probabilistic constrained optimization on flow networks
TRANSCRIPT
Probabilistic Constrained Optimization on Flow Net-works
Martin Gugat, Jens Lang, Elisa Strauch, Michael SchusterFAU Erlangen-Nürnberg, TU Darmstadt18.02.2020
Motivation
Martin Gugat, Jens Lang, Elisa Strauch, Michael Schuster · FAU Erlangen-Nürnberg, TU Darmstadt · Probabilistic Constrained Optimization on Flow Networks 2
Motivation
Consider the optimization problem
min f (x , ξ)
s.t. P(g(x , ξ) ≤ 0) ≥ α
with objective function f , constraint g, decision vector x , random variable ξ (withprobability distribution and density function) and probability level α.
How to compute this probability?
Martin Gugat, Jens Lang, Elisa Strauch, Michael Schuster · FAU Erlangen-Nürnberg, TU Darmstadt · Probabilistic Constrained Optimization on Flow Networks 3
Motivation
Consider the optimization problem
min f (x , ξ)
s.t. P(g(x , ξ) ≤ 0) ≥ α
with objective function f , constraint g, decision vector x , random variable ξ (withprobability distribution and density function) and probability level α.
How to compute this probability?
Martin Gugat, Jens Lang, Elisa Strauch, Michael Schuster · FAU Erlangen-Nürnberg, TU Darmstadt · Probabilistic Constrained Optimization on Flow Networks 4
Spheric Radial Decomposition
Theorem: Spheric radial decomposition (SRD)
Let ξ ∼ N (0,R) be the n-dimensional standardGaussian distribution with zero mean andpositive definite correlation matrix R. Then, forany Borel measurable subset M ⊆ Rn it holdsthat
P(ξ ∈ M) =
∫Sn−1
µχr ≥ 0|rLv ∈ Mdµη(v),
where Sn−1 is the (n − 1)-dimensional sphere inRn, µη is the uniform distribution on Sn−1, µχdenotes the χ-distribution with n degrees offreedom and L is such that R = LLT .
[e.g. Van Ackooij, Henrion: Gradient formulae for nonlinear probabilistic constraints with Gaussian andGaussian-like distributions, SIAM J. Optim. (2014)]
Martin Gugat, Jens Lang, Elisa Strauch, Michael Schuster · FAU Erlangen-Nürnberg, TU Darmstadt · Probabilistic Constrained Optimization on Flow Networks 5
Kernel Density Estimator
Definition: Kernel density estimator (KDE)
Let Y = y1, · · · , yN be independent andidentically distributed samples of the randomvariable Y , which has a absolutely continuousdistribution function with probability densityfunction %. Let K be a kernel function. Then, thekernel density estimator %N corresponding to thebandwidth h ∈ (0,∞) is defined as
%N(z) =1
Nh
N∑i=1
K(z − yi
h
).
[e.g. Parzen: On Estimation of a Probability Density Function and Mode, Ann. Math. Stat. (1962)]
Martin Gugat, Jens Lang, Elisa Strauch, Michael Schuster · FAU Erlangen-Nürnberg, TU Darmstadt · Probabilistic Constrained Optimization on Flow Networks 6
Kernel Density Estimator
Martin Gugat, Jens Lang, Elisa Strauch, Michael Schuster · FAU Erlangen-Nürnberg, TU Darmstadt · Probabilistic Constrained Optimization on Flow Networks 7
Outline
1) A stationary setting: Gas transport
1.1) A single pipe1.2) Necessary optimality conditions
2) A dynamic setting: Contamination of water
2.1) Dynamic probabilistic constraints2.2) A single pipe2.3) Necessary optimality conditions
Martin Gugat, Jens Lang, Elisa Strauch, Michael Schuster · FAU Erlangen-Nürnberg, TU Darmstadt · Probabilistic Constrained Optimization on Flow Networks 8
A stationary setting: Gas transport
1) A stationary setting: Gas transport
Martin Gugat, Jens Lang, Elisa Strauch, Michael Schuster · FAU Erlangen-Nürnberg, TU Darmstadt · Probabilistic Constrained Optimization on Flow Networks 9
Gas transport in a single pipe
For x ∈ [0, L], consider the stationary, semilinear, isothermal Euler equations
qx = 0
c2px = − λ
2D(RST )2 |q|q
p
with pressure p(x), flow q(x), sound speed in the gas c ∈ R, pipe friction coefficientλ ∈ R+ and pipe diameter D ∈ R+.
Let boundary conditionsp(0) = p0
q(L) = bD
be given.
Martin Gugat, Jens Lang, Elisa Strauch, Michael Schuster · FAU Erlangen-Nürnberg, TU Darmstadt · Probabilistic Constrained Optimization on Flow Networks 10
Gas transport in a single pipe
Solution of the Semi-Linear IIE for Horizontal Pipes
Using the Ideal Gas-Equation pe(x) = RSTρe(x), a solution of the upper IIE is givenby:
q(x) = bD
p(x) =
√p2
0 − (RST )2 λ
a2Dq(x)|q(x)|x
Definition: Set of feasible loads
The setM := b ∈ R≥0 | p(L; b) ∈ [pmin, pmax] .
