Georgian Scientists/ . 6  N 2, 2024 286

Georgian Scientists

Vol. 6  Issue 2, 2024

https://doi.org/10.52340/gs.2024.06.02.30

Selection of Mathematical Optimization Methods for Solving Engineering Practice
Problems

Nona Otkhozoria1, Nino Tsiklauri2,  Vano Otkhozoria3,
1Georgian Technical Univesity ORCID ID: 0000-0002-5837-5345 2Georgian Technical University 3Georgian Technical

University ORCID ID: 0009-0008-6028-3758

Abstract

Optimization theory plays a pivotal role in contemporary scientific and technical endeavors,
permeating various engineering disciplines. From fine-tuning chemical-technological systems to
optimizing production processes, the application of optimal management techniques is widespread,
particularly in the context of complex automation and sophisticated technical setups. The primary goal
of optimization is to identify the most optimal solution among numerous potential outcomes,
employing diverse strategies ranging from analytical methodologies to numerical simulations. This
paper explores the efficacy of the fastest ascent method in approaching the extremum of the
Rosenbrock function, emphasizing the importance of selecting appropriate starting coordinates.
Furthermore, the study investigates the impact of errors introduced through random variables,
highlighting the need for robust methodologies capable of navigating uncertainties. Through
comprehensive analysis and experimentation, this research contributes to the ongoing discourse
surrounding optimization methodologies, shedding light on their effectiveness and applicability in
diverse engineering contexts.

Key words: Optimization theory, engineering practice, optimal management techniques,
fastest ascent method, Rosenbrock function.

Introduction
At the modern stage, the theory of optimization makes a significant contribution to scientific

and  technical  progress.  It  is  difficult  to  find  a  field  of  engineering  activity  where  the  problem  of
optimization in the tasks to be performed is not solved. It can be the task of determining the effective
mode of operation of the chemical-technological system, the operation of various technical devices,
the tasks of solving the problem of the production organization, and others.

Optimal management is widely used in conditions of complex automation of technological and
industrial processes or complex technical equipment.



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The goal of optimization is to select the best solution among potentially possible outcomes
using appropriate performance criteria. Optimization can be carried out using many different strategies,
which can begin with the use of complex analytical and quantitative procedures and end with the
judicious use of simple arithmetic.

Numerical optimization is one of the central methods of machine learning. For many problems
it is difficult to determine the best direct solution, but it is relatively easy to determine an error function
that measures how accurately the chosen method is, and then the task of minimizing the parameters
of this function in order to find the best solution.

Researchers and engineers are often faced with the challenge of predicting the behavior of
certain systems or processes in order to control them. This task can be solved through mathematical
models [1,2] and numerical modeling [3]. Although numerical simulations usually provide a good
prediction of the behavior and properties of a certain system [3], initially, it is impossible to determine
which of the many alternatives is the best choice [2].

Since the research activity is aimed at finding an alternative with the best properties, engineers
and researchers in the field of engineering optimization actively use a mathematical approach, optimal
control  methods  [3].  In  engineering  practice,  it  often  happens  like  this  -  the  goal  of  optimization  is
mathematically determined by the objective function - which is formulated taking into account
technical or economic requirements , which is based on trials and research, which allows us to get the
system with the best data. However, when using a scientific approach to solve a real problem, we are
faced with an infinite number of optimization methods and corresponding software for the formulation
and solution of the optimization problem. Since there is no universal optimization algorithm that can
be used to solve any problem, it is important to evaluate the effectiveness of these methods in different
conditions.

Main Part

Our primary objective is to explore the most efficient access method, especially when the bid
length is contingent upon the characteristics of the optimization function. The swiftest approach
involves an iterative algorithm that navigates towards the extremum within the specified range of
argument values. This process entails moving from a chosen point towards the direction of the
function's minimum value. This direction is essentially the opposite of that indicated by the gradient
vector ( f(x)) of the optimization function f(x).

