Distribution of the Optimal Value of a Stochastic Mixed Zero-One Linear Optimization Problem under Objective Uncertainty Karthik Natarajan ∗ Chung-Piaw Teo † Zhichao Zheng ‡ June 20, 2011 Abstract This paper is motivated by the question to approximate the distribution of the completion time of a project network with random activity durations. In general, we consider the mixed zero-one linear optimization problem under objective uncertainty, and develop an approach to approximate the distribution of its optimal value when the random objective coefficients follow a multivariate normal distribution. Linking our model to the classical Stein’s Identity, we show that the best normal approximation of the random optimal value, under the L 2 -norm, can be computed by solving the persistency problem, first introduced by Bertsimas et al. (2006). We further extend our method to the minimum quadratic regret problem, and show that for any general mixed zero-one linear optimization problem, the minimum quadratic regret solution can be computed by solving a related persistency problem. Extensive computational studies on are presented to demonstrate the advantages of the new method. Key words : stochastic mixed zero-one linear optimization; persistency; distribution approx- imation; regret; completely positive programming; project management; portfolio selection 1 Introduction One of the fundamental problems in project management is to identify the project completion time when the activity durations are random. It is well known that any project can be rep- resented as a directed acyclic graph. In this paper, we adopt the conventional activity-on-arc representation of the project network, where activities are represented by arcs and nodes rep- resent the milestones that indicate the starting or ending of the activities. The length of an * Department of Management Sciences, City University of Hong Kong, Hong Kong. Email: knatara- [email protected]† Department of Decision Sciences, NUS Business School, National University of Singapore. Email: biz- [email protected]‡ Department of Decision Sciences, NUS Business School, National University of Singapore. Email: [email protected]1
30
Embed
Distribution of the Optimal Value of a Stochastic Mixed ...
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Distribution of the Optimal Value of a Stochastic Mixed
Zero-One Linear Optimization Problem under Objective
Uncertainty
Karthik Natarajan∗ Chung-Piaw Teo† Zhichao Zheng‡
June 20, 2011
Abstract
This paper is motivated by the question to approximate the distribution of the completion
time of a project network with random activity durations. In general, we consider the
mixed zero-one linear optimization problem under objective uncertainty, and develop an
approach to approximate the distribution of its optimal value when the random objective
coefficients follow a multivariate normal distribution. Linking our model to the classical
Stein’s Identity, we show that the best normal approximation of the random optimal value,
under the L2-norm, can be computed by solving the persistency problem, first introduced
by Bertsimas et al. (2006). We further extend our method to the minimum quadratic
regret problem, and show that for any general mixed zero-one linear optimization problem,
the minimum quadratic regret solution can be computed by solving a related persistency
problem. Extensive computational studies on are presented to demonstrate the advantages
of the new method.
Key words: stochastic mixed zero-one linear optimization; persistency; distribution approx-
Assume that the constraint µ1x1 +µ2x2 ≥ τ is not tight, i.e. τ −µ1x1 −µ2x2 < 0. Then θ = 0,
and we can obtain x1 =σ22−σ12
σ21−2σ12+σ2
2=
σ22−σ12
V ar(r1−r2)
x2 =σ21−σ12
σ21−2σ12+σ2
2=
σ21−σ12
V ar(r1−r2)
if σ21 ≥ σ12 and σ2
2 ≥ σ12{x1 = 0x2 = 1
if σ22 < σ12{
x1 = 1x2 = 0
if σ21 < σ12
which exactly resembles the minimum variance portfolio, i.e., the solution to the following prob-
10
lem:min σ2
1x21 + 2σ12x1x2 + σ2
2x22
s.t. x1 + x2 = 1x1, x2 ≥ 0
Then the condition τ − µ1x1 − µ2x2 < 0 is just τ < τ∗. Note that this solution makes sense
only if
V ar (r1 − r2) = 0. (6)
In another situation when the constraint µ1x1 + µ2x2 ≥ τ is tight, together with the budget
condition x1 + x2 = 1, we can easily find the optimal portfolio:{x1 =
τ−µ2
µ1−µ2
x2 =µ2−τµ1−µ2
which is valid under the assumption that
τ ∈ [µ1, µ2] or [µ2, µ1] , and µ1 = µ2. (7)
Those situations when some of the conditions in (6) and (7) do not hold correspond to the
extreme cases that are trivial to analyze. For example, V ar (r1 − r2) = 0 implies the two
assets are perfectly correlated and their variances are the same, i.e., σ12 = σ1σ2 and σ1 = σ2.
