Chance constrained problems: reformulation using penalty functions - - PowerPoint PPT Presentation

Chance constrained problems: reformulation using penalty functions and sample approximation technique

Martin Branda

Charles University in Prague Faculty of Mathematics and Physics Department of Probability and Mathematical Statistics

25th European Conference on Operational Research 8-11 July 2012, Vilnius

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Contents

1 Stochastic programming formulations 2 Relations between formulations 3 Sample approximations using Monte-Carlo techniques

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Stochastic programming formulations

Contents

1 Stochastic programming formulations 2 Relations between formulations 3 Sample approximations using Monte-Carlo techniques

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Stochastic programming formulations

Program with a random factor Stochastic ւ ↓ ց 1. programming formulation Chance Problem Integrated constrained ⇔ with penalty ⇔ chance constrained problem

• bjective

problem (CCP) (PPO) (ICC) Sample 2. approximation ↓ ↓ ↓ (S.A.) S.A. CCP S.A. PPO S.A. ICC 3. Solution ↓ ↓ ↓ validation Reliability Reliability Reliability check check check

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Stochastic programming formulations

Optimization problem with a random factor

Program with a random factor ξ min {f (x) : x ∈ X, gi(x, ξ) ≤ 0, i = 1, . . . , k} , where gi, i = 0, . . . , k, are real functions on Rn × Rn′, X ⊆ Rn and ξ ∈ 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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Stochastic programming formulations

Chance constrained problem (CCP)

Chance constrained problem ψǫ = minx∈X f (x), s.t. P

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

1 − ε1, . . . P

• gm1(x, ξ) ≤ 0, . . . , gmkm(x, ξ) ≤ 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.

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Stochastic programming formulations

Solving chance constrained 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.

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Stochastic programming formulations

Penalty functions

Penalty functions ϑj : Rm → R+, j = 1, . . . , m, are 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. Let pj denote the penalized constraints pj(x, ξ) = ϑj(gj1(x, ξ), . . . , gjkj(x, ξ)), ∀j.

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Stochastic programming formulations

Penalty function problems

Problem with penalties in the objective function ϕN = min

x∈X

• f (x) + N ·

m

• j=1

E[pj(x, ξ)]

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

Problem with generalized integrated chance constraints ϕICC

L

= min

x∈X

• f (x) : s.t. E[pj(x, ξ)] ≤ Lj, j = 1, . . . , m
• for some prescribed bounds Lj ≥ 0, L = (L1, . . . , Lm)′, with an optimal

solution xICC

L

(originally defined using u2, cf. Klein Haneveld (1986)).

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Stochastic programming formulations

Penalty function problems

Problem with penalties in the objective function ϕN = min

x∈X

• f (x) + N ·

m

• j=1

E[pj(x, ξ)]

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

Problem with generalized integrated chance constraints ϕICC

L

= min

x∈X

• f (x) : s.t. E[pj(x, ξ)] ≤ Lj, j = 1, . . . , m
• for some prescribed bounds Lj ≥ 0, L = (L1, . . . , Lm)′, with an optimal

solution xICC

L

(originally defined using u2, cf. Klein Haneveld (1986)).

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Stochastic programming formulations

History and applications of the penalty approach in SP

Pr´ ekopa (1973): CPP and penalization Dupaˇ cov´ a et al (1991): Water management (empirical study) Ermoliev et al (2000): Managing exposure to catastrophic risks (asymptotic equivalence with particular penalty) M.B. and Dupaˇ cov´ a (2008, 2012): Contamination technique for CCP (asymptotic equivalence using general penalty functions) ˇ Zampachov´ a (2009): Beam design (reliability problem with partial differential equations - nonlinear - significant reduction of computational time) M.B. (2009, 2012A): Value at Risk optimization with transaction costs and integer allocations (general penalty functions and several CC) M.B (2011): Blending problem (asymptotic equivalence with generalized integrated chance constraints)

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Relations between formulations

Contents

1 Stochastic programming formulations 2 Relations between formulations 3 Sample approximations using Monte-Carlo techniques

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Relations between formulations

