A SELECTION PORTFOLIO PROBLEM WITH FUZZY LINEAR EXPONENTIAL DISTRIBUTION

FUZZY LINEAR EXPONENTIAL DISTRIBUTION

ILIE MARINESCU and IOAN M. STANCU-MINASIAN

We consider the optimal selection problem of the rebalancing portfolio with chance constraints, where some of the parameters are uncertain. We model these un- certainties using fuzzy numbers. The random variable of the chance constraints follows fuzzy linear exponential distribution.

AMS 2010 Subject Classification: 60A86, 90C70, 90C26.

Key words: portfolio optimization, concave tranzaction cost, rebalancing, fuzzy probability.

1. INTRODUCTION

The selection problem of the rebalancing portfolio, under nonconvex transactional costs, was defined in Konno et al. [7], [8], which proposed a branch and bound algorithm for calculating an optimal solution of the mini- mum cost rebalancing problem under concave transactional costs.

In order to purchase (invest) and/or sell (disinvest) assets, sometimes the investor has to pay certain fees.

In this paper, the fees associated with x = (x1, x2, . . . , xn) are named transaction cost, where x_j represents the amount of investment (or disinvest- ment) of the asset j (j= 1, . . . , n). The transaction cost of the entire invest- ment is Pn

j=1c_j(x_j),wherec_j(x_j) is a non-decreasing nonconvex function up to certain point x_j [7].

Let us now consider a time horizon, composed fromT moments of time;

we denote by (r_1t, r_2t, . . . , r_nt) the vector of the rates of return of thenassets at thetmoment, andpt=P r{(R₁, R2, . . . , Rn) = (r1t, r2t, . . . , rnt)},t= 1, . . . , T is known and where R_j is a random variable, which represents the rate of the return of the j asset (Pr stands for probability) ([5], [13], [14]).

Let rj = PT

t=1ptrjt be the expected rate of the return of the asset j without transaction costs, and Pn

j=1r_jx_j the expected rate of the return of the portfolio x= (x₁, x₂, . . . , x_n).

MATH. REPORTS13(63),4 (2011), 347–362

Because the current portfolio may deviate from the present efficient fron- tier, the investors are inclined to “rebalance” the portfolio due to the change of investment environment.

Letx⁰= (x⁰₁, x⁰₂, . . . , x⁰_n) [7], [8] be the portfolio at time 0. The investor wants to change the portfolio at a certain later point, say one year later. Let x= (x1, x2, . . . , xn) the new portfolio, with the condition that its expected rate of returnPn

j=1r_jx_j is greater than a constantg₂ and smaller than a constant g₁, because the investor wants to obtain an minimum guaranteed expected rate of return, and also a maximum expected rate of return, if it is possible.

LetW[r(x)] =p_t

Pn

j=1(r_jt−r_j)x_j

be the risk of the problem, assumed to be bounded, which means that the investor does not want a very big risk W0, but neither a small riskw0, which gives a less expected rate of return.

The investor is limited at a minimumM2 (maximumM1) capital which he wishes to invest in the rebalancing problem.

Let us introduce the new portfolio at a certain later point x = y+ x⁰, with x⁰ = (x⁰₁, x⁰₂, . . . , x⁰_n) being the portfolio at time point 0, made up by all the operations resulted from the rebalancing, a portfolio that has the following meanings:

– ifyj >0,j= 1, . . . , n, thencj(yj) is the associated cost with purchasing yj units of the assetj;

– if y_j <0, j = 1, . . . , n, then c_j(y_j) is the associated cost with selling

|y_j|units of the assetj.

In this paper, we model the chance constraints from the minimal cost rebalancing problem using fuzzy theory approach. Then, we solve the deter- ministic equivalent of the fuzzy chance constrained minimal cost rebalancing problem.

This paper is organized as follows. In Section 2, we review the theore- tical background concerning the triangular fuzzy numbers. In Section 3, we state the minimum cost rebalancing problem under the mean-absolute devia- tion (MAD) model. In Section 4, we model the programming problem under fuzzy linear exponential distribution with different choices of the distribution parameters. In the last section, we solve the deterministic problem with modi- fied subgradient algorithm.