Consequently, the covariance matrix is not positive definite. In this case, the variance of the
portfolio would be constant and equal to σ21. Then the only concern is to meet the target return
requirement.
One salient feature of the Markowitz model is the separation on the treatment of mean and
variance of the portfolios. Especially in this example with only two assets, the target expected
return constraint and the budget constraint are enough to determine the optimal portfolio under
certain situations as discussed above. Whereas, the quadratic regret approach is able to capture
more distributional information of the random returns through an integrated framework. The
detailed performance of these two models will be analyzed later in Section 5.3.
5 Computational Study
In this section, we first review some approaches for persistency estimation and then demonstrate
the performance of our approximation model through two classes of problems, i.e., project and
portfolio management problems. Simulation is conducted and serves as a benchmark in all the
experiments, which is possible because the deterministic versions of Problem (1) considered in
this section are all polynomial time solvable. If the deterministic problem is NP-hard, imple-
menting a simulation method would require the solution to a number of NP-hard problems,
which can be very resource consuming.
11
5.1 Persistency Models
As analyzed in the previous section, the success of our approximation method hinges on the
accuracy of the estimation on the expected objective value of the random optimization problem
as well as the persistency values.
In literature, the problem of estimating the expected objective value of random optimization
problems has been studied for a long time. In case of the project management problem, the
search for the expected project completion time started half century ago (cf. Fulkerson (1962))
and is still an active research topic (cf. Yao & Chu (2007)). On the other hand, the persistency
problem has only been brought into the optimization area since Bertsimas et al. (2006). As
reviewed before, there have been several models developed for the purpose of estimating the per-
sistency values (cf. Natarajan et al. (2009), Mishra et al. (2011), Natarajan et al. (2011), Kong
et al. (2011) etc.). In this section, we will review the most recent progress on the persistency
estimation mainly contributed by Natarajan et al. (2011).
Natarajan et al. (2011) consider the following stochastic optimization problem:
ZP := supc∼(µ,Σ)+
E [Z(c)] ,
where c ∼ (µ,Σ)+ means that the set of distributions of the random coefficient vector c (as-
sumed to be nonempty) is defined by the nonnegative support Rn+, finite mean vector µ and
finite second moment matrix Σ, i.e., c ∈ {X : E[X] = µ,E[XXT ] = Σ,P(X ≥ 0) = 1}. Theyproved that ZP can be solved as the following convex conic optimization problem:
ZC = max∑n
j=1 Yj,js.t. aT
i Xai − 2biaTi x+ b2i = 0, ∀i = 1, . . . ,m
Xj,j = xj , ∀j ∈ B 1 µT xT
µ Σ Y T
x Y X
≽cp 0
(i.e., ZP = ZC) where the decision variables are x ∈ Rn, X ∈ Rn×n, and Y ∈ Rn×n, and for
a matrix A ∈ Rn×n, A ≽cp 0 means that A lies in the cone of completely positive matrices of
dimension n defined as
CPn :={A ∈ Rn×n | ∃V ∈ Rn×k
+ , such that A = V V T}.
The linear program over the convex cone of the completely positive matrices is called a com-
pletely positive program (CPP), and ZC is a typical CPP. The authors named this model as
CPCMM, which stands for Completely Positive Cross Moment Model. In a subsequent paper,
Kong et al. (2011) re-developed CPCMM from a different perspective based on the general conic
optimization theory, which significantly generalizes CPCMM such that more support constraints
can be incorporated, e.g., some ellipsoid bounding constraints on the random vector.