is called the set of feasible loads
Martin Gugat, Jens Lang, Elisa Strauch, Michael Schuster · FAU Erlangen-Nürnberg, TU Darmstadt · Probabilistic Constrained Optimization on Flow Networks 11
Gas transport in a single pipe
Solution of the Semi-Linear IIE for Horizontal Pipes
Using the Ideal Gas-Equation pe(x) = RSTρe(x), a solution of the upper IIE is givenby:
q(x) = bD
p(x) =
√p2
0 − (RST )2 λ
a2Dq(x)|q(x)|x
Definition: Set of feasible loads
The setM := b ∈ R≥0 | p(L; b) ∈ [pmin, pmax] .
is called the set of feasible loads
Martin Gugat, Jens Lang, Elisa Strauch, Michael Schuster · FAU Erlangen-Nürnberg, TU Darmstadt · Probabilistic Constrained Optimization on Flow Networks 11
Gas transport in a single pipe
Motivated by applications, the gas demand is random. For mean value µ > 0, astandard deviation σ > 0 and a suitable probability space (Ω,A,P), let a Gaussiandistributed random variable
ξb ∼ N (µ, σ)
be given. We setb := ξb(ω)
for ω ∈ Ω.
How to compute P(b ∈ M) ?
Martin Gugat, Jens Lang, Elisa Strauch, Michael Schuster · FAU Erlangen-Nürnberg, TU Darmstadt · Probabilistic Constrained Optimization on Flow Networks 12
Gas transport in a single pipe
Motivated by applications, the gas demand is random. For mean value µ > 0, astandard deviation σ > 0 and a suitable probability space (Ω,A,P), let a Gaussiandistributed random variable
ξb ∼ N (µ, σ)
be given. We setb := ξb(ω)
for ω ∈ Ω.
How to compute P(b ∈ M) ?
Martin Gugat, Jens Lang, Elisa Strauch, Michael Schuster · FAU Erlangen-Nürnberg, TU Darmstadt · Probabilistic Constrained Optimization on Flow Networks 12
Gas transport in a single pipe (SRD)
1. Rewrite the feasible set:
b ∈ M ⇔ (pmin)2 ≤ p20 − φ | b | b ≤ (pmax)2 with φ =
λ
c2D(RST )2L.
2. Sample the sphere Sn−1 = S0 = −1, 1 and for every v ∈ Sn−1 set
bv(r ) = rσv + µ.
3. Define the regular range:
Rv ,reg := r ≥ 0 | bv(r ) ≥ 0.4. Compute the one dimensional sets
Mv = r ∈ Rv ,reg | bv(r ) ∈ M =
s⋃j=1
[av ,j , av ,j ].
5. Compute the probability
P(b ∈ M) ≈ 12
∑v∈−1,1
s∑j=1
Fχ(av ,j)−Fχ(av ,j),
where Fχ is the cumulative distribution function of the χ-distribution.
Martin Gugat, Jens Lang, Elisa Strauch, Michael Schuster · FAU Erlangen-Nürnberg, TU Darmstadt · Probabilistic Constrained Optimization on Flow Networks 13
Gas transport in a single pipe (SRD)
1. Rewrite the feasible set:
b ∈ M ⇔ (pmin)2 ≤ p20 − φ | b | b ≤ (pmax)2 with φ =
λ
c2D(RST )2L.
2. Sample the sphere Sn−1 = S0 = −1, 1 and for every v ∈ Sn−1 set
bv(r ) = rσv + µ.
3. Define the regular range:
Rv ,reg := r ≥ 0 | bv(r ) ≥ 0.4. Compute the one dimensional sets
Mv = r ∈ Rv ,reg | bv(r ) ∈ M =
s⋃j=1
[av ,j , av ,j ].
5. Compute the probability
P(b ∈ M) ≈ 12
∑v∈−1,1
s∑j=1
Fχ(av ,j)−Fχ(av ,j),
where Fχ is the cumulative distribution function of the χ-distribution.
Martin Gugat, Jens Lang, Elisa Strauch, Michael Schuster · FAU Erlangen-Nürnberg, TU Darmstadt · Probabilistic Constrained Optimization on Flow Networks 13
Gas transport in a single pipe (SRD)
1. Rewrite the feasible set:
b ∈ M ⇔ (pmin)2 ≤ p20 − φ | b | b ≤ (pmax)2 with φ =
λ
c2D(RST )2L.
2. Sample the sphere Sn−1 = S0 = −1, 1 and for every v ∈ Sn−1 set
bv(r ) = rσv + µ.
3. Define the regular range:
Rv ,reg := r ≥ 0 | bv(r ) ≥ 0.
4. Compute the one dimensional sets
Mv = r ∈ Rv ,reg | bv(r ) ∈ M =
s⋃j=1
[av ,j , av ,j ].
5. Compute the probability
P(b ∈ M) ≈ 12
∑v∈−1,1
s∑j=1
Fχ(av ,j)−Fχ(av ,j),
where Fχ is the cumulative distribution function of the χ-distribution.
Martin Gugat, Jens Lang, Elisa Strauch, Michael Schuster · FAU Erlangen-Nürnberg, TU Darmstadt · Probabilistic Constrained Optimization on Flow Networks 13
Gas transport in a single pipe (SRD)
1. Rewrite the feasible set:
b ∈ M ⇔ (pmin)2 ≤ p20 − φ | b | b ≤ (pmax)2 with φ =
λ
c2D(RST )2L.
2. Sample the sphere Sn−1 = S0 = −1, 1 and for every v ∈ Sn−1 set
bv(r ) = rσv + µ.
