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Center of training and analysis in risk's engineering

International Journal of Risk Theory

Vol 6 (no.2)

Alexandru Myller

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Iaşi, 2016

Center of training and analysis in risk's engineering

International Journal of Risk Theory

ISSN: 2248 – 1672

ISSN-L: 2248 – 1672

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Giuseppe D'ASCENZO, "La Sapienza" University, Roma

Gabriel Dan CACUCI, University of Karlsruhe, Germany

Ovidiu CÂRJĂ, "Al.I. Cuza" University, Iaşi

Ennio CORTELLINI, CeFAIR, "Al.I.Cuza" University, Iaşi

Marcelo CRUZ, New York University

Maurizio CUMO, National Academy of Sciences, Italy

Franco EUGENI, University of Teramo, Italy

Alexandra HOROBET, The Bucharest Academy of Economic Studies

Ovidiu Gabriel IANCU, "Al.I.Cuza" University, Iaşi

Vasile ISAN, "Al.I.Cuza" University, Iaşi

Dumitru LUCA, "Al.I.Cuza" University, Iaşi

Henri LUCHIAN, "Al.I.Cuza" University, Iaşi

Christos G. MASSOUROS, TEI Chalkis, Greece

Antonio NAVIGLIO, "La Sapienza" University, Roma

Gheorghe POPA , "Al.I. Cuza" University, Iaşi

Vasile PREDA, University of Bucharest, Romania

Aniello RUSSO SPENA, University of Aquila, Italy

Dănuţ RUSU, CeFAIR, "Al.I. Cuza" University, Iaşi

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Content

Mathematics and Informatics for Risk Theory

V. Preda, I. Băncescu1, M. Drăgulin, M.-C. Țuculan (Diaconu)

Compound Generalized Lindley distributions: Poisson, Binomial and Geometric

type

1

V. Cornaciu

Some optimality necessary conditions for optimization problems based on

Pseudo-Avriel-Ben-Tal algebraic operations

19

Author Guidelines 37

International Journal of Risk Theory, Vol 6 (no.2), 2016 1

Compound Generalized Lindley distributions: Poisson,Binomial and Geometric type

Vasile Preda,1∗ Irina Bancescu,2 Mircea Dragulin3,Tuculan (Diaconu) Maria-Crina4

1 Faculty of Mathematics and Computer Science, University of Bucharest,2 Doctoral School of Mathematics, University of Bucharest3 Doctoral School of Mathematics, University of Bucharest4 Doctoral School of Mathematics, University of Bucharest

∗ E-mail: [email protected]

Abstract

Three new extentions of well-known families of distributions are proposed. This extentions areobtained by compounding a new generalized Lindley distribution with the discret distributions: Pois-son, binomial and geometric. By doing so, we obtain more flexible distributions in respect to thehazard rate shape which can be increasing, decreasing or unimodal. Some characteristics and prop-erties are discussed.

1 Introduction

The compounding of different distributions has led to the extension of many well-known families of dis-tributions obtaining more flexible distributions for modeling lifetime data. In recent years, the study ofthe Lindley distribution [15, 16] has increased due to the necessity of finding a more suitaible distributionfor lifetime data analysis. With that in mind Ghitany et al. [11] showed that the Lindley distribution isa better model and has more flexible mathematical properties than those of the exponential distributionwhich is frequently used as a lifetime distribution. One disadvantage concerning the exponential dis-tribution is the memory loss property. The Lindley distribution has a increasing failure rate while theexponential has a constant one. The generalization proposed by M. Dragulin [8] is a better fit for model-ing lifetime data. The generalized Lindley does not only reduceses to the Lindley distribution, but also tothe exponential and gamma distribution, being a mixture of this last two. This gives us a great advantagein modeling data, an advantage that the Lindley distribution does not have, being only a mixture of theexponential and gamma distribution.

Compounding of the generalized Lindley distribution with the Poisson, binomial and geometric dis-tributions, giving the fact that it is a better distribution than the Lindley distribution and therefore betterthan the exponential one [11], one should expect that this new derived distributions provide a betterfit. Another reason for the compounding of the generalized Lindley distribution is the flexibility of thehazard rate function, an important tool for the study of a system’s reliability. Most real systems have


a increasing/ decreasing or most often a unimodal/ bath-tub hazard rate, but not a constant hazard rate.The distributions presented in this paper have different failure rate shapes.

The contents of this paper are organized as follows. In Section 1 we present the generalized Lindleydistribution with its hazard function. In the next sections 2-4 we introduce the extented compoundingdistribution: type Poisson, type binomial and type geometric and some characteristics and properties. InSection 5 are dedicated to conclusions.

Let T∼ GL(α, θ) be a random variable of generalized Lindley distribution type [8]. Then the proba-bility density function (pdf) of the generalized Lindley distribution is

fT(x) =θ2

αθ + 1e−θx(α + x), x > 0, α > 0, θ > 0

The corresponding cumulative distribution function (cdf) is

FT(x) = 1 −αθ + 1 + θxαθ + 1

e−θx x > 0, α, θ > 0

Figure 1: The probability density and the cumulative function of the generalized Lindley distribution

The failure rate is h(x) =θ2(α + x)αθ + 1 + θx

which is an increasing failure rate (IFR). In terms of reliabil-ity, this is associated with a system whose lifetime decreases as the time passes.

In reliability, the applicability of the compounding of distributions can be easily seen. If we con-sider a serie system with K components and X1,X2, ...,XK the lifetimes of the components then X =min(X1,X2, ...,XK) denotes the lifetime of the system. K can be view as a random variable because asystem’s lifetime can be reduced by the intervention of man, by the surroundings, by weather (in the caseof systems operating outside) etc. X = min(X1,X2, ...,XK) also denotes the time of first failure whichmany distributions can not model; this being a subject of interest for statisticians. Similar, a paralelsystem’s lifetime is denoted by X = max(X1,X2, ...,XK). In the following section we present three newfamily distributions. Some characteristics and properties are discussed.

Definition 1. Let(H(x,Y)

)be a family of probability distributions where Y has the cumulative distribu-

tion function G(y). Then the cumulative distribution function F(x) is a continuous mixture of the family(H(x,Y)

)with the cumulative distribution function of Y if

F(x) =

∫Dom(Y)

H(x, y)dG(y) (1)


In terms of densities, the formulae is

f (x) =

∫Dom(Y)

h(x, y)g(y)dy (2)

Let W be a random variable with a cumulative distribution function G(x) and probability densityfunction g(x). Let also W1,W2, ...,Wn be n random variables independent, identically distributed withthe same distribution as W.

