0.00E+00 0
0 0 0.01 0.24 0.02 0.02 0.01 0.01

- - - -
1.03E+0 1.03E+0 1.56E+0 - 1.03E+0
f8 -1.03 -1.03E+00 0 -0.47 0 4 1.03E+00 0
0 1.79E-06 6.58E-16 0.79 4.50E-11 4623.8 2.28E-09 6.78E-16

2.20E+0 0.00E+0
f9 0 0 0 2 2.62E+0 0 0.02 0.00E+00 0 0.00E+0
0 0 0 2 0 0.06 0.00E+00 0

1.79E+0 0.00E+0 0.00E+0
f10 0 0.00E+00 0 2 3.23E+0 5.55E-18 0 0.00E+0 0.00E+00 0
0 0 0 2 1.69E-17 0 0.00E+00 0

-1.87E+0 -1.87E+0 - -1.87E+0
f11 -186.73 -1.86E+02 2 -89.07 5.29E+0 2 -Inf 1.85E+02 2
0 0.27 2.24E-14 1 4.08E-05 NaN 1.15E+01 3.62E-14

6.00E+0
f12 0 1.12 6.67E-15 2 7.46E+0 0.78 4.89 1.39 0.19
f13 0 2.22 2.66E-14 2 0.92 4.32 2.18 1.02

-4.69 -4.24E+00 -4.41 -2.1 -3.67 -Inf -4.25 -4.67
0 0.34 0.24 0.42 0.62 NaN 0.48 0.04

1.05E-
f14 0 7.06E-71 1.26E-06 66.55 1.00E-02 2.23E-17 114 4.90E- 2.16E-09
f15 0 2.28E-70 1.18E-06 22.88 0.03 1.27E-18 114 6.18E-09

-9.66 -6.33 -8.05 -3.46 -5.96 -Inf -7.83 -9.61
0 0.57 1.03 0.74 0.98 NaN 0.94 0.12

2.06E+0
f16 0 2.14 0 4 3.89E+0 0.09 6.24 0.85 3.21E-29
0 0.45 0 3 0.16 0.66 0.41 4.06E-29

1.80E+0 4.59E- 4.61E-
f17 0 9.01E-74 0 4 4.82E+0 268 2.29E-17 103 2.03E- 2.17E-29
0 2.96E-73 0 3 0 9.63E-19 102 2.88E-29

3.04E+0 5.03E- 1.25E-
f18 0 2.79E-72 0.21 3 9.01E+0 260 2.26E-17 103 5.62E- 4.70E-30
f19 0 7.29E-72 0.16 2 0 1.13E-18 103 4.79E-30

0 8.45E-05 0.03 0.34 0 0 0 0.01

f20 f21
f22
f23
f24
f25 f26 0 8.29E-05 0.01 0.98 0 0 0 0

0 0 1.69E-37 4.46E-37 1.91 0.81 97.75 26.31 6.23E-163 0 1.75E-14 3.28E-15 1.36E-60 1.93E-60 3.09E-18 1.84E-18

0 0 1.82E-73 9.87E-73 30.35 31.04 2.84E+0
5 6.63E+0
4 9.89E-263
0 2.36E-17 1.14E-18 3.66E-102
1.15E-101 3.31E-28 3.31E-28