3. Define the regular range:
Rv ,reg := r ≥ 0 | bv(r ) ≥ 0.4. Compute the one dimensional sets
Mv = r ∈ Rv ,reg | bv(r ) ∈ M =
s⋃j=1
[av ,j , av ,j ].
5. Compute the probability
P(b ∈ M) ≈ 12
∑v∈−1,1
s∑j=1
Fχ(av ,j)−Fχ(av ,j),
where Fχ is the cumulative distribution function of the χ-distribution.
Martin Gugat, Jens Lang, Elisa Strauch, Michael Schuster · FAU Erlangen-Nürnberg, TU Darmstadt · Probabilistic Constrained Optimization on Flow Networks 13
Gas transport in a single pipe (SRD)
1. Rewrite the feasible set:
b ∈ M ⇔ (pmin)2 ≤ p20 − φ | b | b ≤ (pmax)2 with φ =
λ
c2D(RST )2L.
2. Sample the sphere Sn−1 = S0 = −1, 1 and for every v ∈ Sn−1 set
bv(r ) = rσv + µ.
3. Define the regular range:
Rv ,reg := r ≥ 0 | bv(r ) ≥ 0.4. Compute the one dimensional sets
Mv = r ∈ Rv ,reg | bv(r ) ∈ M =
s⋃j=1
[av ,j , av ,j ].
5. Compute the probability
P(b ∈ M) ≈ 12
∑v∈−1,1
s∑j=1
Fχ(av ,j)−Fχ(av ,j),
where Fχ is the cumulative distribution function of the χ-distribution.Martin Gugat, Jens Lang, Elisa Strauch, Michael Schuster · FAU Erlangen-Nürnberg, TU Darmstadt · Probabilistic Constrained Optimization on Flow Networks 13
Gas transport in a single pipe (KDE)
Let B = bS,1, · · · , bS,N ⊆ R≥0 be independent and identically distributed samplesbS,i := ξb(ωi) (i = 1, · · · ,N, ξb ∼ N (µ, σ)).
Let PB = pS,1, · · · , pS,N ⊆ R be the pressures pS,i = p(L, bS,i) at the end of thepipe for the different loads bS,i ∈ B.
With a Gaussian kernel function
K (t) =1√2π
exp
(−1
2t2)
and a bandwidth h ∈ R+, we get an approximation of the probability density functionof the pressure
%p,N(z) =1
Nh
N∑i=1
1√2π
exp
(−1
2
(z − pS,i
h
)).
It follows
P(b ∈ M) ≈∫ pmax
pmin
%p,N(z)dz.
Martin Gugat, Jens Lang, Elisa Strauch, Michael Schuster · FAU Erlangen-Nürnberg, TU Darmstadt · Probabilistic Constrained Optimization on Flow Networks 14
Gas transport in a single pipe (KDE)
Let B = bS,1, · · · , bS,N ⊆ R≥0 be independent and identically distributed samplesbS,i := ξb(ωi) (i = 1, · · · ,N, ξb ∼ N (µ, σ)).
Let PB = pS,1, · · · , pS,N ⊆ R be the pressures pS,i = p(L, bS,i) at the end of thepipe for the different loads bS,i ∈ B.
With a Gaussian kernel function
K (t) =1√2π
exp
(−1
2t2)
and a bandwidth h ∈ R+, we get an approximation of the probability density functionof the pressure
%p,N(z) =1
Nh
N∑i=1
1√2π
exp
(−1
2
(z − pS,i
h
)).
It follows
P(b ∈ M) ≈∫ pmax
pmin
%p,N(z)dz.
Martin Gugat, Jens Lang, Elisa Strauch, Michael Schuster · FAU Erlangen-Nürnberg, TU Darmstadt · Probabilistic Constrained Optimization on Flow Networks 14
Gas transport in a single pipe (KDE)
Let B = bS,1, · · · , bS,N ⊆ R≥0 be independent and identically distributed samplesbS,i := ξb(ωi) (i = 1, · · · ,N, ξb ∼ N (µ, σ)).
Let PB = pS,1, · · · , pS,N ⊆ R be the pressures pS,i = p(L, bS,i) at the end of thepipe for the different loads bS,i ∈ B.
With a Gaussian kernel function
K (t) =1√2π
exp
(−1
2t2)
and a bandwidth h ∈ R+, we get an approximation of the probability density functionof the pressure
%p,N(z) =1
Nh
N∑i=1
1√2π
exp
(−1
2
(z − pS,i
h
)).
It follows
P(b ∈ M) ≈∫ pmax
pmin
%p,N(z)dz.
Martin Gugat, Jens Lang, Elisa Strauch, Michael Schuster · FAU Erlangen-Nürnberg, TU Darmstadt · Probabilistic Constrained Optimization on Flow Networks 14
Gas transport in a single pipe (KDE)
Choice of bandwidth
A good heuristic choice for the bandwidth is
h = 1.06σN5√
N.
[Gramacki: Nonparametric Kernel Density Estimation and Its Computational Aspects; Springer (2018)Turlach: Bandwidth Selection in Kernel Density Estimation: A Review ; Technical Report (1999)]
L∞-convergence
If the probability density function %p is uniformly continuous, then NadarayasTheorem guarantees
supz|%p(z)− %p,N|
N→∞−−−→ 0 P-almost surely.