2. FUZZY NUMBERS THEORY. SOME PRELIMINARIES

The purpose of this section is to recall some concepts which will be needed in the sequel. The programming problem with chance constraints has some uncertain informations which can be modeled through different methods involving randomness and fuzziness in different scenarios. Buckley [1], [2], [3]

has defined fuzzy probability using fuzzy numbers as parameters in probability

density function. The fuzzy numbers are obtained from the set of confidence interval.

According to Dashet al.[4], the fuzzy chance constrained programming problem is a chance constrained programming problem in the presence of am- biguous information, where the random variable follows different fuzzy distri- butions. In this paper, fuzzy random variables for chance constrained rebal- ancing problem follow a linear exponential distribution, which is very efficient from the practical point of view.

We place a “ e ” over a symbol to denote a fuzzy set. All our fuzzy sets will be fuzzy subsets of the real numbers. So, aei,A,e xeall represent fuzzy subsets of the real numbers.

Given the fuzzy number Fe with membership function µ_Fe : R → [0,1], the set {b | µF˜(b) ≥ α, ∀0 ≤ α ≤ 1} = ˜F[α] is α-cut of Fe. F˜[0] is sepa- rately defined as the closure of the union of all the ˜F[α], 0 < α ≤ 1. So, a fuzzy number Fe is a fuzzy subset of the real numbers satisfying the following conditions:

–µ_F˜(b) = 1 for some b (normalized);

– ˜F[α] is a closed and bounded interval for 0≤α≤1.

A triangular fuzzy numberFeis a triplet (y¹, y², y³)∈R³. The member- ship function ofFe is defined in connection with the real numbersy¹, y², y³ as follows:

Fe(x) =















0 x∈(−∞, y¹),

x−y¹

y²−y¹ x∈[y¹, y²],

x−y³

y²−y³ x∈(y², y³], 0 x∈(y³,∞).

Fe(x) represents a number in [0,1], which is the membership function of Fe evaluated in x. For any fuzzy number F, thee α-cut of Fe is a closed and bounded interval for 0 ≤ α ≤ 1, i.e., Fe([α]) = [q∗(α), q^∗(α)]. We define a partial order relation between two fuzzy numbers Fe₁ and Fe₂ using α- cuts Fe1[α] and Fe2[α]. Let Fe1[α] =

q_∗¹(α), q^1∗(α)

and Fe2[α] =

q_∗²(α), q^2∗(α) . Then, Fe₁ Fe₂ iff q_∗¹(α)≥q^2∗(α), for eachα∈[0,1].

3. PROBLEM STATEMENT

We assume that for the expected rate of return to have chance constraints with the probabilities p⁰_t = P r{(R₁, R2, . . . , Rn) = (r⁰_1t, r_2t⁰ , . . . , r⁰_nt)}, t = 1, . . . , T and p⁰⁰_t = P r{(R₁, R₂, . . . , R_n) = (r⁰⁰_1t, r⁰⁰_2t, . . . , r_nt⁰⁰)}, t = 1, . . . , T are known. The minimum cost rebalancing problem under the mean-absolute

deviation (MAD) model can be formulated as follows min

n

X

j=1

c_j(y_j) (P)

subject to

P r n

X

j=1

rj(yj+x⁰_j)≤g1≤

n

X

j=1

r_j⁰(yj +x⁰_j)

≥q1,

P r n

X

j=1

r⁰⁰_j(y_j+x⁰_j)≤g₂ ≤

n

X

j=1

r_j(y_j+x⁰_j)

≥q₂,

M₂ ≤

n

X

j=1

(y_j +x⁰_j)≤M₁, γ_j⁰ ≤yj+x⁰_j ≤γj, j= 1, n, w0 ≤

T

X

t=1

pt

n

X

j=1

(rjt−rj)yj+ (rjt−rj)x⁰_j

≤W0.

Let us consider a set of nonnegative variables z_t,z_t⁰, t= 1, . . . , T, satis- fying the following conditions

zt−z_t⁰ =pt n

X

j=1

(rjt−rj)yj+ (rjt−rj)x⁰_j , z_tz_t⁰ = 0, z_t≥0, z_t⁰ ≥0, t= 1, . . . , T.

Then we have z_t+z⁰_t=

p_t

n

X

j=1

(r_jt−r_j)y_j+ (r_jt−r_j)x⁰_j

, t= 1, . . . , T.

