Solving portfolio selection models with uncertain returns using an artificial neural network scheme

Appl IntellDOI 10.1007/s10489-014-0616-z

Solving portfolio selection models with uncertain returnsusing an artificial neural network scheme

Alireza Nazemi · Behzad Abbasi · Farahnaz Omidi

© Springer Science+Business Media New York 2014

Abstract This paper presents a neural network model forsolving two models for portfolio selection in which thesecurities are assumed to be uncertain variables. The mainidea is to replace the portfolio selection models with linearprogramming (LP) problems. According to the convex opti-mization theory and some concepts of ordinary differentialequations, a neural network model for solving LP problemsis presented. The equilibrium point of the proposed modelis proved to be equivalent to the optimal solution of theoriginal problem. It is also shown that the proposed neu-ral network model is stable in the sense of Lyapunov andit is globally convergent to an exact optimal solution ofthe portfolio selection problem with uncertain returns. Twoillustrative examples are provided to show the feasibility andthe efficiency of the proposed method in this paper.

Keywords Uncertain variables · Portfolio selection ·Mean-variance · Crisp equivalent programming · Neuralnetwork · Stability · Convergent

1 Introduction

Portfolio selection is concerned with optimization of cap-ital allocation to a large number of securities. Since the

A. NazemiDepartment of Mathematics, School of Mathematical Sciences,University of Shahrood, P.O. Box 3619995161-316,Shahrood, Irane-mail: [email protected]

B. Abbasi · F. Omidi (�)Department of Mathematics, Semnan University, Semnan, Irane-mail: omidi [email protected]

B. Abbasie-mail: [email protected]

introduction of mean-variance theory by [48, 49], portfo-lio selection has been one of the hottest research topics infinancial area. Traditionally, security returns are assumedto be random and probability theory is the most powerfultool to help select the optimal portfolio. For example, recentworks [1, 9, 25–27, 39], etc. However, the security marketis complex and randomness is not the only type of uncer-tainty in reality. It is well known that the security returnsare sensitive to various factors including economic, social,political and very importantly, people’s psychological fac-tors. It is found that many security returns, especially shortterm security returns are strongly affected by people’s psy-chological expectation and are hard to be well reflected bythe historical data. With the introduction [75] and develop-ment [10, 11, 34, 36, 51, 76, 80] of fuzzy set theory, scholarsbegan to employ fuzzy set theory to describe and study thefuzziness contained in the portfolio investment. Employingthe possibility theory, fuzzy portfolio selection problem wasresearched from 1990’s. Much work was focused on extend-ing Markowitz’s mean-variance selection idea and a varietyof fuzzy mean-variance models have been developed, e.g.,[2, 5, 6, 20, 32, 33, 37, 38, 63–65, 67, 69, 77, 78].

One great limitation of possibility measure to use in port-folio selection is that it is not self-dual which is definedlater in Section 2. Using possibility measure which has noself-duality property, we find that paradoxes will appear ifwe use fuzzy variable to describe the subjective estimationsof security returns [28]. For example, if a security returnis regarded as a fuzzy variable, then we have a member-ship function to characterize it. Suppose it is a triangularfuzzy variable θ = (−0.7, 0.2, 1.1). Based on the mem-bership function, it is known from possibility theory orcredibility theory that the return is exactly 0.2 with beliefdegree 1 in possibility measure or 0.5 in credibility mea-sure. However, this conclusion is hard to accept because

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A. Nazemi et al.

the belief degree of exactly 0.2 should be almost zero. Inaddition, it is known from possibility theory or credibilitytheory that the return being exactly 0.2 and not exactly 0.2have the same belief degree in either possibility measure orcredibility measure, which implies that the two events willbe equally likely to happen. In order to solve these problemsand to model the subjective imprecise quantity, [40] pro-posed an uncertain measure and developed an uncertaintytheory which can be used to handle subjective imprecisequantity. Much research work has been done on the devel-opment of uncertainty theory and related theoretical work.For example, [74] proved some convergence theorems ofuncertain sequences, and [17] proved some properties ofcontinuous uncertain measure. Liu studied uncertain pro-gramming [41]. [18] discussed the inference rule for uncer-tain systems. [60] gave a sufficient and necessary conditionof uncertainty distribution, and [7] proved the existence anduniqueness theorem for uncertain differential equations, etc.When we use uncertain variable to describe the experts’ esti-mations of security returns, the above mentioned paradoxesdisappear immediately. Based on uncertainty theory, [79]has solved an uncertain optimal control problem and appliedit to a portfolio selection model, and [29] has defined arisk curve and has given a new selection method for uncer-tain portfolio selection and further proposed the uncertainmean-variance and mean-semivariance selection methods[30, 31].

In this paper, we are concerned with a portfolio opti-mization problem with uncertain returns and we must solveit. The dynamic system approach is one of the impor-tant methods for solving optimization problems. Artificialrecurrent neural networks for solving constrained optimiza-tion problems can be considered as a tool to transfer theoptimization problems into a specific dynamic system offirst-order differential equations. Furthermore, it is expectedthat the dynamic system will approach it’s static state (oran equilibrium point), which corresponds to the solutionfor the underlying optimization problem, starting from aninitial point. In addition, neural networks for solving opti-mization problems are hardware-implementable; that is, theneural networks can be implemented by using integratedcircuits.