LetX = min

i=1,nWi, Y = max

i=1,nWi

We have that

GX(x) = P(X ≤ x) = 1 − P(Y > x) = 1 − (1 − G(x))n

gX(x) = (GX(x))′

= ng(x)(1 − G(x))n−1

andGY(x) = P(Y ≤ x) = (G(x))n

gY(x) = (GY(x))′

= ng(x)(G(x))n−1

2 Compound distribution. Poisson type

2.1 Generalized Lindley Poisson Max

Let (Wi)i=1,n be independent, identically, Wi ∼ GL(α, θ),

fGL(α,θ)(x) = e−θx θ2

αθ + 1(α + x), x > 0, α, θ > 0

andFGL(α,θ)(x) = 1 − (1 +

θxαθ + 1

)e−θx, x > 0, α, θ > 0

We have that n is a random variable, zero truncated Poisson distributed of parameter λ > 0

P(n = k) =e−λ

1 − e−λλk

k!, k ≥ 1

Let Y = maxi=1,n

Wi. We have

FY(y) =(1 −

αθ + 1 + θxαθ + 1

e−θx)n


and the probability density function is

hn(y) = nθ2e−θy(α + y)

αθ + 1

(1 −

αθ + 1 + θyαθ + 1

e−θy)n−1

Theorem 1. Let X ∼ generalized Lindley Poisson Max(α, θ, λ). Then the pdf of X is

f (x) =λθ2e−θx(x + α)(αθ + 1)(eλ − 1)

eλ[1− αθ+1+θxαθ+1 e−θx]

and the corresponding cdf is

F(x) =eλ[1− αθ+1+θx

αθ+1 e−θx]− 1

eλ − 1, x > 0, α, θ, λ > 0

Figure 2: The probability density and cumulative function of generalized Lindley Poisson Max distribu-tion

Proposition 1. The failure rate function of X ∼ generalizedLindleyPoissonMax(α, θ, λ) is

h(y) =λθ2e−θy(y + α)

αθ + 1eλ[1− αθ+1+θy

αθ+1 e−θy]

eλ − eλ[1− αθ+1+θyαθ+1 e−θy]

2.1.1 Characteristics and some properties

Theorem 2. The rth moment of the generalized Lindley Poisson Max distribution is

E(Xr) =θ2

e−λ − 1

∑k≥1

(−λ)k

(k − 1)!(αθ + 1)k

{θk−1α

(αθ + 1θ

)r+kΓ(r + 1)Ψ(r + 1, r + k + 1, k(αθ + 1))

+ θk−1(αθ + 1

θ

)r+k+1Γ(r + 2)Ψ(r + 2, r + k + 2, k(αθ + 1))

}


Figure 3: The failure rate function of generalized Lindley Poisson Max/Min distribution

Corollary 1. The mean is

E(X) =θ2

e−λ − 1

∑k≥1

(−λ)k

(k − 1)!(αθ + 1)k

{θk−1α

(αθ + 1θ

)k+1Ψ(2, k + 2, k(αθ + 1))

+ θk−1(αθ + 1

θ

)k+22Ψ(3, k + 3, k(αθ + 1))

}Corollary 2. The variance is

Var(X) =θ2

e−λ − 1

∑k≥1

(−λ)k

(k − 1)!(αθ + 1)k

{θk−1α

(αθ + 1θ

)k+22Ψ(3, k + 3, k(αθ + 1))

+ θk−1(αθ + 1

θ

)k+36Ψ(4, k + 4, k(αθ + 1))

}−

{θ2

e−λ − 1

∑k≥1

(−λ)k

(k − 1)!(αθ + 1)k

·

[θk−1α

(αθ + 1θ

)k+1Ψ(2, k + 2, k(αθ + 1)) + θk−1

(αθ + 1θ

)k+22Ψ(3, k + 3, k(αθ + 1))

]}2

where Ψ(a, b; u) = 1Γ(a)

∫∞

0 ta−1(1+t)b−a−1e−utdt denotes the Kummer function [19] and Γ(s) =∫∞

0 xs−1e−xdx

Proposition 2. The Laplace Stieltjes transformation of the generalized Lindley Poisson Max distributionis

ϕ(s) =∑k≥1

λkθ2

(αθ + 1)k

e−λ

1 − e−λ(−θ)k−1

(k − 1)!αk+1Γ(k)Ψ(k, k + 2, α(s + θk))

Proposition 3. The Renyi entropy

JR(γ) =1

1 − γln

{λγθ2γe−γλ

(1 − e−λ)γ(αθ + 1)λ∑k≥1

k−1∑i=0

γ∑j=0

Cik−1C j

γαγ− j(−1)i

·(λγ)k−1θi

(k − 1)!(αθ + 1)i

(αθ + 1θ

)i+ j+1Γ( j + 1)Ψ( j + 1, j + i + 2, (γ + i)(αθ + 1))

}, γ > 0, γ , 1


2.2 Generalized Lindley Poisson Min

Let (Wi)i=1,n be independent, identically distributed, Wi ∼ GL(α, θ), and let the random variable n bezero truncated Poisson distributed of parameter λ > 0.

Let Y = mini=1,n

Wi. We have

FY(x) = 1 −(αθ + 1 + θx)n

(αθ + 1)n e−θnx

Using the following

hn(x) =nθ2(αθ + 1 + θx)n−1

(αθ + 1)n (α + x)e−θnx

we can determin the density function of the generalized Lindley Poisson Min distribution.

Theorem 3. Let X ∼ generalized Lindley Poisson Min(α, θ,n). Then the pdf is

f (x) =λθ2(α + x)αθ + 1

e−θx 1eλ − 1

eλ[αθ + 1 + θx

αθ + 1e−θx

]and the corresponding cdf is

F(x) =1

eλ − 1

[eλ − e

λαθ + 1 + θxαθ + 1

e−θx], x > 0, α, θ, λ > 0

Figure 4: The pdf and cdf of the generalized Lindley Poisson Min

Proposition 4. The hazard function is

h(x) =λθ2(α + x)e−θx

αθ + 1eλ

αθ+1+θxαθ+1 e−θx

eλαθ+1+θxαθ+1 e−θx

− 1



Theorem 4. The rth moment is

E(Xr) =∑k≥1

θ2

(αθ + 1)k

e−λ

1 − e−λλk

(k − 1)!

{αθk−1

(αθ + 1θ

)r+kΓ(r + 1)Ψ(r + 1, r + k + 1, k(αθ + 1))

+ θk−1(αθ + 1

θ

)r+k+1Γ(r + 2)Ψ(r + 2, r + k + 2, k(αθ + 1))

}where Γ(s) =

∫∞

0 xs−1e−xdx, s > 0 denotes the gamma function and Ψ(·, ·, ·) the Kummer function.


E(X) =∑k≥1

θ2

(αθ + 1)k

e−λ

1 − e−λλk

(k − 1)!

{αθk−1

(αθ + 1θ

)1+kΨ(2, k + 2, k(αθ + 1))

+ θk−1(αθ + 1

θ

)k+22Ψ(3, k + 3, k(αθ + 1))

}Corollary 4. The variance is

Var(X) =∑k≥1

θ2

(αθ + 1)k

e−λ

1 − e−λλk

(k − 1)!