[Nadaraya: On Non-Parametric Estimates of Density Functions and Regression Curves; Theory Probab.Appl. (1965)]
Martin Gugat, Jens Lang, Elisa Strauch, Michael Schuster · FAU Erlangen-Nürnberg, TU Darmstadt · Probabilistic Constrained Optimization on Flow Networks 15
Gas transport in a single pipe (KDE)
Choice of bandwidth
A good heuristic choice for the bandwidth is
h = 1.06σN5√
N.
[Gramacki: Nonparametric Kernel Density Estimation and Its Computational Aspects; Springer (2018)Turlach: Bandwidth Selection in Kernel Density Estimation: A Review ; Technical Report (1999)]
L∞-convergence
If the probability density function %p is uniformly continuous, then NadarayasTheorem guarantees
supz|%p(z)− %p,N|
N→∞−−−→ 0 P-almost surely.
[Nadaraya: On Non-Parametric Estimates of Density Functions and Regression Curves; Theory Probab.Appl. (1965)]
Martin Gugat, Jens Lang, Elisa Strauch, Michael Schuster · FAU Erlangen-Nürnberg, TU Darmstadt · Probabilistic Constrained Optimization on Flow Networks 15
Gas transport in a single pipe (KDE)
L1-convergence
If the distribution of the pressure is absolute continuous with probability densityfunction %p, then
‖%p − %p,N‖L1N→∞−−−→ 0 P-almost surely.
[Devroye, Gyorfi: Nonparametric density estimation: the L1 view ; Wiley Ser. Probab. Stat. (1985)]
L1-convergence II
If the distribution of the pressure is absolute continuous with probability densityfunction %p, then from Scheffé’s lemma it follows
|P(b ∈ M)− PN(b ∈ M)| ≤ 12‖%p − %p,N‖L1
N→∞−−−→ 0 P-almost surely.
[Devroye, Gyorfi: Nonparametric density estimation: the L1 view ; Wiley Ser. Probab. Stat. (1985)]
Martin Gugat, Jens Lang, Elisa Strauch, Michael Schuster · FAU Erlangen-Nürnberg, TU Darmstadt · Probabilistic Constrained Optimization on Flow Networks 16
Gas transport in a single pipe (KDE)
L1-convergence
If the distribution of the pressure is absolute continuous with probability densityfunction %p, then
‖%p − %p,N‖L1N→∞−−−→ 0 P-almost surely.
[Devroye, Gyorfi: Nonparametric density estimation: the L1 view ; Wiley Ser. Probab. Stat. (1985)]
L1-convergence II
If the distribution of the pressure is absolute continuous with probability densityfunction %p, then from Scheffé’s lemma it follows
|P(b ∈ M)− PN(b ∈ M)| ≤ 12‖%p − %p,N‖L1
N→∞−−−→ 0 P-almost surely.
[Devroye, Gyorfi: Nonparametric density estimation: the L1 view ; Wiley Ser. Probab. Stat. (1985)]
Martin Gugat, Jens Lang, Elisa Strauch, Michael Schuster · FAU Erlangen-Nürnberg, TU Darmstadt · Probabilistic Constrained Optimization on Flow Networks 16
Gas transport in a single pipe (KDE)
Example: We choose the following values:
p0 pmin pmax µ σ φ
60 40 60 4 0.5 100Table: Values for the example with one edge.
⇒ M = [0,√
20]
Test 1 Test 2 Test 3 Test 4 Test 5 Test 6 Test 7 Test 8MC 82.99% 82.88% 82.83% 82.95% 83.09% 82.77% 82.83% 82.58%
KDE 82.83% 82.75% 82.69% 82.74% 82.91% 82.58% 82.70% 82.43%
SRD 82.75%
Table: Results for the example with one edge.
Martin Gugat, Jens Lang, Elisa Strauch, Michael Schuster · FAU Erlangen-Nürnberg, TU Darmstadt · Probabilistic Constrained Optimization on Flow Networks 17
Necessary optimality conditions: stationary states
Let B = bS,1, · · · , bS,N ⊆ Rn≥0 be independent and identically distributed samples
bS,i := ξb(ωi) (i = 1, · · · ,N, ξb ∼ N (µ,Σ)).
Let PB = pS,1, · · · , pS,N ⊆ Rn be the pressures with pS,ij = pj(L, bS,i), where
pj(L, bS,i) is the pressure at the end of pipe j (j = 1, · · · , n) for the load bS,i ∈ B.
Then for bandwidths hj (j = 1, · · · , n), we get an approximation of the probabilitydensity function of the pressure
%p,N(z) =1
N∏n
j=1 hj
N∑i=1
n∏j=1
1√2π
exp
(−1
2
(zj − pj(bS,i)
hj
)2)
and we can compute the probability
PN(b ∈ M) =
∫Pmaxmin
%p,N(z)dz.