0 0 28.73 0.03 78.54 84.65 1.80E+0
7 6.75E+0
6 26.82 0.48 28.94 0.04 27 0.81 60.15 34.35

0 0 0.79 0.03 1.01 0.6 1.48E+0
5 5.66E+0
4 0.67 7.94E-05 0.99 0.01 0.67 4.35E-07 0.68 0.04

0 0 0 0 33.27 10.56 2.75E+0
2 1.11E+0
2 0 0 37.95 77.47 3.79E-15 1.44E-14 21.32 11.41

0 0 0 0 0.02 0.01 4.23E+0
2
84.63 3.70E-18 2.03E-17 0 0 0 0.01 0 0

0 0 8.88E-16 0 4.49 1.04 19.62 0.12 4.20E-15 2.27E-15 1.78E-14 5.41E-15 9.77E-15 2.60E-15 1.42E-14 2.07E-15
[042] The number of iterations used was 2000 which was empirically determined as all algorithms converge before that for all functions. It was noted that the method 200 of the present disclosure performs best in most of the cases and in some cases, it performs comparably with the existing approaches in arriving at the best solution.
[043] The second metric used to analyze the performance of the method 200 of the present disclosure is the Area Under the Curve (AUC) of the convergence curve for the given algorithm over the predefined number of iterations (bounded by the maximum value obtained in the first run and the global minima of that particular algorithm). FIG.3 illustrates the AUC metric to analyze performance of the method in accordance with some embodiments of the present disclosure. The AUC is the region bounded by the curve (for a given algorithm) with coordinates (global minimum, initial best value) and (number of iterations, final best solution). In the FIG.3, the global minimum is at 0 and the number of iterations is 2000. Each

algorithm starts with an initial arbitrary value and ends with its final best value. The AUC metric provides an insight into how fast the algorithm can converge towards the global minimum as it incorporates the essence of both the time taken (iterations) and the solution obtained.
[044] Table 3 provides the average AUC for 30 runs for different optimization algorithms tested over the 26 functions. Some of the column headers of Table 3 are modified for brevity as explained with reference to Table 2 above. Table 3: Mean (SD) of AUC values across 30 runs. SD>Mean may indicate many/far off outliers.

F-id Global min Method 200 Crow Bat Whale Butterfly Grey Wolf
f1 12.99 14.51 1.4E+09 6.07E+09 1773.74 3662.86 167.31 389.57 717.92 588.68 58.95 277.44 8.07 7.22
f2 25.32 8.38 31.38 5.69 1828.04 520.25 30.74 29.14 1811.54 721.15 9.75 2.32 57.66 21.03
f3 0.43 0.29 16.23 29.08 189.95 200.56 0.78 0.71 14.72 12.22 0.83 0.62 0.98 0.71
f4 693.01 514.62 935.28 803.64 396401.4 816218.3 1311.66 1313.8 13309.67 11892.7 637.78 492.37 878.79 941.35
f5 567.1 365.39 29091.6 72612.56 8295.47 9580.77 653.97 600.8 1414.77 870.2 663.48 513.14 483.25 504.22
f6 4.89 3.13 12.16 4.37 1132.42 560.95 13.17 19.36 -Inf NaN 63.76 290.53 1.81 1.29
f7 12.95 9.07 16.39 19.29 480.18 461.55 39.19 42.19 203.29 122.33 9.16 21.73 18.09 25.52
f8 3.22 1.82 21180.05 81706.26 1272.34 1632.38 5.85 5.83 -1.3E+07 4165757 3.6 2.32 3.77 3.24
f9 489.83 374.88 1116.77 817.78 447771.1 524201.5 1007.97 1097.42 12240.03 10847.48 767.45 633.46 820.4 554.4
f10 481.6 340.13 988.96 789.75 364702.1 646525.6 1884.76 1862.31 13838.06 13264.11 683.18 555.79 859.26 701.19
f11 3798.69 1723.36 2557.4 726.55 199352.7 103017.5 1158.98 1603.11 -Inf NaN 6190.25 22872.84 1224.63 615.88
f12 7356.85 6992.45 949080.5 2658756 1337235 1573365 23714.99 23516.33 68306.47 50963.12 8718.38 6395.61 14794.19 11884.22
f13 1447.36 531.32 673.28 470.48 5187.5 840.21 2294.52 1132.65 -Inf NaN 1717.61 700.24 71.55 74.76
f14 44485.7 99044.43 11629560 27726330 11325578 12797968 89687.58 117749.4 200089 440769.7 26880.28 85418.22 21795.63 38171.18

