Modified Generalized Way for Optimization Problem

1ALEXANDER ZEMLIAK, 2CHRISTIAN SERRANO

1Dept. of Physics and Mathematics, Autonomous University of Puebla, Puebla, MEXICO

2Dept. of Electronic, Autonomous University of Puebla, Puebla, MEXICO

Abstract: The solution to the problem of circuit optimization is obtained on the basis of a combination of a genetic

algorithm (GA) and the idea of generalized optimization, developed earlier for the deterministic case. It is shown that

such a GA modification allows one to overcome premature convergence to local minima and to increase the

minimization accuracy by several orders of magnitude. In this case, GA forms a set of populations determined by the

fitness function, given in different way, depending on the strategy chosen within the framework of the idea of

generalized optimization. The way of setting fitness functions as well as the length and structure of chromosomes, are

determined by a control vector artificially introduced within the framework of generalized optimization. This vector

determines the number of independent variables of the optimization problem and the method for calculating the

fitness function. It allows you to build compound strategies that significantly increase the accuracy of the resulting

solution. This, in turn, makes it possible to reduce the number of generations required during the operation of the GA

and minimize the processor time for solving the problem of circuit optimization.

Keywords: generalized optimization, GA, circuit optimization, control vector, set of strategies

Received: June 25, 2021. Revised: June 28, 2022. Accepted: July 25, 2022. Published: Septemebr 9, 2022.

1. Introduction

One of the major challenges in designing a large system is

the excessive computing time required to reach the optimum

point in the design process. This problem is important as it has

many applications, for example, in the design of VLSI circuits.

The design process starts with an initial approximation that is

provided by analysis of circuit for the initial point and then the

process is continued till adjusting of the system parameters to

obtain the necessary performance features defined in the

specification. The process of setting parameters is usually

based on the optimization procedure. So, the process of design-

by-analysis can be realized instead of the difficult problem of

synthesis of a complex system. Mathematically, this process is

defined as the minimization of a special objective function that

includes necessary properties of the designed circuit. It means

that any circuit design strategy includes two main blocks:

analysis of the mathematical model of the circuit and an

optimization procedure that reaches the minimum point of the

objective function. The minimum value of this function can

ensure that the required circuit characteristics are obtained. The

interaction of the circuit analysis block and the optimization

procedure block forms the circuit optimization process.

Optimization methods for systems of various natures can be

divided into two main groups: deterministic optimization

algorithms and stochastic search algorithms. Some of the

weaknesses of classical deterministic optimization algorithms

are the requirement for a good starting point in the parameter

space, the difficulty of finding the global minimum, and a long

execution time. To overcome these problems some special

methods were developed. For example, a method that

determines initial point of the optimization process by

centering [1], geometric programming methods [2] that

guarantee the convergence to a global minimum, but, on the

other hand, this require a special formulation of the design

equations to which additional difficulties accompany. Other

approach based on the idea of space mapping technique [3-4],

which achieves a satisfactory solution. This technology

successfully used for optimization of microwave systems but

there are no experience for solution of other problems.

Some alternative stochastic search algorithms, especially

evolutionary computation algorithms, can solve the problem of

finding the global minimum and have been developed in recent

years. An analysis of various stochastic algorithms for system

optimization allowed select some groups: simulated annealing

method [5-7], evolutionary computing techniques that produce

some different approaches as evolutionary algorithms [8-11]

particle swarm optimization (PSO) method, GA, differential

evolution, genetic programming. A PSO technique [12-15] is

one of the evolutionary algorithms that competes with genetic

algorithms. This method has been successfully used to solve

electromagnetic problems and to optimize microwave systems.

Separately, we highlight GA that is used to solve nonlinear

programming problems both for optimizing systems of various

nature [16-22], and, in particular, for optimizing and designing

electronic systems [23-24]. GA has been used as an

optimization procedure for analog circuits due to the ability to

find a satisfactory solution. The disadvantages of these

methods include a premature convergence to a local minimum

and an increase in computer operation time when setting a

sufficiently high accuracy for obtaining the minimum. To

prevent this, we propose to use the approach underlying the

generalized optimization method defined for the deterministic

case of circuit optimization in [25]. In this formulation of the

problem, an artificially introduced control vector produces

many different optimization strategies and sets a different type

of the objective function for each new strategy.

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This control vector is introduced into the system of

equations describing the optimization process of a certain

function, determines the structure of these equations and leads

to the emergence of many different optimization strategies that

differ in the number of operations and CPU time. The set of

different strategies that appear in this case depends on the

dimension of the control vector, which, in turn, is determined

by the number of circuit nodes. At the same time, this

dependence is exponential, i.e. if the number of circuit nodes is

M, then the number of different optimization strategies is 2M.