In the formulation of ZC , the variables x, Y and X attempt to encode the information
xj = E [xj(c)], Yi,j = E [cjxi(c)] and Xi,j = E [xi(c)xj(c)]. Thus, through solving ZC , the
12
optimal objective value gives the value of E [Z(c)], and the optimal value of x is simply the
persistency value. However, CPCMM ignores the distributional information. Hence, when
c is normally distributed, CPCMM only gives an upper bound to E [Z(c)] and estimates on
the persistency values. Another issue is that although the completely positive cone is closed,
convex and pointed, it is widely believed that CPPs are NP-hard to solve. Fortunately, there
are various hierarchies of tractable approximations for the completely positive cone (cf. Bomze
et al. (2000), Parrilo (2000) and Klerk et al. (2002) etc.). In the computational study, when
we need to solve CPCMM, we will exploit the simple SDP approximation of the completely
positive constraint, i.e., A ≽cp 0 is relaxed to A ≽ 0 and A ≥ 0, where A ≽ 0 means that A is
positive semidefinite.
Despite all these numerical inaccuracies, we show in the next section that our approximation
method is still practically attractive due to the use of persistency in the approximation and the
ability to capture correlations through CPCMM.
5.2 Project Management Problem
In this section, we present the results for approximating the distribution of the project com-
pletion time when the activity durations are stochastic. We try to compare our approach with
as many existing approximation methods as possible, except some approaches that require the
activity duration distribution to be either discrete or at least represented in some discrete form,
e.g., bounding distribution method by Kleindorfer (1971), and the finite-state discrete approx-
imation with cascading collapse method by Ord (1991)2, etc.
Another critical issue is that almost all the approximated distributions derived using previous
methods do not reside in the same probability space as Z(c), which makes the computation of the
squared deviation impossible. This problem arises since the traditional approaches solely focus
on the distribution (like tail probabilities, etc.) but overlook the approximation error between
the approximated completion time and the true completion time under a specific realization of
the random activity durations. For example, Cox (1995) assumed the project completion time
to be normally distributed at first, and then tried to estimate the moments of the completion
time. Thus the final approximated distribution does not admit a calculation of the squared
deviation from the true project completion time distribution. Hence, we have to resort to other
measures to compare the performance of different approximation methods, e.g., some descriptive
statistics, like mean and standard deviation. In addition, we also deploy the following measure
to quantify the distance between two distributions:
Square Norm Distance (F,G) :=
∫ 1
0
[F−1 (y)−G−1 (y)
]2dy
where F and G are the cumulative distribution functions of two distributions.
2Although the approach developed in Ord (1991) is based on the finite-state discrete characterization of allthe activity durations as well as the completion time of each milestone, it could still be applied to the casewhen the activity durations are continuously distributed. The only concern is to find an appropriate discreteapproximation of each activity duration distribution.
13
On the other hand, PERT approach simply considers the expected duration of each activity
when choosing the critical path (i.e., the path with longest expected completion time), and
then use the mean and variance of this critical path to approximate the mean and variance
of the project completion time, respectively. Finally, resorting to the Central Limit Theorem,
PERT assumes the project completion time is normally distributed (c.f. MacCrimmon & Ryavec
(1964)). Therefore, when the project activities follow the multivariate normal distribution, the
PERT estimation admits the computation of squared deviation and thus can be compared to
our approach in a greater detail.
The most important advantage of our approximation approach using CPCMM is its ability
to capture the correlations among the random variables. To the best of our knowledge, none of
the previous studies address the issues of correlated activities for the project management prob-
lem. However, potential positive correlations among the activity duration may exist in many
project networks. For example, if some troubles are encountered in the delivery of concrete
for a task in a construction project, this problem is likely to influence the duration of numer-
ous activities involving concrete pours in this project. With the following example network,
besides comparing the performance of different approximation methods, we also illustrate the
importance of considering the correlations among activities.
Example 2 The network consists of four nodes and five arcs as shown in Figure 1. All the
activities are independent and normally distributed with mean and variance both equal to one.
