Telecommunication Network Design under Uncertainty: A ... · Viet Anh Nguyen Telecommunication Network Design under Uncertainty: A Distributionally Robust Optimization Approach30th

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Telecommunication Network Designunder Uncertainty:

A Distributionally Robust Optimization Approach

Viet Anh Nguyen

30th October 2012

Viet Anh Nguyen () Telecommunication Network Design under Uncertainty: A Distributionally Robust Optimization Approach30th October 2012 1 / 25

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Outline

.. .1 Introduction

.. .2 Linear decision rule approximation

.. .3 Approximations of linear probabilistic constraints

.. .4 The Abilene case

.. .5 Conclusion


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Introduction

Outline

.. .1 Introduction




.. .5 Conclusion


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Introduction

Motivation

The network design problem consists of finding the lowest costconfiguration for the network, and the routing for each commodity.

Extensive works have been done on deterministic demand.

Stochastic/robust optimization approach for stochastic demand.

In this work, we consider a distributionally robust formulation of theproblem.


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Introduction

Problem descriptions

Bidirectional graph G = (N,E ). Edge e connecting node i and j canbe equivalently referred to as (i , j) or (j , i).

Let C = 1, 2, ...,C be the set of commodities. Each commodity chas origin o(c) and destination d(c).

Capacity on each edge e has to be installed in batch of size B > 0with cost of κe per batch


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Introduction

Problem descriptions

Let z = z0, z1, z2, ..., zK be a vector of (K + 1) random variablesdefined over the probability space (Ω,P,F). WLOG, z0 = 1 w.p.1.

No exact probability knowledge about z is available, except that thetrue joint probability measure P lies in a set FThe demand for each commodity is modeled as a function of z:

dc = dc(z)

The flow of commodity c on each edge e has two components:

f cij (z) ≥ 0 and f cji (z) ≥ 0


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Introduction

Mathematical programming model

z∗ =minimize∑e∈E

κexe

subject to∑j :(i,j)∈E

f cij (z)−∑

j :(j,i)∈E

f cji (z) =

dc(z) if i = o(c)−dc(z) if i = d(c)0 otherwise

∀c ∈ C, ∀i ∈ N

P(∑c∈C

(f cij (z) + f cji (z)) ≤ Bxe) ≥ β ∀e ∈ E

f cij (z) ≥ 0 ∀(i , j) ∈ E

x ∈ Z|E |+

Difficulty: How to tackle the probability constraints?


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Linear decision rule approximation

Outline

.. .1 Introduction




.. .5 Conclusion


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Linear decision rule (LDR) approximation

We assume that the demand for each commodity c is a linearcombination of zk , k = 0, 1, ...,K .

dc = dc(z) = dc′z =K∑

k=0

dck zk , dck ∈ R

We use linear decision rule to approximate the flow of commodity con edge e:

f cij (z) = fcij′z =

K∑k=0

f ckij zk , f ckij ∈ R


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LDR vs Oblivious Routing

In oblivious routing, for each realization of the demand for commodityc , the same set of path from o(c) to d(c) and the same percentageof flow are used.

Oblivious routing is a special instance of LDR approximation.Indeed, if C = K , and we use:

dc = zc

f cij (z) = f cij zc , f cij ≥ 0

we will get the problem under oblivious routing.


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LDR vs Oblivious Routing vs Dynamic Routing

.Proposition..

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Let z∗LDR and z∗OR be the optimal value for the problem under LDRapproximation and Oblivious Routing, respectively. We have

z∗ ≤ z∗LDR ≤ z∗OR


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LDR Optimization Problem

z∗LDR =minimize∑e∈E

κexe

subject to

∑j :(i,j)∈E

f ckij −∑

j :(j,i)∈E

f ckji =

dck if i = o(c)−dck if i = d(c)0 otherwise

∀c ∈ C, ∀i ∈ N,k = 0, 1, ...K

ye =∑c∈C

(fcij + fcji) ∀e ∈ E

fcij′z ≥ 0 ∀(i , j) ∈ E

P(ye′z ≤ Bxe) ≥ β ∀e ∈ E

x ∈ Z|E |+

Still, there are probabilistic constraints

... but, linear probabilistic constraints can be handled efficiently.


