A New Cone Programming Approach for Robust

Portfolio Selection

Zhaosong Lu∗

1st version: December 10, 20062nd version: May 18, 2008

Abstract

The robust portfolio selection problems have recently been studied by several re-searchers (e.g., see [15, 14, 16, 24]). In their work, the “separable” uncertainty sets ofthe problem parameters (e.g., mean and covariance of the random returns) were con-sidered. These uncertainty sets share two common properties: i) the actual confidencelevel of the uncertainty sets is unknown, and it can be much higher than the desired one;and ii) the uncertainty sets are fully or partially box-type. The associated consequencesare that the resulting robust portfolios can be too conservative, and moreover, they areusually highly non-diversified as observed in the computational experiments conductedin this paper and [24]. To combat these drawbacks, we consider a factor model for therandom asset returns. For this model, we introduce a “joint” ellipsoidal uncertainty setfor the model parameters and show that it can be constructed as a confidence regionassociated with a statistical procedure applied to estimate the model parameters. Wefurther show that the robust maximum risk-adjusted return (RMRAR) problem withthis uncertainty set can be reformulated and solved as a cone programming problem.Computational experiments are also conducted to compare the performance of the ro-bust portfolios determined by the RMRAR model with our “joint” uncertainty set andGoldfarb and Iyengar’s “separable” uncertainty set proposed in the seminal paper [15].The computational results demonstrate that our robust portfolio outperforms Goldfarband Iyengar’s in terms of wealth growth rate and transaction cost; and moreover, oursis fairly diversified, but Goldfarb and Iyengar’s is surprisingly highly non-diversified.

Key words: Robust optimization, mean-variance portfolio selection, maximum risk-adjusted return portfolio selection, cone programming, linear regression.

MSC 2000 subject classification. 91B28, 90C20, 90C22.

∗Department of Mathematics, Simon Fraser University, Burnaby, BC, V5A 1S6, Canada. (email:zhaosong@sfu.ca). This author was supported in part by SFU President’s Research Grant and NSERCDiscovery Grant.

1

1 Introduction

Portfolio selection problem is concerned with determining a portfolio such that the “return”and “risk” of the portfolio have a favorable trade-off. The first mathematical model forportfolio selection problem was developed by Markowitz [19] five decades ago, in which anoptimal or efficient portfolio can be identified by solving a convex quadratic program. In hismodel, the “return” and “risk” of a portfolio are measured by the mean and variance of therandom portfolio return, respectively. Thus, the Markowitz portfolio model is also widelyreferred to as the mean-variance model.

Despite the theoretical elegance and importance of the mean-variance model, it continuesto encounter skepticism among the investment practitioners. One of the main reasons isthat the optimal portfolios determined by the mean-variance model are often sensitive toperturbations in the parameters of the problem (e.g., expected returns and the covariancematrix), and thus lead to large turnover ratios with periodic adjustments of the problemparameters; see for example Michaud [20]. Various aspects of this phenomenon have also beenextensively studied in the literature, for example, see [9, 7, 8].

As a recently emerging modeling tool, robust optimization can incorporate the perturba-tions in the parameters of the problems into the decision making process. Generally speaking,robust optimization aims to find solutions to given optimization problems with uncertainproblem parameters that will achieve good objective values for all or most of realizations ofthe uncertain problem parameters. For details, see [2, 3, 4, 11, 13]. Recently, robust op-timization has been applied to model portfolio selection problems in order to alleviate thesensitivity of optimal portfolios to statistical errors in the estimates of problem parameters.For example, Goldfarb and Iyengar [15] considered a factor model for the random portfolioreturns, and proposed some statistical procedures for constructing uncertainty sets for themodel parameters. For these uncertainty sets, they showed that the robust portfolio selec-tion problems can be reformulated as second-order cone programs. Subsequently, Erdoganet al. [14] extended this method to robust index tracking and active portfolio managementproblems. Alternatively, Tutuncu and Koenig [24] (see also Halldorsson and Tutuncu [16])considered a box-type uncertainty structure for the mean and covariance matrix of the assetsreturns. For this uncertainty structure, they showed that the robust portfolio selection prob-lems can be formulated and solved as smooth saddle-point problems that involve semidefiniteconstraints. In addition, for finite uncertainty sets, Ben-Tal et al. [1] studied the robust for-mulations of multi-stage portfolio selection problems. Also, El Ghaoui et al. [12] consideredthe robust value-at-risk (VaR) problems given the partial information on the distribution ofthe returns, and they showed that these problems can be cast as semidefinite programs. Zhuand Fukushima [26] showed that the robust conditional value-at-risk (CVaR) problems can bereformulated as linear programs or second-order cone programs for some simple uncertaintystructures of the distributions of the returns. Recently, DeMiguel and Nogales [10] proposeda novel approach for portfolio selection by minimizing certain robust estimators of portfoliorisk. In their method, robust estimation and portfolio optimization are performed by solvinga single nonlinear program.

2

The structure of uncertainty set is an important ingredient in formulating and solvingrobust portfolio selection problems. The “separable” uncertainty sets have been commonlyused in the literatures. For example, Tutuncu and Koenig [24] (see also Halldorsson andTutuncu [16]) proposed the box-type uncertainty sets Sm = {µ : µL ≤ µ ≤ µU} and Sv ={Σ : Σ � 0, ΣL ≤ Σ ≤ ΣU} for the mean µ and covariance Σ of the asset return vector r,respectively. Here A � 0 (resp. ≻ 0) denotes that the matrix A is symmetric and positivesemidefinite (resp. definite). In the seminal paper [15], Goldfarb and Iyengar studied a factormodel for the asset return vector r in the form of

r = µ + V T f + ǫ,

where µ is the mean return vector, f is the random factor return vector, V is the factor loadingmatrix and ǫ is the residual return vector (see Section 2 for details). They also proposed someuncertainty sets Sm and Sv for µ and V , respectively; in particular, Sm is a box (see (2))and Sv is a Cartesian product of a bunch of ellipsoids (see (3)). It shall be mentioned thatall “separable” uncertainty sets above share two common properties: i) viewed as a jointuncertainty set, the actual confidence level of Sm × Sv is unknown though Sm and/or Sv

may have known confidence levels individually; and ii) Sm ×Sv is fully or partially box-type.The associated consequences are that the resulting robust portfolio can be too conservativesince the actual confidence level of Sm × Sv can be much higher than the desired one, orequivalently, the uncertainty set Sm × Sv can be too noisy; and moreover it is highly non-diversified as observed in the computational experiments conducted in this paper and [24].These drawbacks will be further addressed in details in Sections 2 and 5.