( ) = , , … , ,                 (1)

The formula for determining the argument   with the value  on the k-th bit using the
fastest approach method is as follows: = +

where is a unit vector pointing in the opposite direction of the gradient ( )at the
specified point .

( )= ( ( ))
( ( ))



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We tested the algorithm using the minimization of a function of two variables, specifically the
Rosenbrock function, as an example. We established the necessary parameters, with the notations
aligning with those used in the Mathcad software system:

n:=20 - the maximum number of iterations on the x and y axes;
i: 0..n,  j: 0n - the sequence number of calculations;
a1:= 0.2, a2:= 0.2;
b1:=0.06, b2:=0.06;

xi=a1+b1 i, yj=a2+b2 j;
Formulas for computing the arguments of the i-th and j-th order:
Mij:=f(xi,yj) - matrix of Rosenbrock function values.
As depicted in the figure, the function exhibits a flat bottom. Near this bottom, the gradient

assumes small values. Hence, it's apparent from the outset that an algorithm unaffected by the
function's shape will be less efficient in approaching the minimum.

Figure 2 Graphic representation of the Rosenbrock function

Now, let's proceed with the computation of the elements of the gradient vector for the
Rosenbrock function:

a) Partial derivatives with respect to the arguments x and y: g_x(X,Y), g_y(X,Y);
b)  Second-order  partial  derivatives  with  respect  to x and y: gxx(X,Y), gyy(X,Y); c) Mixed

second-order derivatives with respect to x and y g_xy(X,Y), g_yx(X,Y).
The following parameters are necessary for implementing the fastest ascent method algorithm:
vmax:=200 - the maximum number of iterations required for approaching the minimum;
V:=0…vmax - the range of iteration changes;
x0:=0,y0:=0 - the coordinates of the starting approach point;
0:= _f(x0,y0) - the value corresponding to the initial approximation;

sx0:=s_x(x0,y0), sy0:=s_y(x0,y0)  -  the  values  of  the  bit  corresponding  to  the  initial
approximation;



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f0:=ff(x0,y0) -  the  value  of  the  optimization  function  corresponding  to  the  initial
approximation.

Vector of initial values for the iterative process:
0
0
0

Please provide the image of the function coordinates calculation and their corresponding
values for further analysis.

=
+ _ ( , ). _ ( , )
+ _ ( , ). _ ( , )

( + _ ( , ). _ ( , ), + _ ( , ). _ ( , ))

Using the obtained values, we constructed a diagram (Fig. 2). Upon examining the values, it
becomes evident that there is oscillation near the minimum.

The diagram illustrates the dependency of the function coordinates on the iteration number
as the extremum is approached.

x

0
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251

0.93900212
0.93921365

0.9393041
0.93951411
0.93960314
0.93981165

0.9398993
0.94010632
0.94019262
0.94039818
0.94048316
0.94068728
0.94077096
0.94097366
0.94105607
0.94125737

y

0
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251

0.88185501
0.88182281
0.88242049
0.88238872
0.88298067
0.88294931
0.88353562
0.88350468
0.88408546
0.88405492
0.88463026
0.88460012
0.88517011
0.88514036

0.8857051
0.88567573

ff

0
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252

3.70394858·10 -3

3.68563878·10 -3

3.667426·10 -3

3.6493836·10 -3

3.63143631·10 -3

3.61365553·10 -3

3.59596798·10 -3

3.5784432·10 -3

3.56100985·10 -3

3.54373563·10 -3

3.52655106·10 -3

3.50952212·10 -3

3.49258112·10 -3

3.47579233·10 -3

3.4590898·10 -3

0 50 100 150 200 250
0

0.5

11

0

xv

yv

2500 v

Figure 3 Dependence of function coordinates on iteration order for approximation to extremum



Georgian Scientists/ . 6 N 2, 2024 290

We carried out research for different initial values.
  a) x0:=1.5 y0:=2 - coordinates of the starting point of approach;

Table 1 fragment of the table of obtained values:

As evident from the graph, the approach to the extremum with starting points x0=1.5 and y0
=2 is notably coarse, even under ideal conditions. Thus, we sought comparisons with other initial
coordinates.