Figure 1: Network for Example 2
The network in Example 2 is the “Wheatstone bridge” network from Lindsey (1972) and
later regarded as the “forbidden graph” by Dodin (1985) since it is the basic evidence of graph
irreducibility. This network has been studied in almost every piece of the research work in this
field. Ord (1991) summarized the results for this graph with normally distributed activity du-
rations, and also provided the results from his discrete approximation method with a parameter
k indicating the number of discrete points used to approximate the normal distribution3. All
these results are presented in Table 1, where T denotes the project completion time and σ(.)
3Note that the approximated distributions obtained by Ord (1991) should be a discrete distribution as well.However, we modified his theory a bit in computing the square norm distance by assuming the final approximateddistribution follows a normal distribution with the moments derived from his original procedure.
14
denotes its standard deviation. The new result from our method is also presented in Table 1 un-
der “CPCMM”, since we use CPCMM to find the estimates on the expected project completion
Table 3: Results of random project networks for Example 4
18
For general random correlation, as the size of the network increases, the mixture of positive
and negative correlations among activities tend to cancel out and PERT produces competitively
good estimate on the expected project completion time for most instances. Meanwhile, for some
graphs with dominant critical paths, PERT gives quite accurate estimates on both mean and
variance of the project completion time. On the other hand, since the approximation error of
CPP gets larger as the size of the problem increases, the upper bounds on the expected project
completion time from CPCMM gets worse. Thus, we observe that sometimes PERT gets a
lower expected squared deviation than CPCMM. Nevertheless, the persistency estimates from
CPCMM still perform very well, so the approximated distribution designed using the mean
from PERT and the persistency from CPCMM (shown as “Combo” in Table 3) tends to give
much lower expected squared deviation. For projects with sparsely and positively correlated
activities, PERT substantially underestimates the mean and overestimates the variance, and
the performance of CPCMM dominates PERT.
5.3 Portfolio Management Problems
In this section, we test the performance of the portfolio selected based on the minimum quadratic
regret criterion using our method developed in Section 4 with the constraint Ax = b specified
by∑n
j=1 xj = 1.
Data
We collect the daily return data of n industry portfolios (n = 10 and 174) over the past twenty
years between 1991 and 2010 from Fama/French Data Library5. The portfolios are constructed
by the following manner:
Each NYSE, AMEX, and NASDAQ stock is assigned to an industry portfolio at the
end of June of year t based on its four-digit SIC code at that time. (The Compustat
SIC codes are used for the fiscal year ending in calendar year t − 1. Whenever
Compustat SIC codes are not available, the CRSP SIC codes are used for June of
year t.) Then the returns are computed from July of t to June of t + 1. All the
stocks are equally weighted in the portfolio.
The mean (AVG), standard deviation (STD) and coefficient of variation (CV) of the daily
returns for each industry portfolio are summarized in Table 4. We also examine the frequency
plots of the daily returns for each industry portfolio and confirm the applicability of the normal
distribution assumption. See Figure 4 for two example frequency plots (on the top row) of
“Food” and “Util” industry portfolio from 17 industry portfolios data6. Although the time
4Similar experiments are conducted for n = 30 and the findings are the same as what will be presented later.We omit the detailed reports for n = 30 case due to the space constraint.
5http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data library.html6The reason for demonstrating these two industry portfolios is that they represent two classes of portfolios in
our data set. The food industry is relatively more stable, while the “Util” is more volatile, which is particularlyreflected on their return fluctuations during the last financial crisis period from 2007 to 2010.
Daily Return of "Util" Industry Portfolio from 1991 to 2010
−5 −4 −3 −2 −1 0 1 2 3 4 50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Daily Return
Pro
babi
lity
Cumulative Distribution Function of the Daily Return of "Food" Industry Portfolio
1991 − 20101991 − 2001
−5 −4 −3 −2 −1 0 1 2 3 4 50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Daily Return
Pro
babi
lity
Cumulative Distribution Function of the Daily Return of "Util" Industry Portfolio
1991 − 20101991 − 2001
(a) Food (b) Other
Figure 4: Distribution analysis of the daily returns of two portfolios picked from 17 industryportfolios
from this information set is then tested on Jan 2nd, 2001, i.e., the return of the investment based
on the actual portfolio returns on Jan 2nd, 2001 is recorded. Such experimental framework will
shift forward day by day with a fixed length of history in terms of the number of days. If we
choose to utilize only the past 5-year information to support the current investment decision,
then the data from Jan 2nd, 1996 to Dec 29th, 2000 would be the initial history.