In this paper, we also consider the same factor model for asset returns as studied in [15]. Tocombat the aforementioned drawbacks, we propose a “joint” ellipsoidal uncertainty set for themodel parameters (µ, V ), and show that it can be constructed as a confidence region associatedwith a statistical procedure applied to estimate (µ, V ) for any desired confidence level. Wefurther show that for this uncertainty set, the robust maximum risk-adjusted return (RMRAR)problem can be reformulated and solved as a cone programming problem. Computationalexperiments are also conducted to compare the performance of the robust portfolio determinedby the RMRAR model with our “joint” uncertainty set and Goldfarb and Iyengar’s “separable”uncertainty set proposed in the seminal paper [15]. The computational results demonstratethat our robust portfolio outperforms Goldfarb and Iyengar’s in terms of wealth growth rateand transaction cost; and moreover, our robust portfolio is fairly diversified, but Goldfarb andIyengar’s is surprisingly highly non-diversified.

The rest of this paper is organized as follows. In Section 2, we describe the factor modelfor asset returns that was studied in Goldfarb and Iyengar [15], and briefly review the statis-tical procedure proposed in [15] for constructing a “separable” uncertainty set of the modelparameters. The associated drawbacks of this uncertainty set are also addressed. In Section3, we introduce a “joint” uncertainty set for the model parameters, and propose a statisticalprocedure for constructing it for any desired confidence level. Several robust portfolio selectionproblems for this uncertainty set are also discussed. In Section 4, we show that for our “joint”uncertainty set, the RMRAR problem can be reformulated and solved as a cone programming

3

problem. In Section 5, we report the computational results for the RMRAR models with our“joint” uncertainty set and Goldfarb and Iyengar’s “separable” uncertainty set, respectively.Finally, we give some concluding remarks in Section 6.

2 Factor model and separable uncertainty sets

In this section, we first describe the factor model for asset returns that was studied in Goldfarband Iyengar [15]. Then we briefly review the statistical procedure proposed in [15] for con-structing a “separable” uncertainty set for the model parameters. The associated drawbacksof this uncertainty set are also addressed.

The following factor model for asset returns was studied in [15]. Suppose that a discrete-time market has n traded assets. The vector of asset returns over a single market period isdenoted by r ∈ ℜn. The returns on the assets in different market periods are assumed to beindependent. The single period return r is assumed to be a random vector given by

r = µ + V T f + ǫ, (1)

where µ ∈ ℜn is the vector of mean returns, f ∼ N (0, F ) ∈ ℜm denotes the returns of them factors driving the market, V ∈ ℜm×n denotes the factor loading matrix of the n assets,and ǫ ∼ N (0, D) ∈ ℜn is the vector of residual returns. Here x ∼ N (µ, Σ) denotes thatx is a multivariate normal random variable with mean µ and covariance Σ. Further, it isassumed that D is a positive semidefinite diagonal matrix, and the residual return vector ǫ isindependent of the factor return vector f . Thus, it can be seen from the above assumptionthat r ∼ N (µ, V T FV + D).

Goldfarb and Iyengar [15] also studied a robust model for (1). In their model, the meanreturn vector µ is assumed to lie in the uncertainty set Sm given by

Sm = {µ : µ = µ0 + ξ, |ξi| ≤ γi, i = 1, . . . , n}, (2)

and the factor loading matrix V is assumed to belong to the uncertainty set Sv given by

Sv = {V : V = V0 + W, ‖Wi‖G ≤ ρi, i = 1, . . . , n}, (3)

where Wi is the ith column of W and ‖w‖G =√

wT Gw denotes the elliptic norm of w withrespect to a symmetric, positive definite matrix G.

Two analogous statistical procedures were proposed in [15] for constructing the aboveuncertainty sets Sm and Sv. As observed in our computational experiments, the behavior ofthese uncertainty sets is almost identical. Therefore, we only briefly review their first statisticalprocedure as follows. Suppose the market data consists of asset returns {rt : t = 1, . . . , p} andfactor returns {f t : t = 1, . . . , p} for p trading periods. Then the linear model (1) implies that

rti = µi +

m∑

j=1

Vjiftj + ǫt

i, i = 1, . . . , n, t = 1, . . . , p. (4)

4

As in the typical linear regression analysis, it is assumed that {ǫti : i = 1, . . . , n, t = 1, . . . , p}

are all independent normal random variables and ǫti ∼ N (0, σ2

i ) for all t = 1, . . . , p. Now, letB = (f 1, f 2, . . . , f p) ∈ ℜm×p denote the matrix of factor returns, and let e ∈ ℜp denote anall-one vector. Further, define

yi = (r1i , r

2i , . . . , r

pi )

T , A = (e BT ), xi = (µi, V1i, V2i, . . . , Vmi)T , ǫi = (ǫ1

i , . . . , ǫpi )

T (5)

for i = 1, . . . , n. Then (4) can be rewritten as

yi = Axi + ǫi, ∀i = 1, . . . , n. (6)

Due to p ≫ m usually in practice, it was assumed in [15] that the matrix A has full columnrank. As a result, it follows from (6) that the least-squares estimate xi of the true parameterxi is given by

xi = (AT A)−1AT yi. (7)

The following result has played a crucial role in constructing the uncertainty sets Sm andSv in [15]. It will also be used to build a ‘joint” ellipsoidal uncertainty set in Section 3.

Theorem 2.1 Let s2i be the unbiased estimate of σ2

i given by

s2i =

‖yi − Axi‖2

p − m − 1(8)

for i = 1, . . . , n. Then the random variables

Yi =

1

(m + 1)s2i

(xi − xi)T AT A(xi − xi), i = 1, . . . , n, (9)

are distributed according to the F -distribution with m+1 degrees of freedom in the numeratorand p − m − 1 degrees of freedom in the denominator. Moreover, {Y i}n

i=1 are independent.

Proof. The first statement was shown in [15] (see page 16 of [15]). We now prove thesecond statement. In view of (6) and (7), we obtain that

xi − xi = (AT A)−1AT ǫi, yi − Axi = [I − A(AT A)−1AT ]ǫi.

Using these relations and the assumption that {ǫi : i = 1, . . . , n} are independent, we concludethat {Y i}n

i=1 are independent.

