Chance constrained problems: penalty reformulation and performance - - PowerPoint PPT Presentation

Chance constrained problems: penalty reformulation and performance of sample approximation technique

Martin Branda, Jitka Dupaˇ cov´ a

Charles University in Prague Faculty of Mathematics and Physics

SP XII Conference August 16-20, 2010, Halifax

M.Branda, J.Dupaˇ cov´ a (Charles University) 2010 1 / 44

Contents

1 Reformulations of chance constrained problems 2 Asymptotic equivalence 3 Sample approximations using Monte-Carlo techniques 4 Numerical study and comparison

M.Branda, J.Dupaˇ cov´ a (Charles University) 2010 2 / 44

Reformulations of chance constrained problems

Contents

1 Reformulations of chance constrained problems 2 Asymptotic equivalence 3 Sample approximations using Monte-Carlo techniques 4 Numerical study and comparison

M.Branda, J.Dupaˇ cov´ a (Charles University) 2010 3 / 44

Reformulations of chance constrained problems

Optimization problem with uncertainty

In general, we consider the following program with a random factor ω min {f (x) : x ∈ X, gi(x, ω) ≤ 0, i = 1, . . . , k} , (1) where gi, i = 0, . . . , k, are real functions on Rn × Rn′, X ⊆ Rn and ω ∈ Rn′ is a realization of a n′-dimensional random vector defined on the probability space (Ω, F, P). If P is known, we can use chance constraints to deal with the random constraints...

M.Branda, J.Dupaˇ cov´ a (Charles University) 2010 4 / 44

Reformulations of chance constrained problems

Multiple chance constrained problem

ψǫ = minx∈X f (x), s.t. P

• g11(x, ω) ≤ 0, . . . , g1k1(x, ω) ≤ 0
• ≥

1 − ε1, . . . P

• gm1(x, ω) ≤ 0, . . . , gmkm(x, ω) ≤ 0
• ≥

1 − εm, with optimal solution xǫ, where we denoted ǫ = (ε1, . . . , εm) with levels εj ∈ (0, 1). The formulation covers the joint (k1 > 1 and m = 1) as well as the individual (kj = 1 and m > 1) chance constrained problems as special cases.

M.Branda, J.Dupaˇ cov´ a (Charles University) 2010 5 / 44

Reformulations of chance constrained problems

Solving chance constraned problems

In general, the feasible region is not convex even if the functions are convex, it is even not easy to check feasibility because it leads to computations of multivariate integrals. Hence, we will try to reformulate the chance constrained problem using penalty functions.

M.Branda, J.Dupaˇ cov´ a (Charles University) 2010 6 / 44

Reformulations of chance constrained problems

Consider the penalty functions ϑj : Rm → R+, j = 1, . . . , m, continuous nondecreasing, equal to 0 on Rm

− and positive otherwise, e.g.

ϑ1,p(u) =

k

• i=1
• [ui]+p, p ∈ N

ϑ2(u) = max

1≤i≤k[ui]+,

= min

• t ≥ 0 : ui − t ≤ 0, i = 1, . . . , k}

where u ∈ Rm.

M.Branda, J.Dupaˇ cov´ a (Charles University) 2010 7 / 44

Reformulations of chance constrained problems

Penalty function problem

Let pj denote the penalized constraints pj(x, ω) = ϑj(gj1(x, ω), . . . , gjkj(x, ω)), ∀j. Then the penalty function problem is formulated as follows ϕN = min

x∈X

• f (x) + N ·

m

• j=1

E[pj(x, ω)]

• with an optimal solution xN.

In Y.M. Ermoliev, et al (2000) for ϑ1,1 and m = 1.