Portfolio Selection Strategies:
The abundance of the historical return data forms a natural sample of the stochastic port-
folio returns. Hence, Besides of relying on the persistency models, we can also estimate the
persistency from the historical data. The strategies based on both approaches are considered.
The following six different portfolio selection strategies are investigated including our minimum
quadratic regret approach:
1. Expected Return
All the budget is devoted in the portfolio with the highest expected return in history.
21
2. Uniform
At the beginning of every period, the budget is reallocated and divided evenly among the
n portfolios. Since the portfolios are always rebalanced to a fixed proportion, this is also
called a constant-rebalanced portfolio strategy.
3. Markowitz
The Markowitz mean/variance model is used to compute the optimal portfolio, and the
target return is the average of the overall historical returns, i.e., the average of the average
historical returns of each industry portfolio. Effectively, the constraint requires the ex-
pected return to exceed the expected return of the “Uniform” strategy when the historical
returns are used to predict the future returns.
4. Persistency
(a) Empirical: For each portfolio, the percentage of all the trading days that it outper-
forms the rest portfolios is calculated from the historical data, and then the budget is
allocated according to these percentages, i.e., the empirical persistency.
(b) CPCMM: The budget is allocated according to the persistency estimated from CPCM-
M.
5. Conditional Value-at-Risk (CVaR)
CVaR of a random variable X (representing a loss) at a given probability level β ∈ (0, 1)
is defined as the conditional expectation of X exceeding a certain threshold, called Value-
at-Risk(VaR), which is the percentile of X at β, i.e.,
V aRβ
(X)= inf
{x ∈ R : P
(X ≤ x
)≥ β
}, and
CV aRβ
(X)= E
[X∣∣∣X > V aRβ
(X)]
.
As a coherent risk measure, CVaR has gained a lot interest over the past ten years.
Rockafellar & Uryasev (2000) showed that CVaR can be determined by minimizing a
more tractable auxiliary function without predetermining VaR first as follows:
CV aRβ
(X)= min
α∈R
{α+
1
1− βE
[(X − α
)+]}Then the budget allocation can be chosen to minimize CVaR of loss at a given tolerance
level β, i.e.,
(C) minα∈R,
∑nj=1 γj=1,γ≥0
{α+
1
1− βE[(−rTγ − α
)+]}This is a risk averse approach since it aims to minimize the conditional expectation of loss
exceeding VaR and meanwhile with probability 1− β, the loss is below α. Please refer to
Rockafellar & Uryasev (2000) and Rockafellar & Uryasev (2004) for more details on the
development of CVaR, and Zhu & Fukushima (2009) for the most recent review in this
area of research. We tested β = 0.8, 0.9 and 0.95 in the experiments, and since the results
22
are almost the same, only the results corresponds to β = 0.9 are reported.
(a) Empirical: Solve Problem (C) as a stochastic programming problem with the historical
returns as the sample.
(b) CPCMM: Solve Problem (C) using the extensions of CPCMM developed in Natarajan
et al. (2011).
6. Quadratic Regret
(a) Empirical: The investment decision is obtained by solving Problem (R) with parame-
ters empirically determined from the history returns as “Persistency (a)”, which include
the first two moments of the returns, E [x(c)] as well as E [Z (c)].
(b) CPCMM: The investment decision is obtained by solving Problem (R) with E [x(c)]
and E [Z (c)] estimated from solving CPCMM.
Performance Measures:
We impose the following measures to gouge the performance of different portfolio selection
strategies.
1. Regret
Two types of regret are considered, i.e., absolute regret (or difference regret) and quadratic
regret. For each type of regret, the total regret over the ten years investment horizon
(from 2001 to 2010) is recorded, which also reflects the average level of regret the investor
may face everyday. Note that although our investment decision is developed based on
minimizing the quadratic regret, the actual performance on the quadratic regret measure
from our strategy may not be the best, because the test we conducted is out-of-sample
and the existence of the numerical errors either from the return moment estimation or
solving the tractable approximation of CPCMM is unavoidable.
