Optimization in Identification of Logical-Probabilistic Risk Models

Automation and Remote Control, Vol. 64, No. 7, 2003, pp. 1063–1073. Translated from Avtomatika i Telemekhanika, No. 7, 2003, pp. 51–63.Original Russian Text Copyright c© 2003 by Rybakov, Solozhentsev.

RISKS IN ENGINEERING AND TECHNOLOGY

Optimization in Identification

of Logical-Probabilistic Risk Models

A. V. Rybakov and E. D. Solozhentsev

Institute of Machine Science Problems, Russian Academy of Sciences, St. Petersburg, RussiaReceived October 15, 2002

Abstract—Identification of logical-probabilistic risk models with groups of incompatible eventsis investigated by the random search method. The integral multi-extremal multi-parametricaim function of the optimization problem in investigated as a function of the number of opti-mizations, maximal and minimal amplitudes of increments of parameters, initial value of theaim function, choice of identical or different amplitudes of increments for different parameters,and risk distribution. An effective method for finding the global extremum in identifying alogical-probabilistic risk model within reasonable computation time is developed.

1. INTRODUCTION

Identification of logical-probabilistic risk models is regarded as an extension of reliability theoryto systems with elements and output characteristics having several value levels. This researchtrend is known as the reliability analysis and optimization of multi-state systems) [1]. The theoryof identification of logical-probabilistic risk models with groups of incompatible events developedin [2] is based on Ryabinin’s theory of reliability and safety [3, 4] widely used in technology.

We study structurally complex systems, in which risk is a common phenomenon and sufficientstatistical data on risky objects are available. Banks, business institutions, and diagnostic systemsare examples of such systems. By risk, we mean the probability of failure.

Logical-probabilistic risk models are twice more accurate and seven-fold robust than other well-known methods [2, 5, 6]. But multi-parameter multi-criteria optimization (identification) for deter-mining the probabilities of a logical-probabilistic risk model from statistical data is an extremelydifficult problem [5–7].

Below we present the results of identification of logical-probabilistic risk models with groupsof incompatible events obtained by the random search method. We studied the dependence ofthe integral multi-extremal multi-parameter aim function of optimization on the number of opti-mizations, maximal and minimal amplitudes of increments of parameters, initial value of the aimfunction, choice of identical or different amplitudes for different parameters, and risk distributionsof objects.

2. RELATIONSHIP OF PROBABILITIES IN A GROUP OF INCOMPATIBLE EVENTS

An object of risk is described by attributes, each of which has several gradations. Attributesand gradations are defined by random events that end in a failure [2, 5, 6]. Attributes (j = 1, n)are linked by logical connections. Gradations (r = 1, Nj) of an attribute are incompatible events.

Let us associate every event with a random variable or a logical variable. An L-function offailure is constructed from a risk scenario or a structural failure model (Fig. 1). For example,for the “node-type” structural risk model (Fig. 1a), arrows denote the logical links “OR” and

0005-1179/03/6407-1063$25.00 c© 2003 MAIK “Nauka/Interperiodica”

1064 RYBAKOV, SOLOZHENTSEV

f

f

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f

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ff

f

h

h

h

h

h

h

h

f

h

h

h

h

h

h

h

h

h

h

h

Fig. 1. Structural risk models: (a) node; (b) bridge.

attributes X1 have three gradations X11, X12, and X13, which form a group of incompatible events.Therefore, only one gradation is used for an attribute in a concrete object.

The structural risk model of failure may be equivalent to a real system (for example, an electricalsystem), and may be associative if it is sensibly constructed or mixed. In the “node-type” failurerisk model (Fig. 1a), a failure occurs if one, two, . . . , or all initiating events occur. Structuralschemes of failure risk are defined by L-functions of different complexities having AND, OR, andNOT links with cycles and repeated elements (Fig. 1b).

The binary logical variable Xj takes the value 1 with probability Pj if the jth attribute leads to afailure and 0 with probability Qj = 1−Pj otherwise. The binary logical variable Xjr correspondingto the rth gradation of the jth attribute is equal to 1 with probability Pjr, and 0 with probabilityQjr = 1 − Pjr otherwise. The vector X(i) = (X1,X2, . . . ,Xj , . . . ,Xn) describes the ith object instatistical data.