M.Branda, J.Dupaˇ cov´ a (Charles University) 2010 8 / 44

Asymptotic equivalence

Contents

1 Reformulations of chance constrained problems 2 Asymptotic equivalence 3 Sample approximations using Monte-Carlo techniques 4 Numerical study and comparison

M.Branda, J.Dupaˇ cov´ a (Charles University) 2010 9 / 44

Asymptotic equivalence

Assumptions (brief)

Under the following assumptions, the asymptotic equivalence of the problems can be shown: Continuity of the constraints and the probabilistic functions. Compactness of the fixed set of feasible solutions. Existence of integrable majorants. Existence of a permanently feasible solution. No assumption on linearity or convexity!

M.Branda, J.Dupaˇ cov´ a (Charles University) 2010 10 / 44

Asymptotic equivalence

Assumptions

Assume that X = ∅ is compact, f (x) is a continuous function and (i) gji(·, ω), i = 1, . . . , kj, j = 1, . . . , m, are almost surely continuous; (ii) there exists a nonnegative random variable C(ω) with E[C 1+κ(ω)] < ∞ for some κ > 0, such that |pj(x, ω)| ≤ C(ω), j = 1, . . . , m, for all x ∈ X; (iii) E[pj(x

′, ω)] = 0, j = 1, . . . , m, for some x ′ ∈ X;

(iv) P(gji(x, ω) = 0) = 0, i = 1, . . . , kj, j = 1, . . . , m, for all x ∈ X.

M.Branda, J.Dupaˇ cov´ a (Charles University) 2010 11 / 44

Asymptotic equivalence

Denote η = κ/(2(1 + κ)), and for arbitrary N > 0 and ǫ ∈ (0, 1)m put εj(x) = P

• pj(x, ω) > 0
• ,

j = 1, . . . , m, αN(x) = N ·

m

• j=1

E[pj(x, ω)], βǫ(x) = ε−η

max m

• j=1

E[pj(x, ω)], where εmax denotes the maximal component of the vector ǫ = (ε1, . . . , εm) and [1/N1/η] = (1/N1/η, . . . , 1/N1/η) is the vector of length m. THEN for any prescribed ǫ ∈ (0, 1)m there always exists N large enough so that minimization of the penalty objective generates optimal solutions xN which also satisfy the chance constraints with the given ǫ.

M.Branda, J.Dupaˇ cov´ a (Charles University) 2010 12 / 44

Asymptotic equivalence

Bounds on optimal values

Moreover, bounds on the optimal value ψǫ based on the optimal value ϕN and vice versa can be constructed: ϕ1/εη

max(xN) − βǫ(xN)(xǫ(xN)) ≤ ψǫ(xN) ≤ ϕN − αN(xN),

ψǫ(xN) + αN(xN) ≤ ϕN ≤ ψ[1/N1/η] + β[1/N1/η](x[1/N1/η]), with lim

N→+∞ αN(xN) =

lim

N→+∞ εj(xN) =

lim

εmax→0+ βǫ(xǫ) = 0

for any sequences of optimal solutions xN and xǫ.

M.Branda, J.Dupaˇ cov´ a (Charles University) 2010 13 / 44

Sample approximations using Monte-Carlo techniques

Contents

1 Reformulations of chance constrained problems 2 Asymptotic equivalence 3 Sample approximations using Monte-Carlo techniques 4 Numerical study and comparison

M.Branda, J.Dupaˇ cov´ a (Charles University) 2010 14 / 44

Sample approximations using Monte-Carlo techniques

Let ω1, . . . , ωS be an independent Monte Carlo sample of the random vector ω. Then, the sample version of the chance constraint is defined to be ˆ qS

j (x) := S−1 S

• s=1

I(0,∞)

• pj(x, ωs)
• ≤ γj,

(2) γj ∈ (0, 1).

M.Branda, J.Dupaˇ cov´ a (Charles University) 2010 15 / 44

Sample approximations using Monte-Carlo techniques

Sample approximated chance constrained problem

Finally, the sample version of the multiple jointly chance constrained problem is defined as ˆ ψS

γ

= minx∈X f (x), s.t. ˆ qS

1 (x)

≤ γ1, . . . ˆ qS

m(x)

≤ γm, (3) where the levels γj are allowed to be different from the original levels εj. The sample approximation of the chance constrained problem can be reformulated as a large mixed-integer nonlinear program.

