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A MCMC/Bernstein Approach to Chance Constrained Programs
Zinan Zhao and Mrinal Kumar
Abstract This paper presents an extension of convex Bern-stein approximations to non-affine and dependent chance con-strained optimization problems. The Bernstein approximationtechnique transcribes probabilistic constraints into conservativeconvex deterministic constraints, relying heavily upon the eval-uation of exponential moment generating functions. This is acomputationally burdensome task for non-affine probabilisticconstraints involving dependent random variables. In thispaper, the theoretical framework of Bernstein approximationsis combined with the practical benefits of Markov chain MonteCarlo (MCMC) integration for its use in a range of highdimensional applications. Numerical results for the combinedBernstein/MCMC approach are compared with scenario ap-proximations.
I. INTRODUCTION ANDP ROBLEMS TATEMENT
Chance constrained programs arise in numerous appli-
cations including portfolio optimization (Ref.[1]), water
management (Ref.[2]) and chemical processes optimization
(Refs.[3], [4]). We consider the following optimization prob-
lem involving a deterministic cost function and a mixture of
probabilistic and deterministic constraints:
minxX
J(x)
subject to Prob{F(x,) 0} G(x) 0H(x) =0
, (1)
where x is the decision variable in a nonempty set X Rn,J : Rn R is a real valued cost function and is arandom variable with probability measure Z() supportedon a set Rd. Also, is the prescribed risk parameterwhich should be no less than the violation rate of the
joint probabilistic constraints represented by the vector func-
tion F= (f1, . . . ,fm): Rn Rm. The vector functions
G= (g1, . . . ,gp): Rn Rp and H= (h1, . . . ,hq): R
n Rq
represent deterministic constraints. Finally, Prob{} denotesprobability and / denote vector inequalities.
Chance constrained programs were first introduced by
Charnes et al. (Ref.[5]) and have been studied extensively in
the stochastic programming literature (Ref.[6]). Except forsome special cases, e.g., linear chance constraint Prob{Ax 0} , where A is a deterministic matrix and islogarithmically concave distributed, they remained largely
intractable and little progress had been made until recently.
This work was not supported by any organizationZ. Zhao is a graduate student in the Department of Mechanical and
Aerospace Engineering, University of Florida, Gainesville, FL 32608, [email protected]
M. Kumar is with the Department of Mechanical and AerospaceEngineering, University of Florida, Gainesville, FL 32608, [email protected]
Recent developments can be grouped into two distinct cate-
gories. The first category includes sampling-based methods
that use Monte Carlo samples to approximate the probabilis-
tic constraints and the final approximation is stochastic (i.e.
a random variable). Essentially, the probabilistic constraints
are enforced in a deterministic manner at a set of (stochastic)
collocation points, which in turn are sampled from the
underlying random variable ( in Eq.1). Notable methods
include sample average approximation (Ref.[7], [8]) and
scenario approximation (Ref.[9], [10]). Ref.[10] shows that
the sample size for scenario approximation should be at least
inversely proportional to the risk parameter .
In the second category, at least part of the solution isdeveloped analytically. The probabilistic constraints are ana-
lytically transcribed into equivalent deterministic constraints,
so that the resulting problem can be numerically solved
using existing nonlinear programming (NLP) solvers. One
suite of methods, developed in the field of reliability based
design optimization include the first-order and second-order
reliability methods (Ref.[11], [12]). These are based on the
first and second order approximations of the constraining
function F and can be inaccurate in highly nonlinear situ-
ations. Another approach, called the Bernstein approxima-
tion was proposed by Nemirovski et al. (Ref.[13]). Here,
the exponential moment generating function is utilized for
transcription of the probabilistic constraints into conservativeconvex deterministic constraints. Care must be taken to avoid
over-conservatism during the transcription or else severely
suboptimal solutions may result.
