Reformulation of chance constrained problems using penalty functions - - PowerPoint PPT Presentation

Reformulation of chance constrained problems using penalty functions

Martin Branda

Charles University in Prague Faculty of Mathematics and Physics

EURO XXIV July 11-14, 2010, Lisbon

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Contents

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

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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

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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...

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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, (2) 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.

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Reformulations of chance constrained problems

Solving CCP

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.

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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 ∈ Rk.

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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)]

• (3)

with an optimal solution xN. In Y.M. Ermoliev, et al (2000) for ϑ1,1 and m = 1.

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Reformulations of chance constrained problems

Deterministic vs. stochastic penalty method

Deterministic - penalizes the infeasibility with respect to the decision vector, cf. M.S. Bazara, et al. (2006). Stochastic - penalizes violations of the constraints jointly with respect to the decision vector and to the random parameter...

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Asymptotic equivalence

Contents

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

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Asymptotic equivalence

Assumptions (brief)

Under the following assumptions, the asymptotic equivalence of the problems can be shown: Continuity of constraints and 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!

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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.

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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 maximum 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 (3) generates optimal solutions xN which also satisfy the chance constraints (2) with the given ǫ.

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Asymptotic equivalence

Moreover, bounds on the optimal value ψǫ of (2) based on the optimal value ϕN of (3) 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.

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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

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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 function qj is defined to be ˆ qS

j (x) = S−1 S

• s=1

I(0,∞)

• pj(x, ωs)
• .

(4)

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Sample approximations using Monte-Carlo techniques

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, (5) 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.

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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 ...

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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.

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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 . (6) If we set m = 1, we get the same inequality as J. Luedtke, et al (2008).

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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).

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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) Martin Branda (MFF UK) Reformulation of CCP 2010 22 / 39

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

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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 problem. Similar study was performed by J. Dupaˇ cov´ a, et al. (1991).

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Numerical study and comparison

Integer VaR and penalty function problems

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 means and the variance matrix. 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.

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Numerical study and comparison

We denote Qi the quotation of the ”mini-lot” of security i, fi the fixed transaction costs (not depending on the investment amount), ci the proportional transaction costs (depending on the investment amount), Ri the random return of the security i, xi the number of ”mini-lots”, yi binary variables which indicate, whether the security i is bought or not. Then, the random loss function depending on our decisions and the random returns has the following form −

n

• i=1

(Ri − ci)Qixi +

n

• i=1

fiyi.

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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.

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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,

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

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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 + . (8) Setting N = 1/(1 − ε) we obtain the CVaR problem, c.f. R.T. Rockafellar,

• S. Uryasev (2002).

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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

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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

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Numerical study and comparison

Monte-Carlo simulation study

Truncated normal distribution was used to model the random returns. 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 and CPLEX... Reliability (feasibility) of the optimal solutions was verified on an independent sample of 10 000 realizations.

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Numerical study and comparison

Reliability 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

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Numerical study and comparison

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

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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.

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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.

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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.

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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.

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Numerical study and comparison

Thank you for your attention.

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