Let FJ denote the cumulative distribution function of the F -distribution with J degreesof freedom in the numerator and p−m−1 degrees of freedom in the denominator. Given anyω ∈ (0, 1), let cJ(ω) be its ω-critical value, i.e., FJ(cJ(ω)) = ω. Also, we let

Si(ω) ={

xi : (xi − xi)T AT A(xi − xi) ≤ (m + 1)cm+1(ω)s2

i

}

, i = 1, . . . , n, (10)

S(ω) = S1(ω) × S2(ω) × · · · × Sn(ω).

5

Using Theorem 2.1, Goldfarb and Iyengar showed that S(ω) is an ωn-confidence set of (µ, V ).Let Sm(ω) and Sv(ω) denote the projection of S(ω) along µ, V , respectively. Their explicitexpressions can be found in Section 5 of [15], and they are in the form of (2)-(3). Goldfarband Iyengar [15] set Sm := Sm(ω) and Sv := Sm(ω), and viewed them as the ωn-confidencesets of µ and V , respectively. However, we immediately observe that

P (µ ∈ Sm(ω)) ≥ P ((µ, V ) ∈ S(ω)) = ωn,

and similarly, P (V ∈ Sv(ω)) ≥ ωn. Hence, Sm and Sv have at least ωn-confidence levels, buttheir actual confidence levels are unknown and can be much higher than ωn. Further, in viewof the relation S(ω) ⊆ Sm(ω) × Sv(ω), one has

P ((µ, V ) ∈ Sm(ω) × Sv(ω)) ≥ P ((µ, V ) ∈ S(ω)) = ωn.

Thus Sm(ω) × Sv(ω), as a joint uncertainty set of (µ, V ), has at least ωn-confidence level.However, its actual confidence level is unknown and can be much higher than the desiredone that is ωn. One immediate consequence is that the robust portfolio selection modelsbased on such uncertainty sets Sm(ω) and Sv(ω) can be too conservative. In addition, weobserve from computational experiments that the resulting robust portfolios are highly non-diversified, in other words, they concentrate only on a few assets. One possible interpretationof this phenomenon is that Sm(ω) has a box-type structure. To combat these drawbacks, weintroduce a “joint” ellipsoidal uncertainty set for (µ, V ) in Section 3, and show that it can beconstructed by a statistical approach.

Before ending this section, we shall remark that in Goldfarb and Iyengar’s robust factormodel, some uncertainty structures were also proposed for the parameters F and D thatare the covariances of factor and residual returns. Nevertheless, they are often assumed to befixed in practical computations, and can be estimated by some standard statistical approaches(see Section 7 of [15]). For the brevity of presentation, we assume that F and D are fixedthroughout the rest of paper. But we shall mention that the results of this paper can beextended to the case where F and D have the same uncertainty structures as described in[15].

3 Joint uncertainty set and robust portfolio selection

models

In this section, we consider the same factor model for asset returns as described in Section 2.In particular, we first introduce a “joint” ellipsoidal uncertainty set for the model parameters(µ, V ). Then we propose a statistical procedure for constructing it. Finally, we discuss severalrobust portfolio selection models for this uncertainty set.

Throughout this section, we assume that all notations are given in Section 2, unless ex-plicitly defined otherwise.

6

Recall from Section 2 that the “separable” uncertainty set Sm(ω)×Sv(ω) of (µ, V ) proposedin [15] has several drawbacks. To overcome these drawbacks, we consider a “joint” ellipsoidaluncertainty set of (µ, V ) with ω-confidence level in the form of

Sµ,v(ω) =

{

(µ, V ) ∈ ℜn × ℜm×n :n∑

i=1

(xi − xi)T (AT A)(xi − xi)

s2i

≤ (m + 1)c(ω)

}

(11)

for some c(ω), where xi = (µi, V1i, V2i, . . . , Vmi)T for i = 1, . . . , n. Using the definition of

{Y i}ni=1 (see (9)), we can observe that the following property holds for Sµ,v(ω).

Proposition 3.1 Sµ,v(ω) is an ω-confidence uncertainty set of (µ, V ) for some c(ω) if and

only if P(n∑

i=1

Y i ≤ c(ω)) = ω, that is, c(ω) is the ω-critical value ofn∑

i=1

Y i.

We are now ready to propose a statistical procedure for constructing the aforementioned“joint” ellipsoidal uncertainty set for the parameters (µ, V ). First, we know from Theorem2.1 that the random variables {Y i}n

i=1 have the F -distribution with m+1 degrees of freedomin the numerator and p − m − 1 degrees of freedom in the denominator. It follows from astandard statistical result (e.g., see [22]) that their mean and standard deviation are

µF =p − m − 1

p − m − 3, σF =

√

2(p − m − 1)2(p − 2)

(m + 1)(p − m − 3)2(p − m − 5), (12)

respectively, provided that p > m + 5, which often holds in practice. In view of Theorem 2.1,we also know that {Y i}n

i=1 are i.i.d.. Using this fact and the central limit theorem (e.g., see[18]), we conclude that the distribution of the random variable

Zn =

n∑

i=1

Y i − nµF

σF

√n

converges towards the standard normal distribution N (0, 1) as n → ∞. Notice that when napproaches a couple dozen, the distribution of Zn is very nearly N (0, 1). Given an ω ∈ (0, 1),let c(ω) be the ω-critical value for a standard normal variable Z , i.e., P (Z ≤ c(ω)) = ω.Then we have lim

n→∞P (Zn ≤ c(ω)) = ω. Hence, for a relatively large n,

P

n∑

i=1

Y i − nµF

σF

√n

≤ c(ω)

≈ ω,

and hence, P(n∑

i=1

Y i ≤ c(ω)) ≈ ω, where c(ω) = c(ω)σF

√n + nµF . In view of this result

and Proposition 3.1, one can see that the set Sµ,V given in (11) with such a c(ω) is an ω-confidence uncertainty set of (µ, V ) when n is relatively large (say a couple dozen). We next

7

consider the case where n is relatively small. Recall that {Y i}ni=1 are i.i.d. and have F -

distribution. Thus, we can apply simulation techniques (e.g., see [21]) to find a h(ω) such

that P(n∑

i=1

Y i ≤ h(ω)) ≈ ω. In view of this relation and Proposition 3.1, we immediately

see that the set Sµ,V given in (11) with c(ω) = h(ω) is an ω-confidence uncertainty set of(µ, V ). Therefore, for every ω ∈ (0, 1), one can apply the above statistical procedure to buildan uncertainty set of (µ, V ) in the form of (11) with ω-confidence level.