With this approach, each strategy is determined by its objective

function, which depends on the structure of the control vector.

In this case, the optimization process is generalized and, in

fact, is a dynamic controlled system with a control vector.

A detailed analysis of various optimization strategies in the

deterministic case showed the prospects and advantages of this

approach when solving the problem of reducing the time spent

on the optimization and design of electronic systems. It would

like to find out the validity of this approach when solving

optimization problems by stochastic methods. In this paper, the

approach of generalized optimization is included into the

implementation of a standard GA. This means that one of the

most important steps is the setting of the GA fitness function,

which now includes the control vector.

2. GA and Generalized Optimization

Approach

The process of circuit optimization can be defined as the

problem of minimization of objective function C(X),

RX 

with additional conditions. It is supposed that the minimum of

the objective function C(X) corresponds to achievement of all

the necessary design goals of the circuit, and the system of

constraints is a mathematical model of the electronic circuit.

A typical formulation of a circuit optimization problem can

be defined mathematically as a constrained optimization

problem for the objective function C(X). The process to

minimize the objective function C(X) is conventionally defined

by the following equation:



,...2,1),(

1

sXX ss

 

where Λ is the operator of transition from iteration s to

iteration s+1. The constraints are determined by the circuit

equations and can be described by a system of nonlinear

equations:



MjXgj,...,2,1,0)( 

 

We will declare some of the variables as independent, and

the other part as dependent, the value of which is determined

from the constraint equations (2): X = (X ', X "), X ' ϵ RK is a

vector of independent variables, X " ϵ RM is a vector of

dependent variables, K is the number of independent variables,

М is the number of the circuit’s dependent variables, N is the

total number of variables (N=K+M). Traditionally, resistance

(conductivity) of resistors are defined as independent variables

of optimization procedure (vector

X

). Other variables are

defined as dependent variables (currents or nodal voltages).

However, this partition is conditional, since any variable may

be considered independent or dependent.

To calculate the function C(X), it is required to solve a

system of nonlinear equations (2) at each step of the

optimization process. This approach can be named as

traditional strategy of optimization (TSO).

Let us accept the following statement that there is no need

to fulfil condition (2) at each step of the optimization

procedure, and that it is enough to fulfil it at the final point of

the optimization process. We use the approach [25] leading to

a generalization of the optimization process. Let’s define as

independent all the variables included in the vector X", and

previously declared dependent. In this case, the constraint

equations (2) can be removed, but to fulfill all the constraints

(2), at least at the end point of the optimization process, we

introduce a new, generalized, objective function F(X), which

can be defined as follows:



)()()( XXCXF





 

where φ(X) is an additional penalty function, the equality of

which to zero, at the end point of the optimization process,

ensures the fulfilment of conditions (2). This function can be,

for example, the following form:



).()(

2XgX M









Generalizing this approach, it is possible to declare

independent only a part of the previously dependent variables,

for example, Z variables, where Zϵ[0, M]. In this case, Z

equations are removed from system (2), and the formula (4)

contains Z terms. This approach generalizes the optimization

problem by introducing a special control vector U=(u1, u2,…,

uM), that changes the structure of all equations and formulas of

the optimization procedure. In this case, system (2) is

transformed into the following:



MjXgu jj ,...,2,1,0)()1( 

 

where uj is the jth component of the control vector

U=(u1, u2,…, uM), uj ϵ Ω, Ω={0;1}. Formulas (3) and (4) are

transformed into the following:



),()(),( UXXCUXF





 



)(

),(

2XguUX M

jjj









 

where σ is a special adjusting parameter.

Thus, the control vector U allows one to change both the

structure of the basic equations of the constraints (5) and the

form of the generalized objective function F(X,U). Zero values

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of components of the vector U determine the TSO. In this case,

the system (2) is solved at each step of the optimization

procedure, and the generalized objective function F(X,U)

coincides with the function C(X). Further, the penalty function

φ(X,U) is equal to zero. If some components of the vector U

are equal to 1, then the corresponding equations disappear from

system (5), but information about them appears in the penalty

function and in the function F(X,U). If all components of the

vector U are equal to 1, then the optimization is determined by

the modified traditional strategy (MTS). This means that

system (5) disappears, and the penalty function includes

complete information about system (5).

It is also important to note the necessary changes in the

optimization procedure. When using the deterministic

approach, the optimization procedure is specified by

differential (8) or difference (9) equations:



,,...,2,1),,( NiUXf

dx i

i





ss HtXX 

1

 

where fi(X,U) or H are determined by a specific optimization

method. A change in the components of the control vector U

from 0 to 1 corresponds to a transformation of the

corresponding dependent component of the vector X into an

independent one, leading to a change in the number of

independent variables and the number of equations both in the

optimization procedure (8) or (9) and in the system of

constraints (5). The control vector U defines strategies of the

structural basis of generalized optimization. The number of

these strategies is 2M.