Therefore, instead of Problem (P) we consider the following problem min

n

X

j=1

c_j(y_j) (P₁)

subject to

P r n

X

j=1

rj(yj+x⁰_j)≤g1≤

n

X

j=1

r_j⁰(yj +x⁰_j)

≥q1,

P r n

X

j=1

r⁰⁰_j(yj+x⁰_j)≤g2 ≤

n

X

j=1

rj(yj+x⁰_j)

≥q2,

M2 ≤

n

X

j=1

(yj +x⁰_j)≤M1,

γ_j⁰ ≤y_j+x⁰_j ≤γ_j, j= 1, . . . , n, w0 ≤

T

X

t=1

(zt+z_t⁰)≤W0, z_t−z_t⁰ =p_t

n

X

j=1

(r_jt−r_j)y_j+ (r_jt−r_j)x⁰_j , ztz_t⁰ = 0, zt≥0, z⁰_t ≥0, t= 1, . . . , T.

As in [7] we can prove a similar result for problem (P₁).

Theorem 1. The complementarity constraintztz_t⁰= 0, zt≥0, z_t⁰≥0, t=

1,2, . . . , T, can be eliminated from problem (P1). Moreover,PT

t=1(zt−z⁰_t) = 0.

Proof. Let (y₁^∗, . . . , y_n^∗, z^∗₁, . . . , z^∗_T, z^0∗₁, . . . , z^0∗_T) be an optimal solution of the problem (P1) without complementarity constraint and let us assume that z_t^∗z^0∗_t >0, z_t^∗≥0,z^0∗_t ≥0, ∀t∈I ⊂ {1, . . . , T}.

We see that

– if z_t^∗ ≥ z^0∗_t ≥ 0, ∀t ∈ I then consider a solution of problem (P₁) (ze_t,zf⁰_t) = (z_t^∗−z^0∗_t,0),∀t∈I;

– if 0 ≤ z_t^∗ ≤ z^0∗_t, ∀t ∈ I then consider a solution of problem (P1) (zet,zf⁰t) = (0, z^0∗_t −z_t^∗),∀t∈I.

Therefore, (y^∗₁, . . . , y^∗_n,ze₁, . . . ,fz_T,zf⁰₁, . . . ,zf⁰_T) satisfies all the constraint of (P₁), while the objective function value in (y^∗₁, . . . , y_n^∗,ze1, . . . ,fzT,zf⁰1, . . . ,zf⁰T) is equal with the objective function value calculated in (y^∗₁, . . . , y^∗_n, z₁^∗, . . . , z_T^∗, z^0∗₁, . . . , z^0∗_T). Moreover, it checks that

T

X

t=1

zt−z⁰t

=

T

X

t=1

pt n

X

j=1

(rjt−rj)yj+ (rjt−rj)x⁰_j

=

T

X

t=1 n

X

j=1

p_t(r_jt−r_j)y_j +p_t(r_jt−r_j)x⁰_j

= 0.

The theorem is proved.

Thus, the mean-absolute deviation (MAD) model from Marinescu [10], can be represented as a crisp chance constrained programming problem for

the minimal cost of the portfolio transaction of the form min

n

X

j=1

cj(yj) (P2)

subject to

P r n

X

j=1

r_j(y_j+x⁰_j)≤g₁≤

n

X

j=1

r_j⁰(y_j +x⁰_j)

≥q₁,

P r n

X

j=1

r⁰⁰_j(y_j +x⁰_j)≤g₂ ≤

n

X

j=1

r_j(y_j+x⁰_j)

≥q₂, γ_j⁰ ≤yj+x⁰_j ≤γj, j= 1, . . . , n,

w0 ≤2

T

X

t=1

zt≤W0,

pt n

X

j=1

(rjt−rj)yj−zt≤ −p_t

n

X

j=1

(rjt−rj)x⁰_j, t= 1, . . . , T, where g_i, i= 1,2, is uncertain and Pn

j=1r_j(y_j +x⁰_j) represents the expected rate of return for rebalancing portfolio.

We suppose that g1 and g2 are fuzzy random variables and the others real parameters M₁, M₂, γ_t, γ_t⁰, W₀, w₀ are given. Their significations and definitions are given in [9], [10], [11].