From now on, we assume that for every ω ∈ (0, 1), Sµ,v(ω) given in (11), simply denotedby Sµ,v, is a “joint” ellipsoidal uncertainty set of (µ, V ) with ω-confidence level. In view of (9),one has

∑ni=1 Y i ≥ 0. Moreover, we know from Theorem 2.1 that

∑ni=1 Y i is a continuous

random variable. Using these facts and Proposition 3.1, one can observe that the followingproperty holds for Sµ,v:

c(ω) > 0, if ω > 0. (13)

We next consider several robust portfolio selection problems for the “joint” ellipsoidaluncertainty set Sµ,v. Indeed, an investor’s position in market can be described by a portfolioφ ∈ ℜn, where the ith component φi represents the fraction of total wealth invested in ithasset. The return rφ on the portfolio φ is given by

rφ = rT φ = µTφ + fT V φ + ǫT φ ∼ N (φTµ, φT (V T FV + D)φ), (14)

and hence, the mean and variance of rφ are

E[rφ] = φT µ, Var[rφ] = φT (V T FV + D)φ, (15)

respectively. For convenience, we assume that short sales are not allowed, i.e., φ ≥ 0. Let

Φ = {φ : eT φ = 1, φ ≥ 0}. (16)

The objective of the robust maximum return problem is to maximize the worst case expectedreturn subject to a constraint on the worst case variance, i.e., to solve the following problem

maxφ

min(µ,V )∈Sµ,v

E[rφ]

s.t. max(µ,V )∈Sµ,v

Var[rφ] ≤ λ,

φ ∈ Φ.

(17)

A closely related problem, the robust minimum variance problem, is the “dual” of (17). Theobjective of this problem is to minimize the worst case variance of the portfolio subject to aconstraint on the worst case expected return on the portfolio. It can be formulated as

minφ

max(µ,V )∈Sµ,v

Var[rφ]

s.t. min(µ,V )∈Sµ,v

E[rφ] ≥ β,

φ ∈ Φ.

(18)

8

We now address a drawback associated with problem (17). Let

Sµ = {µ : (µ, V ) ∈ Sµ,v for some V }, SV = {V : (µ, V ) ∈ Sµ,v for some µ}.

In view of (15), we observe that (17) is equivalent to

maxφ

min(µ,V )∈Sµ×SV

E[rφ]

s.t. max(µ,V )∈Sµ×SV

Var[rφ] ≤ λ,

φ ∈ Φ.

(19)

Hence, the uncertainty set of (µ, V ) used in problem (17) is essentially Sµ × SV . Recall thatSµ,v has ω-confidence level, and hence, we have

P ((µ, V ) ∈ Sµ × SV ) ≥ P ((µ, V ) ∈ Sµ,v) = ω.

Using this relation, we immediately conclude that Sµ×SV has at least ω-confidence level, butits actual confidence level is unknown and can be much higher than the desired ω. Hence,problem (17) can be too conservative even though Sµ,v has the desired confidence level ω.Using a similar argument, we see that problem (18) also has this drawback.

To combat the drawback of problem (17), we establish the following two propositions.

Proposition 3.2 Letλl = min

φ∈ΦmaxV ∈SV

Var[rφ],

and let φ∗λ denote an optimal solution of problem (17) for any λ > λl. Then, φ∗

λ is also anoptimal solution of the following problem

maxφ∈Φ

min(µ,V )∈Sµ×SV

E[rφ] − θVar[rφ], (20)

for some θ ≥ 0.

Proof. Recall that problem (17) is equivalent to problem (19). It implies that φ∗λ is also

an optimal solution of (19). Let

f(φ) = min(µ,V )∈Sµ×SV

E[rφ], g(φ) = max(µ,V )∈Sµ×SV

Var[rφ].

In view of (15), we see that f(φ) is concave and g(φ) is convex over the convex compact setΦ. For any λ > λl, we can observe that: i) problem (19) is feasible and its optimal valueis finite; and ii) there exists a φ0 ∈ Φ such that g(φ0) < λ. Hence, by Proposition 5.3.1 ofBertsekas [5], we know that there exists at least one Lagrange multiplier θ ≥ 0 such that φ∗

λ

solves problem (20) for such θ, and the conclusion follows.

We observe that the converse of Proposition 3.2 also holds.

9

Proposition 3.3 Let φ∗θ denote an optimal solution of problem (20) for any θ ≥ 0. Then, φ∗

θ

is also an optimal solution of problem (17) for

λ = maxV ∈SV

Var[rφ∗

θ]. (21)

Proof. We first observe that φ∗θ is a feasible solution of problem (17) with λ given by (21).

Now assume for a contradiction that there exits a φ ∈ Φ such that

max(µ,V )∈Sµ,v

Var[rφ] ≤ λ, min(µ,V )∈Sµ,v

E[rφ] > min(µ,V )∈Sµ,v

E[rφ∗

θ],

or equivalently,maxV ∈SV

Var[rφ] ≤ λ, minµ∈Sµ

E[rφ] > minµ∈Sµ

E[rφ∗

θ]

due to (15). These relations together with (15) and (21) imply that for such φ,

min(µ,V )∈Sµ×SV

E[rφ] − θVar[rφ] = minµ∈Sµ

E[rφ] − θ maxV ∈SV

Var[rφ],

> minµ∈Sµ

E[rφ∗

θ] − θ max

V ∈SV

Var[rφ∗

θ],

= min(µ,V )∈Sµ×SV

E[rφ∗

θ] − θVar[rφ∗

θ],

which is a contradiction to the fact that φ∗θ is an optimal solution of problem (20). Thus, the

conclusion holds.

In view of Propositions 3.2 and 3.3, we conclude that problems (17) and (20) are equivalent.We easily observe that the conservativeness of problem (20) (or equivalently (17)) can bealleviated if we replace Sµ × SV by Sµ,v in (20). This leads to the robust maximum risk-adjusted return (RMRAR) problem with the uncertainty set Sµ,v:

maxφ∈Φ

min(µ,V )∈Sµ,v

E[rφ] − θVar[rφ], (22)

where θ ≥ 0 represents the risk-aversion parameter.In contrast with problems (17) and (18), the RMRAR problem (22) has a clear advantage

that the confidence level of its underlying uncertainty set is controllable. In next section, weshow that problem (22) can be reformulated as a cone programming problem.