M.Branda, J.Dupaˇ cov´ a (Charles University) 2010 16 / 44

Sample approximations using Monte-Carlo techniques

Rates of convergence, sample sizes

We will draw our attention to the case when the set of feasible solutions is finite, i.e. |X| < ∞, which appears in the bounded integer programs, or infinite bounded. Using slight modification of the approach by S. Ahmed, J. Luedtke, A. Shapiro, et al. (2008, 2009), we obtain ...

M.Branda, J.Dupaˇ cov´ a (Charles University) 2010 17 / 44

Sample approximations using Monte-Carlo techniques

Lower bound for the chance constrained problem

We will assume that it holds γj > εj for all j. we can choose the sample size S to obtain that the feasible solution x is also feasible for the sample approximation with a probability at least 1 − δ, i.e. S ≥ 2 minj∈{1,...,m}(γj − εj)2/εj ln m δ , which corresponds to the result of S. Ahmed, et al (2008) for m = 1. (The estimate is based on Chernoff and Bonferroni inequalities.)

M.Branda, J.Dupaˇ cov´ a (Charles University) 2010 18 / 44

Sample approximations using Monte-Carlo techniques

Feasibility - finite |X|

We will assume that it holds γj < εj for all j. Then it is possible to estimate the sample size S such that the feasible solutions of the sample approximated problems are feasible for the original problem, i.e. x ∈ Xǫ, with a high probability 1 − δ S ≥ 1 2 minj∈{1,...,m}(γj − εj)2 ln m|X \ Xǫ| δ . (4) If we set m = 1, we get the same inequality as J. Luedtke, et al (2008). (The estimate is based on Hoeffding and Bonferroni inequalities.)

M.Branda, J.Dupaˇ cov´ a (Charles University) 2010 19 / 44

Sample approximations using Monte-Carlo techniques

Feasibility - bounded |X|

We will assume that it holds γj < εj for all j and Lipschitz continuity of the penalized constraints, i.e. |pj(x, ξ) − pj(x′, ξ)| ≤ Mj

• x − x′

, ∀x, x′ ∈ X, ∀ξ ∈ Ξ, ∀j, for some moduli Mj > 0. Then it is possible to estimate the sample size S such that the feasible solutions of the sample approximated problems are feasible for the original problem, i.e. x ∈ Xǫ, with a high probability 1 − δ S ≥ 2 minj∈{1,...,m}(εj − γj)2

• ln m

δ + ln

• 2

minj∈{1,...,m}(εj − γj)

• + n ln

2MmaxD τ . If we set m = 1, we get the same inequality as J. Luedtke, et al (2008). (The estimate is based on Hoeffding and Bonferroni inequalities.)

M.Branda, J.Dupaˇ cov´ a (Charles University) 2010 20 / 44

Sample approximations using Monte-Carlo techniques 1. 2. 3. Stochastic Sample Solution programming approximation validation formulation (SA) Program with a random − → Chance constrained − → SA CCP − → Reliability factor problem (CCP) ց ↓ Penalty function − → SA PFP − → Reliability problem (PFP) M.Branda, J.Dupaˇ cov´ a (Charles University) 2010 21 / 44

Numerical study and comparison

Contents

1 Reformulations of chance constrained problems 2 Asymptotic equivalence 3 Sample approximations using Monte-Carlo techniques 4 Numerical study and comparison

M.Branda, J.Dupaˇ cov´ a (Charles University) 2010 22 / 44

Numerical study and comparison

Numerical study and comparisons

We will compare the ability of both sample approximated problems to generate a feasible solution of the original chance constrained problem. Similar study was performed by J. Dupaˇ cov´ a, et al. (1991).