An object Xi is defined by the gradations of its attributes. The maximal number of differentobjects is Nmax = N1 × N2 × . . . × Nj × . . . × Nn, where N1, N2, . . . , Nj , . . . are the numbers ofgradations of attributes. Let us express the logical risk function of failure for an object as

Y = Y (X1,X2, . . . ,Xj , . . . ,Xn) (1)

and the probability function of failure risk of an object defined by the vector X(i) as

Pi{Y = 1 | X(i)} = Ψ(P1, P2, . . . , Pj , . . . , Pn), i = 1, N. (2)

For every gradation in a group of incompatible events, Wjr denotes the relative frequency of agradation for the objects in statistical data, P1jr is the probability of a gradation in the group ofincompatible events, Pjr is the probability of the gradation in formula (2) instead of Pj . Thesecharacteristics for the jth group of incompatible events are

Wjr = P{Xjr = 1},Nj∑r=1

Wjr = 1, r = 1, Nj , (3)

P1jr = P{Xjr = 1 | Xj = 1},Nj∑r=1

P1jr = 1, r = 1, Nj , (4)

Pjr = P{Xj = 1 | Xjr = 1}, r = 1, Nj . (5)

AUTOMATION AND REMOTE CONTROL Vol. 64 No. 7 2003

OPTIMIZATION IN IDENTIFICATION 1065

The mean values of Wjr, P1jr, and Pjr for the gradations in a group of incompatible events are

W j = 1/Nj , P1j =Nj∑r=1

P1jrWjr, P j =Nj∑r=1

PjrWjr. (6)

The average a priori risk Pav and mean computed risk P of objects are

Pav = Nb/N, P =

(N∑i=1

Pi

)/N, (7)

where N and Nb are the total number and number of “bad” objects in the statistics.A relation between the probabilities Pjr and P1jr of the gradations at each step of iterative

learning of the B risk model is constructed from statistical data using expressions (3)–(5) by theBayes formula [2]:

Pjr = P1jr × (P j/P1j), r = 1, Nj , j = 1, n. (8)

3. IDENTIFICATION OF LOGICAL-PROBABILISTIC RISK MODELS

Information on risk objects is defined by a database as an ‘object-attribute’ table, in whichobjects and attributes are listed in rows and columns, respectively, and cells contain gradations ofattributes [2]. For example, if the jth attribute has four gradations 1, 2, 3, and 4, then the cell atthe ith row (object) and jth column (attribute) contains one of the numbers 1, 2, 3, 4.

Let us define the concept of an admissible risk Pad, to classify objects into “good” and “bad”[2, 5]: an object is bad if Pi > Pad and, good if Pi < Pad. If the objects are grouped in n classes,then n admissible risks Pad1, Pad2, . . . , Padn are introduced.

Identification of a B-risk model consists in determining the optimal probabilities Pjr of grada-tions. The problem of identification (learning) of a B-risk model is formulated as follows.

Given an ‘object-attribute’ table containing Ng “good” andNb “bad” objects and a B-risk model,find the probabilities Pjr of gradations and admissible risk Pad separating objects into “good” and“bad” by their risk values.

The aim function: the number of correctly classified objects must be maximal

F = Nbs +Ngs = MAX, (9)

where Ngs and Nbs are the number of objects that are classified as good and bad by the statisticsand B-model, respectively (coinciding estimates). Expression (9) implies that the errors or indexesof accuracy of the B-risk model in the classification of “good” objects Eg and “bad” objects Eband the total error Em are equal:

Eg = (Ng −Ngs)/Ng, Eb = (Nb −Nbs)/Nb, Em = (N − F )/N. (10)

Constraints:(1) the probabilities Pjr must satisfy the condition

0 < Pjr < 1, j = 1, n, r = 1, Nj , (11)

(2) the mean risks Pm of objects for the B-model and Pav in the table Pav must be equal. Inteaching a B-risk model, the probabilities Pjr are corrected at each iterative learning step by theformula

Pjr = Pjr × (Pav/P ), r = 1, Nj , j = 1, n, (12)



F

max

P

1

2

P

1

2

Fig. 2. Step variation of the aim function Fmax.

(3) the admissible risk Pad must be defined for a given ratio of badly classified “good” and “bad”objects, because of nonequivalent loss under incorrect classification:

Egb = (Ng −Ngs)/(Nb −Nbs). (13)

Usually, this ratio, according to [9], is equal to or greater than three.The aim function Fmax shown in Fig. 2 is a function of only two parameters. It is a step function

with steps equal to 2. The areas have different dimensions. The parameters P11 and P12 belongto the interval [0, 1], but may differ by one order in magnitude. The dimensions of areas decreaseupon approaching the extremum.