In order to ensure solvability of the transcribed opti-
mization problem in the traditional Bernstein approximation
approach, it is assumed that the cost function J() andthe search space X are both convex. In addition, there are
three key assumptions about the nature of the probabilistic
constraints: (1) the function F(x, ) (assume scalar, can beextended to vector functions in similar manner) is convex in
x; (2) function F(,) is affine in and; (3) random vari-able (if multivariate) has independent components. Under
these assumptions, the approach was applied successfully tochance constraints of the form:
F(x,) = f0(x) +d
j=1
fj(x)j. (2)
If the affinity and/or independence assumptions stated above
are violated, the approach requires the evaluation of multi-
dimensional integrals, which is computationally burdensome.
Unfortunately, several real-life optimization problems do not
satisfy the three assumptions on the chance constraints. To
the best of knowledge of the authors, the performance of
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Bernstein approximations in these problems has not been
investigated. The current paper presents a numerical adap-
tation of the Bernstein approximation approach for opti-
mization problems with general probabilistic constraints not
satisfying the above listed assumptions. Markov chain Monte
Carlo (MCMC) numerical integration is utilized to solve
high dimensional problems lacking affinity and convexity
of F(, ) as well as independence of . The transcribednonconvex optimization problem is solved using off-the-shelf
NLP solvers.
The rest of the paper is organized as follows: In Section II,
the scenario approach is reviewed and the proposed MCMC
variant of the Bernstein approximation is developed. Section
III discusses the need for multi-dimensional integration in the
Bernstein approximation. Section IV, the MCMC-Bernstein
approach is applied to several optimization problems with
non-affine constraints involving dependent random variables,
and the results are compared with the scenario approximation
technique. Concluding remarks are provided in Section VI
along with directions for future research.
I I . SCENARIO ANDB ERNSTEIN A PPROXIMATION
METHODS
In this section, a brief review of the scenario and Bern-
stein approximation techniques is provided. Under the three
assumptions on probabilistic constraints discussed in Section
I, Bernstein approximation is able to circumvent the need
for multi-dimensional integration, which can be a significant
roadblock in high dimensional applications.
A. Scenario Approximation
Consider the scalar case for the constraining function
F, with m = 1. Assuming that N samples are availablefor the random variable, i.e., {i}Ni=1 Z(), the scenarioapproximation enforces the probabilistic constraint of Eq.1
as a set of Ndeterministic constraints
F(x,i) 0, i=1, ,N. (3)
Although Eq.3 is a set of deterministic constraints, the
scenario approximation method itself is non-deterministic.
This approach is over-conservative in the sense that each
constraint listed in Eq.3 must be satisfied as a hard constraint.
In other words, there is zero tolerance (i.e. zero violation)
for the samples considered. At the same time, there is no
gaurantee for any other realization of the random variable .A question often faced in the scenario approach is determin-
ing the size of the sample (N). It was shown in Ref.[10] that
N should be at least inversely proportional to the acceptable
risk parameter , which can be extremely large. While thisis unattractive, rapid advancements in our computational
capabilities have rendered it less of a concern. The fact
however remains that the scenario approximation is still
rigorously valid only for the particular samples drawn and
gross statistical estimates about the approximation cannot be
inferred with ease.
B. Bernstein Approximation
In this semi-analytical approach, the idea is to determine a
deterministic upper bound for the probabilistic constraint in
Eq.1. Nemirovski (Ref.[13]) achieved this via the exponen-
tial moment generating function, to construct the so-called
Bernstein approximation of the probabilistic constraint
inf>0
{log{E[exp(1F(x,))]}log()} 0. (4)
Note that the parameter appearing in Eq.4 is different fromthe original risk parameter and in general >. This ison account of the fact that the approximation of Eq.4 is more
conservative than the original constraint of Eq.1. If =is used, suboptimal results will be obtained. However, there
are no guidelines for the actual margin in this relaxation and
a numerical study is presented in Sec IV. For >0, theexpectation term in Eq.4 represents the exponential moment
generating function of F. Note that Prob{F(x,) 0} =E[1[0,+)(
1F(x, ))] E[exp(1F(x,))] (> 0).
Thereby, Eq.4 is a conservative approximation of the original
probabilistic constraint in Eq.1. The benefit of using therefore is that it serves as an auxiliary variable minimizing
the moment generating function and hence achieving the
tightest upper bound. This minimization process can also
be jointly carried out in the optimization routine by treating
[x,] as an augmented decision variable, hence the currentpaper uses a variant of Eq.4, given below:
B(x,) 0, (5)
where B(x,) = log{E[exp(1F(x, ))]} log().