Martin Gugat, Jens Lang, Elisa Strauch, Michael Schuster · FAU Erlangen-Nürnberg, TU Darmstadt · Probabilistic Constrained Optimization on Flow Networks 18
Necessary optimality conditions: stationary states
∫Pmaxmin
%p,N(z)dz =
∫Pmaxmin
1N∏n
j=1 hj
N∑i=1
n∏j=1
K
(zj − pS,i
j
hj
)dz
=1
N∏n
j=1 hj
N∑i=1
∫ pmax1
pmin1
· · ·∫ pmax
n
pminn
n∏j=1
K
(zj − pS,i
j
hj
)dz1 · · · dzn
=1
N∏n
j=1 hj
N∑i=1
n∏j=1
∫ pmaxj
pminj
K
(zj − pS,i
j
hj
)dzj
=1N
N∑i=1
n∏j=1
∫ ϕi,j(pmaxj )
ϕi,j(pminj )
1√π
exp(−τ2
i ,j
)dτi ,j
=1N
12n
N∑i=1
n∏j=1
[erf(ϕi ,j(pmax
j ))− erf(ϕi ,j(pminj ))
]
Martin Gugat, Jens Lang, Elisa Strauch, Michael Schuster · FAU Erlangen-Nürnberg, TU Darmstadt · Probabilistic Constrained Optimization on Flow Networks 19
Necessary optimality conditions: stationary states
∫Pmaxmin
%p,N(z)dz =
∫Pmaxmin
1N∏n
j=1 hj
N∑i=1
n∏j=1
K
(zj − pS,i
j
hj
)dz
=1
N∏n
j=1 hj
N∑i=1
∫ pmax1
pmin1
· · ·∫ pmax
n
pminn
n∏j=1
K
(zj − pS,i
j
hj
)dz1 · · · dzn
=1
N∏n
j=1 hj
N∑i=1
n∏j=1
∫ pmaxj
pminj
K
(zj − pS,i
j
hj
)dzj
=1N
N∑i=1
n∏j=1
∫ ϕi,j(pmaxj )
ϕi,j(pminj )
1√π
exp(−τ2
i ,j
)dτi ,j
=1N
12n
N∑i=1
n∏j=1
[erf(ϕi ,j(pmax
j ))− erf(ϕi ,j(pminj ))
]
Martin Gugat, Jens Lang, Elisa Strauch, Michael Schuster · FAU Erlangen-Nürnberg, TU Darmstadt · Probabilistic Constrained Optimization on Flow Networks 19
Necessary optimality conditions: stationary states
∫Pmaxmin
%p,N(z)dz =
∫Pmaxmin
1N∏n
j=1 hj
N∑i=1
n∏j=1
K
(zj − pS,i
j
hj
)dz
=1
N∏n
j=1 hj
N∑i=1
∫ pmax1
pmin1
· · ·∫ pmax
n
pminn
n∏j=1
K
(zj − pS,i
j
hj
)dz1 · · · dzn
=1
N∏n
j=1 hj
N∑i=1
n∏j=1
∫ pmaxj
pminj
K
(zj − pS,i
j
hj
)dzj
=1N
N∑i=1
n∏j=1
∫ ϕi,j(pmaxj )
ϕi,j(pminj )
1√π
exp(−τ2
i ,j
)dτi ,j
=1N
12n
N∑i=1
n∏j=1
[erf(ϕi ,j(pmax
j ))− erf(ϕi ,j(pminj ))
]
Martin Gugat, Jens Lang, Elisa Strauch, Michael Schuster · FAU Erlangen-Nürnberg, TU Darmstadt · Probabilistic Constrained Optimization on Flow Networks 19
Necessary optimality conditions: stationary states
∫Pmaxmin
%p,N(z)dz =
∫Pmaxmin
1N∏n
j=1 hj
N∑i=1
n∏j=1
K
(zj − pS,i
j
hj
)dz
=1
N∏n
j=1 hj
N∑i=1
∫ pmax1
pmin1
· · ·∫ pmax
n
pminn
n∏j=1
K
(zj − pS,i
j
hj
)dz1 · · · dzn
=1
N∏n
j=1 hj
N∑i=1
n∏j=1
∫ pmaxj
pminj
K
(zj − pS,i
j
hj
)dzj
=1N
N∑i=1
n∏j=1
∫ ϕi,j(pmaxj )
ϕi,j(pminj )
1√π
exp(−τ2
i ,j
)dτi ,j
=1N
12n
N∑i=1
n∏j=1
[erf(ϕi ,j(pmax
j ))− erf(ϕi ,j(pminj ))
]
Martin Gugat, Jens Lang, Elisa Strauch, Michael Schuster · FAU Erlangen-Nürnberg, TU Darmstadt · Probabilistic Constrained Optimization on Flow Networks 19
Necessary optimality conditions: stationary states
Consider the optimization problem
(?)
min f (pmax)
s.t. gα(pmax) := α− P(b ∈ M(pmax)) ≤ 0.
∂gα(pmax)
∂pmaxk
= − 1N
12n
N∑i=1
n∏j = 1j 6= k
[erf(ϕi ,j(pmax
j ))− erf(ϕi ,j(pminj ))
]·√
2√πhk
exp(−ϕ2
i ,k(pmaxk ))< 0
Necessary optimality conditions
Let p∗,max ∈ Rn be a optimal solution of (?). Then there exists a multiplier µ∗ ≥ 0, s.t.
∇f (p∗,max) + µ∗∇gα = 0gα(p∗,max) ≤ 0µ∗gα(p∗,max) = 0.
Martin Gugat, Jens Lang, Elisa Strauch, Michael Schuster · FAU Erlangen-Nürnberg, TU Darmstadt · Probabilistic Constrained Optimization on Flow Networks 20
Necessary optimality conditions: stationary states
Consider the optimization problem
(?)
min f (pmax)
s.t. gα(pmax) := α− P(b ∈ M(pmax)) ≤ 0.