Over the years, neural networks for optimization andtheir engineering applications have been widely investi-gated. Tank and Hopfield applied the Hopfield networkfor solving linear programming problems [21, 62], whichmotivated the development of neural networks for solvinglinear programming [43, 68, 70–72], variational inequal-ities [8, 19, 23, 24], nonlinear programming [4, 14, 15,22, 44–47, 52, 73] and so on. These neural networks areessentially governed by a set of dynamic systems charac-terized by an energy function, which is the combinationof the objective function and constraints of the original

optimization problem, and three common techniques, suchas penalty functions, Lagrange functions and primal anddual functions. In this paper, we focus on neural networkapproach to the portfolio selection problems with uncertainreturns. Our neural network will be aimed to solve an equiv-alent optimality system whose solutions are candidates oforiginal problem.

With motivation from the above discussions and follow-ing the properties of uncertain variables [30, 31], we firsttransform the portfolio selection problem which the secu-rities are assumed to be uncertain variables into a linearprogramming problem. According to the Karush-Kuhn-Tacker (KKT) optimality conditions [13], a neural networkmodel can be constructed. It is shown that the limit equi-librium points sequence of the proposed neural network canapproximately converge to an optimal solution of linear pro-gramming problem. Simulation results on two numericalexamples of the portfolio selection problem show the effec-tiveness and performance of the neural network model. Thisneural network model has been also successfully used forsolving the minimax problems in [53] and maximum flowand shortest path problems in [57, 58].

The remainder of this paper is organized as follows. Inthe next section, the preliminaries relevant to uncertain vari-ables are introduced. In Section 3, the mean variance modeland deterministic equivalents with various reformulationsare presented. In Section 4, a neural network is derived.The convergence of the proposed modelling framework isproved in Section 5. Simulation results on two numericalexamples of portfolio selection problem with normal uncer-tain variables and rectangular uncertain variables are givenin Section 6 to demonstrate the performance of the model.Finally, Section 7 concludes this paper.

2 Necessary knowledge about uncertain variables

Let � be a nonempty set, and let L be a σ -algebra over�. Each element � ∈ L is called an event. A set func-tion M(�) is called an uncertain measure if it satisfies thefollowing three axioms [40]:

(i): (Normality) M(�) = 1;(ii): (Self-duality) M(�) + M(�c) = 1 for every event

�;(iii): (Countable subadditivity) For every countable

sequence of events {�i}, we have

M{ ∞⋃

i=1

�i

}≤

∞∑i=1

M{�i}. (1)

The triplet (�,L,M) is called an uncertainty space.It can be proven [42] that any uncertain measure M

is increasing. That is, for any events �1 ∈ �2, we have

Solving portfolio selection models with uncertain returns using an artificial neural network scheme

M(�1) ≤ M(�2). In order to define product uncertainmeasure, [40] proposed the fourth axiom as follows:

(iv): (Product measure) [41]. (Product Measure Axiom)Let (�k,Lk,Mk) be uncertainty spaces for k =1, 2, ... The product uncertain measure M is anuncertain measure satisfying

M{ ∞∏

k=1

�k

}=

∞∧k=1

Mk{�k}, (2)

where �k are arbitrarily chosen events from Lk fork = 1, 2, ..., respectively.

Definition 2.1 [40]. An uncertain variable is a measurablefunction ξ from an uncertainty space (�,L,M) to the set ofreal numbers, i.e., for any Borel set B of real numbers, theset

{ξ ∈ B} = {γ ∈ � | ξ(γ ) ∈ B}, (3)

is an event.

An uncertainty distribution function is used to characterizean uncertain variable and is defined as follows.

Definition 2.2 [40].The uncertainty distribution � : R →[0, 1] of an uncertain variable ξ is defined by

�(t) = M{ξ ≤ t}. (4)

For example, by a normal uncertain variable, we mean thevariable that has the following normal uncertainty distribution

�(t) =(

1 + exp

(π(e − t)√

3σ

))−1

, t ∈ R, (5)

where e and σ are real numbers and σ > 0. For conve-nience, it is denoted in the paper by ξ ∼ N (e, σ ).

We call an uncertain variable the linear uncertain variableif it has the following linear uncertainty distribution

�(t) =⎧⎨⎩

0, if t < a

(t − a)/(b − a), if a ≤ t ≤ b

1, if t > b.

(6)

For convenience, it is denoted in the paper by ξ ∼ L(a, b)

where a < b.When the uncertain variables ξ1, ξ2, ..., ξn are representedby uncertainty distributions, the operational law is given byLiu (2010) as follows:

Theorem 2.3 Let ξ1, ξ2, ..., ξn be independent uncertainvariables with uncertainty distributions �1, �2, ..., �n,respectively. Let f (t1, t2, ..., tn) be strictly increasing withrespect to t1, t2, ..., tn. Then

ξ = f (ξ1, ξ2, ..., ξn), (7)

is an uncertain variable with uncertainty distribution

(t) = supf (t1,t2,...,tn)=t

(min

1≤i≤n�i(ti)

), t ∈ R, (8)

whose inverse function is

−1(α)=f (�−11 (α), �−1

2 (α), ..., �−1n (α)), 0<α<1,(9)

if �−11 (α), �−1

2 (α), ..., �−1n (α) are unique for each

α ∈ (0, 1).

To tell the size of an uncertain variable, Liu defined theexpected value of uncertain variables.

Definition 2.4 [40]. Let ξ be an uncertain variable. Thenthe expected value of ξ is defined by

E[ξ ] =∫ ∞

0M{ξ ≥ r}dr −

∫ 0

−∞M{ξ ≤ r}dr, (10)

provided that at least one of the two integrals is finite.