Equations (5)-(9) define a set of different strategies, each of

which is determined by the corresponding value of the control

vector. In this case, as was shown in [26], there are

opportunities for a significant (by several orders of magnitude)

acceleration of the optimization process due to the different

behavior of the trajectories of different strategies and the

combination of these strategies in the process of optimization.

Let us consider the application of the idea of generalized

optimization in the case of using a GA as the main

optimization procedure. Instead of using equation (8) or (9),

the optimization procedure was carried out on the basis of a

GA. Let us consider the classic version of GA [27], in which

the selection of chromosomes is carried out by a tournament

method and two main genetic operators are used: crossover and

mutation. Variants with one-, two-, and four-point crossover

operators with a probability of 0.95 and mutation operators

with a probability of 0.05 to 0.1 were analyzed. Let's define NN

as the number of chromosomes in a population, and X is a

special matrix with N rows and NN columns, provided that

each column corresponds a specific value of the vector X.

Let's define the fitness function according to the following

generalized formula:



   

UFUP ,/1, XX 

 

Taking into account the concepts of generalized

optimization, the structure of GA can be represented in Fig. 1.

A new element of this algorithm is the control vector U, which

provides the implementation of various GA variants with

different objective functions (fitness functions in GA

terminology). Thus, the fitness function also depends on the

vector U.

Fig. 1. Modified GA flowchart.

The presence of the control vector U is reflected in the

corresponding blocks, since in these blocks either the fitness

function is calculated or it is used. The presence of the control

vector U in the blocks of the diagram ensures the

determination and change of the structure of both the initial

generation of chromosomes and the current generations during

the operation of the algorithm. For this stochastic algorithm,

we can also introduce a vector X consisting of N components,

where each component is calculated as the arithmetic mean of

all values of this component in the generation:







iNN

 

where xij is the element of matrix X.

For the analyzed examples, the length of chromosomes (L)

in GA varied from 20 to 80 for each of the variables, and the

number of chromosomes (NN) in the population varied from

40 to 400 depending on the length of the chromosomes.

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3. Results

3.1 Example 1

Minimize C(X)



 

4xxxXC 

 

Subject to:



   

012 2

1 xx

 

In this example, parameter M=1 because there is only one

constraint equation (13). Define variable

as independent,

and variable

as dependent, the value of which is

determined from equation (14). Based on the generalized

approach, equation (13) is transformed into the following

equation:



     

 

0121 2

1 xxu

 

where u is a control vector consisting, in this particular case, of

one component. Consider two basic strategies: the TSO with

control vector u = 0 and the MTS with control vector u = 1.

Here, we analyse the results of optimization by means of a GA

for these strategies. However, it was shown that in the case of a

deterministic optimization process, a combination of several

strategies can reduce both the number of steps in the

optimization procedure and the computing time of the

optimization process.

Table I shows the number of generations required to

achieve the minimum of the function C(X) with accuracy



for the strategies TSO (U=(0)), MTS (U=(1)) and for the

third, combined strategy determined by the control vector

(1),(0) with one switching point Sp = 2 between strategies (1)

and (0). That is, the first two iterations correspond to strategy

(1) and the next ones correspond to strategy (0).

TABLE I. NUMBER OF GENERATIONS G FOR STRATEGIES (0), (1) AND

COMBINED STRATEGY (1)(0) WITH SWITCH POINT SP FOR DIFFERENT

PRECISION δ

Precision

Control

vector

(0)

Control

vector

(1)

Control

vector

(1)(0)

Sp=2

10-1

10-2

10-3

10-4

10-5

10-6

10-7

10-8

10-9

102

10-10

114

It can be seen that when using the TSO, the number of

generations is less than for MTS up to a certain level of

accuracy (10-5). If the required error is reduced to 10-6 or less,

no solution based on the traditional strategy is found. The

MTS with control vector (1) finds a solution up to an accuracy

of 10-10. At the same time, a combined strategy consisting of

two, (1) and (0) with a switching point between them Sp = 2,

also finds a solution to the problem with no errors up to an

accuracy of 10-10 and, importantly, for a lower number of

generations. The final result for a combined strategy clearly

depends on the switching point from one strategy to another.

Table II shows the dependence of the number of generations

on the switching point for a given error



=10-5.

TABLE II. NUMBER OF GENERATIONS G FOR COMBINED STRATEGY

(1)(0) FOR DIFFERENT SWITCHING POINT SP.