{αθk−1

(αθ + 1θ

)2+k2Ψ(3, k + 3, k(αθ + 1))

+ θk−1(αθ + 1

θ

)k+36Ψ(4, k + 4, k(αθ + 1))

}−

{∑k≥1

θ2

(αθ + 1)k

e−λ

1 − e−λλk

(k − 1)!

·

[αθk−1

(αθ + 1θ

)1+kΨ(2, k + 2, k(αθ + 1))

+ θk−1(αθ + 1

θ

)k+22Ψ(3, k + 3, k(αθ + 1))

]}2

Proposition 5. The Laplace Stieltjes transformation of the generalized Lindley Poisson Min is

ϕ(s) =∑k≥1

λk

(αθ + 1)k

e−λ

1 − e−λθk+1

(k − 1)!

{α(αθ + 1

θ

)kΨ(1, k + 1, (s + θk)

αθ + 1θ

)

+(αθ + 1

θ

)k+1Ψ(2, k + 2, (s + θk)

αθ + 1θ

)}

Proposition 6. The Renyi entropy is

JR(γ) =1

1 − γln

{λγθ2γ

(αθ + 1)γ( 1eλ − 1

)γ∑k≥1

γ∑i=0

Ciγα

γ−i (λγ)k−1θk−1

(αθ + 1)k−1(k − 1)!

·

(αθ + 1θ

)i+kΓ(i + 1)Ψ(i + 1, i + k + 1, (αθ + 1)(γ + k − 1))

}, γ > 0, γ , 1

unde Γ(·) este funct,ia gamma, iar Ψ(·, ·, ·) funct,ia Kummer.


3 Compound distribution. Binomial type

3.1 Generalized Lindley Binomial Max

Let (Wi)i=1,k be independent, identically distributed, Wi ∼ GL(α, θ) and Y = maxi=1,k

Wi, and let the random

variable k be zero truncated binomial distributed K ∼ Binomial(n, p), p > 0, q = 1 − p.

Because

FY(x) =[1 −

αθ + 1 + θxαθ + 1

e−θx]k, x > 0, θ, α > 0

and

hk(x) = k[1 −

αθ + 1 + θxαθ + 1

e−θx]k−1θ2(α + x)

αθ + 1e−θx, x > 0, θ, α > 0

we have

Theorem 5. Let X ∼ generalized Lindley Binomial Max(α, θ,n, p). Then the pdf of X is

f (x) =θ2(α + x)αθ + 1

e−θx np1 − qn

[1 − p

αθ + 1 + θxαθ + 1

e−θx]n−1


F(x) =1

1 − qn

{[1 − p

αθ + 1 + θxαθ + 1

e−θx]n− qn

}, x > 0, θ, α, p > 0

Figure 5: The probability density and cumulative function of the generalized Lindley Binomial Max

Proposition 7. The hazard rate function of the generalized Lindley Binomial Max is

h(y) =θ2e−θy(α + y)np

[1 − pαθ+1+θy

αθ+1 e−θy]n−1

(αθ + 1){1 −

[1 − pαθ+1+θy

αθ+1 e−θy]n}


Figure 6: The hazard rate function of the generalized Lindley Binomial Max/Min


Lemma 1. Let H(a, b, c, d, δ, p) =∫∞

0 xc(d + x)[1 − p 1+bd+bx

1+bd e−bx]a−1

e−δxdx, 1 + bd > 0.

Then

H(a, b, c, d, δ, p) =

∞∑i=0

i∑j=0

j+1∑k=0

Cia−1C j

i Ckj+1

(−1)ipi

(1 + bd)i b jd j+1−k Γ(c + k + 1)(bi + j)c+k+1

Theorem 6. The rth moment is

E(Xr) =θ2np

(αθ + 1)(1 − qn)H(n − 1, θ, r, α, θ, p)


E(X) =θ2np

(αθ + 1)(1 − qn)H(n − 1, θ, 1, α, θ, p)

Corollary 6. The variance is

Var(X) =θ2np

(αθ + 1)(1 − qn)H(n − 1, θ, 2, α, θ, p) −

[ θ2np(αθ + 1)(1 − qn)

H(n − 1, θ, 1, α, θ, p)]2

Proposition 8. The Laplace Stieltjes transformation of the generalized Lindley Binomial Max is

ϕ(s) =npθ2

(αθ + 1)(1 − qn)

n−1∑k=0

Ckn−1

(−1)k

(αθ + 1)kθk

{α(αθ + 1

θ

)k+1Ψ(1, k + 2,

αθ + 1θ

[s + θ(k + 1)])

+(αθ + 1

θ

)k+2Ψ(2, k + 3,

αθ + 1θ

[s + θ(k + 1)])}


Proposition 9. The Renyi entropy of the generalized Lindley Binomial Max distribution is

JR(γ) =1

1 − γln

{( np1 − qn

)γ n−1∑k=0

γ∑i=0

Ckn−1Cγi α

γ−i(−1)k (αθ + 1)i+1−γpk

θi+1−2γ

· Γ(i + 1)Ψ(i + 1, i + k + 2, (αθ + 1)(γ + k))}

3.2 Generalized Lindley Binomial Min

Let (Wi)i=1,K be independent, identically distributed, Wi ∼ GL(α, θ) and Y = mini=1,K

Wi, and let the random

variable K be zero truncated binomial distributed, K ∼ Binomial(n, p), p > 0, q = 1 − p.

We have the following

P(K = k) =1

1 − qn Cknpkqn−k, k = 1,n,n ≥ 2

FY(x) = 1 −(αθ + 1 + θx)k

(αθ + 1)ke−θxk

Theorem 7. Let X ∼ generalized Lindley Binomial Min(α, θ,n, p). Then the cdf of X is

F(x) = 1 −1

1 − qn

{[q + p

αθ + 1 + θxαθ + 1

e−θx]n

− qn}

(3)

and the pdf is

f (x) =np

1 − qn

[q + p

θα + 1 + θxαθ + 1

e−θx]n−1

fGL(α,θ)(x), x > 0, α, θ, p > 0 (4)

where fGL(α,θ) is the generalized Lindley density function.

Proposition 10. The failure rate function of X is

h(x) =

np[q + pθα+1+θx

αθ+1 e−θx]n−1

fGL(α,θ)(x)[q + pθα+1+θx

αθ+1 e−θx]n− qn

or

h(x) =(q + pFGL(α,θ)(x))n−1 fGL(α,θ)(x)np

(q + pFGL(α,θ)(x))n − qn

where FGL(α,θ)(x) is the survival function of the generalized Lindley distribution.