Theorem 2.5 [42]. Let ξ1 and ξ2 be independent uncer-tain variables with finite expected values. Then for any realnumbers a1 and a2, we have

E[a1ξ1 + a2ξ2] = a1E[ξ1] + a2E[ξ2]. (11)

Definition 2.6 [40]. Let ξ be an uncertain variable withfinite expected value e; then the variance of ξ is defined by

V [ξ ] = E[(ξ − e)2].

The detailed exposition on the uncertain theory have beenrecorded in the literature, the interested readers may consultit.

3 Mean-variance model and crisp equivalents

In Markowitz models, security returns were regarded asrandom variables. As discussed in introduction, there existsituations that security returns may be uncertain variableparameters. In this situation, we can use uncertain variablesto describe the security returns. Let xi denote the invest-ment proportion in the ith security, ξi represents uncertainreturn of the ith security, i = 1, 2, ..., n , respectively, anda is the maximum risk level that an investor can tolerate.Following Markowitz’s idea, we quantify investment returnby the expected value of a portfolio, and risk by the vari-ance. Then an optimal portfolio should be one with maximalexpected return for the given variance level. To express itin a mathematical formula, the mean-variance model is asfollows:

maximize E[x1ξ1 + x2ξ2 + ... + xnξn] (12)

A. Nazemi et al.

subject to

⎧⎨⎩

V [x1ξ1 + x2ξ2 + ... + xnξn] ≤ a,

x1 + x2 + ... + xn = 1,

xi ≥ 0, i = 1, 2, ..., n,

(13)

where E denotes the expected value operator, V is the vari-ance operator of the uncertain total return rates, and a isthe maximum risk level that investor can tolerate. When theinvestors preset an expected return level that they feel satis-factory and want to minimize the risk for this given level ofreturn, the optimization model becomes

minimize V [x1ξ1 + x2ξ2 + ... + xnξn] (14)

subject to

⎧⎨⎩

E[x1ξ1 + x2ξ2 + ... + xnξn] ≥ b,

x1 + x2 + ... + xn = 1,

xi ≥ 0, i = 1, 2, ..., n,

(15)

where b denotes the minimum expected investment returnthat the investors can accept. The traditional solution meth-ods are required to convert the objective function and theconstraints to their respective deterministic equivalents. Aswe know, this process is usually hard to perform and onlyis successful for some special cases. Let us consider thefollowing forms of the uncertain return rates.

3.1 Rectangular uncertain variables

Suppose that the return rate ξi of the ith security is a rect-angular uncertain variable, i.e. ξi = (ai, bi), i = 1, 2, ..., n,

where ai < bi . Then∑n

i=1 xiξi is a rectangular uncer-tain variable as

(∑ni=1 xiai,

∑ni=1 xibi

). According to the

properties of the rectangular uncertain variables, we get

E

[n∑

i=1

xiξi

]=(∑n

i=1 xiai +∑ni=1 xibi

)

2

=∑n

i=1 xi(bi + ai)

2,

V

[n∑

i=1

xiξi

]=(∑n

i=1 xibi −∑ni=1 xiai

)28

=[∑n

i=1 xi(bi − ai)]2

8.

Since the term [∑ni=1 xi(bi − ai)]2 is nonnegative,

V [∑ni=1 xiξi] ≤ a is equivalent to

∑ni=1 xi(bi − ai) ≤

2√

2a . Thus, the mean-variance model (12)–(13) and (14)–(15) can be respectively transformed into the following LPproblems:

maximizen∑

i=1

xi(ai + bi) (16)

subject to

⎧⎨⎩

∑ni=1 xi(bi − ai) ≤ 2

√2a,

x1 + x2 + ... + xn = 1,

xi ≥ 0, i = 1, 2, ..., n,

(17)

and

minimizen∑

i=1

xi(bi − ai) (18)

subject to

⎧⎨⎩

∑ni=1 xi(bi + ai) ≥ 2b,

x1 + x2 + ... + xn = 1,

xi ≥ 0, i = 1, 2, ..., n.

(19)

3.2 Trapezoidal uncertain variables

Suppose that the return rate ξi of the ith security is atrapezoidal uncertain variable, i.e. ξi = (ai, bi, ci , di),where ai < bi ≤ ci < di, i = 1, 2, ..., n.Then

∑ni=1 xiξi is a trapezoidal uncertain variable as(∑n

i=1 xiai,∑n

i=1 xibi,∑n

i=1 xici,∑n

i=1 xidi

). Using the

properties of the trapezoidal uncertain variables we get

E

[n∑

i=1

xiξi

]=(∑n

i=1 xiai +∑ni=1 xibi +∑n

i=1 xici +∑ni=1 xidi

)

4

=∑n

i=1 xi (ai + bi + ci + di )

4,

V

[n∑

i=1

xiξi

]= (4α2 + 3αβ + β2 + 9αγ + 3βγ + 6γ 2)

48+ [(α − β − 2γ )+]3

384α,

where

α = max

{(n∑

x=1

xibi −n∑

x=1

xiai

),

(n∑

x=1

xidi −n∑

x=1

xici

)},

β = min

{(n∑

x=1

xibi −n∑

x=1

xiai

),

(n∑

x=1

xidi −n∑

x=1

xici

)},

and

γ =n∑

x=1

xici −n∑

x=1

xibi .

Thus, the mean-variance model (12)–(13) and (14)–(15) canbe respectively converted into the following LP problems:

maximizen∑

i=1

xi(ai + bi + ci + di) (20)

subject to

⎧⎨⎩

V [x1ξ1 + x2ξ2 + ... + xnξn] ≤ a,

x1 + x2 + ... + xn = 1,

xi ≥ 0, i = 1, 2, ..., n,

(21)

and

minimize V [x1ξ1 + x2ξ2 + ... + xnξn] (22)

subject to

⎧⎨⎩

∑ni=1 xi(ai + bi + ci + di) ≥ 4b,

x1 + x2 + ... + xn = 1,

xi ≥ 0, i = 1, 2, ..., n.