4. THE PROGRAMMING PROBLEM UNDER FUZZY LINEAR EXPONENTIAL DISTRIBUTION

Let g₁ and g₂ be continuous random variables with probability density function linear exponential

f(g_i;λ, µ) = (

e^−λgi−

µg2 i

2 (λ+µg_i) ifg_i >0,

0 otherwise,

where λ and µ are uncertain parameters, describing the probability density function. We will study both the case when the parameterλis a fuzzy number and the case when the parameter µ is a fuzzy number.

4.1. The parameterλis a fuzzy number

Let eg1 and eg2 be fuzzy random variables with fuzzy density functions linear exponential, eλ a fuzzy parameter and µ is given parameter. So, the

fuzzy chance constrained programming problem for the minimal cost of the portfolio transaction is of the form

min

n

X

j=1

c_j(y_j) (P₃)

subject to

P re n

X

j=1

rj(yj+x⁰_j)≤eg1≤

n

X

j=1

r_j⁰(yj +x⁰_j)

eq1,

P re n

X

j=1

r⁰⁰_j(yj +x⁰_j))≤eg2≤

n

X

j=1

rj(yj +x⁰_j)

eq2,

M₂≤

n

X

j=1

(y_j+x⁰_j)≤M₁,

γ_j⁰ ≤yj+x⁰_j ≤γj, j= 1, . . . , n, w0 ≤2

T

X

t=1

zt≤W0, p_t

n

X

j=1

(r_jt−r_j)y_j−z_t≤ −p_t

n

X

j=1

(r_jt−r_j)x⁰_j, t= 1, . . . , T.

Letu=Pn

j=1r_j(y_j+x⁰_j),u⁰ =Pn

j=1r⁰_j(y_j+x⁰_j) andu⁰⁰=Pn

j=1r_j⁰⁰(y_j+ x⁰_j). Then u≤eg_i ≤u⁰ is a fuzzy event. P re (u≤eg_i ≤u⁰) is the probability of this event which is a fuzzy number. Its α-cut is the set

P r(ue ≤eg_i≤u⁰)[α] = Z u⁰

u

e^−λgi−

µg2 i

2 (λ+µg_i)dg_i|λ∈λ[α]e

=

q_∗ⁱ(α), q^i∗(α) , for 0≤α≤1.Denoteqei[α] =

qi∗(α), q^∗_i(α)

, theα-cut of the fuzzy numberqei. Lemma 1. If eg₁ is a fuzzy random variable linear exponential distributed with µ real parameter given, and eλ1 is a fuzzy number, then the minimum α-cut of the fuzzy probability of the event (u≤eg₁≤u⁰) is

q¹_∗(α) = minn

e^{−λ1∗(α)u−µu}

2

2 −e^{−λ1∗(α)u0−µu}

02

2 , e^{−λ∗1(α)u−µu}

2

2 −e^{−λ∗1(α)u0−µu}

02 2

o

for α∈[0,1], where eλ1[α] = [λ1∗(α), λ^∗₁(α)].

Proof.

P r ue ≤eg₁ ≤u⁰ [α] =

Z u⁰ u

e^−λ1g1−

µg2 1

2 (λ₁+µg₁)dg₁

λ₁∈eλ₁[α]

=

= n

e^{−λ1u−µu}

2

2 −e^{−λ1u0−µu}

02 2

λe1∗(α)≤λ1 ≤eλ^∗₁(α) o

=

q_∗¹(α), q^1∗(α) . The function

f₁(u, u⁰;λ₁) =e^{−λ1u−µu}

2

2 −e^{−λ1u0−µu}

02 2

is an increasing function of in λ1 until a maximum and it is decreasing at infinite. So, it results that the minimum of the function f₁ is attained in λ1∗(α) for f1 increasing, or if f1 is decreasing then the minimum of f1 is attained in λ^∗₁(α). Thus, the minimum α-cut of the fuzzy probability of the event (u≤eg1 ≤u⁰) is

q¹_∗(α) = min n

e^{−λ1∗(α)u−µu}

2

2 −e^{−λ1∗(α)u0−µu}

02

2 , e^{−λ∗1(α)u−µu}

2

2 −e^{−λ∗1(α)u0−µu}

02 2

o

.