Subsequently, we conducted the research under the specified initial conditions (fig.3 and
fig.4), accounting for errors introduced by random variables following a normal distribution:

g:=rnorm(130,0,0.33).
Incorporating these errors, the iterative procedure takes the following form:

( )

( )

( )

+ _ ( , ). _ ( , ) + ( + _ ( , ). _ ( , )). .
+ _ ( , ). _ ( , ) + ( + _ ( , ). _ ( , )). .

( + _ ( , ). _ ( , ) + ( + _ ( , ). _ ( , )). .
+ _ ( , ). _ ( , ) + ( + _ ( , ). _ ( , )). . )

x

0
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251

1.33074929
1.33001466
1.33033907
1.32960748
1.32993078
1.32920219

1.3295244
1.32879877
1.32911991
1.32839721
1.32871728
1.32799747
1.32831648
1.32759953

1.3279175
1.32720337

y

0
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15

2
2.02390156
2.02550898
2.01787204
2.01734204
2.01055437
2.01004465
2.00390547
2.00341208
1.99778778
1.99730798
1.99210448
1.99163623
1.98678442
1.98632617
1.98177356

ff

0
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15

6.5
0.20481523
0.17888568
0.17766212
0.17646521
0.17537835
0.17431231
0.17332953
0.17236365
0.17146333

0.1705771
0.16974406
0.16892302
0.16814611
0.16737961
0.16665039

Table 1 A fragment of the table of obtained values:



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Figure 4 Fragment of the lookup table and corresponding diagrams x0:=0; y0:=0; =10%;

Figure 5 Fragment of the lookup table and corresponding diagrams x0:=0; y0:=0; =20%;



Georgian Scientists/ . 6 N 2, 2024 292

Conclusion:

In the quest to predict and control system behaviors, researchers and engineers rely on
mathematical models and numerical simulations. While these simulations provide valuable insights,
the initial uncertainty regarding the best choice among alternatives poses a significant challenge.

Engineering optimization endeavors often center around identifying alternatives with optimal
properties, leveraging mathematical approaches and optimal control methods. However, the
abundance of optimization methods and corresponding software necessitates a judicious evaluation of
their effectiveness across various conditions.

Our investigation into the fastest ascent method underscores the importance of selecting
appropriate starting coordinates. The method's efficacy in approaching the extremum of the
Rosenbrock function is evident, albeit varying with the choice of initial points. Furthermore, the
incorporation of error considerations underscores the need for robust methodologies capable of
navigating uncertainties.

In conclusion, optimization theory continues to be a driving force behind technological
innovation, empowering engineers and researchers to tackle complex challenges and unlock new
frontiers in scientific inquiry and technical progress.

References
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Edition1st Edition,, First Published, 2014
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optimization problems." International Journal of Mathematical Modelling and Numerical
Optimization 4.2 (2013)

3. George Lindfield, John Penny Numerical Methods: Using MATLAB· Elsevier Science, 2018
4. Nona Otkhozoria,  Vano Otkhozoria,  & Shorena Khorava. (2022).  Search for an extremum using

the steepest descent method under the conditions of experimental errors. World Science, (2(74).
https://doi.org/10.31435/rsglobal_ws/28022022/7785

5. Z Azmaiparashvili, N Otkhozoria (2021) “Mathematical Model for Studying the Accuracy
Characteristics of Devices for Measuring the Resonant Frequency of Oscillatory Systems New
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Jeeves method in the experiment under error conditions is assessed”, Georgian Scientists, 5(4),
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8. Menabde, T., Otkhozoria, N., & Otkhozoria, V. (2024). Use of the theory of measurement
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