(23)


3.3 Normal uncertain variables

Suppose that the return rate ξi of the ith security is normallydistributed with parameters ei and σi > 0, i = 1, 2, ..., n.i.e. ξi ∼ N (ei, σi). Then we have

E

[n∑

i=1

xiξi

]=

n∑i=1

xiei,

V

[n∑

i=1

xiξi

]=(

n∑i=1

xiσi

)2

.

Thus, the mean-variance model (12)–(13) and (14)–(15) canbe respectively reduced into the following LP problems:

maximizen∑

i=1

xiei (24)

subject to

⎧⎨⎩

∑ni=1 xiσi ≤ √

a,

x1 + x2 + ... + xn = 1,

xi ≥ 0, i = 1, 2, ..., n,

(25)

and

minimizen∑

i=1

xiσi (26)

subject to

⎧⎨⎩

∑ni=1 xiei ≥ b,

x1 + x2 + ... + xn = 1,

xi ≥ 0, i = 1, 2, ..., n.

(27)

From the above analysis, it is seen that the portfolio opti-mization problems with uncertain returns can be convertedto the LP problems. Thus we consider a general form of theLP problems given by

minimize cT x (28)

subject to

Ax − b ≤ 0, (29)

Ex − g = 0, (30)

where A ∈ Rm×n, b ∈ R

m, E ∈ Rl×n, g ∈ R

l , x ∈ Rn

and rank

(A

E

)= m + l (0 ≤ m, l < n). Throughout

this paper, we assume that problem (28)–(30) has a uniqueoptimal solution. In the next section, we will try to proposea high performance neural network model for solving LPproblem (28)–(30) and discuss its architecture.

4 A neural network model

In this section, using standard optimization techniques, wetransform (28)–(30) into a dynamic system. First we intro-duce some notation, definitions, a lemma and two basictheorems in convex optimization theory. Throughout this

paper, Rn denotes the space of n-dimensional real col-

umn vectors and T denotes the transpose. In what follows,‖.‖ denotes the l2-norm of R

n and x = (x1, x2, ..., xn)T .

For any differentiable function G : Rn → R, ∇G(x) ∈

Rn means the gradient of G at x. For any differentiable

mapping F = (F1, ...,Fm)T : Rn → R

m, ∇F =[∇F1(x), ..., ∇Fm(x)] ∈ R

n+m, denotes the transposedJacobian of F at x.

Definition 4.1 A function F : Rn → Rn is said to be Lip-

schitz continuous with constant L on Rn if for each pair of

points x, y ∈ Rn,

‖F(x) − F(y)‖ ≤ L‖x − y‖.F is said to be locally Lipschitz continuous on R

n if eachpoint of R

n has a neighborhood D0 ⊂ Rn such that the

above inequality holds for each pair of points x, y ∈ D0.

Definition 4.2 A mapping F : Rn → Rn is monotone if

(x − y)T (F(x) − F(y)) ≥ 0, ∀x, y ∈ Rn.

F is said to be strictly monotone if the strict inequality holdswhenever x = y.

Lemma 4.3 ([59], p.142) : If a mapping F is continuouslydifferentiable on an open convex set D including , thenF is monotone (strictly monotone) on if and only if itsJacobian matrix ∇F(x) is positive semidefinite (positivedefinite) for all x ∈ .

Definition 4.4 Let x(t) be a solution trajectory of a systemx′ = F(x), and let X∗ denote the set of equilibrium pointsof this equation. The solution trajectory of the system is saidto be globally convergent to the set X∗ if x(t) satisfies

limt→∞ dist(x(t), X∗) = 0,

where dist(x(t), X∗) = infy∈X∗ ‖x −y‖. In particular, if theset X∗ has only one point x∗, then limt→∞ x(t) = x∗, andthe system is said to be globally asymptotically stable at x∗if the system is also stable at x∗ in the sense of Lyapunov.

Theorem 4.5 ([3]) : x∗ ∈ Rn is an optimal solution of

(28)–(30) if and only if there exist u∗ ∈ Rm and v∗ ∈

Rl such that (x∗T , u∗T , v∗T )T satisfies the following KKT

system⎧⎨⎩

u∗ ≥ 0, Ax∗ − b ≤ 0, u∗T (Ax∗ − b) = 0,

c + AT u∗ + ET v∗ = 0,

Ex∗ − g = 0.

(31)

x∗ is called a KKT point of (28)–(30) and a pair(u∗T , v∗T )T is called the Lagrangian multiplier vector cor-responding to x∗.

A. Nazemi et al.

Theorem 4.6 ([3]) : x∗ is an optimal solution of (28)–(30)if and only if x∗ is a KKT point of (28)–(30).

Now, let x(.), u(.) and v(.) be some time dependentvariables. In order to solve the LP problem (28)–(30), weconstruct a continuous-time neural network model. Themotivation for this work arises from the fact that the pro-posed neural network will settle down to an equilibrium,which is the KKT point of the LP problem (28)–(30). Hereis such a system:

dy

dt= κξ(y), (32)

y(t0) = y0, (33)

where κ > 0 is the scale parameter, y = (xT , uT , vT )T ∈R

n+m+l , D∗ is the optimal point set of (28)–(30) and itsdual, and

ξ(y) =⎡⎣

− (c + AT (u + Ax − b)+ + ET v)

(u + Ax − b)+ − u

Ex − g

⎤⎦ ,

(u + Ax − b)+ = ([(u + Ax − b)1]+, [(u + Ax − b)2]+, ..., [(u + Ax − b)m]+) ,[(u + Ax − b)k ]+ = max{(u + Ax − b)k, 0}, k = 1, 2, ..., m.