2. Annual Return
The average (AVG), standard deviation (STD) and signal-to-noise ratios (SNR = AVG/STD)
of the annual returns, as well as the volatility-adjusted annual return (VR = AVG−0.5×STD2) are computed. These measures help capture the risk and return tradeoff for dif-
ferent strategies in a short term.
3. Aggregate Return
We also keep track of the aggregate return for different strategies over the ten years
investment horizon compounded daily. This measure reflects the return from different
strategies in the long run.
Results and Discussion
Table 5 to Table 8 summarize the performance of different portfolio selection strategies on differ-
ent industry portfolios. The top and second best performance for each measure are underscored
S & P 500 IndexUniformExpected ReturnMarkowitzPersistencyCVaRQuadratic Regret
Table 8: Performance of different portfolio selection strategies on 17 industry portfolios using10 years as history
6 Discussion and Conclusion
In this paper, we show that several classes of stochastic optimization problems can be trans-
formed into the related persistency problems, like project completion time distribution ap-
proximation problem and quadratic regret minimization problem. Extensive computational
experiments were presented to demonstrate the advantages of our distribution approximation
method, especially the benefits of introducing persistency into the distribution approximation
problem.
The results in this paper can be developed further in several ways. The computational
experiments on portfolio management problem presented in this paper can be more compre-
hensive. We can further compare the investment strategy with other online portfolio selection
strategies using geometric weights updates etc., often used in the machine learning community
(cf. Helmbold et al. (1998)). With the knowledge on the distribution of the optimal value,
we can now conduct more in-depth risk analysis or parameter calibration for the underlying
stochastic mixed zero-one linear optimization problem. We leave these and other related issues
27
for future research.
Appendix A. Proof of Lemma 1
Proof. The proof is consolidated from Stein (1972), Stein (1981) and Liu (1994).
We begin by showing the univariate version of Stein’ts Identity (cf. Stein (1972) and Stein
(1981)).
Let Y follow a standard normal distribution, N (0, 1), and ϕ (y) denote the standard normal
density with the derivative ϕ′ (y) = −yϕ (y). For any function g : R → R such that g′ exists
almost everywhere and E[|g′(Y )|] < ∞,
E [g′ (Y ))] =∫∞−∞ g′(y)ϕ (y) dy
=∫∞0 g′(y)
[−∫∞y −zϕ (z) dz
]dy +
∫ 0−∞ g′(y)
[∫ y−∞−zϕ (z) dz
]dy
=∫∞0 zϕ (z)
[∫ z0 g′(y)dy
]dz −
∫ 0−∞ zϕ (z)
[∫ 0z g′(y)dy
]dz
=(∫∞
0 +∫ 0−∞
){zϕ (z) [g(z)− g(0)]} dz
=∫∞−∞ zϕ (z) g(z)dz
= E [Y g (Y )]
where the third equality is justified by Fubini’s Theorem. Note that E[Y ] = 0 and V ar(Y ) = 1,
the equality proved above is essentially
Cov (Y, g (Y )) = V ar(Y )E[g′ (Y ))
]. (8)
Next, we generalize the above result into the multivariate case (cf. Stein (1981) and Liu
(1994)).
Let Z = (Z1, . . . , Zn)T , where Zj ’s are independent and identically distributed standard
normal random variables. It is straightforward to show by Equation (8) that for any function
h : Rn → R satisfying the same conditions as h,
E[Zj h (Z) | (Z2, . . . , Zn)
]= E
[∂h (Z)
∂zj| (Z2, . . . , Zn)
], ∀j = 1, . . . , n.
Taking the expectation of both sides, we find that
E[Zj h (Z)
]= E
[∂h (Z)
∂zj
], ∀j = 1, . . . , n,
i.e.,
Cov(Z, h (Z)
)= E
[∇h (Z)
].