Note that in implementation, the convergence of the moment
generating function is a concern, and depends on the nature
of the function F(, ).
C. Numerical Integration for Moment Generating Function
The exponential moment generating function appearing in
Eq.5 involves a d-dimensional integral whose evaluation can
be greatly simplified if the function F is assumed to be
affine (Eq.2) in , which in turn in assumed to be a vector
of independent random variables, i.e. Z() =dj=1Zj(j),whereZj(j)is the marginal pdf ofj with support j. Then,the left hand side of Eq.5 becomes (Ref.[13])
f0+log[1
exp(1f1)Z1 d1] + +
log[d
exp(1fd)Zddd], (6)
whose evaluation involves only a set ofdone-dimensional in-
tegrals. In many applications, the above assumptions are not
applicable, causing the Bernsteing approximation approach
to become computationally tedious. In the next section, we
utilize Markov chain Monte Carlo integration to alleviate this
problem.
III. MARKOVC HAINM ONTE C ARLOI NTEGRATION
Markov chain Monte Carlo (MCMC) is a suite of ran-
domized algorithms used to generate discrete samples from
nonstandard probability density functions. MCMC samples
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are equivalent in measure to the target pdf, partly due to
which they have been highly successful in tackling several
high dimensional numerical problems in engineering, includ-
ing numerical multiple-integration. Let {i}Ni=1 Z() bean MCMC sample. Then the moment generating function
appearing in Eq.5 can be approximated as:
E[exp(1F(x, ))] 1N
N
i=1
exp(1F(x, i)). (7)
Substituting Eq.7 into Eq.5 gives the evaluation Bernstein
approximation B(x,). In addition, it can be seen that thegradientsxB(x,)and B(x,)are also readily available.With these quantities, we can now give the algorithm for
MCMC/Bernstein approach.
Algorithm 1 MCMC/Bernstein approach
1) Generate N MCMC samples {i}Ni=1 following thedistribution Z().
2) Initialize decision variablex and auxiliary variable .Let [x,] be an augmented decision vector.
3) DOgradient-based optimization (e.g. using SNOPTc)
until the breaking condition is reached
a) Cost and constraints: Evaluate B(x,) via in-tegration using MCMC samples {i}Ni=1. Alsocompute J(x), G(x) and H(x);
b) Gradients: Evaluate B(x,) and xB(x,)using MCMC integration. Also compute xJ(x),xG(x) and xH(x);
c) Update the augmented decision vector [x,] perSNOPT c rules
4) END DO
5) Optimal augmented decision vector is [x,].
The accuracy of MCMC integration (Eq.7) depends on
the quality of the samples, which can in turn impact the
optimization results. The current paper uses a variant of the
Metropolis-Hastings (MH) algorithm (Refs. [14], [15], [16])
to generate samples. Adaptive-proposal tuning based on the
history of acceptance probability is used to encourage better
mixing in the chain (Refs.[17], [18]), along with particle
thinning to reduce correlation among samples.
Here, we study the quality of MH-sampling with both
adaptive proposal tuning and thinning for a high-dimensional
pdf used in numerical examples in Sec.IV. Consider atwenty dimensional zero-mean Gaussian distribution with a
covariance matrix containing ones along the diagonal and
0.15 in each off-diagonal location. In adaptive tuning, of the
most recent Q samples generated if more than 70% were
accepted, the covariance matrix of the proposal density is
scaled up by a factor u. If less than 30% are accepted,the covariance is scaled down by the factor d. The test isrepeated one hundred times and in each run, one million
samples are generated after burn-in. The results for Q=10are shown in Table I.
u/d no adaptation 1.1/0.9 1.2/0.8 1.2/0.7e 0.0701 0.0577 0.0611 0.0615e 3.83% 3.16% 3.36% 3.29%
TABLE I
MH-SAMPLING OF A20-D CORRELATEDG AUSSIAN WITH A DAPTIVE
SCALING
Note that Table.I uses two quality metrics:
e= (8a)
e=P P
P 100 (8b)
where is the 2-norm. The metric erepresents the normof the error in the mean ( is the sample mean) and e is the
relative error in the covariance matrix. Table I suggests that
u = 1.1,d=0.9 (i.e. 10% scaling) provides the greatestimprovement over fixed proposal sampling, and is adopted
in the current paper. Combined with a thinning strategy in
which the particles are reduced in number by half, Fig.1
shows the convergence of quality metrics e and e. Settinga threshold of 2% error in e, convergence is seen for N8104 samples.