Without loss of generality we assume κ = 1. An indi-cation on how the neural network (32) and (33) can beimplemented on hardware is provided in Fig. 1.

5 Convergence and stability properties

In this section the stability analysis and convergence prop-erties for (32) and (33) are discussed.

Theorem 5.1 Let y∗ = (x∗T , u∗T , v∗T )T be the equilib-rium point of the neural network (32) and (33). Then x∗ isa KKT point of the problem (28)–(30). On the other hand,if x∗ ∈ R

n is an optimal solution of problem (28)–(30),then there exist u∗ ∈ R

m and v∗ ∈ Rl such that y∗ =

(x∗T , u∗T , v∗T )T is an equilibrium point of the proposedneural network (32) and (33).

Proof Assume y∗ = (x∗T , u∗T , v∗T )T be the equilibriumof the neural network (32) and (33). Then dx∗

dt= 0, du∗

dt= 0

and dv∗dt

= 0. It follows easily that

c + AT (u∗ + Ax∗ − b)+ + ET v∗ = 0, (34)

(u∗ + Ax∗ − b)+ − u∗ = 0, (35)

Ex∗ − g = 0. (36)

It is clear that (u∗ + Ax∗ − b)+ = u∗ if and only if

u∗ ≥ 0, Ax∗ − b ≤ 0, u∗T(Ax∗ − b) = 0. (37)

Substituting (35) into (34) we have

c + AT u∗ + ET v∗ = 0. (38)

From (36)–(38), it is seen that y∗ = (x∗T , u∗T , v∗T )T

satisfies the KKT conditions (31).The converse is straightforward.

Lemma 5.2 (i) For any initial point y(t0) =(x(t0)

T , u(t0)T , v(t0)

T )T , there exists a unique con-tinuous solution y(t) = (x(t)T , u(t)T , v(t)T )T forsystem (32) and (33).

(ii) Let y(t) = (x(t)T , u(t)T , v(t)T )T be the state trajec-tory of (32) and (33) with the initial point y(t0) =(x(t0)

T , u(t0)T , v(t0)

T )T . If u(t0) ≥ 0, then u(t) ≥ 0.

Proof (i) It is easy to verify that c + AT (u + Ax −b)+ + ET v, (u + Ax − b)+ − u and Ex − g arelocally Lipschitz continuous. According to the localexistence of ordinary differential equations [50], neu-ral network (32) and (33) has a unique continuoussolution y(t), t ∈ [t0, τ ) for some τ > t0.

(ii) For the given initial point y(t0) with u(t0) ≥ 0 we have

du

dt+ u = (u + Ax − b)+,

∫ t

t0

(du

dt+ u

)esds =

∫ t

t0

es(u + Ax − b)+ds.

It follows

u(t) = e−(t−t0)u(t0) + e−t

∫ t

t0

es(u + Ax − b)+ds.

Since (u + Ax − b)+ ≥ 0, u(t) ≥ 0 for any t ≥ t0.

Lemma 5.3 Let A ∈ Rm×n be of full rank. Then the Jaco-

bian matrix ∇ξ(y) of the mapping ξ defined in (32) is anegative semidefinite matrix.

Proof Without loss of generality, assume that there exists0 < p < m such that

(u+Ax−b)+ =⎛⎜⎝(u + Ax − b)1, (u + Ax − b)2, ..., (u + Ax − b)p, 0, 0, ..., 0︸︷︷︸

m−p

⎞⎟⎠ .

With a simple calculation, it is clearly shown that

∇ξ(y) =⎛⎝

−(Ap)T Ap −(Ap)T −ET

Ap Tm×m Om×l

E Ol×m Ol×l

⎞⎠ ,


Fig. 1 A simplified blockdiagram for the neural network(32) and (33)

where

Ap =(

Up×n

O(m−p)×n

)=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

A1

A2

...

...

Ap

O1×n

O1×n

...

...

O1×n

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

,

and

Tm×m =(

Op×p Op×(m−p)

O(m−p)×p −I(m−p)×(m−p)

),

where O is the zero matrix. By the assumption of thislemma that A is of full rank, the first p rows of matrixAp, i.e. A1, A2, ..., Ap, are linearly independent [61]. Thusmatrix U is of full rank. Using Lemma 3.5 in [57] and thefact that (Ap)T Ap = UT U, we see that (Ap)T Ap is a pos-itive definite matrix. Moreover, it is clear that matrix Tm×m

is a negative semidefinite matrix. From the stated observa-tions, we can derive that the Jacobian matrix ∇ξ(y) is anegative semidefinite matrix.

If p = m, i.e. (u + Ax − b)+ =((u + Ax − b)1, (u + Ax − b)2, ..., (u + Ax − b)m), then

∇ξ(y) =⎛⎝

−AT A −AT −ET

A Om×m Om×l

E Ol×m Ol×l

⎞⎠ .

Similar to the previous case and the fact that A is of fullrank, it is easily proved that ∇ξ(y) is a negative semidefinitematrix.

Finally, if p = 0, i.e. (u+Ax −b)+ = (0, 0, ..., 0︸︷︷︸m

), then

we obtain

∇ξ(y) =⎛⎝

On×n On×m −ET

Om×n −Im×m Om×l

E Ol×m Ol×l

⎞⎠ .