∂gα(pmax)
∂pmaxk
= − 1N
12n
N∑i=1
n∏j = 1j 6= k
[erf(ϕi ,j(pmax
j ))− erf(ϕi ,j(pminj ))
]·√
2√πhk
exp(−ϕ2
i ,k(pmaxk ))< 0
Necessary optimality conditions
Let p∗,max ∈ Rn be a optimal solution of (?). Then there exists a multiplier µ∗ ≥ 0, s.t.
∇f (p∗,max) + µ∗∇gα = 0gα(p∗,max) ≤ 0µ∗gα(p∗,max) = 0.
Martin Gugat, Jens Lang, Elisa Strauch, Michael Schuster · FAU Erlangen-Nürnberg, TU Darmstadt · Probabilistic Constrained Optimization on Flow Networks 20
A dynamic setting: Contamination of water
2) A dynamic setting: Contamination of water
Martin Gugat, Jens Lang, Elisa Strauch, Michael Schuster · FAU Erlangen-Nürnberg, TU Darmstadt · Probabilistic Constrained Optimization on Flow Networks 21
Dynamic probabilistic constraints
How to add a time dependence to a probabilistic constraint?
P( b ∈ M(t) ∀t ∈ [0,T ] ) ≥ α
P( b ∈ M(t) ) ≥ α ∀t ∈ [0,T ]
1T
∫ T
0P( b ∈ M(t) ) dt ≥ α
Martin Gugat, Jens Lang, Elisa Strauch, Michael Schuster · FAU Erlangen-Nürnberg, TU Darmstadt · Probabilistic Constrained Optimization on Flow Networks 22
Dynamic probabilistic constraints
How to add a time dependence to a probabilistic constraint?
P( b ∈ M(t) ∀t ∈ [0,T ] ) ≥ α
P( b ∈ M(t) ) ≥ α ∀t ∈ [0,T ]
1T
∫ T
0P( b ∈ M(t) ) dt ≥ α
Martin Gugat, Jens Lang, Elisa Strauch, Michael Schuster · FAU Erlangen-Nürnberg, TU Darmstadt · Probabilistic Constrained Optimization on Flow Networks 22
Dynamic probabilistic constraints
How to add a time dependence to a probabilistic constraint?
P( b ∈ M(t) ∀t ∈ [0,T ] ) ≥ α
P( b ∈ M(t) ) ≥ α ∀t ∈ [0,T ]
1T
∫ T
0P( b ∈ M(t) ) dt ≥ α
Martin Gugat, Jens Lang, Elisa Strauch, Michael Schuster · FAU Erlangen-Nürnberg, TU Darmstadt · Probabilistic Constrained Optimization on Flow Networks 23
Dynamic probabilistic constraints
How to get time dependent random boundary data?
Let bD(t) ∈ L2([0,T ]) be given. Define
ψm(t) :=
√2√T
sin
((π2
+ mπ) t
T
)and a0
m :=
∫ T
0bD(t)ψm(t)dt
Then it is
bD(t) =
∞∑m=0
a0mψm(t).
Consider Gaussian distributed random variablesξam ∼ N (µ, σ). For am = ξam(ω), (ω ∈ Ω) considerthe random boundary data
b(t) =
∞∑m=0
ama0mψm(t) ∈ L2([0,T ]) P-a.s.
[Farshbaf-Shaker, Gugat, Heitsch, Henrion: Optimal Neumann Boundary Control of a Vibrating String withUncertain Initial Data and Probabilistic Terminal Constraints (submitted 2019)]
Martin Gugat, Jens Lang, Elisa Strauch, Michael Schuster · FAU Erlangen-Nürnberg, TU Darmstadt · Probabilistic Constrained Optimization on Flow Networks 24
Dynamic probabilistic constraints
How to get time dependent random boundary data?Let bD(t) ∈ L2([0,T ]) be given. Define
ψm(t) :=
√2√T
sin
((π2
+ mπ) t
T
)and a0
m :=
∫ T
0bD(t)ψm(t)dt
Then it is
bD(t) =
∞∑m=0
a0mψm(t).
Consider Gaussian distributed random variablesξam ∼ N (µ, σ). For am = ξam(ω), (ω ∈ Ω) considerthe random boundary data
b(t) =
∞∑m=0
ama0mψm(t) ∈ L2([0,T ]) P-a.s.
[Farshbaf-Shaker, Gugat, Heitsch, Henrion: Optimal Neumann Boundary Control of a Vibrating String withUncertain Initial Data and Probabilistic Terminal Constraints (submitted 2019)]
Martin Gugat, Jens Lang, Elisa Strauch, Michael Schuster · FAU Erlangen-Nürnberg, TU Darmstadt · Probabilistic Constrained Optimization on Flow Networks 24
Dynamic probabilistic constraints
How to get time dependent random boundary data?Let bD(t) ∈ L2([0,T ]) be given. Define
ψm(t) :=
√2√T
sin
((π2
+ mπ) t
T
)and a0
m :=
∫ T
0bD(t)ψm(t)dt
Then it is
bD(t) =
∞∑m=0
a0mψm(t).
Consider Gaussian distributed random variablesξam ∼ N (µ, σ). For am = ξam(ω), (ω ∈ Ω) considerthe random boundary data
b(t) =
∞∑m=0
ama0mψm(t) ∈ L2([0,T ]) P-a.s.