Optimization may get “struck up” on any “area,” without attaining the maximum or overshoot-ing it. The variation of the aim function in a multidimensional space is also of the same pattern.Recall that the dimension of the space of optimization parameters for the logical-probabilistic creditrisk model is ninety four [2, 5].

Multi-parameter multi-extremal optimization for learning of logical-probabilistic models is anextremely difficult problem [5–7]:

(1) the aim function Fmax is the number of correctly recognized good and bad objects, i.e., takesintegral values and is a step function,

(2) the aim function has local extrema and depends on a large number real-valued positiveparameters, and

(3) the derivatives of the aim function with respect to the probabilities P1jr cannot be computed.

4. OPTIMIZATION IN IDENTIFICATION PROBLEMS

Neural network learning principles underlie the identification of logical-probabilistic risk modelsby the random search method [8]. For identification of logical-probabilistic risk models with ex-pression (8), the probabilities in a group of incompatible events are interrelated by the formula forthe variation of probabilities of gradations

dP1jr = K1 × (1/Nt)× tan (K3), r = 1, Nj , j = 1, n, (14)

where K1 is a coefficient, Nt is the current number of optimization of the aim function, and K3 isa random number in the uniform distribution in the interval [−π/2,+π/2].



For formula (14), the aim function is the continuous learning error. The number Nt of opti-mizations when the learning process terminates may be very large. The “tangent” operation is aconsequence of the use of the Cauchy learning error distribution. Theoretically, this error is dis-tributed according to the normal law. But learning error distribution according to the Cauchy lawis used to half the time of computation of tabulated values. Otherwise, computation may prolongindefinitely.

In [5], logical-probabilistic risk models are taught with a modification of formula (14)

dP1jr = K1 × (Nopt −Nt)× tan (K3), r = 1, Nj , j = 1, n, (15)

where Nopt is a given number of optimizations of the aim function. New values of P1jr and Pjrfound for F > Fmax are regarded optimal and used in Nt optimization. If the aim function doesnot increase after a given number of optimizations Nmc, then it is decreased by 2–4 units andoptimization is continued [2, 5, 6].

The aim function in identification of logical-probabilistic risk models is an integer and cannotexceed the total number of objects in statistical data. Formula (15) is quite satisfactory, but thecomputation time is large (about 10 h per optimization session).

The “tangent” operation was excluded from formula (15) to reduce the computation time. Thuswe obtained the expression [2, 5]

dP1jr = K1 × (Nopt −Nt)×K3, r = 1, Nj , j = 1, n. (16)

Though formulas (15) and (16) were applied earlier in [5, 6] for optimization, the problem ofoptimization in identification of logical-probabilistic models is still far from being resolved. This isevidenced by the following fact. In a session with Nopt = 245 000 optimizations and almost constantoptimal increment dP1jr, we obtained Fmax = 824 instead of Fmax = 810 with the usual numberof optimizations Nopt = 245. This compelled us to conduct special experiments, whose results aregiven below.

5. IDENTIFICATION/OPTIMIZATION EXPERIMENTS

For a random number K3 in the interval [−1,+1], the absolute probability increment dP1jr isexpressed in percents (%) upon multiplication by 100. This is convenient, because the accuracy ofprobabilities P1jr can be easily estimated. For example, the increment dP1jr = 0.0005 expressedin percents is 0.05 or the probability P1jr is computed to 0.05% accuracy.

According to formula (16), the maximal amplitude of probability increment at the beginning ofoptimization is

AP1max = K1 ×Nopt. (17)

At the end of optimization, it is 0. Let the current amplitude of increments be AP1. There isan optimal domain OPT for the increment amplitude AP1. Its location and width are unknown(Fig. 3). For large AP1, the probability of increasing Fmax is small, but for small AP1 there is ahigh probability that the local extremum remains at the attained Fmax (Fig. 2).

Optimization (learning of logical-probabilistic risk models) must be confined within the optimaldomain OPT for a sufficiently long time. The duration dNopt of residence in the optimal domainOPT is

dNopt = (OPT ×Nopt)/AP1max (18)

and depends on the number of optimizations Nopt and maximal increment amplitude dP1max. Thelarger the amplitude Nopt and smaller the increment AP1max , the greater the duration dNopt.

Therefore, we investigated the influence of the following parameters of the learning formula (16)on the aim function (accuracy of logical-probability risk models):



dN

opt2

dN

opt3

dN

opt1

dP

1

dP

1

max

0

A

N

opt1

N

opt2

N

opt3

Fig. 3. Number of optimizations Nopt versus increment amplitude AP1.