In this case also, it is easy to verify that ∇ξ(y) is a negativesemidefinite matrix. This completes the proof.

Theorem 5.4 (i) Let the assumption of Lemma 5.3 besatisfied. Then the proposed neural network model in(32) and (33) is stable in the Lyapunov sense and isglobally convergent to y∗ = (x∗T , u∗T , v∗T )T , wherex∗ is the equilibrium point of (32) and (33).

(ii) The convergence rate of the neural network in (32) and(33) increases as κ increases.

Proof (i) ∀ y0 = (x(t0)T , u(t0)

T , v(t0)T )T with u(t0) ≥ 0,

let y(t) be the unique continuous solution of (32) and (33)defined for t0 ≤ t < τ, with y(t0) = y0, and let [t0, τ )

be its maximal interval of existence. From Lemma 5.2, itfollows that y(t) = (x(t)T , u(t)T , v(t)T )T satisfies u(t) ≥0 for any t ∈ [t0, τ ). Consider the Lyapunov function E :R

m+n+l → R as follows

E(y) = ‖ξ(y)‖2 + 1

2‖y − y∗‖2. (39)

A. Nazemi et al.

From (32), it is seen that

dξ

dt= ∂ξ

∂y

dy

dt= ∇ξ(y)ξ(y).

Calculating the derivative of E(y(t)) along the solution y(t)

of the neural network (32) and (33), we have

dE(y(t))

dt=(

dξ

dt

)T

ξ + ξT

(dξ

dt

)+ (y − y∗)T dy(t)

dt(40)

= ξT (∇ξ(y)T + ∇ξ(y))ξ + (y − y∗)T ξ(y).

Employing Lemma 5.3, we attain

ξT (y)(∇ξ(y)T + ∇ξ(y))ξ(y) ≤ 0, ∀y = y∗. (41)

Moreover, from Definiton 4.2 and Lemma 4.3 , we have

(y − y∗)T (ξ(y) − ξ(y∗)) = (y − y∗)T ξ(y) ≤ 0, ∀y = y∗.

ThusdE(y(t))

dt≤ 0. (42)

This means that the neural network (32) and (33) is stable inthe sense of Lyapunov. Next, since

E(y) ≥ 1

2‖y − y∗‖2, (43)

for any initial point y0 = (x(t0)T , u(t0)

T , v(t0)T )T ∈

Rn+m+l with u(t0) ≥ 0, the solution trajectory y(t) is

bounded on [t0, τ ). Thus τ → +∞ and there exists aconvergent subsequence

{(x(tk)T , u(tk)

T , v(tk)T )T |t0 < t1 < ... < tk < tk+1}, and tk → ∞ as k → ∞,

such that limk→∞(x(tk)T , u(tk)

T , v(tk)T )T = (xT , uT ,

vT )T , where (xT , uT , vT )T satisfies

dE(y(t))

dt= 0,

which indicates that (xT , uT , vT )T is an ω-limit point of{(x(t)T , u(t)T , v(t)T )T |t ≥ t0}.

Using the LaSalle invariant set theorem [66], onehas that {(x(t)T , u(t)T , v(t)T )T → M} as t →∞, where M is the largest invariant set in K ={(x(t)T , u(t)T , v(t)T )T | dE(y(t))

dt= 0}. From (32), (33) and

(42), it follows that dxdt

= 0, dudt

= 0 and dvdt

= 0 ⇔dE(y(t))

dt= 0. Thus (xT , uT , vT )T ∈ D∗ by M ⊆ K ⊆ D∗.

Substituting x∗ = x, u∗ = u and v∗ = v in (39), wedefine another Lyapunov function

E(y) = ‖ξ(y)‖2 + 1

2‖y − y‖2. (44)

Then E(y) is continuously differentiable and E(y) = 0.Noting that

limk→∞(x(tk)

T , u(tk)T , v(tk)

T )T = (xT , uT , vT )T ,

we therefore have limk→∞ E(x(tk)T , u(tk)

T , v(tk)T )T =

E(x, u, v).So, ∀ε > 0 there exists q > 0 such that for all t ≥ tq , we

have E(y(t)) < ε. Similarly, we can obtain dE(y(t))dt

≤ 0. Itfollows that for t ≥ tq ,

1

2‖y(t) − y‖2 ≤ E(y(t)) ≤ ε.

It follows that limt→∞ ‖y(t) − y‖ = 0 and limt→∞ y(t) =y. Therefore, the proposed neural network in (32) and(33) is globally convergent to an equilibrium point y =(xT , uT , vT )T , where x is the optimal solution of (28)–(30).(ii) From (40)–(42), it can be seen that for any κ > 0 in (32)and (33)

dE(y)

dt≤ κζ(y)T

(∇ζ(y)T + ∇ζ(y)

)ζ(y) ≤ 0.

Then

E(y(t)) ≤ E(y(t0)) + κ

∫ t

t0

ζ(y(t))T(∇ζ(y(t))T + ∇ζ(y(t))

)ζ(y(t))ds.

Since E(y) ≥ 12‖y − y∗‖2, where y∗ is a KKT point of

(28)–(30), we get

‖y(t) − y∗‖2 ≤ 2E(y(t0)) + 2κ

∫ t

t0

ζ(y(t))T(∇ζ(y(t))T + ∇ζ(y(t))

)ζ(y(t))ds.

Therefore, the convergence rate of the trajectory y(t)

increases as κ increases.

As an immediate corollary of Theorem 5.4, we can getthe following result.