[Farshbaf-Shaker, Gugat, Heitsch, Henrion: Optimal Neumann Boundary Control of a Vibrating String withUncertain Initial Data and Probabilistic Terminal Constraints (submitted 2019)]
Martin Gugat, Jens Lang, Elisa Strauch, Michael Schuster · FAU Erlangen-Nürnberg, TU Darmstadt · Probabilistic Constrained Optimization on Flow Networks 24
Contamination of water in a single pipe
For (t , x) ∈ [0,T ]× [0, L] and constants d < 0, m ≤ 0, we consider the deterministicscalar linear PDE with initial condition and boundary condition
(?)
rt(t , x) + drx(t , x) = mr (t , x),
r (0, x) = r0(x),
r (t , L) = b(t).
Assume C0-compatibility between the initial and the boundary condition. Thisequation e.g. models the flow of contamination in water along a pipe or in a network.
Solution of (?)
A solution of (?) is given by
r (t , x) =
exp(mt) r0(x − dt) if x ≤ L + dt ,exp(mx−L
d
)b(t − x−L
d ) if x > L + dt .
Martin Gugat, Jens Lang, Elisa Strauch, Michael Schuster · FAU Erlangen-Nürnberg, TU Darmstadt · Probabilistic Constrained Optimization on Flow Networks 25
Contamination of water in a single pipe
For (t , x) ∈ [0,T ]× [0, L] and constants d < 0, m ≤ 0, we consider the deterministicscalar linear PDE with initial condition and boundary condition
(?)
rt(t , x) + drx(t , x) = mr (t , x),
r (0, x) = r0(x),
r (t , L) = b(t).
Assume C0-compatibility between the initial and the boundary condition. Thisequation e.g. models the flow of contamination in water along a pipe or in a network.
Solution of (?)
A solution of (?) is given by
r (t , x) =
exp(mt) r0(x − dt) if x ≤ L + dt ,exp(mx−L
d
)b(t − x−L
d ) if x > L + dt .
Martin Gugat, Jens Lang, Elisa Strauch, Michael Schuster · FAU Erlangen-Nürnberg, TU Darmstadt · Probabilistic Constrained Optimization on Flow Networks 25
Contamination of water in a single pipe
Definition: Set of feasible loads
For t∗ ∈ [0,T ], the set
M(t∗) := b ∈ L2([0,T ]);R≥0) | r (t∗, 0) ∈ [rmin, rmax]
is called the set of feasible loads.
Our aim is to compute the probability
P( b ∈ M(t∗) ).
Martin Gugat, Jens Lang, Elisa Strauch, Michael Schuster · FAU Erlangen-Nürnberg, TU Darmstadt · Probabilistic Constrained Optimization on Flow Networks 26
Contamination of water in a single pipe (SRD)
0. Estimate the mean µ(t) and the variance σ(t) for the random boundary data
1. Rewrite the set of feasible loads:
b ∈ M(t∗) ⇔ rmin ≤ r (t∗, 0) ≤ rmax.
2. Sample the sphere Sn−1 and set bv(r , t) = rL(t)v + mu(t).
3. Define the regular range Rv ,reg := r ≥ 0|bv(r , t) ≥ 0.
4. Compute the one dimensional sets Mv(t∗) = r ∈ Rv ,reg|bv(r , t) ∈ M(t∗).
5. Compute the probability P(b ∈ M) ≈ 12
∑v∈−1,1
∑sj=1Fχ(av ,j)−Fχ(av ,j).
Martin Gugat, Jens Lang, Elisa Strauch, Michael Schuster · FAU Erlangen-Nürnberg, TU Darmstadt · Probabilistic Constrained Optimization on Flow Networks 27
Contamination of water in a single pipe (KDE)
Let A = (a1,m)m≥0, · · · , (aN,m)m≥0 be a sampling of N independent and identicallydistributed sequences.
Let BA(t) = bS,1(t), · · · , bS,N(t) be the corresponding sampling of randomboundary functions.
Let R(t) = r (t , 0, bS,1), · · · , r (t , 0, bS,n) be the sampling of solutions at x = 0corresponding to the boundary functions in B.
With a Gaussian kernel function and a bandwidth h+ ∈ R+, we get an approximationof the probability density function of the pressure
%r ,t∗,N(z) =1
Nh
N∑i=1
K(
z − r (t∗, 0, bS,i)
h
)and we can compute the probability
P ( b ∈ M(t∗) ) ≈∫ rmax
rmin
%r ,t∗,N(z) dz.
Martin Gugat, Jens Lang, Elisa Strauch, Michael Schuster · FAU Erlangen-Nürnberg, TU Darmstadt · Probabilistic Constrained Optimization on Flow Networks 28
Contamination of water in a single pipe (KDE)
How to compute P ( b ∈ M(t) ∀t ∈ [0,T ] )?
⇒ Minimal and Maximal value must be feasible
Define the valuesr i := min
t∈[0,T ]r (t , 0, bS,i) (i = 1, · · · ,N),
r i := maxt∈[0,T ]
r (t , 0, bS,i) (i = 1, · · · ,N).
For bandwidths h1, h2 we get an approximation of the probability density function ofthe maximal and minimal pressure
%r ,N(z) =1
Nh1h2
N∑i=1
K(
z1 − r i
h1
)K(
z2 − r i
h2
)and we can compute the probability
P ( b ∈ M(t) ∀t ∈ [0,T ] ) ≈∫[rmin,rmax]×[rmin,rmax]
%r ,N(z) dz.