(1) Number of optimizations Nopt.(2) The minimal increment amplitude AP1min under which optimization can be implemented.(3) The initial value of the aim function Fbeg.(4) Choice of identical or different amplitudes AP1 for different gradations.(5) The maximal increment amplitude AP1max.(6) Distriution of risks of objects in statistical data.The question here is whether the increment amplitudes AP1 must be chosen identical or different

for all gradations? In other words, do the amplitudes AP1jr depend on the probabilities P1jr.In formulas (15) and (16) for learning logical-probabilistic risk models, the increment amplitudesAP1jr are identical for all gradations, irrespective of their probabilities P1jr. The incrementsdP1jr differ only due to modeling of the coefficient K3.

Computer-aided experiments on a logical-probabilistic credit risk model were conducted. Thestructural logical-probabilistic credit risk model has twenty attributes (and a respective num-ber of groups of incompatible events) and ninety four gradations. The L-function of the creditrisk [2, 5, 6] is

Y = X1 ∨X2 ∨ · · · ∨Xj ∨ · · · ∨X20. (19)

It is formulated as follows: a failure occurs if one, two, . . . , or all initiating events occur.Orthogonalizing the L-function (19), we obtain the B-risk model for computing the credit risk

P = P1 + P2Q1 + P3Q1Q2 + P4Q1Q2Q3 + . . . . (20)

The experiment was conducted with 1000 credits, of which 700 were “good” and 300 “bad” [9].Computations were carried out with a software package written in object-oriented Java.

5.1. Choice of Nopt, AP1min, and Fbeg

The probabilities P1jr for the initial variant were taken without the last four digits comparedto those for the optimal variant Fmax = 824. Therefore optimization began with Fbeg = 690–760.This reduced the computation time.

Computations were made for two maximal increment amplitudes AP1max = 0.05 (5%) and0.1 (10%) and number of optimizations Nopt, equal to 150, 300, 500, 750, 1000, 2000, 3000, 4000,5000, 6000, 7000, and 8000.

Using Table 1 (variants 2–21) and Fig. 4, we conclude the following.(1) The aim function Fmax (column 6 in Table 1 and Fig. 4) asymptotically increases with the

number of optimizations Nopt.



Table 1. Results on the choice of optimization parameters

No. Nopt Kj AP1max Fbeg Fmax dPc AP1min Nend Formula

1 2000 0.0001 0.2 776 786 0.204 0.1987 202 300 0.000163 0.05 756 794 0.1969 0.00198 289 (16)3 300 0.00033 0.1 712 790 0.221 0.00429 288 Ditto4 750 0.0000665 0.05 756 802 0.1641 0.00545 669 Ditto5 750 0.000133 0.1 692 790 0.2052 0.01316 652 Ditto6 1000 0.00005 0.05 750 802 0.1867 0.00350 931 Ditto7 1000 0.0001 0.1 708 792 0.2174 0.01580 843 Ditto8 2000 0.000025 0.05 776 808 0.1595 0.00747 1702 Ditto9 2000 0.00005 0.1 724 798 0.1802 0.01405 1720 Ditto

10 3000 0.0000166 0.05 748 806 0.1867 0.00699 2581 Ditto11 3000 0.000033 0.1 708 806 0.1867 0.00501 2849 Ditto12 4000 0.0000125 0.05 744 812 0.1945 0.00791 3368 Ditto13 4000 0.000025 0.1 740 802 0.2121 0.00862 3656 Ditto14 5000 0.00001 0.05 754 806 0.1663 0.00556 4445 Ditto15 5000 0.00002 0.1 738 803 0.1586 0.00400 4801 Ditto16 6000 0.000016 0.1 710 810 0.1598 0.00625 5610 Ditto17 6000 0.0000183 0.109 736 810 0.1618 0.00495 5730 Ditto18 7000 0.0000071 0.05 764 810 0.2096 0.00407 6430 Ditto19 7000 0.0000142 0.1 734 810 0.1692 0.00745 6479 Ditto20 8000 0.0000062 0.05 764 810 0.1755 0.00985 6425 Ditto21 8000 0.0000125 0.1 718 814 0.1802 0.00286 7772 Ditto22 12 000 0.0000075 0.09 772 812 0.1737 0.0025 11754 Ditto23 8000 0.00000375 0.03 780 820 0.1526 0.0025 7662 Ditto24 8000 0.00000875 0.07 744 814 0.1733 0.0025 7801 Ditto25 5000 0.0000043 0.0215 812 820 0.1462 0.0025 23 (25)26 5000 0.00000043 0.0025 810 824 0.1511 0.0025 34 Ditto27 8000 0.00000002 0.0025 810 826 0.1538 0.0025 678 Ditto28 8000 0.0000025 0.00458 806 822 0.1604 0.00609 507 Ditto29 8000 0.00000312 0.00572 806 822 0.1677 0.00452 1757 Ditto