Switch

point Sp

This shows that the switching point affects the number of

generations required to achieve the required accuracy. The

minimum value for this example corresponds to the switching

point Sp = 3. Fig. 2 shows the dependence of the minimized

function F on the number of generations G for three strategies

corresponding to three variants of calculating the fitness

function.

Fig. 2. Minimized function F for strategies (0), (1) and (1)(0) on the number

of generations G.

It can be seen that the best strategy for calculating the

fitness function is the composite strategy (1)(0), which after

the 18th generation solves the problem in the best way

compared to other strategies.

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3.2 Example 2

We need to optimize the nonlinear circuit shown in Fig. 3.

Fig. 3. Two-node nonlinear passive circuit.

Consider a simple nonlinear voltage divider circuit. A

nonlinear element has the following dependency: yn=a+b(V1-

V2)2. The admittances y1, y2, y3 are positive and compose a set

of independent circuit parameters (K=3). The node voltages V1,

V2 are the dependent parameters (M=2). Vector X consists of

the following five components: (x1, x2, x3, x4, x5): x12 =y1, x22

=y2, x32 =y3, x4 =V1, x5 =V2. By defining the components x1, x2,

x3 using the above formulas, we can automatically obtain

positive values of the conductance, which eliminates the issue

of positive definiteness for each conductance and allows us to

perform optimization in the full space of the values of these

variables without any restrictions.

The model of this circuit includes two equations

corresponding to Kirchhoff’s laws. The objective function

C(X) is determined by the formula C(X)=(x5-m1)2+((x4-x5)-

m2)2, where m1 and m2 are predetermined values of the divider

voltages. This circuit is characterised by two (M=2) dependent

parameters (x4, x5), and three (K=3) independent parameters

(x1, x2, x3). The control vector has the next structure:

U=(u1,u2). The structural basis of the various strategies

includes four strategies with the following control vectors:

(00), (01), (10), and (11). The mathematical model of the

circuit is determined by the following equations:

0))()(()1()( 2

5454

141  xxxxbaxxxxXg

(15)

0))()(()( 2

54542  xxxxbaxxXg

It is from the solution of system (15) that the values of the

dependent variables x4, x5 can be determined and then the

value of the objective function C(X) can be calculated. In the

case of the transformation of the two dependent variables x4,

x5 (or at least one of them) into independent ones, it is

necessary to form a generalized objective function F(X,U)

according to the following formula:





/))()(()(),( 2

11 XguXguXCUXF 

 

Consider the optimization problem for the circuit shown in

Fig. 1. Let a = 1, b = 1, m1=0.2, and m2=0.25. For the example

analyzed, the length L of the chromosomes varied from 20 to

80 for each of the five variables, and the number NN of

chromosomes in the population varied from 60 to 320.

Variable limits are specified in a normalized form; for

variables x1, x2, x3, they were set from 10-5 to 2.0, and for

variables x4, x5 from 10-3 to 1.0. Matrix X has five rows (N = 5)

and NN columns. At the same time, we introduce a vector X

consisting of five components, each of which is calculated by

formula (11). This vector is not directly involved in the

calculations, but serves as an informative object for

constructing averaged trajectories of the optimization process.

In this case, we can follow the evolution of the mean values

and build graphs of the trajectories of the optimization process

based on the GA in N dimensions or different projections of

these trajectories. The calculations for each trajectory

continued until the required accuracy δ for the generalized

objective function F(X,U) was achieved. For a given accuracy

δ = 3·10-5, two projections of the four trajectories for strategies

(00), (01), (10), and (11) are shown in Fig. 4. Strategy (00)

corresponds to TSO, and strategy (11) corresponds to MTS.

These dependences are obtained by averaging the stochastic

results of the GA, in contrast to the trajectories obtained by

direct integration of differential equations in the analytical

approach [25]. However, they reflect the behavior of the

optimization trajectory. All strategies start at one point S and

end at approximately one point F, but their behavior during

optimization process are very different.

It is important to note that the required accuracy of

minimising the objective function F(X,U) has a significant

impact on the optimization process and its characteristics.

Each of the four strategies has its own convergence accuracy.

Table III shows the results reflecting the potential accuracy ε

of the optimization process that each strategy can achieve and

the number of generations required to obtain a solution with

precision δ = 10-5.

Fig. 4. x2-x5 and x3-x5 projections for four optimization strategies.