(a) norm of error of mean
(b) norm of relative error of covariance
Fig. 1. Error plots of simulated Gaussian samples
IV. RESULTS
In this section, the MCMC/Bernstein approach is ap-
plied to chance constrained programs that require numerical
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evaluation of multi-dimensional integrals due to the non-
affine nature of the constraints and/or dependence among
random parameters contained therein, as is true for most
real-life problems. The transcribed deterministic nonlinear
optimization problems are solved using the NLP package
SNOPT c. For the computed optimal decision vector, Monte
Carlo tests are performed using 1 million realizations of the
random variable () to compute the constraint violation rate.
A. Chance Program 1: Portfolio Optimization
Consider a static-model portfolio optimization problem in
which capital is allocated among d assets. Let xj be the
fraction invested in asset j, j = 1, ,dsuch that dj=1xj= 1.Excluding short sales, investments in all assets are non-
negative, i.e.,xj 0 for alli. The objective is to maximize the-th quantile of the total return, denoted by t. The decisionvector is x= {x1,x2, . . . ,xd, t}. Letting rj be the return ininvestment for asset j, the optimization problem can be
formulated as (Ref.[13])
maxJ(x) = t, such that,
d
j=1
xj=1, xj 0, j=1, . . .d, and,
Prob
d
j=1
rjxj< t
(9)
Let rj = (1 + 0.06j) + (0.04j)j , j = 1, ,d. Here, ={1, . . .d} is a correlated Gaussian random variable withthe distribution specified in Sec.III, as a result of which
numerical integration is required for transcription. We use
d=20 and set the allowable risk parameter = 0.01 (i.e. athreshold of 1% violation rate). The Bernstein approximation
of the probabilistic constraint using MCMC integration leads
to the following inequality:
log{1
N
N
i=1
exp(1(td
i=1
rijxj))}log() 0, (10)
whererij= (1 +0.06j)+(0.04j)i
j,i = 1, ,N, j= 1, ,d.As mentioned in Sec.II-B, the risk factor used in the
Bernstein approximation can be allowed to be greater than
the originally specified risk, i.e. , and we study fivecases: 1= 0.01,
2= 0.02, . . .,
5= 0.05. The simulation
results are presented in Figure 2 as a function of the number
of MCMC samples used for numerically evaluating the
Bernstein approximation (x-axis). Fig.2(a) shows the optimal
total return (t) obtained for each case and Fig.2(b) theviolation rate at the obtained optimal solution (t), computed
using 106 MC realizations of.
The following observations can be made: It was seen in
Sec.III that the MCMC samples for the twenty dimensional
correlated Gaussian random variable converge for N> 8104. In accordance with this result, the optimization results
(optimal value t and the corresponding violation rate at
t) also show convergence for N> 8 104 (Fig.2). As theBernstein risk parameter is allowed to increase, theoptimal return t improves, albeit with greater associated
(a) Optimal Value (t)
(b) Violation Rate at t
Fig. 2. Simulation results for portfolio optimization using Bernsteinapproximation
violation rate. However, for N> 8 104, the computed
violation rate is under the specified allowable risk (= 0.01)for all five cases. For as high as 0.05, the computedviolation rate is still under 1%, emphasizing the conservatism
of the Bernstein approximation. It is therefore possible to
improve the quality of optimization by tuning .