f15 7761.61 827.05 3825.27 2034.01 12406.24 1490.77 7864.92 1800.02 -Inf NaN 7960.73 851.99 436.98 225.25
f16 43085.64 12055.25 565169.1 92621.04 43490751 7726114 406480.5 113993.4 800862 66441.95 358610.5 55840.78 1886637 195888.5
f17 36837.62 12768.45 542032.4 71620.05 38071684 9666252 376178.2 85330.54 778965.3 47161.57 391808.3 68872.64 1860255 186114.9
f18 5289.22 1337.58 266409.2 216279.7 6114410 1743323 55020.04 18351.08 138544.5 13354.11 47684.41 7755.81 250514.3 26572.02
f19 31.95 9.52 5.75E+08 2.04E+09 5217.11 6576.03 591.59 206.81 1565.76 211.79 455.89 96.56 2226.3 411.5
f20 1.65E+10 6.32E+10 5.69E+42 3.12E+43 208182 39325.78 1.98E+13 4.19E+13 558.28 357.76 6.76E+12 1.92E+13 1.87E+13 4.83E+13
f21 534889.2 167528.9 8333270 1434724 5.93E+08 1.36E+08 5529166 1650880 8762013 814621.5 4965894 807743.2 24789049 2553670
f22 65038247 26701308 4.1E+09 9.9E+09 3.66E+10 1.35E+10 1.11E+09 3.76E+08 6.81E+08 87650243 9.02E+08 2.62E+08 4.48E+09 7.24E+08
f23 444674.9 159837.9 4.33E+08 2.28E+09 2.99E+08 1.07E+08 8050411 3345680 8907776 1398584 6648936 1992828 30721396 6114181
f24 1752.8 695.73 110187.2 21394.4 610470.5 187387.7 15655.02 12571.14 292770.3 102077.1 15982.06 2833.89 146740.9 12483.67
f25 361.38 146.37 1221.29 171.02 900520 154301.6 3555.66 1040.44 10846.48 703.09 3486.91 669.1 17753.05 1343.65
f26 143.13 22.45 10421.61 1730.37 39269.29 225.88 408.51 134.79 1773.67 103.38 444.4 33.21 2359.89 128.27
[045] It is noted that the method 200 is better than the existing algorithms as it converges faster which is evident from the lesser AUC values obtained.
USE CASE:
[046] The method 200 and system 100 of the present disclosure may find practical application in several scenarios. For instance, the method and system of the present disclosure may be used for selecting an optimal subset of features from all the available features in machine learning. This helps in reducing the training period and also the computational requirements. The issues related to redundant features is avoided, thereby enhancing the overall classification accuracy. Another application of the method and system of the present disclosure is in Electroencephalogram (EEG) based optimal channel selection. High end EEGs have multiple channels and using all the channels makes the process time

consuming. In this case, a subset of optimal channels is helpful can be obtained efficiently and in relatively less time using the method and system of the present disclosure.
[047] Provided below is yet another use case of the method and system of the present disclosure to explain the practical application thereof. The use case relates to removing the systematic error from eye tracker data. Eye tracking is a popular means of monitoring gaze behavior which can provide valuable insights in human behavior sensing and medical applications. Eye trackers often suffer from errors in accurately mapping the gaze to the computer screen coordinates. This error is mainly due to the systematic bias or the offset from the target location which is gazed by the participant. This is termed as systematic error. FIG.4 illustrates a systematic error in eye tracker data, as known in the art. Systematic error is caused mainly due to badly performed calibration phase, and/or due to head movements of a subj ect and/or sudden jerk in the position of eye tracker after the initial calibration.
[048] In this use case, the optimization problem is a minimization problem with the objective function being|G - C|, wherein G represents ground truth, C represents corrected gaze coordinates, associated with raw gaze coordinates received from an eye tracker and |G - C| is the Euclidean distance between G and C. Let F represent the raw gaze coordinates received from the eye tracker for a static position viewed on a screen. F is a 2-diminesional vector consisting of the horizontal and vertical screen coordinates and may be represented as shown below. F = [(x1,y1),(x2,y2),….(xN,yN)] where N is the number of points and is a function of the sampling rate of the eye tracker. The corrected gaze coordinates which is desired to be bereft of the systematic error may be obtained as, C = F*T, where T is a transformation matrix which needs to be obtained given the high sampling rates and the variability that is inherent in human gaze behavior even when gazing at a static position.
[049] The normalized screen coordinates were obtained when a subject was gazing at 9 different locations on the screen (analogous to the conventional calibration phase). The optimal transformation matrices for each of these 9 points