Program with a random factor Stochastic ւ ↓ ց 1. programming formulation Chance Problem Integrated constrained ⇐ ⇒ with penalty ⇔ chance constrained problem

• bjective

problem (CCP) (PPO) (ICC) Sample 2. approximation ↓ ↓ ↓ (S.A.) S.A. CCP S.A. PPO S.A. ICC 3. Solution ↓ ↓ ↓ validation Reliability Reliability Reliability check check check

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Relations between formulations

Asymptotic equivalence and bounds

M.B. (2012A): Under the following assumptions, the asymptotic equivalence of the CCP and the PPO 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. 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.

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Relations between formulations

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

• pj(x, ξ) > 0
• ,

j = 1, . . . , m, αN(x) = N ·

m

• j=1

E[pj(x, ξ)], βǫ(x) = ε−η

max m

• j=1

E[pj(x, ξ)], where εmax = maxj εj and [1/N1/η] = (1/N1/η, . . . , 1/N1/η). THEN bounds on the optimal values 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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Relations between formulations

Program with a random factor Stochastic ւ ↓ ց 1. programming formulation Chance Problem Integrated constrained ⇔ with penalty ⇐ ⇒ chance constrained problem

• bjective

problem (CCP) (PPO) (ICC) Sample 2. approximation ↓ ↓ ↓ (S.A.) S.A. CCP S.A. PPO S.A. ICC 3. Solution ↓ ↓ ↓ validation Reliability Reliability Reliability check check check

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Relations between formulations

Asymptotic equivalence and bounds

M.B. (2011): Under the following assumptions, the asymptotic equivalence of the ICC and the PPO problems can be shown: Continuity of the constraints. Compactness of the fixed set of feasible solutions. Existence of integrable majorants. Existence of a permanently feasible solution. THEN for any prescribed Lj ≥ 0 there always exists N large enough so that minimization of the penalty function problem generates the optimal solutions xN which also satisfy the integrated chance constraints with given L = (L1, . . . , Lm)′.

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Relations between formulations

For arbitrary γ ∈ (0, 1), N > 0 and Lj ≥ 0 put Lj(x) = E[pj(x, ξ)], j = 1, . . . , m, αN(x) = N ·

m

• j=1

E[pj(x, ξ)], βL(x) = m

• j=1

Lj γ−1

m

• j=1

E[pj(x, ξ)], and let [N1/(γ−1)/m] = (N1/(γ−1)/m, . . . , N1/(γ−1)/m)′. THEN bounds

• n the optimal values can be constructed:

ϕ m

j=1 Lj(xN)

γ−1 − βL(xN)(xL(xN)) ≤ ϕICC

L(xN) ≤ ϕN − αN(xN),

ϕICC

L(xN) + αN(xN) ≤ ϕN ≤ ϕICC [N1/(γ−1)/m] + β[N1/(γ−1)/m](xICC [N1/(γ−1)/m]),

with lim

N→+∞ αN(xN) =

lim

N→+∞ Lj(xN) =

lim

Lmax→0+ βL(xICC L

) = 0 for any sequences of the optimal solutions xN and xICC

L

where Lmax denotes the maximal component of the vector L.

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Sample approximations using Monte-Carlo techniques

Contents

1 Stochastic programming formulations 2 Relations between formulations 3 Sample approximations using Monte-Carlo techniques

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Sample approximations using Monte-Carlo techniques

Program with a random factor Stochastic ւ ↓ ց 1. programming formulation Chance Problem Integrated constrained ⇔ with penalty ⇔ chance constrained problem

• bjective

problem (CCP) (PPO) (ICC) Sample 2. approximation ↓ ↓ ↓ (S.A.) S.A. CCP S.A. PPO S.A. ICC 3. Solution ↓ ↓ ↓ validation Reliability Reliability Reliability check check check

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Sample approximations using Monte-Carlo techniques

Rates of convergence, sample sizes

Let ξ1, . . . , ξS be an independent Monte Carlo sample of the random vector ξ. By generalization of the results proposed by S. Ahmed, J. Luedtke, A. Shapiro, et al. (2008, 2009) ...