In Fig.3, comparative results for the portfolio optimization
problem is shown using the scenario approach. Figs.3(a) and
3(b) illustrate the maximum variations above and below the
mean observed in the optimal return (t) and violation raterespectively over ten runs of the algorithm. It is clear that
there is significant variation even when a large number of
constraining scenarios are included. Moreover, while one
may talk about the mean optimal value (solid line in
Fig.3(a)), it does not make any sense to talk about the meandecision vector. Finally, as mentioned in Sec.II-A, it is
difficult in the scenario approach to ascertain the level of
risk associated with the obtained approximation, until an MC
simulation is performed to numerically compute the violation
rate (att), shown in Fig.3(b). It is clear that as the number of
scenarios increases, the approximation becomes increasingly
conservative (actual violation rate well below prescribed risk
of 1%), at the cost of suboptimal results (note the decreasing
total return). It is evident that the MCMC/Bernstein approach
is better for this problem.
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(a) Optimal Value (t)
(b) Violation Rate at t
Fig. 3. Simulation results for portfolio optimization using scenarioapproximation
B. Chance Program 2: Flood Control
Described in Ref.[19], this problem involves the minimiza-tion of the construction-cost of a five-reservoir flood control
system designed for retaining flood water. The capacities
of individual reservoirs are denoted as xi, i= 1, ,5 andthe volume of incoming flood water as i, i=1, ,5 (fiveinlets). The variable = {1, . . . ,5} follows a correlateddistribution. The optimization problem can be formulated as
follows
minJ(x) =0.4x1+ 0.5x2+ 0.6x3+ 1.2x4+ 1.8x5, (11a)
s.t. Prob
5x5>0
4+5x4x5>01+4+5x1x4x5>02+4+5x2x4x5>03+4+5x3x4x5>0
1+2+4+5x1x2x4x5>01+3+4+5x1x3x4x5>02+3+4+5x2x3x4x5>0
1+2+3+4+5x1x2x3x4x5>0
,
(11b)
0 xi 1, i=1,2,3
0 x4 2, 0 x5 3. (11c)
The function J(x) represents the cost of construction of thesystem. Eq.11b is a cascading set of probabilistic constraints
capturing various water overflow situations. Note that this
joint constraint begins by looking at the overflow probability
at a single reservoir (#5) and single inlet (5), all the waydown to the combined capacity of all reservoirs filling up
through all inlets. Nemirovski et al. provided a few strategies
in Ref.[13] to separate out joint chance constraints. Cur-
rently, there exist no methods for dealing with the full joint
constraint and no established guidelines for separating them.
Motivated by the finite subadditivity of probability measureP(Mi=1Ai)
Mi=1 P(Ai)
, Ref.[13] suggests the assignment
of individual risk, i to each constraint such that Mi=1 i ,
assuming there areMjoint constraints. The actual assignment
of numerical values to the individual risks is an open problem
(Ref.[13]). In Ref.[19], the supporting hyperplane method
was employed for two cases, =0.1 and = 0.2, i.e. a riskof 10% and 20% respectively, using correlated Gaussian and
Gamma distributions. The Bernstein approach of Ref.[13]
cannot directly be applied here because of the correlated
nature of the random variable .Using the MCMC based Bernstein approximation, opti-
mization results are shown in Fig.4 as a function of the
number of MCMC samples. Four cases of risk assignment
are tested, listed in Table II. In case 1, each constraint
is assigned a Bernstein risk of i =0.2. Compounded bythe fact that the Bernstein approximation is conservative
to begin with, this simple risk assignment strategy results
is excessively conservative estimates (note the computed
joint violation rate in Table II). Motivated by the previously
demonstrated fact that assignment of a high Bernstein risk
() does not violate the originally prescribed risk (), case2 onwards ramps up the individual risk of each constraint as
shown in Table II. Increasing Bernstein risks (i ) uniformlyfrom 20% per constraint to 50% per constraint improves the
optimality of the solution while maintaining the overall joint
violation rate under 15% (see Fig.4).