Figure 7: The probability density and cumulative function of the generalized Lindley Binomial Mindistribution with n=9


Theorem 8. The rth moment of the generalized Lindley Binomial Min distribution is

E(Xr) =npθ2

(1 − qn)(αθ + 1)

n−1∑k=0

pk qn−1−k

(αθ + 1)kCk

n−1

{αθk

(αθ + 1θ

)r+k+1Γ(r + 1)

·Ψ(r + 1, r + 2 + k, θ(k + 1)αθ + 1θ

) + θk(αθ + 1

θ

)r+2+kΓ(r + 2)Ψ(r + 2, r + 3 + k, θ(k + 1)

αθ + 1θ

)}


E(X) =npθ2

(1 − qn)(αθ + 1)

n−1∑k=0

pk qn−1−k

(αθ + 1)kCk

n−1

[αθk

(αθ + 1θ

)k+2Ψ(2, k + 3, θ(k + 1)

αθ + 1θ

)

+ θk(αθ + 1

θ

)k+32Ψ(3, k + 4, θ(k + 1)

αθ + 1θ

)]


Var(X) =npθ2

(1 − qn)(αθ + 1)

n−1∑k=0

pk qn−1−k

(αθ + 1)kCk

n−1

[αθk

(αθ + 1θ

)k+32Ψ(3, k + 4, θ(k + 1)

αθ + 1θ

)

+ θk(αθ + 1

θ

)k+46Ψ(4, k + 5, θ(k + 1)

αθ + 1θ

)]

−

{npθ2

(1 − qn)(αθ + 1)

n−1∑k=0

pk qn−1−k

(αθ + 1)kCk

n−1

[αθk

(αθ + 1θ

)k+2Ψ(2, k + 3, θ(k + 1)

αθ + 1θ

)

+ θk(αθ + 1

θ

)k+32Ψ(3, k + 4, θ(k + 1)

αθ + 1θ

)]}2


Proposition 11. The Laplace-Stieltjes transformation of the generalized Lindley Binomial Min distribu-tion has the following form

ϕ(s) =npθ2

(1 − qn)(αθ + 1)

n−1∑k=0

pk qn−1−k

(αθ + 1)kCk

n−1

{θkα

(αθ + 1θ

)k+1Ψ(1, k + 2,

αθ + 1θ

[s + θ(k + 1)])

+ θk(αθ + 1

θ

)k+2Ψ(2, k + 3,

αθ + 1θ

[s + θ(k + 1)])}


JR(γ) =1

1 − γln

{( np1 − qn

)γ (n−1)γ∑k=0

γ∑i=0

Ck(n−1)γCi

γ

θ2γ−i−1pkαγ−iq(n−1)γ−k

(αθ + 1)γ−i−1

· Γ(i + 1)Ψ(i + 1, i + k + 1, (k + γ)(αθ + 1))}, γ > 0, γ , 1

4 Compound distribution. Geometric type

4.1 Generalized Lindley Geometric Max

Let (Wi)i=1,K be independent, identically distributed, Wi ∼ GL(α, θ) and Y = maxi=1,K

Wi, and let K be a

random variable zero truncated geometric distributed, K ∼ Geometric(p), 0 < p < 1.


P(K = k) = p(1 − p)k−1, k ≥ 1

FY(x) =(1 −

αθ + 1 + θxαθ + 1

e−θx)k

hk(x) =kθ2e−θx(α + x)

αθ + 1

[1 −

αθ + 1 + θxαθ + 1

e−θx]k−1

Theorem 9. Let X ∼ generalized Lindley Geometric Max. Then the pdf of X is

f (x) =pθ2(α + x)e−θx

αθ + 1

[1 − (1 − p)

(1 −

αθ + 1 + θxαθ + 1

e−θx)]−2


F(x) =p

1 − p

[1

1 − (1 − p)(1 − αθ+1+θx

αθ+1 e−θx) − 1

], x > 0, α, θ, p > 0


Figure 8: The pdf and cdf of the generalized Lindley Geometric Max distribution

Figure 9: The failure rate function of the generalized Lindley Geometric Max/Min

Proposition 13. The failure rate function is

h(x) =pθ2(α + x)αθ + 1 + θx

[1 − (1 − p)

(1 −

αθ + 1 + θxαθ + 1e

−θx)]−1

Proposition 14. The failure rate function h(x) is IFR.


Theorem 10. The rth moment of the generalized Lindley Geometric Max distribution is

E(Xr) =∑k≥1

kpθ2(1 − p)k−1

(αθ + 1)k(−θ)k−1αk+r+1Γ(k + r)Ψ(k + r, k + r + 2, αkθ)


E(X) =∑k≥1

kpθ2(1 − p)k−1

(αθ + 1)k(−θ)k−1αk+2Γ(k + 1)Ψ(k + 1, k + 3, αkθ)



Var(X) =∑k≥1

kpθ2(1 − p)k−1


−

[∑k≥1

kpθ2(1 − p)k−1


]2

Proposition 15. The Laplace Stieltjes transformation of the generalized Lindley Geometric Max distri-bution is

ϕ(s) =∑k≥1

kpθ2(1 − p)k−1(−θ)k−1

(αθ + 1)kαk+1Γ(k)Ψ(k, k + 2, α(θk + s))


JR(γ) =1

1 − γln

{ ∞∑k=0

k∑j=0

j∑i=0

CikCi

j(−1) jαi+γ+1 θi+2γpγ

(αθ + 1)i+γ

Γ(2γ + k)Γ(2γ)k!

(1 − p)k

· Γ(i + 1)Ψ(i + 1, i + γ + 2, αθ(γ + j))}, γ > 0, γ , 1

where Γ(·) denotes the gamma function and Ψ(·, ·, ·) the Kummer function.

4.2 Generalized Lindley Geometric Min

Let (Wi)i=1,K be independent, identically distributed, Wi ∼ GL(α, θ) and Y = mini=1,K

Wi, and the random

variable K is zero truncated geometric distributed, K ∼ Geometric(p), 0 < p < 1.


P(K = k) = p(1 − p)k−1, k ≥ 1

Fmini=1,k

Wi(x) = 1 −(αθ + 1 + θx

αθ + 1e−θx

)k

hk(x) =kθ2e−θx(α + x)

αθ + 1

[αθ + 1 + θxαθ + 1

e−θx]k−1

Theorem 11. Let X ∼ generalized Lindley Geometric Min. Then the pdf is

f (x) =pθ2e−θx

αθ + 1(α + x)

[1 − (1 − p)

αθ + 1 + θxαθ + 1

e−θx]−2


F(x) =1

1 − p

[1 −

p

1 − (1 − p)αθ+1+θxαθ+1 e−θx

], x > 0, α, θ, 0 < p < 1


Figure 10: The pdf and the cdf of the generalized Lindley Geometric Min distribution

Proposition 17. The failure rate function of X is

h(x) =θ2(α + x)(αθ + 1)

(αθ + 1 + θx)2

[1 − (1 − p)

αθ + 1 + θxαθ + 1

e−θx]−1


Theorem 12. The rth moment of the generalized Lindley Geometric Min distribution is

E(Xr) =∑k≥1

kp(1 − p)k−1θk−1

(αθ + 1)k

{α(αθ + 1

θ

)r+kΓ(r + 1)Ψ(r + 1, r + k + 1, k(αθ + 1))