Note that the random vector X can be written as X = Σ1/2Z + µ. Consider h (Z) =
h(Σ1/2Z + µ
), then ∇h (Z) = Σ1/2∇h (X). Hence,
Cov (X, h (X)) = Cov(Σ1/2Z, h (Z)
)= Σ1/2E
[∇h (Z)
]= ΣE [∇h (X)] .
28
References
Agrawal, S., Y. Ding, A. Saberi, Y. Ye (2011) Price of correlations in stochastic optimization, Manuscript.
Aissi, H., C. Bazgan, D. Vanderpooten (2009) Min-max and min-max regret versions of combinatorialoptimization problems: A survey , European Journal of Operational Research, 197, pp. 427–438.
Aldous, D., M. Steele (2003) The objective method: Probabilistic combinatorial optimization and localweak convergence, in Probability on Discrete Structures, H. Kesten (ed), Springer, Berlin, 110, pp.1–72.
Bereanu, B. (1963) On stochastic linear programming. I: Distribution problems: A single random variable,Romanian Journal of Pure and Applied Mathematics, 8, pp. 683–697.
Berman, A., N. Shaked-Monderer (2003) Completely Positive Matrices, World Scientific, Singapore.
Bertsimas, D., X. V. Doan, K. Natarajan, C. P. Teo (2010) Models for minimax stochastic linearoptimization problems with risk aversion, Mathematics of Operations Research, 35, pp. 580–602.
Bertsimas, D., K. Natarajan, C. P. Teo (2004) Probabilistic combinatorial optimization: moments,semidefinite programming and asymptotic bounds, SIAM Journal of Optimization, 15, pp. 185–209.
Bertsimas, D., K. Natarajan, C. P. Teo (2006) Persistence in discrete optimization under data uncer-tainty , Mathematical Programming, 108, pp. 251–274.
Bomze, I. M., M. Dur, E. D. Klerk, C. Roos, A. J. Quist, T. Terlaky, (2000) On copositive programmingand standard quadratic optimization problems, Journal of Global Optimization, 18, pp. 301–320.
Boyd, S., L. Vandenberghe (2004) Convex Optimizatioin, Cambridge University Press.
Borkar, V. (1995) Probability Theory: An Advanced Course, S. Axler, F. W. Gehring, P. R. Halmos(eds), Springer, New York..
Brown, G. G., R. F. Dell, R. K. Wood (1997) Optimization and persistence, Interfaces, 27, pp. 15–37.
Burer, S. (2009) On the copositive representation of binary and continuous nonconvex quadratic programs,Mathematical Programming, 120, pp. 479–495.
Cover, T. M. (1991) Universal portfolios, Mathematical Finance, 1, pp. 1–29.
Cox, M. A. (1995) Simple normal approximation to the completion time distribution for a PERT network ,International Journal of Project Management, 13, pp. 265–270.
Dembo, R. S., A. J. King (1992) Tracking models and the optimal regret distribution in asset allocation,Applied Stochastic Models and Data Analysis, 8, pp. 195–207.
Dodin, B. (1985) Bounding the project completion time distribution in PERT networks, OperationsResearch, 33, pp. 862–881.
Dur, M. (2009) Copositive programming: A survey , In: M. Diehl, F. Glineur, E. Jarlebring, W. Michiels(Eds.), Recent Advances in Optimization and its Applications in Engineering, Springer, pp. 3–20.
Ewbank, J. B., B. L. Foote, H. J. Kumin (1974), A method for the solution of the distribution problemof stochastic linear programming , SIAM Journal on Applied Mathematics, 26, pp. 225–238.
Fulkerson, D. R. (1962) Expected critical path lengths in PERT networks, Operations Research, 10, pp.808–817.
Hagstrom, J. N. (1988) Computational complexity of PERT problems, Networks, 18, pp. 139–147.
Helmbold, D. P., R. E. Schapire, Y. Singer, M. K. Warmuth (1997) A comparison of new and oldalgorithms for a mixture estimation problem, Machine Learning, 22, pp. 97–119.
Helmbold, D. P., R. E. Schapire, Y. Singer, M. K. Warmuth (1998) On-line portfolio selection usingmultiplicative updates, Mathematical Finance, 8, pp. 325–347.