Lemma 2. If eg2 is a fuzzy random variable linear exponential distributed with µ real parameter given, and eλ2 is a fuzzy number, then we have the following relation for the minimum α-cut of the fuzzy probability of the event (u⁰⁰≤eg₂ ≤u) is

q²_∗(α) = min n

e^{−λ2∗(α)u00−µu}

002

2 −e^{−λ2∗(α)u−µu}

2

2 , e^{−λ∗2(α)u00−µu}

002

2 −e^{−λ∗2(α)u−µu}

2 2

o

for α∈[0,1], where eλ₂[α] = [λ2∗(α), λ^∗₂(α)].

Proof.

P r ue ⁰⁰≤eg2≤u [α] =

Z u u⁰⁰

e^−λ2g2−

µg2 2

2 (λ2+µg2)dg2

λ2∈eλ2[α]

=

= n

e^{−λ2u00−µu}

002

2 −e^{−λ2u−µu}

2 2

eλ2∗(α)≤λ2≤eλ^∗₂(α) o

=

q_∗²(α), q^2∗(α) . The function

f₂(u⁰⁰, u;λ₂) =e^{−λ2u00−µu}

002

2 −e^{−λ2u−µu}

2 2

is increasing in λ2 until a maximum which is attained for λ2 max = ln(_u^u00)−µ^u2−u₂ ⁰⁰²

u−u⁰⁰ ,

and after this point it is decreasing at infinite. It results that the minimum of the functionf2 is attained in λ2∗(α) ifλ2≤λ2 max or inλ^∗₂(α) forλ2 > λ2 max.

So, the minimum α-cut of the fuzzy probability of the event (u⁰⁰≤eg2 ≤u) is q²_∗(α) = minn

e^{−λ2∗(α)u00−µu}

002

2 −e^{−λ2∗(α)u−µu}

2

2 , e^{−λ∗2(α)u00−µu}

002

2 −e^{−λ∗2(α)u−µu}

2 2

o

.

4.2. THE MODEL WITHµ- A FUZZY NUMBER ANDλ- REAL PARAMETER GIVEN

Letµebe a fuzzy number. Also, let the variableXa fuzzy random variable linear distributed with λreal parameter given, and denote it with X.e

Lemma 3. If eg₁ is a fuzzy random variable linear exponential distributed with λ real parameter given, and µe₁ is a positive fuzzy number, then we have the following relation for the minimum α-cut of the fuzzy probability of the event (u≤eg₁ ≤u⁰):

q_∗¹(α) = minn

e^{−λu−µ1∗(α)u}

2

2 −e^{−λu0−µ1∗(α)u}

02

2 , e^−λu−

µ∗ 1 (α)u2

2 −e^−λu0−

µ∗ 1 (α)u02

2

o

for α∈[0,1], where µe₁[α] = [µ1∗(α), µ^∗₁(α)].

Proof.

P r ue ≤eg₁≤u⁰ [α] =

Z u⁰ u

e^−λg1−

µ1g2 1

2 (λ+µ₁g₁)dg₁

µ₁∈µe₁[α]

=

= n

e^{−λu−µ1u}

2

2 −e^{−λu0−µ1u}

02 2

eµ1∗(α)≤µ1 ≤µe^∗₁(α) o

=

q_∗¹(α), q^1∗(α) . The function

f3(u, u⁰;µ1) =e^{−λu−µ1u}

2

2 −e^{−λu0−µ1u}

02 2

is increasing in µ₁ until a maximum which is attained for µ1 max= 4 ln(^u_u⁰)−2λ(u⁰−u)

u⁰²−u² ,

and after this point it is decreasing at infinite. It results that the minimum of the functionf₃ is attained inµ1∗(α) ifµ₁ ≤µ_{1 max}or inµ^∗₁(α) forµ₁ > µ_{1 max}. So, it results that the minimum α-cut of the fuzzy probability for the event (u≤eg₁≤u⁰) is

q¹_∗(α) = min

e^{−λu−µ1∗(α)u}

2

2 −e^{−λu0−µ1∗(α)u}

02

2 , e^−λu−

µ∗ 1 (α)u2

2 −e^−λu0−µ

∗ 1 (α)u02

2

.