+(αθ + 1

θ

)r+k+1Γ(r + 2)Ψ(r + 2, r + k + 2, k(αθ + 1))

}Corollary 11. The mean is

E(X) =∑k≥1

kp(1 − p)k−1θk−1

(αθ + 1)k

[α(αθ + 1

θ

)k+1Ψ(2, k + 2, k(αθ + 1))

+(αθ + 1

θ

)k+22Ψ(3, k + 3, k(αθ + 1))

]



Var(X) =∑k≥1

kp(1 − p)k−1θk−1

(αθ + 1)k

[α(αθ + 1

θ

)k+22Ψ(3, k + 3, k(αθ + 1))

+(αθ + 1

θ

)k+36Ψ(4, k + 4, k(αθ + 1))

]−

{∑k≥1

kp(1 − p)k−1θk−1

(αθ + 1)k

[α(αθ + 1

θ

)k+1Ψ(2, k + 2, k(αθ + 1))

+(αθ + 1

θ

)k+22Ψ(3, k + 3, k(αθ + 1))

]}2

Proposition 18. The Laplace Stieltjes transformation is

ϕ(s) =∑k≥1

kp(1 − p)k−1

(αθ + 1)kθk−1

[α(αθ + 1

θ

)kΨ(1, k + 1, (s + θk)

αθ + 1θ

)

+(αθ + 1

θ

)k+1Ψ(2, k + 2, (s + θk)

αθ + 1θ

)]

Proposition 19. The Renyi entropy of the generalized Lindley Geometric Min is

JR(γ) =1

1 − γln

{ ∞∑k=0

k∑i=0

CikΓ(2γ + k)Γ(2γ)k!

(1 − p)kpγθ2γ+i

(αθ + 1)i+γ αi+γ+1Γ(i + 1)Ψ(i + 1, i + γ + 2, αθ(γ + k))},

γ > 0, γ , 1

5 Conclusions

The compounding of the generalized Linldey distribution with the Poisson, binomial and geometricgives us some interesting new distribution. This new distributions can be applied in many areas suchas reliability, medicine, insurance and economics. The failure rate shapes are increasing, decreasing,unimodal, and so they are suitable for modeling real life systems.

References

[1] Barlow R.E, Proschan F., Hunter L.C., Mathematical Theory of Reliability. John Wiley & Sons, Inc.,New York, 1965

[2] Barreto-Souza W., A. L. de Morais, s, i G. M. Cordeiro, The Weibull-geometric distribution. J. Statist.Comput. Simul. 81, 645-657, 2011


[3] Barreto-Souza W., s, i H. S. Bakouch, A new lifetime model with decreasing failure rate. Statistics 47,465-476, 2013

[4] Cos, cun Kus, , A new lifetime distribution. Computational Statistics and Data Analysis 51, issue 9, p.4497-4509, 2007

[5] Dominique Lord, Srinivas Reddy Geedipally, The Negative Binomial Lindley Distribution as a Toolfor Analyzing Crash Data Characterized by a Large Amount of Zeros. Accident Analysis & Preven-tion, Elsevier, Volume 43, 1738-1742, 2011

[6] Dragulin M., Preda V., A new family of Lindley-type distributions with applications. Conference onApplied and Industrial Mathematics (CAIM), Bacau, 18-21 septembrie 2014

[7] Dragulin M., Trandafir R., About some Extensions of Lindley Distribution. A 17-a Conferint, a asocietat,ii de probabilitat,i s, i statistica din Romania, Universitatea Tehnica de Construct,ii Bucures, ti,25 aprilie 2014

[8] Dragulin M., Generalized Lindley distribution and Its Power Transformation. International Journalof Risk Theory, Vol 4(no.2), Alexandru Myller Publishing, Ias, i, 2014

[9] Dragulin M., Asupra unor clase de modele statistice de fiabilitate. A 16-a Conferint, a a Societat,ii deProbabilitat,i s, i Statistica din Romania, Bucures, ti, 26 aprilie 2013

[10] Ghitany M. E., D. K. Al-Mutairi, s, i S. Nadarajah, Zero-truncated Poisson-Lindley distribution andits application. Math.Comput. Simul 79 279-287 2008

[11] Ghitany M.E., B. Atieh, S. Nadarajah, Lindley distribution and its application. Mathematics andComputers in Simulation, Elsevier, 78, 493-506, 2008

[12] Gleser, R. E., Bathtub and related failure rate characterizations. Journal of the American StatisticalAssociation, 75, 667-672, 1980

[13] Hassein Zamani s, i Noriszura Ismail, Negative Binomial Lindley Distribution and Its Application.Journal of Mathematics and Statistics, 6 (1), 4-9, ISSN 1549-3644 2010

[14] Jodra P. Computer generation of random variables with Lindley or Poisson-Lindley distribution viathe Lambert W function. Mathematics and Computers in Simulation, vol. 81, Issue 4, pp 851-8592010

[15] Lindley D.V., Fiducial distributions and Bayes theorem. Journal of the Royal Statistical Society,Series B 20, 102-107, 1958

[16] Lindley D.V., Introduction to Probability and Statistics from a Bayesian Viewpoint, Part II: Infer-ence. Cambridge University Press, New York, 1965

[17] Preda, V., Ciumara, R., The Weibull-Logarithmic distribution in lifetime analysis and its propertiesProceedings of the XIII International Conference on Applied Stochastic Models and Data Analysis,56-61, 2009

[18] Preda, V., Panaitescu, E., Ciumara, R., The modified exponential-Poisson distribution. Proceedingsof the Romanian Academy, 12, 1, 22-29, 2011


[19] Prudnikov A.P., Brychkov Yu.A., Marichev O.I., Integrals and series, Volume 1, Elementary Func-tions. Overseas Publishers Association OPA, 1986

[20] Sankaran M., The discrete Poisson-Lindley distribution. Biometrics, 26, 145-149, 1970

[21] Saralees Nadarajah, Hassan S. Bakouch, Rasool Tahmasbi, A generalized Lindley distribution. In-dian Statistical Institute, 2012


Some optimality necessary conditions for optimization problems based on

Pseudo-Avriel-Ben-Tal algebraic operations

Cornaciu Veronica Department of Informatics, Titu Maiorescu University, Bucharest; Romania

E-mail: [email protected]

Abstract

In this paper we introduce new generalized pseudo-operations with one parameter

of the following form: 1x y h h x h y ,where h is an n vector-valued

continuous function, defined on a subset H of Rn and possessing an inverse function h–1,

is a arbitrary but fixed positive real number. Five kinds of cones are introduced,

which are used to establish the constraints qualifications. The generalized Karush-

Kuhn-Tucker optimality necessary conditions are developed for a class of generalized

( , )h -differentiable single-objective programming problems by using this generalized

pseudo-operations, an extension of Avriel-Ben-Tal algebraic operations. The results

obtained in this paper generalize and extend previous results obtained in this field.