M.Branda, J.Dupaˇ cov´ a (Charles University) 2010 23 / 44

Numerical study and comparison

Investment problem

We consider 13 most liquid assets which are traded on the main market (SPAD) on Prague Stock Exchange. Weekly returns from the period 6th February 2009 to 10th February 2010 are used to estimate the parameters of distributions. Suppose that the small investor trades assets on the ”mini-SPAD”

• market. This market enables to trade ”mini-lots” (standardized

number of assets) with favoured transaction costs.

M.Branda, J.Dupaˇ cov´ a (Charles University) 2010 24 / 44

Numerical study and comparison

Loss random variable

The random loss function Z(x, y, R) = −

n

• i=1

(Ri − ci)Qixi +

n

• i=1

fiyi, Qi the quotation of the ”mini-lot” of security i, fi the fixed transaction costs, ci the proportional transaction costs, Ri the random return of the security i, truncated normal distribution and multivariate skewed t-distribution (A. Azzalini et al (2003)) were used to model the random returns, xi the number of ”mini-lots”, yi binary variables which indicate, whether the security i is bought

• r not.

M.Branda, J.Dupaˇ cov´ a (Charles University) 2010 25 / 44

Numerical study and comparison

Set of feasible solutions

The set of feasible solutions contains a budget constraint and the restrictions on the minimal and the maximal number of ”mini-lots” which can be bought, i.e. X = {(x, y) ∈ Nn × {0, 1}n Bl ≤ n

i=1(1 + ci)Qixi + n i=1 fiyi ≤ Bu,

liyi ≤ xi ≤ uiyi, i = 1, . . . , n}, where Bl and Bu are the lower and the upper bound on the capital available for the portfolio investment, li > 0 and ui > 0 are the lower and the upper number of units for each security i.

M.Branda, J.Dupaˇ cov´ a (Charles University) 2010 26 / 44

Numerical study and comparison

Value at Risk problem

The chance constrained portfolio problem can be formulated as follows min

(r,x,y)∈R×X r

P

• −

n

• i=1

(Ri − ci)Qixi +

n

• i=1

fiyi ≤ r

• ≥ 1 − ε,

(5) this is in fact minimization of Value at Risk (VaR).

M.Branda, J.Dupaˇ cov´ a (Charles University) 2010 27 / 44

Numerical study and comparison

Penalty function portfolio problem

Corresponding penalty function problem using the penalty ϑ1,1 is min

(r,x,y)∈R×X r + N · E

• −

n

• i=1

(Ri − ci)Qixi +

n

• i=1

fiyi − r + . (6) Setting N = 1/(1 − ε) we obtain the CVaR problem, c.f. R.T. Rockafellar,

• S. Uryasev (2002).

M.Branda, J.Dupaˇ cov´ a (Charles University) 2010 28 / 44

Numerical study and comparison

Sample approximated chance constrained problem

(SA CCP)

The problem can be formulated as a large mixed-integer linear program using additional binary variables zs, s = 1, . . . , S ˆ ϕCPP,S

γ

= min

(r,x,y,z)∈R×X×{0,1}S r

−

n

• i=1

(Rs

i − ci)Qixi + n

• i=1

fiyi − M(1 − zs) ≤ r, 1 S

S

• s=1

zs ≥ 1 − γ, for some level γ ∈ (0, 1) (γ = ε in general).

M.Branda, J.Dupaˇ cov´ a (Charles University) 2010 29 / 44

Numerical study and comparison

Sample approximated penalty function problem

(SA PFP)

The variables which are necessary to replace the positive parts are nonnegative continuous and the resulting problem is ˆ ϕS

N

= min

(r,x,y,v)∈R×X×RS

+

r + N S ·

S

• s=1

vs vs ≥ −

n

• i=1

(Rs

i − ci)Qixi + n

• i=1

fiyi − r. for some penalty parameter N > 0.