(2) The minimal amplitude AP1min (column 9) is approximately equal to 0.0025 (0.25%). Op-timization does not take place for lesser values of AP1min and the number of the last optimizationNend (column 10) is less than the given number of optimizations Nopt. In the course of learning, aregion with a constant AP1min must be introduced in the law of variation of AP1. This increasesthe probability of obtaining a greater Fmax.

(3) The optimization process invariably worsens if the segment BC (Fig. 5) vanishes.(4) The initial value Fbeg (column 5) of the aim function must not be reduced, because a low

value invariably produces a low final value for Fmax (Fig. 6) due to the unsatisfactory trajectory ofthe optimization process. In our case, we can take Fbeg = 750÷ 760.

Therefore, instead of (16), logical-probabilistic risk models can be taught by the formula

If AP1 < AP1min, then dP1jr = AP1min ×K3,

If AP1 > AP1min, then dP1jr = K1 × (Nopt −Nt)×K3.(21)

Results of optimization by formula (21) for AP1min = 0.0025 (0.25%), different AP1max = 0.098,0.09, 0.03 (9.8%, 9%, 3%), a sufficiently large number of optimizations Nopt = 5000–12 000 and ahigh Fbeg = 745 are shown in Table 1 (variants 22–24). High values of Fmax = 812–822 wereobtained for all variants.

Note that the probabilities P1jr depend on the number of gradations in the group of incompatibleevents, frequency Wjr of gradations in objects, and contributions of gradations to recognition error.



Fig. 4. Aim function Fmax versus number of optimizations Nopt.

AP

1

AP

1

max

0

N

opt

AP

1

min

CD

B

Fig. 5. Graph of variation of increment amplitude AP1.

700 720

790

800

740 760 780680

F

max

810

820

780

F

beg

Fig. 6. Aim function Fmax versus its initial value.

In formula (16), the increment amplitudes AP1jr are taken identical for all gradations, irrespectiveof their probabilities P1jr.

Let us modify this formula such that the probability of each gradation is taken into account

dP1jr = K1 × (Nopt −Nt)×K3)× P1jr, r = 1, Nj , j = 1, n. (22)

Here every gradation has its own amplitude

AP1jr = K1 × (Nopt −Nt)× P1jr (23)

and formula (22) can be written as

AP1jr = AP1jr ×K3. (24)

Let us also modify formula (22) as

dP1jr = K1 × (Nopt −Nt)× ((1 − a) + a× P1jr)×K3, (25)



Table 2. Value of Fmax/APc for a small number of optimizations Nopt and a = 1

Number of optimizations, Nopt KI = 0.00033 KI = 0.00025 KI = 0.00015 KI = 0.0001

600 798/0.248 796/0.225 810/0.180 810/0.149450 802/0.217 804/0.187 814/0.162 819/0.161300 810/0.146 810/0.174 816/0.147 820/0.162225 810/0.154 811/0.152 818/0.148 821/0.146150 816/0.145 820/0.156 822/0.148 822/0.147100 818/0.146 820/0.149 820/0.151 820/0.15350 822/0.151 820/0.146 820/0.152 820/0.148

Table 3. Value of Fmax for variable values of a

Value of a 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 1.0Value of Fmax 802 800 798 804 808 810 808 810 818 820

in which the coefficient a in the interval [0 < a < 1] defines formula (16) for a = 0, formula (22)for a = 1 and all modifications for intermediate values of a.

Incorporating the constraints introduced for formula (21) into formula (25), we obtain the fol-lowing expression for teaching logical-probabilistic risk models:

If AP1jr < AP1min, then dP1jr = AP1min,If AP1jr > AP1min, then dP1jr = K1 × (Nopt −Nt)× ((1− a) + a× P1jr)×K3.