Keywords. Cones, ( , )h -differentiable functions, Karush-Kuhn-Tucker necessary

conditions, constraint qualifications, optimal solutions, effcient solutions.

1. INTRODUCTION

In mathematical programming involving differentiable functions, the Kuhn-Tucker conditions provide

necessary conditions for an optimum, given certain qualifications on the constraints. A problem that

continues to evoke very substantial interest is that of finding sufficient conditions for an optimum.

Many authors studied optimality conditions for vector optimization problems involving constraints are

defined by single-valued mappings and obtained optimality conditions in terms of Lagrange-Kuhn-

Tucker multipliers [3,4,6,8,10-17]. Some pseudo algebraic operations with applications can be found

in [9]. Also, new operations on the set of triangular fuzzy numbers are investigated and the derived

algebraic structures, based on the proposed arithmetic operations, are studied [7,20,21].

Optimality conditions for various optimization problems are ever more, in particular, optimality

sufficient conditions for a class (h, φ)-differentiable optimization problems [1,8,19,22,23].

The main aim of this paper is to study some optimality necessary conditions for optimization

problems based on pseudo-Avriel-Ben-Tal algebraic operations. We introduced new generalized

pseudo-operations with one parameter of the following form: 1x y h h x h y ,where h

is an n vector-valued continuous function, defined on a subset H of Rn and possessing an inverse

function h–1 , is a arbitrary but fixed positive real number. In order to establish the constraints


qualifications, five kinds of cones are introduced. The generalized Karush-Kuhn-Tucker optimality

necessary conditions are derived for a class of generalized ( , )h -differentiable single-objective

programming problems by using this generalized pseudo-operations, an extension of Avriel-Ben-Tal

algebraic operations. The results obtained in this paper generalize and extend the previously known

results in this area [2,5,17].

The paper is organized as follows. Section 2 contains preliminaries and related results that will be used

to obtain the main results of the paper. In Section 3 we introduce a new pseudo-operator on Rn and we

define the concept of differentiable function, relative to the introduced operators. Constraint

qualifications for single-objective problem are obtained in Section 4. Kuhn-Tucker necessary

conditions for ( , )h -differentiable single-objective programming optimization problems are derived

in Section 5.

Throughout the paper, we denote by the set of real numbers and denote by kthe collection of k-

dimensional real vectors, and write

1 2

, ,..., | 0, 1, 2,...,T

k k ix x x x i k

;

0

1 2, ,..., | 0, 1, 2,...,

T

k k ix x x x i k

, there exists a least an 0

0i

x ;

1 2

, ,..., | 0, 1, 2,...,T

k k ix x x x i k

;

2. PRELIMINARIES

1) Let h be an n vector-valued continuous function, defined on a subset H of Rn and

possessing an inverse function h–1. Define the h-scalar multiplication of x H and

as

1y h h x

.

2) Let φ be a real-valued continuous functions, defined on and possessing an inverse

functions 1 .Then the φ-addition of two numbers,

and , is given by

1 ,

and the φ-scalar multiplication of and by

1 .

3) The (h, φ)-inner product of vectors ,x y H is defined as


1

,

TT

hx y h x h y

.

Denote

.

1 21

... , , 1, 2,...,m

i m ii

i m

;

1 .

By Ben-Tal generalized algebraic operation, it is easy to obtain the following conclusions:

1

1 1

m m

i ii i

(2.1)

h x h x

(2.2)

Lemma 2.1 [22] Suppose : is a continuous one-to-one strictly monotone and onto function,

and , . Then

if and only if 0 ,

where 1

0 (0) .

3. A GENERALIZED PSEUDO-OPERATION. SOME LEMAS

We introduce a new pseudo-operation of addition.

Let be arbitrary but fixed positive real number. Let h be an n vector-valued continuous

function, defined on a subset H of Rn and possessing an inverse function h–1. Define the

left - h-vector addition of x H and y H as

1x y h h x h y ,


Denote

1 2

1

... , , 1, 2,...,m

i m i

i

x x x x x H i m

.

It is easy to obtain the following conclusion:

1 1

11

m mi m i

ii

x h h x

(3.1)

Lemma 3.1 The following statements hold:

(i) 1

111

, , , 1,m m

i m i i

i i iii

x h h x x H i m

,

(ii) , , , x x x x ,

particularly, x x x ,

(iii) 1x d h h x h d

,

(iv) 1

1 1

m m

i i i ii i

, ,i i

, for i = 1,2,...,m,

(v) x x , , , x H ,

(vi) x y x y and, in general 1 1

m m

i ii i

x x

Proof. We only prove (i). One can similarly obtain (ii) – (v).

Proof of (i)

From (3.1) we have:

1

1 1

m mi m i i

i ii i

x h h x

(3.2)

From (2.2) we have:

i i

i ih x h x

(3.3)

That, along with (3.2) we obtain:

1

1 1

m mi m i i

i ii i

x h h x


We introduce the following concept, which plays an important role in this article.

Definition 3.1 Let f be a real-valued function defined onn, denote 1ˆ ( )f t f h t

. For

simplicity, write 1ˆ ( )f t fh t . The function f is said to be ( , )h -differentiable at x, if ˆ ( )f t

is

differentiable at t h x . Denote * 1

( )

ˆt h x

f x h f t

.

In addition, f is differentiable onn, if and only if it is ( , )h -differentiable at x, where h(t) = t.

Lemma 3.2 The following assertions hold.

(i) Suppose f is ( , )h -differentiable at x0, k . Then

* 0 * 0k f x k f x .

(ii) Let fi for i = 1,2,...,p be ( , )h -differentiable at x0. Then

* 0 * 0

11

1p p

i ip iii

f x f x

.

(iii) Assume f is ( , )h -differentiable at x0, 0

( ) ( )[ ] ( )c x f x f x .Then

* 0 * 0c x f x .

Proof.

(ii). Let 1

( )p

ii

g x f x

. Then

1

1 1

( ) ( )p p

i ii i

g x f x f x

.

Writing 1( )x h t

, we get

1 1

1

( )p

ii

gh t f h t

.

By hypotheses, we conclude that 1

if h

for i = 1, 2, ..., p are differentiable at 0

0t h x hence

1 1

0 01

p

ii

gh t f h t

.

1 0 1 0

1

p

ii

gh h x f h h x


1 1 0 1 1 0

1

p

ii

hh gh h x hh f h h x

.

1 1 0

01

ˆp

ii

hh g x hh f x

* * 0

01

p

ii

h g x h f x

Thus, applying h–1 in equality , we get

* 1 * 0

01

p

ii

g x h h f x

.

Which, together whith (3.1) we get:

* * 0

01

1m

ip ii

g x f x

.

By Lemma 3.2 (i) and (ii), it is easy to obtain the following theorem, which characterizes the

generalized linearity of ( , )h -differentiable operations.