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Sample approximations using Monte-Carlo techniques

Sample approximated chance constrained problem

...can be reformulated as a large mixed-integer nonlinear program: min(x,u)∈X×{0,1}mS f (x) s.t. g1i(x, ξs) − M(1 − u1s) ≤ 0, i = 1, . . . , k1, s = 1, . . . , S . . . gmi(x, ξs) − M(1 − ums) ≤ 0, i = 1, . . . , km, s = 1, . . . , S,

1 S

S

s=1 u1s

≥ 1 − γ1, . . .

1 S

S

s=1 ums

≥ 1 − γm, u1s, . . . , ums ∈ {0, 1}, s = 1, . . . , S, where M is a large constant and γj ∈ (0, 1).

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Sample approximations using Monte-Carlo techniques

Lower bound for the chance constrained problem

M.B. (2012A): Let γj > εj for all j, where γj are levels used in S.A. problems. Sample size S necessary to obtain that a feasible solution of the original CC problem is also feasible for the sample approximation with a probability at least 1 − δ, δ ∈ (0, 1) small: S ≥ 2 minj∈{1,...,m}(γj − εj)2/εj ln m δ . It corresponds to the result of S. Ahmed, et al (2008) for m = 1. (The estimate is based on Chernoff and Bonferroni inequalities.)

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Sample approximations using Monte-Carlo techniques

Feasibility - finite |X|

M.B. (2012A): Let γj < εj for all j and |X \ Xǫ| denote the number of points from X which are not feasible for the original CC problem. Sample size S necessary for that the feasible solutions of the sample approximated problems are feasible for the original CC problem with a high probability 1 − δ: S ≥ 1 2 minj∈{1,...,m}(γj − εj)2 ln m|X \ Xǫ| δ . 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.) Extended for the bounded infinite and mixed-integer set of feasible solutions, see M.B. (2012A, 2012B)...

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Sample approximations using Monte-Carlo techniques

Mixed-integer CCP

min(x,y)∈Z f (x, y), s.t. P

• g11(x, y, ξ) ≤ 0, . . . , g1k1(x, y, ξ) ≤ 0
• ≥

1 − ε1, . . . P

• gm1(x, y, ξ) ≤ 0, . . . , gmkm(x, y, ξ) ≤ 0
• ≥

1 − εm, (1) where εj ∈ (0, 1), X ⊆ Rn, Y ⊆ Zn′ and Z = {(x, y) ∈ X × Y : h1(x, y) ≤ 0, . . . , hk(x, y) ≤ 0}, gji(x, y, ξ) : Rn × Zn′ × Rn′′ → R, i = 0, . . . , kj, j = 1, . . . , m measurable in ξ for all x ∈ X and y ∈ Y , f (x, y) : Rn × Rn′ → R.

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Sample approximations using Monte-Carlo techniques

Mixed-integer CCP

M.B. (2012B): Let

1 γj < εj, i.e. that the levels of the sample approximated problem are

more restrictive,

2 Y ⊆ Zn′ be finite, 3 X(y) = {x ∈ X : (x, y) ∈ Z} be uniformly bounded for all y ∈ Y ,

i.e. D = supy∈Y sup{x − x′∞ : x, x′ ∈ X(y)} be a finite diameter,

4 functions Gj(x, y, ω) = max{gj1(x, y, ω), . . . , gjkj(x, y, ω)} be

Lipschitz continuous in the real variable x, i.e. for arbitrary y ∈ Y and ξ ∈ Ξ |Gj(x, y, ω) − Gj(x′, y, ω)| ≤ Lj

• x − x′
• ∞ , ∀x, x′ ∈ X(y),

for some Lj > 0.

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Sample approximations using Monte-Carlo techniques

Mixed-integer CCP

M.B. (2012B): It is possible to estimate the sample size S such that the feasible solutions of the relaxed sample-approximated problems are feasible for the original problem with a high probability 1 − δ: S ≥ 1 2 minj(εj − γj − λj)2

• ln m|Y |2

δ + ln 1 λmin

• + n ln

2LmaxD τ , where Lmax = maxj Lj and λmin = minj λj, λj ∈ (0, εj − γj), τ > 0 small. In M.B. (2012B) applied to stochastic vehicle routing problem.