Corollary 5.5 Assume that the hypothesis of Lemma 5.3is satisfied. If D∗ = {(x∗T , u∗T , v∗T )T }, then the neu-ral network (32) and (33) for solving (28)–(30) is globallyasymptotically stable to the unique equilibrium point y∗ =(x∗T , u∗T , v∗T )T .

6 Illustrative examples

In order to demonstrate the effectiveness of the proposedneural network (32) and (33), in this section, we test twoillustrative examples. The simulation is conducted on Mat-lab 7, the ordinary differential equation solver engaged isode45s.

Example 1 Assume that there are 5 securities and thereturns of these securities are all normal uncertain variables


ξi = N (ei, σi), i = 1, ..., 5, respectively. Let the returnrates be

ξ1 = N (3, 3), ξ2 = N (2, 2), ξ3 = N (4, 3),

ξ4 = N (1, 1), ξ5 = N (3, 3).

Thus we haven∑

i=1

xiξi = N (3x1 + 2x2 + 4x3 + x4 + 3x5, 3x1 + 2x2 + 3x3 + x4 + 3x5),

E[x1ξ1 + x2ξ2 + x3ξ3 + x4ξ5] = 3x1 + 2x2 + 4x3 + x4 + 3x5,

V [x1ξ1 + x2ξ2 + x3ξ3 + x4ξ5] = 3x1 + 2x2 + 3x3 + x4 + 3x5.

Suppose that the risk is not allowed to exceed 1.5, and theminimum expected return that investor can accept is 2; thenthe LP models (24)–(25) and (26)–(27) are respectively asfollows:

maximize 3x1 + 2x2 + 4x3 + x4 + 3x5 (45)

subject to

⎧⎨⎩

3x1 + 2x2 + 3x3 + x4 + 3x5 ≤ 1.5,

x1 + x2 + ... + x5 = 1,

xi ≥ 0, i = 1, ..., 5,

(46)

and

minimize 3x1 + 2x2 + 3x3 + x4 + 3x5 (47)

subject to

⎧⎨⎩

3x1 + 2x2 + 4x3 + x4 + 3x5 ≥ 2,

x1 + x2 + ... + x5 = 1,

xi ≥ 0, i = 1, ..., 5.

(48)

The optimal solution of (45) and (46) is x∗ =(0, 0, 0.25, 0.75, 0)T and the value of objective function is1.75. This means that in order to gain maximum expectedreturn with the risk not greater than 1.5, the investor shouldassign his capital according to the optimal solution x∗. Thecorresponding maximum expected return is 1.75. We applythe proposed neural network in (32) and (33) to solve (45)and (46). Simulation results show the trajectory of (32) and(33) with any initial point is always convergent to y∗ =(x∗T , u∗T , v∗T )T . For example, Fig. 2 displays the transientbehavior of (x1(t), x2(t), x3(t), x4(t), x5(t))

T based on (32)and (33) with a random initial point.

The optimal solution of model (47) and (48) is x∗ =(0, 0, 0.33, 0.66, 0)T , and the value of objective function is1.65. This means that in order to minimize the risk with theexpected value greater than 2, the investor should assign hiscapital according to x∗. The corresponding minimum riskis 1.65. Figure 3 shows that the trajectories of the proposedneural network in (32) and (33) with a random initial pointconverge to the optimal solution of the problem (47) and(48).

The question that appears here is: which of the aboveportfolios are more suitable for the investors? The portfolioformed based on the model (45) and (46) or one formed on(47) and (48)?

0 2 4 6 8 10−0.5

0

0.5

1

1.5

x1(t)=x

2(t)=x

5(t)

x3(t)

x4(t)

t

x

Fig. 2 Transient behavior of x1, x2, x3, x4 and x5 of LP (45) and (46)with κ = 10 in Example 1

When we want to compare the risk of different assetswith different expected returns, we need to use the relativemeasure. We propose the coefficient of variation (C.V):

C.Vi = σpi

E(Rpi)

About this example, as shown in Fig. 4 we observe that

E(Rmodel(45)) < E(Rmodel(47)),

σmodel(45) < σmodel(47).

Hence we have

(C.V )model(45) = 1.5

1.75= 0.857,

(C.V )model(47) = 1.65

2= 0.825.

0 2 4 6 8 10−0.5

0

0.5

1

1.5

x1(t)=x

2(t)=x

5(t)

x3(t)

x4(t)

t

x


A. Nazemi et al.

−4 −2 0 2 4 6 80

0.05

0.1

0.15

0.2

0.25

0.3

0.35model(45)−(46)model(47)−(48)

Fig. 4 Compare the risk and expected return of model (45) and (46)with model (47) and (48)

We can consequently say that the portfolio formed by model(45) and (46) is a riskier investment than portfolio formed onthe basis of model (47) and (48). Thus, the portfolio formedbased on model (47) and (48) is more suitable for investors.

Example 2 Assume that there are 6 securities and thereturns of these securities are all rectangular uncertain vari-ables ξi = (ai, bi), i = 1, ..., 6, respectively. Let the returnrates are

ξ1 = (1, 2), ξ2 = (2, 4), ξ3 = (2, 5), ξ4 = (2.5, 6.5),

ξ5 = (3, 5), ξ6 = (2.5, 4.5).

Thus we have

E

[n∑

i=1

xiξi

]=∑n

i=1 xi (bi + ai )

2= 3x1 + 6x2 + 7x3 + 9x4 + 8x5 + 7x6

2,

V

[n∑

i=1

xiξi

]=[∑n

i=1 xi (bi − ai )]2

8= [x1 + 2x2 + 3x3 + 4x4 + 2x5 + 2x6]2

8.