Theorem 3.1 Suppose fi for i = 1, 2, ..., p are ( , )h -differentiable at ,p

x and

1

p

i ii

g x f x

.Then

* *1

1

m

ip ii

g x f x

.

In the rest of the paper, we further assume :n

h is a continuous one-to-one and onto function.

Similarly, suppose : is a continuous one-to-one strictly monotone and onto function.

In the next section, we consider the constraint qualifications for single-objective programming

problems with both inequality and equality constraints.

4. CONSTRAINT QUALIFICATIONS

Consider the following program:


( ) min ( )

s.t. ( ) 0 for 1, 2,..., ,

0 for 1, 2,..., ,

i

j

HFP f x

g x i m

h x j l

.

where 1

0 (0) .

Throughout the remainder of this article let

for 1, 2, ..., , ( )| 0 0 for 1, 2,...,jn i

i m h xX lg jx x

denote the feasible region of problem (HFP), and let

( ) | ( ) 0 , 1, 2,...,i

I x i g x i m

Denote the set of generalized binding constraints, where x X .

Definition 4.1. Let gi for i = 1, 2, ..., m and hj for j = 1, 2, ..., l be ( , )h -differentiable on

,n

x X . The ( , )h -cone of local constraint directions of X at x is defined by

1 *

,.

, | 0T

h n ih

Z X x d d g x

for i I x and *

.0

T

ih

d h x for

1,2,...,j l .

Each nonzero vector 1

,,

hd X x is called an ( , )h -local constraint direction.

Similarly to the definition of the cone of feasible direction of S at x0 ([3, p.127]), we introduce the

following concept.

Definition 4.2. Let S be a nonempty set in n, and

0cl x S . The ( , )h -cone of feasible directions

of S ar x0, denoted by 0

,,

hD S x , is given by

0 0

,, | , 0, , 0

hD S x d x d S .

Every nonzero vector 0

,,

hd D S x is said to be an ( , )h -feasible direction.

Remark 4.1 Each feasible direction of S at x0 is an ( , )h -feasible direction of S at 0x

, where

( ) , h x x x . However, the converse is not true, as is shown in the following.

Example 4.1 Let 3

1 2 2 1 1 2, | , ,

TS x x x x x x , 0

0,0T

x , 31 2 1 2, ,

TTh x x x x ,

1, 1T

d . Then, 1 2 2 1 1 2, | , ,

Th S x x x x x x , 0

0,0T

h x , h d d . We can

verify 1, 1T

d is an ( , )h -feasible direction of S as 0x

, but it is not a feasible direction of S


as x0.

The relationship between the two cones defined above is characterized in the form of the following

lemma:

Lema 4.1. Let gi for i = 1, 2, ..., m and hj for j = 1, 2, ..., l be ( , )h -differentiable on n and

x X , with 1h h x X

.

Then

1

, ,, ,

h hD X x Z X x .

Proof. Without loss of generality, we suppose φ is strictly monotone decreasing on . Let

,,

hd D X x . We need to show 1

,,

hd Z X x , one deduces that at least one of the two cases

holds:

Case 1. There exists a k I x

such that *

,0

T

kh

d g x .

Case 2. There exists a 1,2,...,j l

such that *

,0

T

jh

d h x .

For Case 1, the inequality *

,0

T

kh

d g x gives

1 * 1(0)

T

kh d h g x

,

which along with the strictly monotone decrease of φ leads to

*

0T

kh d h g x

(4.1)

Since gk is ( , )h -differentiable at x , hence1ˆ ( ) ( )

k kg t g h t

is differentiable at ( )t h x , thus

ˆ ˆ ˆ( )

T

k k k kg t h d g t h d g t

(4.2)

where 0k as 0

.

It follows from (4.2) that

1 1 *T

k k k kg h h x h d g h h x h d h g x

(4.3)

On the other hand, by 1h h x X

, we get


1

0 0k

g h h x

(4.4)

Substituting (4.4) into (4.3), one deduces that

1 *T

k k kg h h x h d h d h g x

.

Hence

1

*Tk

k k

g h h x h dh d h g x

(4.5)

Since 0k as 0

, it follows from (4.1) and (4.5) that

10

kg h h x h d

for a sufficiently small positive scalar θ.

By Lemma 3.1 (iii), we obtain

0k

g x d for above θ.

Since φ is strictly monotone decreasing, one derives that

10 0

kg x d


This contradicts ,,

hd D X x .

Therefore, Case 1 does not hold.

For Case 2, without loss of generality, we assume *0

T

jd h x , which gives us:

1 * 10

T

jh d h h x

which together with leads to:

*

0T

jh d h h x

(4.1’)

How hj is ( , )h -differentiable at x , therefore 1ˆj j

h t h h t is differentiable at

( )t h x , we get:

ˆ ˆ ˆT

j j j jh t h d h t h d h t

(4.2’)

were 0j

when 0 .


From (4.2’) we get:

1 1 *T

j j j jh h h x h d h h h x h d h h x

(4.3’)

On the other hand, from 1h h x X

, we get

1

0 0j

h h h x

(4.4’)

Replacing (4.4’) in (4.3’), we conclude:

1 *T

j j jh h h x h d h d h h x

Therefore,

1

*Tj

j j

h h h x h dh d h h x

(4.5’)

How 0j

when 0 , we get from (4.1’) and (4.5’) that

10

jh h h x h d


From lema 3.1 (iii) we have:

0j

h x d for a above θ.

How φ is monotone decreasing, we get 10 0

jh x d

for a sufficiently small

positive scalar θ.

This contradicts ,,

hd D X x .

Therefore, Case 2 does not hold.

Similarly to the foregoing discussion, we conclude that nor does Case 2 hold.

A summary of the above discussions leads to the validity of the lemma.

Analogously to the contingent cone defined in [2, p. 121], we give a definition as follows:

Definition 4.3. Let K be a subset of n and x0 belong to the closure of K. The ( , )h -contingent

cone Th,φ (K, x0) is defined by 0

,,

hd T K x

if and only if 0

nh

and

nd d such that

0,

n nn x h d K .


Definition 3.4. [22] Let K be a nonempty subset of n. The ( , )h -positive polar cone

,hK

of K is

defined by

,,

0 , T

h nh

K d R d y y K

Remark 4.2 The positive polar cone of K is the ( , )h -positive polar cone ,h

K

of K with respect to

h(x) = x. But the converse does not hold. The show this point, let us continue to consider Example 4.1,

we can verify that S , but

,(1,1)

T

hS

.

Similarly to ( , )h -cone of descent directions of f at x ([18]), we give the following definition.

Definition 4.5 Suppose f is ( , )h -differentiable on n, x X . We shall say that

2 *

,,

, 0T

h nh

Z X x d R d f x

is the ( , )h -cone of descent directions of f at x .

We now present the Kuhn-Tucker constraint qualification of X at x that will be used to validate the

Kuhn-Tucker necessary condition in the next section.