Lemma 4. If eg₂ is a fuzzy random variable linear exponential distributed with λreal parameter given, and µe2 is a positive fuzzy number. Then we have

the following relation for the minimum α-cut of the fuzzy probability of the event (u⁰⁰≤eg₂≤u) is

q²_∗(α) = minn

e^{−λu00−µ2∗(α)u}

002

2 −e^{−λu−µ2∗(α)u}

2

2 , e^−λu00−

µ∗ 2 (α)u002

2 −e^−λu−

µ∗ 2 (α)u2

2

o

for α∈[0,1], where µe2[α] = [µ2∗(α), µ^∗₂(α)]. Proof.

P r ue ⁰⁰≤eg₂≤u [α] =

Z u u⁰⁰

e^−λg2−

µ2g2 2

2 (λ+µ₂g₂)dg₂

µ₂ ∈eµ₂[α]

=

= n

e^{−λu00−µ2u}

002

2 −e^{−λu−µ2u}

2 2

eµ2∗(α)≤µ2≤eµ^∗₂(α) o

=

q²_∗(α), q^2∗(α) . The function

f₄(u⁰⁰, u;µ₂) =e^{−λu00−µ2u}

002

2 −e^{−λu−µ2u}

2 2

is increasing in µ₂ until a maximum and and it is decreasing at infinite. So, it results that the minimum of the function f4 is attained in µ2∗(α) for f4

increasing or in µ^∗₂(α) if f₄ is decreasing. Thus, the minimum α-cut of the fuzzy probability for the event (u⁰⁰≤ge2 ≤u) is

q²_∗(α) = min n

e^{−λu00−µ2∗(α)u}

002

2 −e^{−λu−µ2∗(α)u}

2

2 , e^−λu00−

µ∗ 2 (α)u002

2 −e^−λu−µ

∗ 2 (α)u2

2

o

.

5. AN ALGORITHM FOR SOLVING THE DETERMINISTIC EQUIVALENT OF THE FUZZY MINIMUM COST REBALANCING PROBLEM

5.1. The deterministic equivalent of the rebalancing problem

Depending on the choice of the parameters λ_i, i= 1,2 for the functions f1(u, u⁰;λ1) andf2(u, u⁰⁰;λ2), we have more cases for fuzzy chance constraints.

In this paper, we will consider further on the case, where the functions, with unknown parameters λi, i = 1,2 are decreasing. The others cases will be treated likewise.

Theorem2. Assume thateg_i is a fuzzy random variable linear exponen- tial distributed with µreal parameter given, andeλ_i,i= 1,2, is a positive fuzzy number. Also, assume the following conditions λ1 max< λ1 and λ2 max < λ2.

Then the problem P3

is equivalent with the following problem

min

n

X

j=1

cj(yj) (P4)

subject to

e^{−λ∗1(α)u−µu}

2

2 −e^{−λ∗1(α)u0−µu}

02

2 ≥q₁^∗(α), e^{−λ∗2(α)u00−µu}

002

2 −e^{−λ∗2(α)u−µu}

2

2 ≥q₂^∗(α), M2 ≤

n

X

j=1

(yj +x⁰_j)≤M1,

γ_j⁰ ≤y_j +x⁰_j ≤γ_j, j = 1, . . . , n, w0 ≤2

T

X

t=1

zt≤W0, p_t

n

X

j=1

(r_jt−r_j)y_j−z_t≤ −p_t

n

X

j=1

(r_jt−r_j)x⁰_j, t= 1, . . . , T.

Proof. This theorem results from Lemma 1 and Lemma 2.

The deterministic equivalent of the minimum cost rebalancing problem is a nonlinear programming problem. In what follows we will describe a con- vergent algorithm in order to find a solution of this problem.

5.2. Subgradient method

We solve the nonlinear programming problem using the subgradient method. The algorithm is simple, but it has a big running time. We con- sider the triangular fuzzy numbers eλ₁ = (λ⁰₁, λ₁, λ⁰⁰₁), eλ₂ = (λ⁰₂, λ₂, λ⁰⁰₂), qe₁ = (q₁⁰, q1, q₁⁰⁰) andqe2= (q₂⁰, q2, q⁰⁰₂).