Case # i , i =1, . . .9 Violation Rate (%) Optimal Value1 0.2 4.1 7.8
2 0.3 6.7 7.3
3 0.4 9.7 6.8
4 0.5 13.2 6.4
TABLE II
RISK ASSIGNMENT( IN %) T OI NDIVIDUALC HANCEC ONSTRAINTS:
FLOODC ONTROL
In comparison, the scenario approach fails to produce
a result for this optimization problem. Recall that in this
method, probabilistic constraints are imposed as a set of hard
deterministic constraints for a sample of scenarios generated
from the random variable ( in the above problem). In the
flood control problem, this technique becomes infeasible due
to the following reason: note that the maximum possible
value of the sum 5i=1xi is 8 by virtue of the deterministic
constraints on the individual reservoir capacities (Eq.11c).
However, upon sampling the random variable , the value
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(a) optimal value (t)
(b) violation rate at t
Fig. 4. Simulation results for the flood control problem using Bernsteinapproximation
of the sum 5i=1 i often exceeds 8
=max
5i=1xi
. For
these scenarios, the ninth probabilistic constraint in Eq.11b
becomes impossible to satisfy. Since each constraint isimposed in a deterministic manner, the feasible region for
optimization shrinks down to the null set. For example, in
15% of the samples in a small set of 100 realizations of,
5i=1 i > max
5i=1xi
, which leads to a null search space.
Note that as the number of scenarios considered becomes
larger, the likelihood of creating impossible constraints only
becomes greater, causing the method to fail.
Note that in the current problem, it was easy to detect the
incidence of impossible constraints due to the simple form
of Eq.11b. In general, this may not be feasible and therefore
a strategy of simply rejecting the scenarios that lead to
impossible constraints cannot be employed.
V. DISCUSSION
The Bernstein method results in a conservative deter-
ministic approximation of the probabilistic constraints. The
current paper employs MCMC integration (a randomized
technique) to extend the Bernstein approach to the case
of non-affine probabilistic constraints involving correlated
random variables. MCMC integration is a well established,
successful numerical integration technique that generally
scales well to high dimensional problems. We observed that
convergence of optimization follows the same trend as the
convergence of the MCMC integration. On the other hand,
the scenario approximation is a stochastic technique that
implements the probabilistic constraints in a deterministic
manner for a sample of constraining scenarios. There is
significant variation about the mean optimization results and
the approximation becomes increasingly conservative as the
number of constraining scenarios is increased.
One must note however that in very high dimensional
applications, MCMC sampling may require significant tun-
ing. While numerous adaptive MCMC algorithms exist
(Refs.[20], [21]), care is needed to ensure that enough
samples are used for convergence of optimization. In con-
trast, scenario approximation offers speed in implementation.
While sampling is needed in the scenario approach as well
(to generate constraining scenarios), the number of samples
required is significantly less.
Both the Bernstein and scenario approaches are conserva-
tive and a tradeoff exists between optimality and violation
rate. In the Bernstein method, it is straightforward to control
the violation rate by tuning the Bernstein risk, and
generally,
> (the actual prescribed risk). However, thereare no guidelines for the margin in this relaxation and tuningmust be carried out empirically. On the other hand, there
is little control over the risk associated with the scenario
method. Given the large variations in the optimal values
as well as violation rates, one is typically forced to pick
a relatively suboptimal solution.
Finally, neither the MCMC/Bernstein technique nor the
scenario method are applicable to all types of problems. The
Bernstein approach is based on the evaluation of a moment
generating function, which may diverge if the function F is
not integrable, or if during the execution of the algorithm,
the parameter becomes too small (Eq.7). It has already
been demonstrated that the scenario approach is not suitableto problems which involve complex probabilistic constraints
which may lead to impossible deterministic constraints for
certain realizations of the random variable .
VI. CONCLUSIONS
In this paper, the Bernstein approximation method was
combined with MCMC numerical integration to extend it
to optimization problems containing non-affine chance con-
straints with correlated random variables. It was shown
through numerical examples that convergence of the op-
timization procedure follows the same trend as the con-
vergence of numerical integration. Generally, the Bernstein
approach provides greater flexibility and wider applicability
than the popular scenario method, although at the cost of
greater computational effort. Through the numerical ex-
amples considered, it has been shown that it is relatively
straightforward to control the constraint violation rate by
tuning the so-called Bernstein risk parameter.
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