Abstract: - The approach developed earlier, based on generalized optimization, was successfully applied to the

problem of designing electronic circuits using deterministic optimization methods. In this paper, a similar

approach is extended to the problem of optimizing electronic circuits using a genetic algorithm (GA) as the

main optimization method. The fundamental element of generalized optimization is an artificially introduced

control vector that generates many different strategies within the optimization process and determines the

number of independent variables of the optimization problem, as well as the length and structure of

chromosomes in the GA. In this case, the GA forms a set of populations defined by a fitness function specified

in different ways depending on the strategy chosen within the framework of the idea of generalized

optimization. The control vector allows you to generate different strategies, as well as build composite

strategies that significantly increase the accuracy of the resulting solution. This, in turn, makes it possible to

Received: August 15, 2023. Revised: February 11, 2024. Accepted: March 9, 2024. Published: April 18, 2024.

An effective way to solve problems of design and

synthesis of electronic systems is to use

optimization procedures. These procedures are a set

of operators inspired by biological processes in

nature (mutation, crossbreeding, and selection), has

been actively used over the past two decades to find

highly accurate algorithms for solving optimization

problems. One area in which the genetic algorithm

is widely used is the computer-aided design of

cases is offered in [11].

premature convergence. It lies in the fact that the

GA offers, as the best solution over a sufficiently

long sequence of generations, an individual that

provides a certain local minimum of the objective

function, without further movement to the global

minimum. Various ways to solve this problem have

been proposed. One of them is inherent in the very

design of the algorithm: the ability to regulate the

retrieval systems. It is based on the use of single-

point crossing of ordered chromosomes; the result

of this method is a daughter chromosome that

combines the best genes of both parent

chromosomes. The problem of premature

convergence was analyzed in the framework of a

Markov chain in [14], [15]. The authors prove that

the degree of diversity of populations tends to zero

with probability 1, causing a decrease in the

searchability of the GA and, consequently,

population. The authors of [17], propose a

combination of the frequency crossover strategy

with nine different mutation strategies to reduce the

effect of premature convergence using the traveling

salesman problem as an example. In [18], to

overcome premature convergence, an improved

adaptive GA is considered, containing dynamic

adjustment of crossover and mutation operators

during the evolution process, as well as a restart

strategy. Other aspects of the problem of premature

optimization algorithms. Attempts to apply this

methodology to GA have shown its effectiveness in

must be satisfied at the end of the optimization

procedure for the solution to satisfy the problem.

This formulation frees us from the mandatory

division of variables into independent and

dependent when solving the problem (1) – (2). This

idea underlies the generalized optimization

methodology and is implemented through a

generalized objective function, including a so-called

penalty function. This structure of the objective

function allows both to achieve minimization of the

The term “generalized” means that the extraction of

not all equations from the circuit model is

considered, but only parts of them. When starting to

implement the idea of generalized optimization, we

declare all components of the vector X to be

independent. However, to preserve the physical

meaning of the problem and fulfill the

corresponding constraints at the endpoint of the

optimization process, it is necessary to introduce a

new objective function into consideration:

crossover, both in their number and in their location

relative to each other.

variable x1 (K=1), and parameter M=1 because there

is achieved at the point x1 = 3, x2 = 2. At this point,

the goal function C(X) takes the minimum zero

value. These values are the solution to the problem.

Let us find, however, this solution by the developed

approach.

is transformed into the following equation:

where u is the component of the control vector U, in

this case, the only one.

control vector U=(0) and MTOS which has a control

vector U=(1).

using 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 of the

Variables x1 and x2 take the values 3 and 2,

respectively, but with different degrees of accuracy

for different strategies.

Fig. 1: Dependence of the generalized objective

function F under successive generational change for

an independent parameter. In this case, x2 and x3 are

solve an optimization problem with two orders of

magnitude fewer generations than TOS for 10-2

precision and four orders of magnitude less for 5·10-

with a precision of 10-2 and 240.53 s with a

precision of 5·10-3. The composite strategy solves

this problem in 0.047 s with an accuracy of 10-2 and

0.05 s with an accuracy of 5·10-3. In this case, the

CPU time gain is 126 times for 10-2 precision and

composite strategies that include two main strategies

with a control vector (0)(1) are given in Table 3,

Figure 3 and Figure 4.

MTOS and composite strategy find a global

minimum equal to zero with coordinates x1 = 1.10,

optimization process for different accuracy δ of

minimizing the objective function F for MTOS and

a composite strategy with control vector (0)(1) and

switching point Sp=3.

advantage of the composite strategy while

increasing the required accuracy of solving the

(1), and composite strategy (0)(1)

function F decreases to the values 10-8–10-9 giving

a high accuracy of the optimization process

implementation, since it corresponds to the global

minimum.

The minima correspond to the following

coefficients: c1 = 0.15, c2 = 0.1, c3 = 0.3, c4 = 0.2, c5

constraints (16)-(17) for strategies TOS, MTOS, and

with a control vector (0)(1) are given in Table 4.

global minimum with a high degree of accuracy.

conductivities

parameters (M=4). Circuit optimization aims to

obtain the required values of all nodal voltages

Fig. 5: Four-node nonlinear passive circuit

and

the components