was computed using the ground truth locations. FIG.5 illustrates corrected gaze coordinates for the actual eye tracker data, with the transformation matrices computed in accordance with some embodiments of the present disclosure, when training and testing data are the same (for the sake of validation). Similarly, FIG.6 illustrates corrected gaze coordinates for the actual eye tracker data, with the transformation matrices computed.in accordance with some embodiments of the present disclosure, when training and testing data are not the same, i.e. the transformation matrices are derived on the train data and the F is different from the training data. It is noted that the processed data (asterisks) come closer to the ground truth locations (black fixation cross) in comparison to the actual eye tracker data (circles).
[050] FIG. 5 and FIG.6 may be viewed quantitatively using the root mean square error (RMSE) values between the gaze coordinates and the ground truth locations. The RMSE values are computed as,

[051] FIG.7 and FIG.8 illustrate root mean square error (RMSE) values between the gaze coordinates and the ground truth locations corresponding to FIG.5 and FIG.6 respectively, in accordance with some embodiments of the present disclosure. It is noted that the systematic error decreases considerably after using the method and system of the present disclosure for obtaining the transformation matrices for correcting the systematic error.
[052] The written description describes the subject matter herein to enable any person skilled in the art to make and use the embodiments. The scope of the subject matter embodiments is defined by the claims and may include other modifications that occur to those skilled in the art. Such other modifications are intended to be within the scope of the claims if they have similar elements that do not differ from the literal language of the claims or if they include equivalent elements with insubstantial differences from the literal language of the claims.

[053] It is to be understood that the scope of the protection is extended to such a program and in addition to a computer-readable means having a message therein; such computer-readable storage means contain program-code means for implementation of one or more steps of the method, when the program runs on a server or mobile device or any suitable programmable device. The hardware device can be any kind of device which can be programmed including e.g. any kind of computer like a server or a personal computer, or the like, or any combination thereof. The device may also include means which could be e.g. hardware means like e.g. an application-specific integrated circuit (ASIC), a field-programmable gate array (FPGA), or a combination of hardware and software means, e.g. an ASIC and an FPGA, or at least one microprocessor and at least one memory with software processing components located therein. Thus, the means can include both hardware means and software means. The method embodiments described herein could be implemented in hardware and software. The device may also include software means. Alternatively, the embodiments may be implemented on different hardware devices, e.g. using a plurality of CPUs.
[054] The embodiments herein can comprise hardware and software elements. The embodiments that are implemented in software include but are not limited to, firmware, resident software, microcode, etc. The functions performed by various components described herein may be implemented in other components or combinations of other components. For the purposes of this description, a computer-usable or computer readable medium can be any apparatus that can comprise, store, communicate, propagate, or transport the program for use by or in connection with the instruction execution system, apparatus, or device.
[055] The illustrated steps are set out to explain the exemplary embodiments shown, and it should be anticipated that ongoing technological development will change the manner in which particular functions are performed. These examples are presented herein for purposes of illustration, and not limitation. Further, the boundaries of the functional building blocks have been arbitrarily defined herein for the convenience of the description. Alternative boundaries can be defined so long as the specified functions and relationships thereof are