4 Cone programming reformulation

In this section, we show that the RMRAR problem (22) can be reformulated as a cone pro-gramming problem.

In view of (14), the RMRAR problem (22) can be written as

maxφ∈Φ

{

min(µ,V )∈Sµ,v

{

µTφ − θφT V T FV φ}

− θφT Dφ

}

. (23)

10

By introducing auxiliary variables ν and t, problem (23) can be reformulated as

maxφ,ν,t

ν − θt

s.t. min(µ,V )∈Sµ,v

{

µT φ − θφT V T FV φ}

≥ ν,

φTDφ ≤ t,φ ∈ Φ.

(24)

We next aim to reformulate the inequality

min(µ,V )∈Sµ,v

{

µTφ − θφT V T FV φ}

≥ ν (25)

as linear matrix inequalities (LMIs). Before proceeding, we introduce two lemmas that willbe used subsequently.

The following lemma is about S -procedure. For a discussion of S -procedure and itsapplications, see Boyd et al. [6].

Lemma 4.1 Let Fi(x) = xT Aix + 2bix + ci, i = 0, . . . , p be quadratic functions of x ∈ ℜn.Then F0(x) ≤ 0 for all x such that Fi(x) ≤ 0, i = 1, . . . , p, if there exists τi ≥ 0 such that

p∑

i=1

τi

(

ci bTi

bi Ai

)

−(

c0 bT0

b0 A0

)

� 0.

Moreover, if p = 1 then the converse holds if there exists x0 ∈ ℜn such that F1(x0) < 0.

In the next lemma, we state one simple property of the standard Kronecker product,denoted by ⊗. For its proof, see [17].

Lemma 4.2 If H � 0 and K � 0, then H ⊗ K � 0.

We are now ready to show that the inequality (25) can be reformulated as LMIs.

Lemma 4.3 Let Sµ,v be an ω-confidence uncertainty set given in (11) for ω ∈ (0, 1). Then,the inequality (25) is equivalent to the following LMIs

τR − 2θS ⊗(

0 00 F

)

τh + q

τhT + qT τη − 2ν

� 0,

(

1 φT

φ S

)

� 0, τ ≥ 0,

(26)

11

where

R =

AT As2

1

. . .AT As2n

∈ ℜ[(m+1)n]×[(m+1)n], η =n∑

i=1

xTi

(

AT As2

i

)

xi − c(ω), (27)

h =

−AT Ax1

s2

1

...

−AT Axn

s2n

∈ ℜ(m+1)n, q =

φ1

0...

φn

0

∈ ℜ(m+1)n (28)

(here, 0 denotes the m-dimensional zero vector).

Proof. Given any (ν, θ, φ) ∈ ℜ × ℜ× ℜn, we define

H(µ, V ) = −µT φ + θφT V T FV φ + ν.

As in (5), let xi = (µi, V1i, V2i, . . . , Vmi)T for i = 1, . . . , n. Viewing H(µ, V ) as a function of

x = (x1, . . . , xn) ∈ ℜ(m+1)n, we have

∂H

∂xi

=

(

−φi

2θφiFV φ

)

,∂2H

∂xi∂xj

=

(

0 00 2θφiφjF

)

, i, j = 1, . . . , n.

Using these relations and performing the Taylor series expansion for H(µ, V ) at x = 0, weobtain that

H(µ, V ) =1

2

n∑

i,j=1

xTi

(

0 00 2θφiφjF

)

xj +n∑

i=1

(

−φi

0

)T

xi + ν. (29)

Notice that Sµ,v is given by (11). It can be written as

Sµ,v =

{

(µ, V ) :n∑

i=1

xTi

(

AT A

s2i

)

xi − 2n∑

i=1

(

AT Axi

s2i

)T

xi +n∑

i=1

xTi

(

AT A

s2i

)

xi − c(ω) ≤ 0

}

.

(30)In view of (13) and the fact that ω > 0, we have c(ω) > 0. Hence, we see that x = xstrictly satisfies the inequality given in (30). Using (29), (30) and Lemma 4.1, we concludethat H(µ, V ) ≤ 0 for all (µ, V ) ∈ Sµ,v if and only if there exists a τ ∈ ℜ such that

τ

(

R hhT η

)

−(

E −q−qT 2ν

)

� 0, τ ≥ 0, (31)

where R, η, h and q are defined in (27) and (28), respectively, and E is given by

E = (Eij) ∈ ℜ[(m+1)n]×[(m+1)n], Eij =

(

0 00 2θφiφjF

)

∈ ℜ(m+1)×(m+1), i, j = 1, . . . , n.

12

In terms of Kronecker product ⊗, we can rewrite E as

E = 2θ(φφT ) ⊗(

0 00 F

)

.

This identity together with Lemma 4.2 and the fact that F � 0, implies that (31) holds if andonly if

τR − 2θS ⊗(

0 00 F

)

τh + q

τhT + qT τη − 2ν

� 0, S � φφT , τ ≥ 0. (32)

Using Schur Complement Lemma, we further observe that (32) holds if and only if (26) holds.Thus, we see that H(µ, V ) ≤ 0 for all (µ, V ) ∈ Sµ,v if and only if (26) holds. Then theconclusion immediately follows from this result and the fact that the inequality (25) holds ifand only if H(µ, V ) ≤ 0 for all (µ, V ) ∈ Sµ,v.

In the following theorem, we show that the RMRAR problem (22) can be reformulated asa cone programming problem.

Theorem 4.4 Let Sµ,v be an ω-confidence uncertainty set given in (11) for ω ∈ (0, 1). Then,the RMRAR problem (22) is equivalent to

maxφ,S,τ,ν,t

ν − θt

s.t.

τR − 2θS ⊗(

0 00 F

)

τh + q

τhT + qT τη − 2ν

� 0,

(

1 φT

φ S

)

� 0,

1 + t1 − t

2D1/2φ

∈ Ln+2,

τ ≥ 0, φ ∈ Φ,

(33)

where Φ, R, η, h and q are defined in (16), (27) and (28), respectively, and Lk denotes thek-dimensional second-order cone given by

Lk =

z ∈ ℜk : z1 ≥

√

√

√

√

k∑

i=2

z2i

.

