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Center of training and analysis in risk's engineering
International Journal of Risk Theory
Vol 6 (no.2)
Alexandru Myller
Publishing
Iaşi, 2016
Center of training and analysis in risk's engineering
International Journal of Risk Theory
ISSN: 2248 – 1672
ISSN-L: 2248 – 1672
Editorial Board:
Hussein ABBASS, University of New South Wales, Australia
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
Ioan TOFAN, CeFAIR, "Al.I.Cuza" University, Iaşi
Akihiro TOKAI, Osaka University, Japan
Andrea VACCA, University Napoli 2, Italy
Executive Editors:
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e-mail: [email protected]; [email protected]
Ioan TOFAN
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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
International Journal of Risk Theory, Vol 6 (no.2), 2016 2
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)
International Journal of Risk Theory, Vol 6 (no.2), 2016 3
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
International Journal of Risk Theory, Vol 6 (no.2), 2016 4
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))
}
International Journal of Risk Theory, Vol 6 (no.2), 2016 5
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
International Journal of Risk Theory, Vol 6 (no.2), 2016 6
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
International Journal of Risk Theory, Vol 6 (no.2), 2016 7
2.2.1 Characteristics and some properties
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.
Corollary 3. The mean is
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.
International Journal of Risk Theory, Vol 6 (no.2), 2016 8
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
and the corresponding cdf is
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}
International Journal of Risk Theory, Vol 6 (no.2), 2016 9
Figure 6: The hazard rate function of the generalized Lindley Binomial Max/Min
3.1.1 Characteristics and some properties
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)
Corollary 5. The mean is
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)])}
International Journal of Risk Theory, Vol 6 (no.2), 2016 10
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.
International Journal of Risk Theory, Vol 6 (no.2), 2016 11
Figure 7: The probability density and cumulative function of the generalized Lindley Binomial Mindistribution with n=9
3.2.1 Characteristics and some properties
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
)}
Corollary 7. The mean is
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
)]
Corollary 8. The variance is
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
International Journal of Risk Theory, Vol 6 (no.2), 2016 12
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)])}
Proposition 12. The Renyi entropy is
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.
We have the following
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
and the corresponding cdf is
F(x) =p
1 − p
[1
1 − (1 − p)(1 − αθ+1+θx
αθ+1 e−θx) − 1
], x > 0, α, θ, p > 0
International Journal of Risk Theory, Vol 6 (no.2), 2016 13
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.
4.1.1 Characteristics and some properties
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)
Corollary 9. The mean is
E(X) =∑k≥1
kpθ2(1 − p)k−1
(αθ + 1)k(−θ)k−1αk+2Γ(k + 1)Ψ(k + 1, k + 3, αkθ)
International Journal of Risk Theory, Vol 6 (no.2), 2016 14
Corollary 10. The variance is
Var(X) =∑k≥1
kpθ2(1 − p)k−1
(αθ + 1)k(−θ)k−1αk+3Γ(k + 2)Ψ(k + 2, k + 4, αkθ)
−
[∑k≥1
kpθ2(1 − p)k−1
(αθ + 1)k(−θ)k−1αk+2Γ(k + 1)Ψ(k + 1, k + 3, αkθ)
]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))
Proposition 16. The Renyi entropy is
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.
We have the following
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
and the corresponding cdf is
F(x) =1
1 − p
[1 −
p
1 − (1 − p)αθ+1+θxαθ+1 e−θx
], x > 0, α, θ, 0 < p < 1
International Journal of Risk Theory, Vol 6 (no.2), 2016 15
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
4.2.1 Characteristics and some properties
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))
]
International Journal of Risk Theory, Vol 6 (no.2), 2016 16
Corollary 12. The variance is
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
International Journal of Risk Theory, Vol 6 (no.2), 2016 17
[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
International Journal of Risk Theory, Vol 6 (no.2), 2016 18
[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
International Journal of Risk Theory, Vol 6 (no.2), 2016 19
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
International Journal of Risk Theory, Vol 6 (no.2), 2016 20
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
International Journal of Risk Theory, Vol 6 (no.2), 2016 21
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 ,
International Journal of Risk Theory, Vol 6 (no.2), 2016 22
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
International Journal of Risk Theory, Vol 6 (no.2), 2016 23
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
International Journal of Risk Theory, Vol 6 (no.2), 2016 24
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:
International Journal of Risk Theory, Vol 6 (no.2), 2016 25
( ) 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
International Journal of Risk Theory, Vol 6 (no.2), 2016 26
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
International Journal of Risk Theory, Vol 6 (no.2), 2016 27
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
for a sufficiently small positive scalar θ.
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 .
International Journal of Risk Theory, Vol 6 (no.2), 2016 28
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
for a sufficiently small positive scalar θ.
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 .
International Journal of Risk Theory, Vol 6 (no.2), 2016 29
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.
International Journal of Risk Theory, Vol 6 (no.2), 2016 30
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
International Journal of Risk Theory, Vol 6 (no.2), 2016 31
*
*
*
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
International Journal of Risk Theory, Vol 6 (no.2), 2016 32
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
International Journal of Risk Theory, Vol 6 (no.2), 2016 33
*
, ,,
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.
International Journal of Risk Theory, Vol 6 (no.2), 2016 34
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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