M.Branda, J.Dupaˇ cov´ a (Charles University) 2010 30 / 44

Numerical study and comparison

Estimated sample sizes

Estimates for the sample sizes which are necessary to generate a lower bound for the optimal value of the original chance constrained problem, cf. S. Ahmed, A. Shapiro (2008): S ≥ 2ε (γ − ε)2 ln 1 δ , a feasible solution of the original chance constrained problem, slight modification of J. Luedtke, S. Ahmed (2008): S ≥ 2 (ε − γ)2

• ln 1

δ + 13 ln 116 + 13 ln 2 + ln

• 2

(ε − γ)

• + ln

2D τ , which is based on the decomposition of the set of feasible solutions into the integer and real bounded part, where |X| ≤ 11613 · 213, and we set τ = 10−6, D = 2 · 106 (difference between the worst loss and the best profit for the distribution with bounded support).

M.Branda, J.Dupaˇ cov´ a (Charles University) 2010 31 / 44

Numerical study and comparison

Sample sizes - lower bound

Table: Sample sizes - lower bound

ε γ δ S 0.1 0.2 0.01 93 0.05 0.1 0.01 185 0.01 0.02 0.01 9211 0.1 0.2 0.001 139 0.05 0.1 0.001 277 0.01 0.02 0.001 13816

M.Branda, J.Dupaˇ cov´ a (Charles University) 2010 32 / 44

Numerical study and comparison

Sample sizes - feasibility

Bounded losses

Table: Sample sizes - feasibility

ε γ δ S 0.1 0.05 0.01 86496 0.05 0.025 0.01 348199 0.01 0.005 0.01 901792970 0.1 0.05 0.001 88338 0.05 0.025 0.001 355567 0.01 0.005 0.001 920213650

M.Branda, J.Dupaˇ cov´ a (Charles University) 2010 33 / 44

Numerical study and comparison

Monte-Carlo simulation study

Truncated normal distribution was simulated using R 2.10.1. 100 samples for each sample size S = 100, 250, 500, 750, 1000. Decreasing levels ε = 0.1, 0.05, 0.01. Increasing penalty parameters N = 1, 10, 100, 1000. Resulting mixed-integer linear programming problems were solved using GAMS 23.2 and IBM ILOG CPlex 12.1. Reliability (feasibility) of the optimal solutions of the SA problems was verified on an independent sample of 10 000 realizations.

M.Branda, J.Dupaˇ cov´ a (Charles University) 2010 34 / 44

Numerical study and comparison

Reliabilities

Reliability of CCP solutions S γ min max mean st.dev 100 0.1 0.8844 0.9967 0.9592 0.0255 100 0.05 0.9054 0.9869 0.9516 0.0189 100 0.01 0.8939 0.9941 0.9456 0.0250 250 0.1 0.9546 0.9968 0.9824 0.0098 250 0.05 0.9545 0.9950 0.9820 0.0086 250 0.01 0.9555 0.9950 0.9807 0.0115 500 0.1 0.9744 0.9982 0.9903 0.0043 500 0.05 0.9744 0.9982 0.9903 0.0043 500 0.01 0.9726 0.9982 0.9906 0.0043 750 0.1 0.9849 0.9994 0.9952 0.0033 750 0.05 0.9849 0.9994 0.9952 0.0033 750 0.01 0.9866 0.9994 0.9953 0.0032 1000 0.1 0.9870 1.0000 0.9966 0.0025 1000 0.05 0.9870 1.0000 0.9966 0.0025 1000 0.01 0.9870 1.0000 0.9966 0.0025