From problem (P₄) we obtain the following model.

min

n

X

j=1

cj(yj) (P5)

subject to

e

−λ⁰⁰₁−α(λ⁰⁰₁−λ1)_Pn

j=1rj(yj−x⁰_j)

−^µ

Pn

j=1rj(yj−x0 j)

2

2 −

−e

−λ⁰⁰₁−α(λ⁰⁰₁−λ1)_Pn

j=1r⁰_j(yj−x⁰_j)

−^µ

Pn j=1r0

j(yj−x0 j)

2

2 −

−q₁⁰⁰+α q₁⁰⁰−q1

+k1= 0, e

−λ⁰⁰₂−α(λ⁰⁰₂−λ₂)_Pn

j=1r⁰⁰_j(yj−x⁰_j)

−^µ

Pn j=1r00

j(yj−x0 j)

²

2 −

−e

−λ⁰⁰₂−α(λ⁰⁰₂−λ₂)_Pn

j=1rj(yj−x⁰_j)

−^µ

Pnj=1rj(yj−x0 j)

²

2 −

−q₁⁰⁰2 +α q₂⁰⁰−q₂

+k₂= 0,

n

X

j=1

(y_j+x⁰_j)−M₁+k₃= 0,

n

X

j=1

(yj+x⁰_j)−M2−k4= 0,

y_j+x⁰_j −γ_j+k_j+4= 0, j= 1, . . . , n, y_j+x⁰_j −γ_j⁰ −k_j+n+4 = 0, j= 1, . . . , n, 2

T

X

t=1

zt−W0+k2n+5= 0,

2

T

X

t=1

z_t−W₀−k_2n+6= 0, pt

n

X

j=1

(rjt−rj)yj −zt+pt n

X

j=1

(rjt−rj)x⁰_j +kt+2n+6, t= 1, . . . , T.

The solution set of problem (P₅) is denote by S =

y⁰, α, k

| y⁰ = y₁, . . . , y_n, z₁, . . . , z_Tt

, k = k₁, . . . , k_2n+T+6

, 0 ≤ α ≤ 1, k_i ≥ 0 and is a compact subset of R^3n+2T+7. Let us consider g0 : Rⁿ → R the objec- tive function of the problem (P₅) and g : R^3n+2T+7 → R^2n+T+6 the con- straints function of the problem (P5), where we assume that g y⁰, α, k

= g1, . . . , g2n+T+6

t

is a continuous function.

The augmented LagrangianL associated with problem (P₅) is L:R^3n+2T+7×R^2n+T+6×R₊→R^2n+T+6, where

L y⁰, α, k;u, c

=g0 y⁰, α, k +c

g(y⁰, α, k) +

u, g y⁰, α, k , with y⁰, α, k

∈S,u∈R^2n+T+6,c∈R₊and k · kis the Euclidean norm, and h i is the Euclidean inner product.

So,

L y⁰, α, k;u, c

=

=

n

X

j=1

cj(yj) +c

e

−λ⁰⁰₁−α(λ⁰⁰₁−λ1)_Pn

j=1rj(yj−x⁰_j)

−^µ

Pn

j=1rj(yj−x0 j)

2

2 −

−e

−λ⁰⁰₁−α(λ⁰⁰₁−λ₁)_Pn

j=1r⁰_j(yj−x⁰_j)

−^µ

Pn j=1r0

j(yj−x0 j)²

2 −q₁⁰⁰+α q⁰⁰₁−q₁ +k₁

2

+ +

e

−λ⁰⁰₂−α(λ⁰⁰₂−λ2)_Pn

j=1r⁰⁰_j(yj−x⁰_j)

−^µ

Pn j=1r00

j(yj−x0 j)

2

2 −

−e

−λ⁰⁰₂−α(λ⁰⁰₂−λ2)_Pn

j=1rj(yj−x⁰_j)

−^µ

Pn

j=1rj(yj−x0 j)²

2 −q₂⁰⁰+α q⁰⁰₂−q₂ +k₂

2

+ +

ⁿ X

j=1

(yj+x⁰_j)−M1+k3

2

+ ⁿ

X

j=1

(yj+x⁰_j)−M2−k4

2

+ +

n

X

j=1

yj +x⁰_j −γj+kj+4

2

+

n

X

j=1

yj +x⁰_j −γ_j⁰ −kj+n+4

2

+

+

2

T

X

t=1

z_t−W₀+k_2n+5 2

+

2

T

X

t=1

z_t−w₀−k_2n+6 2

+

+

T

X

t=1

pt

n

X

j=1

(rjt−rj)yj−zt+pt n

X

j=1

(rjt−rj)x⁰_j+kt+2n+6

2¹₂ +

u₁

e

−λ⁰⁰₁−α(λ⁰⁰₁−λ1)_Pn

j=1rj(yj−x⁰_j)