Martin Gugat, Jens Lang, Elisa Strauch, Michael Schuster · FAU Erlangen-Nürnberg, TU Darmstadt · Probabilistic Constrained Optimization on Flow Networks 29
Contamination of water in a single pipe (KDE)
How to compute P ( b ∈ M(t) ∀t ∈ [0,T ] )?
⇒ Minimal and Maximal value must be feasible
Define the valuesr i := min
t∈[0,T ]r (t , 0, bS,i) (i = 1, · · · ,N),
r i := maxt∈[0,T ]
r (t , 0, bS,i) (i = 1, · · · ,N).
For bandwidths h1, h2 we get an approximation of the probability density function ofthe maximal and minimal pressure
%r ,N(z) =1
Nh1h2
N∑i=1
K(
z1 − r i
h1
)K(
z2 − r i
h2
)and we can compute the probability
P ( b ∈ M(t) ∀t ∈ [0,T ] ) ≈∫[rmin,rmax]×[rmin,rmax]
%r ,N(z) dz.
Martin Gugat, Jens Lang, Elisa Strauch, Michael Schuster · FAU Erlangen-Nürnberg, TU Darmstadt · Probabilistic Constrained Optimization on Flow Networks 29
Contamination of water in a single pipe (KDE)
How to compute P ( b ∈ M(t) ∀t ∈ [0,T ] )?
⇒ Minimal and Maximal value must be feasible
Define the valuesr i := min
t∈[0,T ]r (t , 0, bS,i) (i = 1, · · · ,N),
r i := maxt∈[0,T ]
r (t , 0, bS,i) (i = 1, · · · ,N).
For bandwidths h1, h2 we get an approximation of the probability density function ofthe maximal and minimal pressure
%r ,N(z) =1
Nh1h2
N∑i=1
K(
z1 − r i
h1
)K(
z2 − r i
h2
)and we can compute the probability
P ( b ∈ M(t) ∀t ∈ [0,T ] ) ≈∫[rmin,rmax]×[rmin,rmax]
%r ,N(z) dz.
Martin Gugat, Jens Lang, Elisa Strauch, Michael Schuster · FAU Erlangen-Nürnberg, TU Darmstadt · Probabilistic Constrained Optimization on Flow Networks 29
Contamination of water in a single pipe (KDE)
Example: We choose the following values:rmin0 rmax
0 µ σ d m L T2 6 1 0.25 −5 −1 1 4
Table: Values for the dynamic example.
Further: 101 time discretization points, 30 terms in the Fourier series.
Test 1 Test 2 Test 3 Test 4 Test 5 Test 6 Test 7 Test 8MC 74.36% 74.42% 74.44% 74.20% 74.16% 74.41% 74.37% 74.35%
KDE 74.26% 74.31% 74.35% 74.11% 74.04% 74.32% 74.27% 74.27%
Table: Results for the dynamic example with one edge
Martin Gugat, Jens Lang, Elisa Strauch, Michael Schuster · FAU Erlangen-Nürnberg, TU Darmstadt · Probabilistic Constrained Optimization on Flow Networks 30
Contamination of water in a single pipe (KDE)
Example: We choose the following values:rmin0 rmax
0 µ σ d m L T2 6 1 0.25 −5 −1 1 4
Table: Values for the dynamic example.
Further: 101 time discretization points, 30 terms in the Fourier series.
Test 1 Test 2 Test 3 Test 4 Test 5 Test 6 Test 7 Test 8MC 74.36% 74.42% 74.44% 74.20% 74.16% 74.41% 74.37% 74.35%
KDE 74.26% 74.31% 74.35% 74.11% 74.04% 74.32% 74.27% 74.27%
Table: Results for the dynamic example with one edge
Martin Gugat, Jens Lang, Elisa Strauch, Michael Schuster · FAU Erlangen-Nürnberg, TU Darmstadt · Probabilistic Constrained Optimization on Flow Networks 30
Necessary optimality conditions: dynamics
Necessary optimality conditions
Let r ∗,max ∈ R be a solution of the optimization problem
min f (rmax)
s.t. gα(rmax) = α− P(b ∈ M(t) ∀t ∈ [0,T ] ) ≤ 0.
Then there exist a multiplier µ∗ ≥ 0, s.t.
f ′(r ∗,max) + µ∗g′α(r ∗,max) = 0gα(r ∗,max) ≤ 0µ∗gα(r ∗,max) = 0.
Martin Gugat, Jens Lang, Elisa Strauch, Michael Schuster · FAU Erlangen-Nürnberg, TU Darmstadt · Probabilistic Constrained Optimization on Flow Networks 31
References
C. Gotzes, H. Heitsch, R. Henrion, R. SchultzOn the quantification of nomination feasibility in stationary gas networks with random loadMath. Meth. Oper. Res. 84 (2016)
M. Gugat, M. SchusterStationary Gas Networks with Compressor Control and Random Loads: Optimization with ProbabilisticConstraintsMathematical Problems in Engineering (2018)
M. GugatContamination Source Determination in Water Distribution NetworksSIAM J. Appl. Math. (2012)
W. Härdle, A. Werwatz, M. Müller, S. SperlichNonparametric and Semiparametric Models; Springer (2004)
A. PrékopaStochastic Programming; Springer (1995)
Thank you for your attention!
Martin Gugat, Jens Lang, Elisa Strauch, Michael Schuster · FAU Erlangen-Nürnberg, TU Darmstadt · Probabilistic Constrained Optimization on Flow Networks 32