M.Branda, J.Dupaˇ cov´ a (Charles University) 2010 35 / 44

Numerical study and comparison

Reliability of PFP solutions S N min max mean st.dev 100 1 0.7622 0.9480 0.8770 0.0303 100 10 0.8967 0.9976 0.9581 0.0220 100 100 0.8967 0.9976 0.9581 0.0219 100 1000 0.8967 0.9976 0.9581 0.0218 250 1 0.8330 0.9290 0.8888 0.0199 250 10 0.9495 0.9950 0.9788 0.0101 250 100 0.9571 0.9973 0.9841 0.0089 250 1000 0.9571 0.9973 0.9840 0.0089 500 1 0.8716 0.9270 0.9016 0.0134 500 10 0.9723 0.9955 0.9871 0.0044 500 100 0.9813 0.9996 0.9935 0.0033 500 1000 0.9813 0.9995 0.9934 0.0033 750 1 0.8697 0.9330 0.8990 0.0108 750 10 0.9785 0.9950 0.9878 0.0036 750 100 0.9890 0.9995 0.9957 0.0026 750 1000 0.9890 0.9993 0.9956 0.0026 1000 1 0.8739 0.9253 0.8976 0.0097 1000 10 0.9753 0.9964 0.9886 0.0038 1000 100 0.9900 0.9999 0.9966 0.0023 1000 1000 0.9900 0.9999 0.9966 0.0023

M.Branda, J.Dupaˇ cov´ a (Charles University) 2010 36 / 44

Numerical study and comparison

Optimal values

Optimal values ˆ ϕCPP,S

γ

S γ min max mean st.dev 100 0.1 29739.36 66854.82 41784.66 7525.69 100 0.05 29739.36 66854.82 41821.60 7465.46 100 0.01 29680.35 69513.05 42312.34 7612.11 250 0.1 37609.63 121252.72 52429.77 9887.54 250 0.05 37609.63 121252.72 52431.23 9884.16 250 0.01 38260.62 121972.21 52626.23 9909.60 500 0.1 45085.97 125638.34 67824.32 15849.91 500 0.05 45085.97 125638.34 67824.32 15849.91 500 0.01 45085.97 125638.34 67942.02 15757.14 750 0.1 48562.73 160984.79 74655.08 19435.11 750 0.05 48562.73 160984.79 74652.82 19436.71 750 0.01 48562.73 155469.46 74679.40 19187.28 1000 0.1 59129.41 187831.95 93390.26 28293.28 1000 0.05 59129.41 187831.95 93414.25 28269.13 1000 0.01 59129.41 187831.95 93384.85 28264.63

M.Branda, J.Dupaˇ cov´ a (Charles University) 2010 37 / 44

Numerical study and comparison

Optimal values ˆ ϕPFP,S

γ

S N min max mean st.dev 100 1 24011.20 45692.02 33403.52 4311.27 100 10 30739.36 67854.82 42830.41 7489.58 100 100 30739.36 67854.82 42902.79 7484.36 100 1000 30739.36 67854.82 42903.93 7474.20 250 1 29745.59 83386.57 37382.48 6017.16 250 10 37848.38 118742.60 52156.49 9360.82 250 100 39630.90 122252.72 53493.47 9862.21 250 1000 39630.90 122252.72 53458.34 9898.87 500 1 32345.07 71348.70 43537.39 8424.45 500 10 45481.55 110479.11 63886.92 13472.75 500 100 46085.97 126638.34 68995.38 15851.31 500 1000 46085.97 126638.34 68914.67 15748.83 750 1 33415.20 94959.23 44922.49 9914.34 750 10 47249.47 150732.32 68251.45 17167.97 750 100 49562.73 157103.91 75669.31 19379.62 750 1000 49562.73 157103.91 75541.31 19234.11 1000 1 34061.30 98653.13 51840.01 12169.11 1000 10 51622.86 162568.42 82550.78 23493.47 1000 100 59121.39 182075.76 94331.08 27977.78 1000 1000 59121.39 182561.86 94357.45 28209.17