−^µ

Pn

j=1rj(yj−x0 j)²

2 −

−e

−λ⁰⁰₁−α(λ⁰⁰₁−λ₁)_Pn

j=1r⁰_j(yj−x⁰_j)

−^µ

Pn j=1r0

j(yj−x0 j)

2

2 −q₁⁰⁰+α q₁⁰⁰−q1

+k1

+ +u₂

e

−λ⁰⁰₂−α(λ⁰⁰₂−λ2)_Pn

j=1r⁰⁰_j(yj−x⁰_j)

−^µ

Pn j=1r00

j(yj−x0 j)

2

2 −

−e

−λ⁰⁰₂−α(λ⁰⁰₂−λ₂)_Pn

j=1rj(yj−x⁰_j)

−^µ

Pn

j=1rj(yj−x0 j)

²

2 −q₂⁰⁰+α q₂⁰⁰−q2

+k2

+ +u₃

n

X

j=1

(y_j+x⁰_j)−M₁+k₃

+u₄ n

X

j=1

(y_j+x⁰_j)−M₂−k₄

+ +

n

X

j=1

uj+4

yj+x⁰_j −γj+kj+4

+

n

X

j=1

uj+n+4

yj+x⁰_j−γ_j⁰ −kj+n+4

+

+u2n+5

2

T

X

t=1

zt−W0+k2n+5

+u2n+6

2

T

X

t=1

zt−w0−k2n+6

+

+

T

X

t=1

ut+2n+6

pt

n

X

j=1

(rjt−rj)yj−zt+pt n

X

j=1

(rjt−rj)x⁰_j +kt+2n+6

. The dual functionH is defined as

H(u, c) = min

(y⁰,α,k)∈SL(y⁰, α, k;u;c), u∈R^2n+T+6, c∈R₊. Then the dual problem of (P₅) is given by

(P₅^∗) max

(u,c)∈R^2n+T+6×R+

H(u, c).

The modified subgradient algorithm is devised for solving the dual prob- lem. Let us outlines the steps of the algorithm:

Step 0. Choose a vector (u¹, c¹) with c¹ ≥0,let p = 1 and go to the Step 1.

Step 1. Given (u^p, c^p), solve the following subproblem min

(y⁰,α,k)∈SL(y⁰, α, k;u^p;c^p)

and let (y^0p, α^p, k^p) any optimal solution. We have two situations:

• Ifg y^0p, α^p, k^p

= 0, then STOP; by Theorem 3, (u^p, c^p) is a solution to the dual problem and y^0p, α^p, k^p

is a solution to the primal problem;

• Otherwise, go to Step 2.

Step 2. Let

u^p+1=u^p−s^pg y^0p, α^p, k^p

, c^p+1 =c^p+ (s^p+^p)

g y^0p, α^p, k^p , where s^p and ^p are positive scalar stepsizes; replace pby p+ 1 set p=p+ 1 and repeat Step 1.

For proving the stoping step of the algorithm, we need the following theorem from [11].

Theorem 3 (Theorem 5 from [6]). Let inf(P5) = sup(P₅^∗) and suppose that for some

(¯u,¯c)∈R^2n+T+6×R₊ and ( ¯y⁰,α,¯ k)¯ ∈R^3n+2T+7, min

(y⁰,α,k)∈SL(y⁰, α, k; ¯u; ¯c) =g₀( ¯y⁰,α,¯ ¯k) + ¯c

g( ¯y⁰,α,¯ ¯k) +

¯

u, g( ¯y⁰,α,¯ k)¯ . Then( ¯y⁰,α,¯ ¯k)is a solution to (P₅)and(¯u,c)¯ is a solution to(P₅^∗)if and only if g( ¯y⁰,α,¯ ¯k) = 0.

The next proposition demonstrates that for the certain values of step- sizes ^p and s^p, the distance between the points (u^p+1, c^p+1) generated by