TABLE III. POTENTIAL ACCURACY ε AND NUMBER OF GENERATIONS

Control

vector

Potential

accuracy ε

Number of

generations

for δ =10-5

(00)

1.69·10-5

No solution

(01)

2.04·10-5

No solutiion

(10)

6.05·10-6

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(11)

2.87·10-5

No solution

Table III reveals that for an error of 1.6·10-5 or less in

obtaining the solution, only one of the strategies, namely

strategy (10), will allow solving the problem. In this case, 77

generations are required. Other strategies, including the

traditional strategy (00), cannot find solutions for any number

of steps in the optimization procedure. The presence of

various strategies determined by the control vector U allows

one to formulate the problem of constructing a complex

optimization strategy consisting of several different strategies.

It is possible to propose the structure of a composite strategy

consisting of two, three, or more different strategies, where

each composite strategy is determined by the control vector U.

In this case, it is important to obtain the optimal position of

the switching points from one strategy to another, which

ensures a decrease in the parameter ε, i.e., an increase in the

accuracy of solving the optimization problem. Table IV shows

the results of some composite strategies with optimal

switching point Sp that can significantly improve the accuracy

of solving the problem. This table contains data for six

composite strategies, each of which consists of two strategies

in Table III. As can be seen, the accuracy of the solution was

improved by 2 to 3 orders of magnitude. The table shows that

a decrease in the required error leads at the same time to a

slight increase in the number of required populations. We have

seen that only one strategy from Table III can solve the

problem with an accuracy of 10-5.

TABLE IV. POTENTIAL ACCURACY ε AND NUMBER OF GENERATIONS

FOR COMPLEX SINGLE SWITCHING POINT STRATEGIES

Control

vector

Number of generations G for

various precision δ

10-5

10-6

10-7

4·10-8

(01) (00)

3.94·10-8

(10) (00)

3.94·10-8

(11) (00)

3.94·10-8

(00) (01)

10-7

(00) (10)

6.75·10-7

(00) (11)

7.22·10-7

However, the results presented in Table IV show that

complex strategies allow solving the problem with much more

stringent requirements for the accuracy of the solution

obtained. This table provides data on the potential accuracy

achievable for various compound strategies at the optimum

position of switching point. The results of the optimization

process are also presented in the form of the required number

of GA populations at which the required accuracy



achieved. The composite strategies allow solving the

optimization problem with a significantly higher accuracy than

the original strategies of Table III. Some of these strategies can

solve the problem up to 4·10-8 accuracy. In this case, as can be

seen from Table IV, number of generations increases

insignificantly with an increase in the required accuracy of

solving the problem. This happens until the potential strategy

error exceeds the required precision for solving the problem.

Fig. 5 shows the dependence of the minimized function F

on the number of generations G for three strategies

corresponding to three variants of calculating the fitness

function for precision δ = 10-6.

These dependences are plotted for two scales - large (Fig.

5(a)) and small, which corresponds to the inner region of the

ellipse in the first figure (Fig. 5(b)). The composite strategy

includes two strategies (01) and (00) with the switching point

between them Sp=13. Simple strategies (00) and (11) do not

achieve the required accuracy δ = 10-6, but the composite

strategy solves the problem in 35 generations.

(a) Large scale

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(b) Small scale

Fig. 5. Minimized function F for strategies (00), (11) and (01)(00) on the

number of generations G.

The results of the analysis of compound strategies with two

switching points are presented in Table V. This table shows

similar results for compound strategies with three parts. Again,

a significant improvement in the accuracy of the solution by 2

to 3 orders of magnitude was obtained in comparison with the

strategies in Table III. In this case, the possible number of

compound strategies that allow solving the problem with high

accuracy also increases.

TABLE V. POTENTIAL ACCURACY ϵ AND NUMBER OF GENERATIONS

FOR COMPLEX STRATEGIES WITH TWO SWITCHING POINTS SP1, SP2

Control

vector

Sp1,

Sp2

Number of generations for

various precision δ

10-5

10-6

10-7

4·

10-8

(00)(01)(00)

2, 4

3.94

·10-8

(00)(10)(00)

4, 6

7.53

·10-7

(00)(11)(00)

4, 11

7.53

·10-7

(00)(11)(10)

15,

9.03

·10-8

(01)(11)(00)

3, 6

3.94

·10-8

(10)(11)(00)

3, 4

3.94

·10-8

(11)(01)(00)

3, 4

3.94

·10-8

(11)(10)(00)

4, 9

7.53

·10-7

(11)(00)(01)

2, 11

6.84

·10-8

(01)(10)(00)

4, 7

3.94

·10-8

The obtained result shows that the change in the structure

of the fitness function in the course of the optimization

algorithm allows us to bypass local minima and overcome

premature convergence. Such an improvement in the accuracy

of the solution leads to a significant reduction in the number

of GA generations needed to obtain the required accuracy of

the solution to the optimization problem.