appropriately performed. Alternatives (including equivalents, extensions,
variations, deviations, etc., of those described herein) will be apparent to persons skilled in the relevant art(s) based on the teachings contained herein. Such alternatives fall within the scope of the disclosed embodiments. Also, the words “comprising,” “having,” “containing,” and “including,” and other similar forms are intended to be equivalent in meaning and be open ended in that an item or items following any one of these words is not meant to be an exhaustive listing of such item or items, or meant to be limited to only the listed item or items. It must also be noted that as used herein and in the appended claims, the singular forms “a,” “an,” and “the” include plural references unless the context clearly dictates otherwise.
[056] Furthermore, one or more computer-readable storage media may be utilized in implementing embodiments consistent with the present disclosure. A computer-readable storage medium refers to any type of physical memory on which information or data readable by a processor may be stored. Thus, a computer-readable storage medium may store instructions for execution by one or more processors, including instructions for causing the processor(s) to perform steps or stages consistent with the embodiments described herein. The term “computer-readable medium” should be understood to include tangible items and exclude carrier waves and transient signals, i.e., be non-transitory. Examples include random access memory (RAM), read-only memory (ROM), volatile memory, nonvolatile memory, hard drives, CD ROMs, DVDs, flash drives, disks, and any other known physical storage media.
[057] It is intended that the disclosure and examples be considered as exemplary only, with a true scope of disclosed embodiments being indicated by the following claims.

We Claim:
1. A processor implemented method (200) comprising the steps of:
defining, via one or more hardware processors, an M-dimensional search space for an optimal solution to an optimization problem being a minimization problem or a maximization problem, wherein M is based on the optimization problem (202);
employing N entities, N being empirically determined, via the one or more hardware processors, in the defined M-dimensional search space, wherein each of the N entities acts as a caterpillar in search of food being representative of the optimal solution, in the defined M-dimensional search space, and wherein a change in each of the N entities is governed by a periscopic property, a pheromone persistence property and a fractal dimension (204);
randomly initializing, via the one or more hardware processors, positions of the N entities in an N * M matrix represented as a matrix W (206);
iteratively updating the matrix W, via the one or more hardware processors, until a termination criterion is met (208), using the steps of: performing, for each of the N entities, the steps of:
computing a fitness value associated with each of the N entities in the matrix W by evaluating an objective function associated with the optimization problem (208a);
evaluating, using a Katz Fractal Dynamics (FD) method, the fractal dimension of a time series that comprises the computed fitness value associated with each of the N entities, wherein the fractal dimension is indicative of roughness of the defined M-dimensional search space (208b);
computing new positions of each of the N entities in the matrix W to obtain a matrix U representative of a new

state corresponding to movement of each of the N entities, based on (i) a randomly generated waving probability ri associated with each of the N entities in the matrix W, ri being greater than a pre-defined randomly generated initial waving probability Pw, (ii) position of a neighboring entity therein using the pheromone persistence property and (iii) the evaluated fractal dimension (208c);
updating the matrix W using the computed new positions of each of the N entities in the matrix U by comparing the fitness value associated with each of the N entities in the matrix W with the fitness value associated with each of the N entities in the matrix U and replacing the entity wi in the matrix W with the entity ui in the matrix U, if the fitness value of ui is optimal compared to the fitness value of wi (208d); and
identifying the entity wi in the matrix W having an
optimal fitness value compared to the fitness values of the
rest of the N entities in the matrix W, and designating a
position of the identified entity wi as an interim optimal
solution (208e),
wherein the termination criterion is one of (i) completion of a
predefined number of iterations and (ii) if the designated interim optimal
solution is lower than a tolerance value for a predetermined global minimum
for the minimization problem or higher than the tolerance value for a
predetermined global maximum for the maximization problem; and
designating the interim optimal solution associated with a last iteration as the optimal solution for the optimization problem (210).
The processor implemented method of claim 1, wherein the periscopic property is the ability of each of the N entities to move such that an associated vicinity is viewed beyond line of sight in a rest position; and the