Proof. We observe that the inequality φTDφ ≤ t is equivalent to the third constraint of(33), which together with Lemma 4.3, implies that (24) is equivalent to (33). The conclusionimmediately follows from this result and the fact that (22) is equivalent to (24).

The following theorem establishes the solvability of problem (33). Its proof is given in theappendix.

13

Theorem 4.5 Assume that 0 6= F � 0, ω ∈ (0, 1) and θ > 0. Then, problem (33) and itsdual problem are both strictly feasible, and hence, both problems are solvable and the dualitygap is zero.

In view of Theorem 4.5, we conclude that problem (33) can be suitably solved by primal-dual interior point solvers (e.g., SeDuMi [23] and SDPT3 [25]).

5 Computational results

In this section, we present computational experiments on the RMRAR models. We conducttwo types of computational tests. The first type of tests are based on simulated data, and thesecond type of tests use real market data. The main objective of these computational tests isto compare the performance of the RMRAR models with our “joint” uncertainty set (11) andGoldfarb and Iyengar’s “separable” uncertainty set (2)-(3). All computations are performedusing SeDuMi V1.1R2 [23]. Throughout this section, the symbols “LROB” and “GIROB” areused to label the robust portfolios determined by the RMRAR models with our “joint” andGoldfarb and Iyengar’s “separable” uncertainty sets, respectively. The following terminologywill also be used in this section.

Definition 1 The diversification number of a portfolio is defined as the number of its com-ponents that are above 1%.

5.1 Computational results for simulated data

In this subsection, we conduct computational tests for simulated data. The data is generatedin the same manner as described in Section 7 of [15]. Indeed, we fix the number of assetsn = 50 and the number of factors m = 5. A symmetric positive definite factor covariancematrix F is randomly generated, and it is assumed to be certain. The nominal factor loadingmatrix V is also randomly generated. The covariance matrix D of the residual returns ǫ isassumed to be certain and set to D = 0.1 diag(V T FV ), that is, the linear model explains 90%of the asset variance. The nominal asset returns µ ∈ ℜn are chosen independently accordingto a uniform distribution on [0.5%, 1.5%]. Finally, we generate a sequence of asset and factorreturn vectors r and f according to the normal distributions N (µ, V T FV + D) and N (0, F )for an investment period of length p = 90, respectively. We also randomly generate the assetreturns, denoted by R ∈ ℜn×p, for next period of length p according to N (µ, V T FV + D). Inaddition, given a desired confidence level ω > 0, our “joint” uncertainty set Sµ,v is built asin Section 3, and Goldfarb and Iyengar’s “separable” uncertainty set Sm × Sv is built as inSection 2 with ω = ω1/n. As discussed in Section 2, Sm × Sv has at least ω-confidence level,but its actual confidence level is unknown.

Let φr be a robust portfolio computed from the data of the current period. Suppose thatφr is held constant for the investment over next period. The wealth growth rate of φr overnext period is defined as

(Π1≤k≤p(e + Rk))T φr − 1, (34)

14

0 5 100

5

10

15

20

25

30ω=0.05

risk aversion

dive

rsifi

catio

n nu

mbe

r

LROBGIROB

0 5 102.8

2.85

2.9

2.95

3ω=0.05

risk aversion

wea

lth g

row

th r

ate

LROBGIROB

0 5 100

5

10

15

20

25

30ω=0.50

risk aversion

dive

rsifi

catio

n nu

mbe

r

LROBGIROB

0 5 10

2.7

2.75

2.8

2.85

2.9ω=0.50

risk aversion

wea

lth g

row

th r

ate

LROBGIROB

0 5 100

5

10

15

20

25

30ω=0.95

risk aversion

dive

rsifi

catio

n nu

mbe

r

LROBGIROB

0 5 102.6

2.65

2.7

2.75

2.8

2.85ω=0.95

risk aversion

wea

lth g

row

th r

ate

LROBGIROB

Figure 1: Performance of portfolios for ω = 0.05, 0.50 and 0.95.

where e ∈ ℜn denotes the all-one vector and Rk ∈ ℜn denotes the kth column of R fork = 1, . . . , p.

We next report the performance of the RMRAR models with our “joint” uncertaintyset (11) and Goldfarb and Iyengar’s “separable” uncertainty set described in (2)-(3) as therisk aversion parameter θ ranges from 0 to 10. The computational results averaged over 10randomly generated instances are shown in Figure 1 that consists of three groups of plotsfor ω = 0.05, 0.50, 0.95, respectively. In each of these three groups, the left plot is about thediversification number of the robust portfolios, and the right plot is about the wealth growthrate of the robust portfolios over next period. We first observe that our robust portfoliois fairly diversified, but Goldfarb and Iyengar’s is highly non-diversified. Indeed, for ω =0.05, 0.50, 0.95, the diversification number of our robust portfolio is around 26, and that ofGoldfarb and Iyengar’s is around one or two. One possible interpretation of this phenomenon isthat our uncertainty set Sµ,v is ellipsoidal, but Goldfarb and Iyengar’s uncertainty set Sm×Sv

is partially box-type. It seems that the ellipsoidal uncertainty structure tends to producemore diversified robust portfolio than does the fully or partially box-type one. In addition, weobserve that for ω = 0.05 or ω = 0.50 with a relatively small θ, the wealth growth rate of ourrobust portfolio is lower than that of Goldfarb and Iyengar’s. But for ω = 0.95 or ω = 0.50with a relatively large θ, our wealth growth rate is higher than Goldfarb and Iyengar’s. Thisphenomenon is actually not surprising. Indeed, we know that Sµ,v has confidence ω whileSm ×Sv has at least ω confidence, but its actual confidence level can be much higher than ω.Hence for a small ω, the associated model with Sm × Sv can be more robust than that withSµ,v. However, for a relatively large ω, the uncertainty set Sm × Sv can be over confident,and its corresponding robust model can be conservative. This phenomenon becomes moreprominent as the risk aversion parameter θ gets larger.