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Sample approximations using Monte-Carlo techniques

Program with a random factor Stochastic ւ ↓ ց 1. programming formulation Chance Problem Integrated constrained ⇔ with penalty ⇔ chance constrained problem

• bjective

problem (CCP) (PPO) (ICC) Sample 2. approximation ↓ ↓ ↓ (S.A.) S.A. CCP S.A. PPO S.A. ICC 3. Solution ↓ ↓ ↓ validation Reliability Reliability Reliability check check check

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Sample approximations using Monte-Carlo techniques

Set of feasible solutions ICC

The set of feasible solutions of the original ICC problem XL =

• x ∈ X : pj(x) := E[pj(x, ξ)] ≤ Lj, j = 1, . . . , m
• ,

the (relaxed) set of feasible solutions of the sample approximated problem, τj ∈ R, τ = (τ1, . . . , τm), X S

L+τ =

• x ∈ X : pS

j (x) := 1

S

S

• s=1

pj(x, ξs) ≤ Lj + τj, j = 1, . . . , m

• .

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Sample approximations using Monte-Carlo techniques

Finite |X|

M.B. (2011): Let the moment generating function of pj(x, ξ) − pj(x) is finite and τj > 0. Estimated sample size S such that the feasible solutions of the original problem are feasible for the relaxed sample approximated problem with a high probability 1 − δ: S ≥ 1 minj∈{1,...,m},x∈X τ 2

j /2σ2 jx

ln m|X| δ , where σ2

jx = Var[pj(x, ξ) − pj(x)] < ∞. If we set m = 1, we get simlar

inequality as W. Wang, S. Ahmed (2008). (Based on Large Deviation Theory and Bonferroni inequality.) Extended for the bounded infinite set of feasible solutions, see M.B. (2011).

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Sample approximations using Monte-Carlo techniques

Program with a random factor Stochastic ւ ↓ ց 1. programming formulation Chance Problem Integrated constrained ⇔ with penalty ⇔ chance constrained problem

• bjective

problem (CCP) (PPO) (ICC) Sample 2. approximation ↓ ↓ ↓ (S.A.) S.A. CCP S.A. PPO S.A. ICC 3. Solution ↓ ↓ ↓ validation Reliability Reliability Reliability check check check

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Sample approximations using Monte-Carlo techniques

M.B. (2011). Stochastic programming problems with generalized integrated chance

• constraints. Accepted to Optimization, DOI: 10.1080/02331934.2011.587007.

M.B. (2012A). Chance constrained problems: penalty reformulation and performance of sample approximation technique. Kybernetika 48(1) 105-122. (SPEPS 2010-2) M.B. (2012B). Sample approximation technique for mixed-integer stochastic programming problems with several chance constraints. Operations Research Letters 40(3) 207-211. M.B. and J. Dupaˇ cov´ a (2012). Approximations and contamination bounds for probabilistic programs. Annals of Operations Research 193(1) 3-19. (SPEPS 2008-13)

• J. Dupaˇ

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

• ptimization of insurance portfolios for managing exposure to catastrophic risks.

Annals of Operations Research 99, 207-225. W.K. Klein Haneveld (1986). Duality in stochastic linear and dynamic programming. Lecture Notes in economics and mathematical systems, vol. 274, Springer.

• J. Luedtke and S. Ahmed (2008). A sample approximation approach for optimization

with probabilistic constraints. SIAM Journal on Optimization 19, 674-699.

• A. Pr´

ekopa (1973). Contributions to the theory of stochastic programming. Mathematical Programming, 4, 202-221.

• W. Wang, S. Ahmed (2008). Sample average approximation of expected value

constrained stochastic programs. Operations Research Letters 36, 515–519.

• E. ˇ

Zampachov´ a (2009). Approximations in stochastic optimization and their

• applications. Ph.D. thesis, Brno University of Technology.

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