A. Example 3

Let's analyze the optimization process of the nonlinear

circuit shown in Fig. 6.

Fig. 6. Single-stage amplifier.

The conductivities y1, y2, y3 are positive and compose the

set of non-dependent parameters of the circuit (K=3). Nodal

voltages V1, V2, V3 for nodes 1, 2 and 3 are the dependent

parameters (M=3). Let's define a vector of variables X ϵ R6,

including six components (x1, x2, x3, x4, x5, x6): x12 =y1, x22 =y2,

x32 =y3, x4 =V1, x5 =V2, x6 =V3. A static Ebers-Moll model of

transistor was used [28].

The objective function C(X) of the optimization process

was determined as the sum of the squares of the differences

between the previously specified and current values of the

nodal voltages:



 







iiiK VxXC

0)(



where V10,V20,V30 are the before-defined values of nodal

voltages.

The circuit model is defined by Kirchhoff's law as:



   

1401  xxEIXg B

 



 

22  xxIXg E

 



   

3613  xxEIXg C

   

where IB, IE, IC – are the base, emitter and collector currents,

respectively. This system serves as a system of constraints for

minimizing the objective function C(X). The control vector

includes three components U=(u1,u2,u3). Using the generalized

approach, we transform system (18) into system (19).



 

01  Xgu jj

j 

The generalized objective function is defined by the

following formula:



 







2)(

)(,

jjj XguXCUXF



 

Table VI shows the number of generations required to

achieve the minimum of the function F with accuracy δ for

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three strategies. The first two are TSO with control vector U =

(0,0,0) and MTS with control vector U = (1,1,1). The third

strategy is a combined strategy defined by the control vector

(000)(111) with one switching point Sp = 9. That is, the first

nine iterations correspond to the traditional strategy (000), and

the next ones correspond to the strategy (111).

It can be seen that when using the traditional strategy, the

required number of generations is much larger than in the case

of the modified traditional strategy and the combined strategy

to achieve the same accuracy. In addition, the traditional

strategy does not provide a good accuracy of achieving the

minimum of the objective function. When the required error is

10-4 or less, no solution based on the traditional strategy is

found. At the same time, the modified strategy (111) and the

combined strategy, consisting of two strategies (000) and

(111), find a solution with an accuracy of 5·10-11. Note that the

combined strategy finds a solution to the problem in a

significantly fewer generations.

TABLE VI. NUMBER OF GENERATIONS G FOR STRATEGIES (000), (111)

AND (000)(111) FOR DIFFERENT PRECISION δ

Precision

Control

vector

(000)

Control

vector

(111)

Control

vector

(000)(111)

Sp=9

10-1

137

10-2

20706

10-3

348514

10-4

10-5

10-6

102

10-7

109

10-8

123

10-9

144

10-10

172

5·10-11

210

108

10-11

It can be seen that the gain in the number of iterations

(number of generations) for the combined strategy and MTS is

three to four orders of magnitude compared to the traditional

one, if the traditional strategy as a whole is able to find a

solution. The reduction in CPU time is even greater, since the

time of one iteration of the modified strategy is much less than

the traditional one.

It is clear that the final result of the combined strategy

depends on the switching point from one strategy to another.

Table VII shows the dependence of the number of generations

on the switching point for a given error δ =10-5. It can be seen

that the switching point significantly affects the number of

generations required to achieve the necessary accuracy. The

minimum value for this combined strategy corresponds to the

switching point Sp = 9.

TABLE VII. NUMBER OF GENERATIONS G FOR COMBINED STRATEGIES

(000)(111) FOR DIFFERENT SWITCHING POINT SP

Control

vector

(000),(111)

Table VIII contains information for the three considered

strategies, summarizing their comparative characteristics when

achieving an accuracy of 10-2 and 10-3 for the minimized

objective function. The numerical values of the number of

generations, the CPU time of all strategies, as well as the

comparative gain for the MTS and for the combined strategy,

both in the number of generations and in the CPU time

compared to TSO, are given. Note that TSO does not allow

finding a solution to the problem with the required error less

than 10-3. It can be seen that both the MTS and the combined

strategy provide a large gain over TSO.

With an error of 10-2, the gain in terms of the number of

generations is more than two orders of magnitude, and in

terms of CPU time, more than three orders of magnitude. With

a given error of 10-3, the gain in terms of the number of

generations is 3-4 orders of magnitude, and in terms of CPU

time, it is almost five orders of magnitude.