5.2 Computational results for real market data

In this subsection, we perform experiments on real market data for the RMRAR models withour “joint” and Goldfarb and Iyengar’s “separable” uncertainty sets. The universe of assetsthat are chosen for investment are those ranked at the top of each of 10 industry categoriesby Fortune 500 in 2006. In total there are n = 47 assets in this set (see Table 1). The set of

15

Table 1: Assets

Aerospace and Defense TelecommunicationsBA Boeing Corp. VZ Verizon CommunicationsUTX United Technologies T AT&TLMT Lockheed Martin S Sprint NextelNOC Northrop Grumman CMCSK ComcastHON Honeywell Intl. BLS BellSouthSemiconductors and Other Electronic Components Computer SoftwareINTC Intel Corp. MSFT MicrosoftTXN Texas Instruments ORCL OracleSANM Sanmina-SCI CA CASLR Solectron ERTS Electronic ArtsJBL Jabil Circuit SYMC Symantec

Computers and Office Equipment PharmaceuticalsIBM Intl. Business Machines PFE PfizerHPQ Hewlett-Packard JNJ Johnson & JohnsonDELL Dell ABT Abbott LaboratoriesXRX Xerox MRK MerckAAPL Apple Computer BMY Bristol-Myers SquibbNetwork and Other Communications Equipment Chemicals

MOT Motorola DOW Dow ChemicalCSCO Cisco Systems DD DuPonLU Lucent Technologies LYO Lyondell ChemicalQCOM Qualcomm PPG PPG Industries

Electronics and Electrical Equipment Utilities (Gas & Electric)EMR Emerson Electric DUK Duke EnergyWHR Whirlpool D Dominion ResourcesROK Rockwell Automation EXC ExelonSPW SPX SO Southern

PEG Public Service Enterprise Group

factors are 10 major market indices (see Table 2). The data sequence consists of daily assetreturns from July 25, 2002 through May 10, 2006. It shall be mentioned that the data used inthis experiment was collected on May 11, 2006. The most recent data available at that timewas the one on May 10, 2006.

A complete description of our experimental procedure is as follows. The entire data se-quence is divided into investment periods of length p = 90 days. For each investment periodt, the factor covariance matrix F is computed based on the factor returns of the previous ptrading days, and the variance di of the residual return is set to di = s2

i , where s2i is given

in Section 1. In addition, given a desired confidence level ω > 0, our “joint” uncertainty setSµ,v is built as in Section 3, and Goldfarb and Iyengar’s “separable” uncertainty set Sm × Sv

is built as in Section 2 with ω = ω1/n. The robust portfolios are then obtained by solving theRMRAR models with these uncertainty sets, and they are held constant for the investmentat each period t.

Since a block of data of length p = 90 is required to construct uncertainty sets or estimatethe parameters, the first investment period indexed by t = 1 starts from (p + 1)th day. Thetime period July 25, 2002 – May 10, 2006 contains 11 periods of length p = 90, and hence inall there are 10 investment periods. Given a sequence of portfolios {φt}10

t=1, the correspondingoverall wealth growth rate is defined as

Π1≤t≤10

[

Πtp≤k≤(t+1)p(e + rk)]T

φtr − 1,

16

Table 2: Factors

DJCMP65 Dow Jones Composite 65 Stock AverageDJINDUS Dow Jones IndustrialsDJUTILS Dow Jones UtilitiesDJTRSPT Dow Jones TransportationFRUSSL2 Russell 2000NASA100 Nasdaq 100NASCOMP Nasdaq CompositeNYSEALL NYSE CompositeS&PCOMP S&P 500 CompositeWILEQTY Dow Jones Wilshire 5000 Composite

0 5 100

5

10

15

20

25

30

35ω=0.05

risk aversion

aver

age

dive

rsifi

catio

n nu

mbe

r

LROBGIROB

0 5 100.1

0.2

0.3

0.4

0.5

0.6ω=0.05

risk aversion

over

all w

ealth

gro

wth

rat

e

LROBGIROB

0 5 100

5

10

15

20

25

30

35ω=0.50

risk aversion

aver

age

dive

rsifi

catio

n nu

mbe

r

LROBGIROB

0 5 100.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6ω=0.50

risk aversion

over

all w

ealth

gro

wth

rat

e

LROBGIROB

0 5 100

5

10

15

20

25

30

35ω=0.95

risk aversion

aver

age

dive

rsifi

catio

n nu

mbe

r

LROBGIROB

0 5 100.1

0.2

0.3

0.4

0.5

0.6ω=0.95

risk aversion

over

all w

ealth

gro

wth

rat

e

LROBGIROB

Figure 2: Performance of portfolios for ω = 0.05, 0.50 and 0.95.

and the average diversification number is defined as∑10

t=1 I(φt)/10, where I(φt) denotes thediversification number of the portfolio φt.

We now report the performance of the RMRAR models with our “joint” uncertainty setSµ,v, and Goldfarb and Iyengar’s “separable” uncertainty set Sm × Sv as the risk aversionparameter θ ranges from 0 to 104. The computational results for the confidence level ω =0.05, 0.50, 0.95 are shown in Figure 2 that consists of three group of plots. In each of thesegroups, the left plot is about the average diversification number of robust portfolios, and thesecond plot is about the overall wealth growth rate over next 10 periods of the investmentusing robust portfolios. We observe that our robust portfolio is fairly diversified, but Goldfarband Iyengar’s is highly non-diversified. Also, the overall wealth growth rate of the investmentbased on our robust portfolio is higher than that using Goldfarb and Iyengar’s robust portfolio.

The realization cost is another natural concern for any investment strategy. We nextcompare the cost of implementing the above investment strategies. For a sequence of portfolios{φt}10

t=1, its average transaction cost is defined as∑10

t=2 ‖φt − φt−1‖1/9 (see also the discussionin [15]). In Figure 3 we report the average transaction costs of the investments using therobust portfolios for the confidence levels ω = 0.05, 0.50, 0.95, respectively. We observe thatthe investment based on our robust portfolio incurs lower average transaction cost than thatusing Goldfarb and Iyengar’s robust portfolio.

17

0 5 100

0.5

1

1.5

2ω=0.05

risk aversion

aver

age

tran

sact

ion

cost

LROBGIROB

0 5 100.5

1

1.5

2ω=0.50

risk aversion

aver

age

tran

sact

ion

cost

LROBGIROB

0 5 100.5

1

1.5

2ω=0.95

risk aversion

aver

age

tran

sact

ion

cost

LROBGIROB

Figure 3: Average cost of portfolios for ω = 0.05, 0.50 and 0.95.