pheromone persistence property is the ability of each of the N entities to leave a trail of the position thereof.
3. The processor implemented method of claim 2, wherein the step of computing new positions of each of the N entities in the matrix W is represented by ui = wi + [rand() × Hw] × [wj - (wi × a )], wherein ui represents an entity in the matrix U, wi represents a corresponding entity in the matrix W, Hw represents a waving height of an associated entity, wj represents a neighboring entity in the matrix W, α represents the fractal dimension and wherein rand() is a function configured to generate values uniformly distributed between 0 and 1.
4. The processor implemented method of claim 3, wherein the waving height Hw of each of the N entities is related to an empirically determined fixed length Lw and is represented by Hw = Lw + [rand() + 1] × 0.5.
5. The processor implemented method of claim 4, wherein the optimization problem is the minimization problem with the objective function
being|G - C|, wherein G represents ground truth, C represents corrected gaze coordinates, associated with raw gaze coordinates received from an eye tracker and |G - C| is the Euclidean distance between G and C.
6. The processor implemented method of claim 1, wherein the randomly generated waving probability ri has a value between 0 and 1.
7. A system (100) comprising:
one or more data storage devices (102) operatively coupled to one or more hardware processors (104) and configured to store instructions configured for execution via the one or more hardware processors to:
define an M-dimensional search space for an optimal
solution to an optimization problem being a minimization problem

or a maximization problem, wherein M is based on the optimization problem;
employ N entities, N being empirically determined in the defined M-dimensional search space, wherein each of the N entities acts as a caterpillar in search of food being representative of the optimal solution, in the defined M-dimensional search space, and wherein a change in each of the N entities is governed by a periscopic property, a pheromone persistence property and a fractal dimension;
randomly initialize positions of the N entities in an N * M matrix represented as a matrix W;
iteratively update the matrix W until a termination criterion is met, using the steps of:
perform, for each of the N entities, the steps of:
computing a fitness value associated with each of the N entities in the matrix W by evaluating an objective function associated with the optimization problem;
evaluating, using a Katz Fractal Dynamics (FD) method, the fractal dimension of a time series that comprises the computed fitness value associated with each of the N entities, wherein the fractal dimension is indicative of roughness of the defined M-dimensional search space;
computing new positions of each of the N entities in the matrix W to obtain a matrix U representative of a new state corresponding to movement of each of the N entities, based on (i) a randomly generated waving probability ri associated with each of the N entities in the matrix W, ri being greater than a pre-defined randomly generated initial

waving probability Pw, (ii) position of a neighboring entity therein using the pheromone persistence property and (iii) the evaluated fractal dimension;
updating the matrix W using the computed new positions of each of the N entities in the matrix U by comparing the fitness value associated with each of the N entities in the matrix W with the fitness value associated with each of the N entities in the matrix U and replacing the entity wi in the matrix W with the entity ui in the matrix U, if the fitness value of ui is optimal compared to the fitness value of wi; and
identifying the entity wi in the matrix W
having an optimal fitness value compared to the
fitness values of the rest of the N entities in the matrix
W, and designating a position of the identified entity
wi as an interim optimal solution,
wherein the termination criterion is one of (i) completion of
a predefined number of iterations and (ii) if the designated interim
optimal solution is lower than a tolerance value for a predetermined
global minimum for the minimization problem or higher than the
tolerance value for a predetermined global maximum for the
maximization problem; and
designate the interim optimal solution associated with a last iteration as the optimal solution for the optimization problem.
The system of claim 7, wherein the periscopic property is the ability of each of the N entities to move such that an associated vicinity is viewed beyond line of sight in a rest position; and the pheromone persistence property is the ability of each of the N entities to leave a trail of the position thereof.

9. The system of claim 8, wherein the new positions of each of the N entities in the matrix W is represented as ui = wi + [rand() × Hw] × [wj -(wi × α)], wherein ui represents an entity in the matrix U, wi represents a corresponding entity in the matrix W, Hw represents a waving height of an associated entity, wj represents a neighboring entity in the matrix W, α represents the fractal dimension and wherein rand() is a function configured to generate values uniformly distributed between 0 and 1.
10. The system of claim 9, wherein the waving height Hw of each of the N entities is related to an empirically determined fixed length Lw and is represented by Hw = Lw + [rand() + 1] × 0.5.
11. The system of claim 10, wherein the optimization problem is the minimization problem with the objective function being |G - C|, wherein
G represents ground truth, C represents corrected gaze coordinates associated with raw gaze coordinates received from an eye tracker and |G - C| is the Euclidean distance between G and C.
12. The system of claim 7, wherein the randomly generated waving probability
ri has a value in between 0 and 1.