TABLE VIII. GENERALIZED COMPARATIVE CHARACTERISTICS FOR THREE

DIFFERENT STRATEGIES

Precision

Control

vector

(000)

(111)

(000)(111)

Sp=9

10-2

Number of

generations

20706

Gain in the

number of

generations

440

647

CPU time (s)

1178.15

0.266

0.244

Time gain

4429

4828

10-3

Number of

generations

348514

Gain in the

number of

generations

5532

9171

CPU time (s)

20518.2

0.353

0.276

Time gain

58124

74340

The information presented in this table is the main practical

result of the work. It can be stated that the use of a generalized

approach that changes the structure of the vector of basic

variables X and the shape of the fitness function makes it

possible to overcome the problem of the GA's premature

convergence to a local minimum. In this case new strategies

appear that can substantially increase the accuracy of solving

the problem and significantly speed up the optimization

procedure.

In Fig. 7 shows the dependence of the function F to be

minimized on the number of generations G for the three

analyzed strategies at a given accuracy of 10-5.

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Fig. 7. Minimized function F for strategies (000), (111) and

(000)(111) on the number of generations G.

Traditional strategy of optimization cannot find a solution

to the problem with the required accuracy. On the contrary, it

is obvious that the modified traditional strategy and the

combined strategy find a solution to the task rather quickly.

Thus, we can conclude that new optimization strategies that

appear within the framework of the presented generalized

approach have good prospects for improving the optimization

process of electronic circuits.

4. Conclusion

A generalized approach in terms of control theory to

solving the problem of optimizing electronic circuits using

deterministic optimization methods was developed earlier. The

obtained algorithms have shown high efficiency in comparison

with the traditional approach in terms of both accuracy and

speed.

This paper demonstrates the possibility of embedding the

idea of generalized optimization into the body of stochastic

optimization methods. It was shown that this approach can be

built into GA, which leads to the formation of a set of different

optimization strategies and a significant improvement in the

main characteristics of GA.

The studied examples demonstrate the practical

implementation of a modified GA based on a generalized

approach for solving the problem of optimizing electronic

circuits. The emerging new optimization strategies make it

possible to increase the accuracy of the problem solution by

several orders of magnitude. It should also be emphasized that

the real gain of these strategies in CPU time compared to the

traditional approach is much higher than the gain in the

number of GA populations. This is due to the fact that the

processor time for evaluating the fitness function for new

strategies is much less than in the traditional case.

REFERENCES

[1] G. Stehr, M. Pronath, F. Schenkel, H. Graeb, and K. Antreich K, “Initial

sizing of analog integrated circuits by centering within topology-given

implicit specifications”, Proceedings of the IEEE/ACM International

Conference on Computer-Aided Design, 2003, pp. 241–246.

[2] M. Hershenson, S. Boyd, and T. Lee, “Optimal design of a CMOS op-

amp via geometric programming”, IEEE Transactions on Computer-

Aided Design of Integrated Circuits, vol. 20, no.1, 2001, pp.1–21.

[3] J. W. Bandler, R. M. Biernacki, S. H. Chen, P. A. Grobelny, and R.

Hemmers, “Space mapping technique for electromagnetic optimization”,

IEEE Transactions on Microwave Theory and Techniques, vol. 42,

1994, pp. 2536–2544.

[4] S. Koziel, J. W. Bandler, and K. Madsen, “Space-mapping-based

interpolation for engineering optimization”, IEEE Transactions on

Microwave Theory and Techniques, vol. 54, 2006, pp. 2410–2421.

[5] S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by

Simulated Annealing”, Science, vol. 220, 1983, pp. 671–680.

[6] M. Moutchou, H. Mahmoudi, and A. Abbou, “Induction Machine

identification based on a new technique of simulated annealing

optimization”, Proceedings of International Conference on Renewable

and Sustainable Energy, 2014, pp. 873–878.

[7] P. Grabusts, J. Musatovs, and V. Golenkov, “The application of

simulated annealing method for optimal route detection between

objects”, Procedia Computer Science, vol.149, 2019, pp. 95–101.

[8] D. Nam, Y. Seo, L. Park, C. Park, and B. Kim, “Parameter optimization

of an on-chip voltage reference circuit using evolutionary

programming”, IEEE Transactions on Evolutionary Computations, vol.

5, no. 4, 2001, pp. 414–421.

[9] G. Alpaydin, S. Balkir, and G. Dundar, “An evolutionary approach to

automatic synthesis of high performance analog integrated circuits”,

IEEE Transactions on Evolutionary Computation, vol. 7, no. 3, 2003,

pp. 240–252.

[10] B. Liu, Y. Wang, Z. Yu, L. Liu, M. Li, Z. Wang, J. Lu, and F. V.

Fernandez, “Analog circuit optimization system based on hybrid

evolutionary algorithms”, Integration the VLSI journal 2009, vol. 42, no.