Kamburowski, J. (1985) A Note on the Stochastic Shortest Route Problem, Operations Research, 33,pp. 696–698.
29
King, A. J., D. L. Jensen (1992) Linear-quadratic efficient frontiers for portfolio optimization, AppliedStochastic Models and Data Analysis, 8, pp. 195–207.
Klerk, E. de, D. V. Pasechnik (2002) Approximation of the stability number of a graph via copositiveprogramming , SIAM Journal on Optimization, 12, pp. 875–892.
Kleindorfer, G. B. (1971) Bounding distributions for a stochastic acyclic network , Operations Research,19, pp. 1586–1601.
Kong, Q., C. Y. Lee, C. P. Teo, Z. Zheng (2011) Scheduling arrivals to a stochastic service deliverysystem using copositive cones, Manuscript.
Liu, J. S. (1994) Siegel’s formula via Stein’s identities, Statistics & Probability Letters, 21, pp. 247–251.
Lindsey, J. H. (1972) An estimate of expected critical-path length in PERT networks, OperationsResearch, 20, pp. 800–812.
MacCrimmon, K. R., C. A. Ryavec (1964) An analytical study of the PERT assumptions, OperationsResearch, 12, pp. 16–37.
Markowitz, H. M. (1952) Portfolio selection, Journal of Finance, 7, pp. 77–91.
Markowitz, H. M. (1959) Portfolio Selection: Efficient Diversification of Investments, Wiley, New York.
Mishra, V. K., K. Natarajan, H. Tao, C. P. Teo (2011) Choice prediction with semidefinite optimizationwhen utilities are correlated , Manuscript.
Natarajan, K., M. Song, C. P. Teo (2009) Persistency model and its applications in choice modeling ,Management Science, 55, pp. 453–469.
Natarajan, K., C. P. Teo, Z. Zheng (2011a) Mixed zero-one linear programs under objective uncertainty:a completely positive representation, Forthcoming in Operations Resaerch.
Ord, J. K. (1991) A simple approximation to the completion time distribution for a PERT network , TheJournal of the Operational Research Society, 42, pp. 1011–1017.
Papadatos, N., V. Papathanasiou (2003) Multivariate covariance identities with an application to orderstatistics, Sankhya: The Indian Journal of Statistics, 65, pp. 307–316.
Parrilo, P. A. (2000) Structured Semidefinite Programs and Semi-algebraic Geometry Methods in Ro-bustness and Optimization, Ph.D. Dissertation, California Institute of Technology.
Prekopa, A. (1966) On the probability distribution of the optimum of a random linear program, SIAMJournal on Control and Optimization, 4, pp. 211–222.
Rockafellar, R. T., S. Uryasev (2000) Optimization of conditional value-at-risk , Journal of Risk, 2, pp.21–42.
Rockafellar, R. T., S. Uryasev (2004) Conditional value-at-risk for general loss distributions, Journal ofBanking and Finance, 26, pp. 1443–1471.
Sculli, D. (1983) The completion time of PERT networks, The Journal of the Operational ResearchSociety, 34, pp. 155–158.
Siegel, A. F. (1993) A surprising covariance involving the minimum of multivariate normal variables,Journal of the American Statistical Association, 88, pp. 77–80.
Stein, C. M. (1972) A bound for the error in the normal approximation to the distribution of a sum ofdependent random variables, Proceedings of the Berkeley Symposium on Mathematical Statisticsand Probability, 2, pp. 583–602.
Stein, C. M. (1981) Estimation of the mean of a multivariate normal distribution, The Annals ofStatistics, 9, pp. 1135–1151.
Vandenberghe, L., S. Boyd, K. Comanor (1972) Generalized Chebyshev bounds via semidefinite program-ming , SIAM Review, 49, pp. 52–64.
Yao, M. J., W. M. Chu (2007) A new approximation algorithm for obtaining the probability distributionfunction for project completion time, Computers & Mathematics with Applications, 54, pp. 282–295.
Zhu, S., M. Fukushima (2009) Worst-case Conditional Value-at-Risk with application to robust portfoliomanagement , Operations Research, 57, pp. 1155–1168.