Kuhn-Tucker constraint qualification: let gi for i = 1,2,...,m and hj for j = 1, 2, ..., l be ( , )h -

differentiable on Rn, and 1

, ,, ,

, ,h h

h hZ X x T X x

, where x X .

5. NECESSARY CONDITIONS FOR ( , )h -SINGLE-

OBJECTIVE PROGRAMMING

We continue to consider the programming problem (HFP) described in Section 4.

The ( , )h -Lagrangian function associated with (HFP) is given by

,

1 1

, ,m l

h i i j ji j

L x u v f x u g x v h x

(5.1)

where n

u ,l

v , whose components ui for i = 1, 2, ..., m and vj for j = 1, 2, ..., l are called the

( , )h -Lagrangian multipliers.

The following lemma is needed later.


Lemma 5.1 Let f, gi for i = 1, 2, ..., m and hj for j = 1, 2, ..., l be ( , )h -differentiable on Rn, x X ,

namely, x is a feasible solution for (HFP). Then 1 2

, ,, ,

h hZ X x Z X x implies there exist

vectors m

u

and l

v such that

* * * *

, 2 11 1

1, , 0

m li i

h i j hm i l ji j

u vL x u v f x g x h x

0i i

u g x for i = 1,2,...,m,

where 10 0

hh

.

Before proving the lemma, we prezent Motzkin’s alternative theorem:

Theorem 5.1 [22] Let A be a nonzero m n matrix, B be an r n matrix and C be an s n matrix.

Then exactly one of the following two systems has a solution:

System 1: m

Ax

,r

Bx

, Cx = 0, for some n

x .

System 2: 1 2 3

0T T T

A u B u C u ,for some0

1 2 3, .

m r su u u

.

Now we start to prove Lemma 5.1.

Proof of Lemma 5.1. Without loss of generality, we assume φ is strictly monotone decreasing on .

From 1 2

, ,, ,

h hZ X x Z X x , we conclude that the following system:

*

,

*

,

*

,

0 ,

0 , for

0 , for 1, 2,...,

T

h

T

ih

T

jh

d f x

d g x i I x

d h x j l

has no solution.

This, by the strictly monotone decrease of φ, is equivalent to that the following system:

*

*

*

0

0, for

0, for 1, 2,...,

T

T

i

T

j

h d h f x

h d h g x i I x

h d h h x j l

is inconsistent.

Because :n n

h is a one-to-one and onto function, there does not exist a z satisfying


*

*

*

0

0, for

0, for 1, 2,...,

T

T

i

T

j

z h f x

z h g x i I x

z h h x j l

Denote *T

A h f x ,

*T

ii I x

B h g x

, in other words, B is a matrix whose rows are

*T

ih g x for i I x , and C is a matrix whose rows are *

T

ih h x for j = 1,2,...,l,

namely,

* * *

1 2, ,...,

T

lC h h x h h x h h x ,

It follows from the above discussion that the system

1, , 0

I xAz Bz Cz

is inconsistent, where I x denote the number of elements in I x . By Theorem 5.1, there exist

0

1, ,

lI xu v

such that

* * *

( ) 1

0l

i i j i

i I x j

h f x u h g x v h h x

(5.2)

According to the definition of 0

1

, one has 0 . Division of (5.2) by α leads to

* * *

1

0l

i i j j

i I x j

h f x u h g x v h h x

(5.3)

Where ,i i

i i

u vu v

.

Letting h–1 act on (5.3), we obtain

1 * * * 1

1

0l

i i j j

i I x j

h h f x u h g x v h h x h

(5.4)

Let 0i

u for i I x . Then we arrive at

1 * * * 1

1 1

0m l

i i j j

i j

h h f x u h g x v h h x h

1 * 1 * 1 * 1

1 1

0m l

i i j j

i j

h h f x hh u h g x hh v h h x h


which together with Lemma 3.1 (i),(v), (vi) and 10 0

hh

leads to

1 * * *

1 1

0m l

ji

i j hm i l ji j

vuh h f x h g x h h x

.

* * *

21 1

1 10

m lji

i j hm i l ji j

vuf x g x h x

.

* * *

2 11 1

10

m lji

i j hm i l ji j

vuf x g x h x

.

On the other hand,

* *

, 11 1

, ,m l

h i j ji j

L x u v f x u g x v h x

* * *

21 1

1 1 m l

i i j ji j

f x u g x v h x

* * *

2 11 1

1 1 m lji

i jm l ji j

vuf x g x h x

.

Combining the definition of I x and i

u , one observes that

0 or 0 for each 1,2,...,i i

u g x i m ,

Which means 0i i

u g x . Therefore

10 .

i i i iu g x u g x

Thus we complete the proof.

In the rest of the paper let *ˆ | 0,Df h x h d d be the domain of 1

f fh

Lemma 5.2 Let f, gi for i =1, 2, ..., n and hj for j=1, 2, ..., l be ( , )h -differentiable on n , and

suppose x is an optimal solution for (HFP) , 1h h x X

and f reaches its maximum in

t h x . Then


*

, ,,

h hf x T X x

Proof. It suffices to show that for an arbitrary vector , ,,

h hd T X x

, one has

*

,0

T

hd f x

Let , ,,

h hd T X x

. Then there exist sequences n

d d and 0n

t

such that

n n

x t d X

(5.5)

By the ( , )h -differentiability of f at x , we conclude that 1f t fh t

is differentiable at

t h x , hence

ˆ ˆ ˆT

n n n n n n nf t t h d f t t h d f t t h d

(5.6)

where 0n

as n .

How f reaches its maximum in t h x , we get that ˆ ˆ 0n n

f t t h d f t .

Thus, ˆ ˆ

0n n

n

f t t h d f t

t

, witch toghether with (4.6) and with the fact that 0

n when

n gives:

*0

Th f x h d (5.7)

Applying 1 in the above inequality, we get:

1 * 10

Th f x h d

Therefore

*

,

0T

h

f x d

Thus

*

, ,,

h hf x T X x

In the remainder of this section, we present the necessary condition for a feasible x to be optimal for

(HFP) in the form of the following theorem.


Theorem 5.2. Let the hypotheses of Lemma 5.2 be satisfied, and

1

, , ,,, ,

h h hhZ X x T X x

Then there exist vectors mu

and l

v such that

* * * *

, 2 1 21 1

1, , 0

m lji

x h i j hm i l ji j

vuL x u v f x g x h x

,

0 for 1,2,...,i i

u g x i m ,

where .10 0

hh

Proof. If follows from Theorem 3.1 that

* * * *

, 2 1 21 1

1, ,

m lji

x h i jm i l ji j

vuL x u v f x g x h x

By Lemma 5.2 and hypothesis 1

, , ,,, ,

h h hhZ X x T X x

, we derive that

* 1

,,

,h

hf x Z X x

Hence, for an arbitrary 1

,,

hd Z X x , one gets

*

,0

T

hd f x

Consequently,

1 2

, ,, ,

h hZ X x Z X x

Using Lemma 5.1, it follows the conclusion.

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