M.Branda, J.Dupaˇ cov´ a (Charles University) 2010 38 / 44

Numerical study and comparison

Penalty term S N min max mean st.dev 100 1 2622.61 15554.88 7731.06 2530.95 100 10 0.00 309.34 3.09 30.93 100 100 0.00 0.00 0.00 0.00 100 1000 0.00 0.00 0.00 0.00 250 1 5787.60 20394.96 9673.10 2115.72 250 10 162.04 12197.59 2570.37 2255.62 250 100 0.00 0.00 0.00 0.00 250 1000 0.00 0.00 0.00 0.00 500 1 6682.76 27887.84 11865.88 3520.77 500 10 886.09 12248.70 5109.98 2719.34 500 100 0.00 0.00 0.00 0.00 500 1000 0.00 0.00 0.00 0.00 750 1 7304.04 43597.69 13191.21 5100.68 750 10 1995.04 18457.88 7328.08 3405.62 750 100 0.00 0.00 0.00 0.00 750 1000 0.00 0.00 0.00 0.00 1000 1 8886.75 42469.68 16647.91 7013.17 1000 10 2795.02 29325.36 9591.71 5108.62 1000 100 0.00 0.00 0.00 0.00 1000 1000 0.00 0.00 0.00 0.00

M.Branda, J.Dupaˇ cov´ a (Charles University) 2010 39 / 44

Numerical study and comparison

Conclusion

The penalty term decreases with increasing penalty parameter N and reduces violations of the stochastic constraint. The reliability of the obtained solutions increases with increasing levels γ and penalty parameters N for each sample size S. Both problems are also able to generate comparable solutions for the same sample sizes. Comparable computing times. Time saving significant for more demanding problem, e.g. nonlinear, cf. E. ˇ Zampachov´ a (2009).

M.Branda, J.Dupaˇ cov´ a (Charles University) 2010 40 / 44

Numerical study and comparison

References

• S. Ahmed and A. Shapiro (2008). Solving chance-constrained stochastic

programs via sampling and integer programming. In Tutorials in Operations Research, Z.-L. Chen and S. Raghavan (eds.), INFORMS.

• E. Angelelli, R. Mansini, M.G. Speranza (2008). A comparison of MAD

and CVaR models with real features. Journal of Banking and Finance 32, 1188-1197. M.S. Bazara, H.D. Sherali, C.M. Shetty (2006). Nonlinear programming: theory and algorithms. Third Edition, John Wiley & Sons, New Jersey.

M.Branda, J.Dupaˇ cov´ a (Charles University) 2010 41 / 44

Numerical study and comparison

References

M.B., J. Dupaˇ cov´ a (2008). Approximations and contamination bounds for probabilistic programs. SPEPS 2008-13. M.B. (2010). Reformulation of general chance constrained problems using the penalty functions. SPEPS 2010-2.

• J. Dupaˇ

cov´ a, A. Gaivoronski, Z. Kos, T. Szantai (1991). Stochastic programming in water management: A case study and a comparison

• f solution techniques. European Journal of Operational Research 52,

28-44. Y.M. Ermoliev, T.Y. Ermolieva, G.J. Macdonald, and V.I. Norkin (2000). Stochastic optimization of insurance portfolios for managing exposure to catastrophic risks. Annals of Operations Research 99, pp. 207-225.

M.Branda, J.Dupaˇ cov´ a (Charles University) 2010 42 / 44

Numerical study and comparison

References

• J. Luedtke and S. Ahmed (2008). A sample approximation approach for
• ptimization with probabilistic constraints. SIAM Journal on

Optimization, vol.19, pp.674-699. R.T. Rockafellar, S. Uryasev (2002). Conditional Value-at-Risk for General Loss Distributions. Journal of Banking and Finance, 26, 1443-1471.

• A. Shapiro (2003). Monte Carlo Sampling Methods. In Stochastic

Programming (A. Ruszczynski and A. Shapiro eds.), Handbook in Operations Research and Management Science Vol. 10, Elsevier, Amsterdam, 483-554.

• E. ˇ

Zampachov´ a (2009). Approximations in stochastic optimization and their applications. Ph.D. thesis, Brno University of Technology.

M.Branda, J.Dupaˇ cov´ a (Charles University) 2010 43 / 44

Numerical study and comparison

Thank you for your attention.

e-mail: branda@karlin.mff.cuni.cz, dupacova@karlin.mff.cuni.cz

M.Branda, J.Dupaˇ cov´ a (Charles University) 2010 44 / 44