6 Concluding Remarks

In this paper, we considered the factor model of the random asset returns. By exploringthe correlations of the mean return vector µ and factor loading matrix V , we proposed astatistical approach for constructing a “joint” ellipsoidal uncertainty set Sµ,v for (µ, V ). Wefurther showed that the robust maximum risk-adjusted return (RMRAR) problem with sucha uncertainty set can be reformulated and solved as a cone programming problem. Thecomputational results reported in this paper demonstrate that the robust portfolio determinedby the RMRAR model with our “joint” uncertainty set outperforms that with Goldfarb andIyengar’s “separable” uncertainty set [15] in terms of wealth growth rate and transactioncost; and moreover, our robust portfolio is fairly diversified, but Goldfarb and Iyengar’s issurprisingly highly non-diversified. It would be interesting to extend the results of this paperto other robust portfolio selection models, e.g., robust maximum Sharpe ratio and robustvalue-at-risk models (see [15]).

Acknowledgements

The author is grateful to Antje Berndt for stimulating discussions on selecting suitable factorsand providing me the real market data. The author is also indebted to Garud Iyengar formaking his code available to me. We also gratefully acknowledge comments from DimitrisBertsimas, Victor DeMiguel, Darinka Dentcheva, Andrzej Ruszczynski and Reha Tutuncu atthe 2006 INFORMS Annual Meeting in Pittsburgh, USA.

Appendix

In this section, we provide proof for Theorem 4.5.Proof of Theorem 4.5: Let ri(·) denote the relative interior of the associated set. We

first show that problem (33) is strictly feasible. In view of (16), we immediately see that

18

ri(Φ) 6= ∅. Let φ0 ∈ ri(Φ), and let t0 ∈ ℜ such that t0 > (φ0)TDφ0. Then we can observe that

1 + t0

1 − t0

2D1/2φ0

∈ ri(Ln+2)

Let S0 ∈ ℜn×n be such that S0 ≻ φ0(φ0)T . Then by Schur Complement Lemma, one has(

1 (φ0)T

φ0 S0

)

≻ 0.

Using the assumption that A has full column rank, we observe from (27) that R ≻ 0. Hence,there exists a sufficiently large τ 0 > 0 such that

M ≡ τ 0R − 2θS0 ⊗(

0 00 F

)

≻ 0. (35)

Now, let ν0 be sufficiently small such that

τ 0η − 2ν0 − (τ 0h + q0)T M−1(τ 0h + q0) > 0,

where q0 = (φ01, 0, . . . , φ

0n, 0)T ∈ ℜ(m+1)n (here, 0 denotes the m-dimensional zero vector). This

together with (35) and Schur Complement Lemma, implies that

τ 0R − 2θS0 ⊗(

0 00 F

)

τ 0h + q0

(τ 0h + q0)T τ 0η − 2ν0

≻ 0.

Thus, we see that (φ0, S0, τ 0, ν0, t0) is a strictly feasible point of problem (33).We next show that the dual of problem (33) is also strictly feasible. Let

X1 =

(

X111 X1

12

X121 X1

22

)

, X2 =

(

X211 X2

12

X221 X2

22

)

, x3 =

x31

x32

x33

(36)

be the dual variables corresponding to the first three constraints of problem (33), respectively,where X1

11 ∈ ℜ[(m+1)n]×[(m+1)n], X112 ∈ ℜ(m+1)n, X2

22 ∈ ℜn×n, X221, x3

3 ∈ ℜn, X122, X2

11, x31,

x32 ∈ ℜ. Also, let x4 ∈ ℜ be the dual variable corresponding to the constraint eT φ = 1. Then,

we see that the dual of problem (33) is

minX1,X2,x3,x4

X211 + x3

1 + x32 + x4

s.t. −2Ψ(X112) − 2X2

21 − 2D1/2x33 + x4e ≥ 0,

2θ

(

0 00 F

)

⊙ X111 − X2

22 = 0,

−(

R hhT η

)

• X1 ≥ 0,

−x31 + x3

2 = −θ,

2X122 = 1,

X1 � 0, X2 � 0, x3 ∈ Ln+2,

(37)

19

where Ψ : ℜ(m+1)n → ℜn is defined as Ψ(x) = (x1, xm+2, . . . , x(n−2)(m+1)+1, x(n−1)(m+1)+1)T for

every x ∈ ℜ(m+1)n, and(

0 00 F

)

⊙ X ≡((

0 00 F

)

• Xij

)

∈ ℜn×n (38)

for any X = (Xij) ∈ ℜ[(m+1)n]×[(m+1)n] with Xij ∈ ℜ(m+1)×(m+1) for i, j = 1, . . . , n. We nowconstruct a strictly feasible soultion (X1, X2, x3, x4) of the dual problem (37). Let x3 =(θ, 0, . . . , 0) ∈ ℜn+2. It clearly satisfies the constraint −x3

1 + x32 = −θ, and moreover, x3 ∈

ri(Ln+2) due to θ > 0. Next, let

X1 =1

2(1 + γ)

x1...

xn

1

x1...

xn

1

T

+ γI

. (39)

In view of this identity and (36), one has X122 = 1/2. Since ω > 0, we know from (13) that

c(ω) > 0. This together with (27), (28) and (39), implies that

−(

R hhT η

)

• X1 = − 12(1+γ)

R •

x1...

xn

x1...

xn

T

+ 2hT

x1...

xn

+ η + γ

(

R hhT η

)

• I

= − 12(1+γ)

[

−c(ω) + γ

(

R hhT η

)

• I

]

≻ 0

and X1 ≻ 0 for sufficiently small positive γ. Now, let

X222 = 2θ

(

0 00 F

)

⊙ X111. (40)

We next show that X222 ≻ 0. Indeed, using (38) and the assumption that 0 6= F � 0, we

obtain(

0 00 F

)

⊙ I ≻ 0. (41)

Further, we have for every u ∈ ℜn,

uT

(

0 00 F

)

⊙

x1...

xn

x1...

xn

T

u =

∑

i,j

(

0 00 F

)

• (uiujxixTj ),

=

(

0 00 F

)

•(

∑

i,j

uiujxixTj

)

,

=

(

0 00 F

)

•(

(∑

i

uixi)(∑

i

uixi)T

)

≥ 0,

20

and hence,

(

0 00 F

)

⊙

x1...

xn

x1...

xn

T

� 0.

This together with (39)-(41) and the assumption that θ > 0, implies that X222 ≻ 0. Letting

X212 = 0 and X2

11 = 1, we immediately see that X2 ≻ 0. We also observe that for sufficientlylarge x4, (X1, X2, x3, x4) also strictly satisifes the first constraint of (37). Hence, it is a strictlyfeasible solution of the dual problem (37). The remaining proof directly follows from strongduality.

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