(26)

The results of optimization by formula (26) for a = 1 (AP1max = 2.15%, 0.25%, 0.45%, 0.57%)listed in Table 1 (variants 25–29) show that the aim function takes high values Fmax = 822–826for a restricted number of optimizations Nend (column 10). At the first optimization, we obtainFbeg = 806–810. The optimization process terminates at Nend = 23–1750, but not at the predefinednumber of optimizations Nopt = 5000–8000 (column 6). It seems that the number of optimizationsNopt is rather undervalued. Additional experiments were conducted to verify this assumption.

Using Nopt = 600, 450, 300, 150, 100, 50, and K1 = 0.00033, 0.00025, 0.00015, and 0.0001, weconducted our experiments. The maximal increment amplitude AP1max was varied in the interval0.5–20% of P1jr. Table 2 shows the values of Fmax and difference between the maximal and minimalrisks of objects in the statistic APc. The results are good (Fmax = 810–822) and corroborate theeffectiveness of formulas (22), (25), and (26).

We also studied the influence of the coefficient a on optimization for a small number of opti-mizations Nopt = 150, K1 = 0.00015, and maximal increment amplitude AP1max = 0.0225× P1jr.

The results shown in Table 3 also corroborate the effectiveness of formulas (22), (25) and (26) fora = 1. Indeed, the aim function is equal to Fmax = 820 and 802 for a = 1 and a = 0, respectively.

5.2. Amplitude AP1max and Global Extremum Fmax

Let us once again examine the choice of the maximal amplitude AP1max for probability incre-ments. The variations of Fmax with the amplitude AP1max = K1 × Nopt in the interval 0.5–20%of P1jr listed in Table 2 show that the larger the amplitude AP1max, the lesser the value of Fmax.Figure 7 shows the dynamics and optimization results for five variants with Nopt = 2000.

Variant 1 AP1max = 0.05(5%) Formula (16) Fmax = 808 Variant 8, Table 1Variant 2 AP1max = 0.1(10%) Formula (16) Fmax = 798 Variant 9, Table 1Variant 3 AP1max = 0.05(5%) Formula (26), a = 1 Fmax = 820Variant 4 AP1max = 0.1(10%) Formula (26), a = 1 Fmax = 804Variant 5 AP1max = 0.2(20%) Formula (26) Fmax = 786 Variant 1, Table 1



Fig. 7. Dynamics and optimization results as a function of the parameter AP1max.

Fig. 8. Fmax versus APc.

The learning dynamics and results for variants 4 and 5 with high AP1max, despite the applicationof the effective formula (26) for a = 1, are bad. Their aim functions are equal to 786 and 804,respectively. The optimization process terminates rather early at (Nend = 1608 and Nend = 20).The aim function did not increase in other optimizations (Nopt −Nend). This example shows thatthe amplitude AP1max must not be taken greater than 0.02–0.05 (2–5%). The global extremumFmax was determined from the graph of variation of Fmax for the function of difference between themaximal and minimal objects in the statistics APc (Fig. 8). This graph, according to [5, 6], has anextremum. The graph plotted for different variants (Table 1 and Table 2) shows that solutions arerobust (stabile) if the scatter of APc near the global extremum Fmax is small.

6. CONCLUSIONS

(1) An effective method for determining the global extremum of the aim function in identify-ing logical-probabilistic risk models is elaborated. It is helpful in solving many-parameter multi-extremal optimization problems with integer aim function within reasonable computation time.



(2) The random number K3 in the learning formula must be generated in the interval [−1,+1]in order to express the absolute increments dP1jr in percents and estimate the accuracy of theprobabilities P1jr.

(3) The method is based on the use of the following behavior of variation of the aim function:The aim function asymptotically increases with the number of optimizations Nopt.The minimal admissible amplitude AP1min of increments of probabilities P1jr is determined by

two or three computations; for the values of AP1min less than 0.25%, optimization does not takeplace.

—The initial aim function Fbeg must not be undervalued since low values invariably yield lowfinal values for Fmax due to the unsatisfactory optimization trajectory.

—The maximal amplitude of increments of the probabilities AP1max must not be taken greaterthan 0.02–0.05 (2–5%), since learning dynamics worsens and the aim function Fmax is undervalued.

(4) New effective learning formulas (22), (25), and (26) designed for determining the globalextremum of the aim function Fmax use different increment amplitudes for different gradations.

(5) The determination of the global extremum of the aim function must be verified with thehelp of the graph of variation of Fmax in the function of the difference between the maximal andminimal risks of objects in the statistic APc.

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This paper was recommended for publication by A.I. Kibzun, a member of the Editorial Board


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