Suppose that the risk is not allowed to exceed 1.2, andthe minimum expected return that investor can accept is 2.4;then the LP models (16)–(17) and (18)–(19) are respectivelyas follows:

maximize 3x1 + 6x2 + 7x3 + 9x4 + 8x5 + 7x6 (49)

subject to

⎧⎨⎩

x1 + 2x2 + 3x3 + 4x4 + 2x5 + 2x6 ≤ 2√

2(1.2),x1 + x2 + ... + x6 = 1,xi ≥ 0, i = 1, ..., 6,

(50)

and

minimize x1 + 2x2 + 3x3 + 4x4 + 2x5 + 2x6 (51)

0 1 2 3 4 5−1

−0.5

0

0.5

1

1.5

x2(t)=x

3(t)=x

4(t)=x

6(t)

x1(t)

x5(t)

t

x


subject to

⎧⎨⎩

3x1 + 6x2 + 7x3 + 9x4 + 8x5 + 7x6 ≥ 2(2.4),x1 + x2 + ... + x6 = 1,xi ≥ 0, i = 1, ..., 6.

(52)

The optimal solution of (49) and (50) is x∗ =(0.64, 0, 0, 0, 0.36, 0)T and the value of objective functionis 8.54. This means that in order to gain maximum expectedreturn with risk less than 1.2, the investor should assignhis money according to x∗. The corresponding maximumexpected return is 8.54. Figure 5 shows that the outputtrajectories of the proposed neural network to solve theproblem (49)–(50) with a random initial point converge tothe optimal solution of this problem.

The optimal solution of model (51) and (52) is x∗ =(0, 0, 0, 0.55, 0.45, 0)T , and the value of objective func-tion is 1.36. This means that in order to minimize the riskwith the expected value greater than 2.4, the investor shouldassign his money according to x∗. The corresponding mini-mum risk is 1.36. Figure 6 depicts the convergence behaviorof x(t) with a random initial point. This vector converges toit’s exact solution of (51) and (52).

To end this section, we answer a natural question: arethere advantages of our proposed neural network comparedto the existing ones? To answer this, we summarize what wehave observed from numerical experiments and theoreticalresults as below.

• Compared with traditional numerical optimization algo-rithms, the neural networks have a fast convergence ratein real-time solutions.

• We compare the offered network (32) and (33) with theFriesz et al. model [16], the Kennedy and Chua’s model


0 0.5 1 1.5 2 2.5 3−2

−1

0

1

2

3

4

5

x1(t)=x

2(t)=x

3(t)=x

6(t)

x4(t)

x5(t)

t

x


[35], the proposed model in [54], Example 5.1 in [58]and Example 4.1 in [56]. At first glance, these modelslook to have lower complexity. However, we observethat the difference of the numerical performance is verymarginal by testing some special optimization problemsin these Examples.

• Compared with some existing gradient neural networkssuch as the models in [12] and [55], the proposed neuralnetwork frame (32) and (33) can be utilized without apenalty parameter. Thus this model is wholly suitable tobe implemented in hardware.

• Changing initial points may not have much effect forour neural network model, whereas it does for someexisting models. The reason is that the model is globallyconvergent to the optimal solution of the problem.

According to the above discussions, proposing anefficient neural network for solving portfolio selectionmodels with good stability properties and convergenceresults is very necessary and meaningful.

7 Conclusion

In this paper, we have proposed a high-performance neuralnetwork model for solving the portfolio selection problemwith uncertain returns that are neither random nor fuzzy.Based on some concepts of linear programming theory andordinary differential equations, we strictly prove the asymp-totic stability of the proposed network. From any initialpoint, the trajectory of this network converges to an optimalsolution of the original programming problem. The struc-ture of the proposed network is reliable and efficient. Theother advantages of the proposed neural network are that it

can be implemented without a penalty parameter and canbe convergent to an exact solution of the problem. It shouldalso be noted that Theorem 5.4 and Corollary 5.5 guaranteethat the stated model in (32) and (33) converges globally tothe unique optimal solution. The results obtained are highlyvaluable in both theory and practice for solving the portfo-lio selection with uncertain returns problems in economicsand financial mathematics.

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Alireza Nazemi received theB.Sc. degree from Sharif Uni-versity of Technology, Tehran,Iran in 2001, and the M. Sc.degree from Sabzevar TarbiatMoallem University, Sabze-var, Iran in 2003, and thePh.D degree from FerdowsiUniversity of Mashhad, Mash-had, Iran in 2008, all inapplied mathematics. He iscurrently an associate profes-sor at University of Shahrood,Shahrood, Iran. His currentresearch interests include con-trol and optimization, mathe-

matical finance, distributed parameter systems, and neural networktheory.

Behzad Abbasi received theB.Sc. degree from Universityof Kurdistan, Sanandaj, Iranin 2011, in pure mathemat-ics, and the M. Sc. degreefrom Semnan University,Semnan, Iran in 2013, infinancial mathematics. Hiscurrent research interestsinclude mathematical finance,optimization, neural networktheory, option pricing methodsand Monte-Carlo simulation.

Farahnaz Omidi received theB.Sc. degree from Universityof Shahrood, Shahrood, Iranin 2011, in applied mathe-matics, and the M. Sc. degreefrom Semnan University,Semnan, Iran in 2013, infinancial mathematics. Hercurrent research interestsinclude mathematical finance,optimization, neural networktheory and option pricingmethods.

Solving portfolio selection models with uncertain returns using an artificial neural network scheme

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