2, 2009, pp. 137–148.

[11] H. Nenavath and R. Kumar Jatoth, “Hybridizing sine cosine algorithm

with differential evolution for global optimization and object tracking”,

Applied Soft Computing, vol. 62, 2018, pp. 1019–1043.

[12] J. Robinson and Y. Rahmat-Samii, “Particle swarm optimization in

electromagnetic”, IEEE Transactions on Antennas and Propagation, vol.

52, 2004, pp. 397–407.

[13] M. A. Zaman, M. Gaffar, M. M. Alam, S. A. Mamun, and M. Abdul

Matin, “Synthesis of antenna arrays using artificial bee colony

optimization algorithm”, International Journal of Microwave and Optical

Technology, vol. 6, no. 8, 2011, pp.234–241.

[14] A. Sallem, B. Benhala, M. Kotti, M. Fakhfakh, A. Ahaitouf, and M.

Loulou, “Application of swarm intelligence techniques to the design of

analog circuits: evaluation and comparison”, Analog Integrated Circuits

and Signal Processing, vol. 75, no. 3, 2013, pp. 499-516.

[15] F. Li, X. Cai, and L. Gao, “Ensemble of Surrogates Assisted Particle

Swarm Optimization of Medium Scale Expensive Problems”, Applied

Soft Computing, vol. 74, 2019, pp. 291-305.

[16] M. L. Carneiro, P. H. P. de Carvalho, N. Deltimple, L. da C Brito, L.R.

de Menezes, E. Kerherve, S. G. de Araujo, and A. S. Rocira, “Doherty

amplifier optimization using robust genetic algorithm and unscented

transform”, Proceedings of Annual IEEE Northeast Workshop CAS,

2011, pp. 77–80.

[17] M. Z. R. Khan and A. K. Bajpai, “Genetic algorithm and its application

in mechanical engineering”, International Journal of Engineering

Research & Technology, vol. 2, no. 5, 2013, pp. 677–683.

[18] M. Quiroz-Castellanos, L. Cruz-Reyes, J. Torres-Jimenez, C. Gomez, H.

J. Fraire-Huacuja, and A. C. Alvim, “A grouping genetic algorithm with

controlled gene transmission for the bin packing problem”, Computer

Operation Research, vol. 55, 2015, pp. 52–64.

[19] Y. C. Chuang, C. T. Chen, and C. Hwang, “A simple and efficient real-

coded genetic algorithm for constrained optimization”, Applyed

Software Computing, vol. 38, no. C, 2016, pp. 87–105.

[20] L. L. Corso, A. L. Gasparin, and H. M. Gomes, “Reliability based

design optimization using a genetic algorithm: application to bonded

thin films areas of copper/polypropylene”, Ingeniare. Revista chilena de

ingeniería, vol. 24, no. 3, 2016, pp. 510–519.

WSEAS TRANSACTIONS on SYSTEMS

DOI: 10.37394/23202.2022.21.18

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Volume 21, 2022

[21] A. A. Khan and R. N. Mir, “Optimization of Constrained Function

Using Genetic Algorithm”, Computer Engineering and Intelligent

Systems, vol. 8, no. 2, 2017, pp. 11–15.

[22] A. A. R. Hosseinabadi, J. Vahidi, B. Saemi, A. K. Sangaiah, and M.

Elhoseny, “Extended Genetic Algorithm for solving open-shop

scheduling problem”, Soft Computing, vol. 23, 2019, pp. 5099–5116.

[23] C. Goh and Y. Li, “GA automated design and synthesis of analog

circuits with practical constraints”, Proceedings of IEEE Congress on

Evolutionary Computations, vol. 1, 2001, pp. 170–177.

[24] N. F. Paulino, J. Goes, and A. Steiger-Garcao, “Design methodology for

optimization of analog building blocks using genetic algorithms”,

Proceedings of Symposium on Circuits and Systems, 2001, pp. 435–438.

[25] A. Zemliak, “Analog circuit optimization on basis of control theory

approach”, COMPEL: The International Journal for Computation and

Mathematics in Electrical and Electronic Engineering, vol. 33, no. 6,

2014, pp. 2180–2204.

[26] A. Zemliak, and T. Markina, “Behaviour of Lyapunov´s function for

different strategies of circuit optimization”, International Journal of

Electronics, vol. 102, no. 4, 2015, pp. 619–634.

[27] L. Davis (ed.), Handbook of Genetic Algorithms. Van Nostrand

Reinhold: New York, 1991.

[28] G. Massobrio and P. Antognetti, Semiconductor Device Modeling with

SPICE, Mc. Graw-Hill, Inc.: New York, 1993.

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