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κexe

subject to

∑j :(i,j)∈E

f ckij −∑

j :(j,i)∈E

f ckji =


∀c ∈ C, ∀i ∈ N,k = 0, 1, ...K

ye =∑c∈C


fcij′z ≥ 0 ∀(i , j) ∈ E


x ∈ Z|E |+




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κexe

subject to

∑j :(i,j)∈E

f ckij −∑

j :(j,i)∈E

f ckji =


∀c ∈ C, ∀i ∈ N,k = 0, 1, ...K

ye =∑c∈C


fcij′z ≥ 0 ∀(i , j) ∈ E


x ∈ Z|E |+




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Approximations of linear probabilistic constraints

Outline

.. .1 Introduction




.. .5 Conclusion


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CVaR approximation of linear probabilistic constraints

Goal: We need an efficient way to approximate P(ye′z ≤ Bxe) ≥ β

.Theorem..

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Given that the true distribution P of z lies in the set (of measures) F, wehave:

infve∈R

ve +1

1− βsupP∈F

EP[(ye′z− ve)

+]︸︷︷︸β−CVaRF(z)

≤ Bxe =⇒ P(ye′z ≤ Bxe) ≥ β

How to evaluate supP∈F

EP[(ye′z− ve)

+]?

Ideas: Bounds are available for certain type of F


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.Theorem..

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infve∈R

ve +1

1− βsupP∈F

EP[(ye′z− ve)




EP[(ye′z− ve)

+]?



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.Theorem..

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.


infve∈R

ve +1

1− βsupP∈F

EP[(ye′z− ve)




EP[(ye′z− ve)

+]?



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.Theorem..

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infve∈R

ve +1

1− βsupP∈F

EP[(ye′z− ve)




EP[(ye′z− ve)

+]?



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Model of uncertainty for z

We assume that we know the support W , the mean support W, and thecovariance Σ of z.

F = P : z = EP[z] ∈ W,P(z ∈ W) = 1,EP[(z− z)(z− z)′] = Σ

and W, W are second-order cone representable (can be written assecond-order cone constraints).


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Bounds taken from Goh and Sim (2010)

.Theorem..

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Given F, the following are upper bounds for supP∈F

EP[(ye′z− ve)

+]

π1(−ve , ye) = infs∈RK

(supz∈W

s′z+ supz∈W

(max−ve + ye′z− s′z,−s′z)

)

π2(−ve , ye) = supz∈W

1

2(−ve + ye

′z) +1

2

√(−ve + ye′z)2 + ye′Σye

π2(−ve , ye) is second order cone representable


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.Theorem..

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.

Given F, the following are upper bounds for supP∈F

EP[(ye′z− ve)

+]

π1(−ve , ye) = infs∈RK

(supz∈W

s′z+ supz∈W

(max−ve + ye′z− s′z,−s′z)

)

π2(−ve , ye) = supz∈W

1

2(−ve + ye

′z) +1

2

√(−ve + ye′z)2 + ye′Σye

π2(−ve , ye) is second order cone representable


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.Theorem..

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Using infimal convolution, we have an improved upper bound forsupP∈F

EP[(ye′z− ve)

+]

π(−ve , ye) = minimize π1(−ve1, ye1) + π2(−ve2, ye2)

s.t ve = ve1 + ve2

ye = ye1 + ye2

If W and W are second-order cone (SOC) representable, π(−ve , ye) is alsoSOC representable.


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.Theorem..

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.

Using infimal convolution, we have an improved upper bound forsupP∈F

EP[(ye′z− ve)

+]

π(−ve , ye) = minimize π1(−ve1, ye1) + π2(−ve2, ye2)

s.t ve = ve1 + ve2

ye = ye1 + ye2

If W and W are second-order cone (SOC) representable, π(−ve , ye) is alsoSOC representable.


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Appoximation of probabilistic constraints

P(ye′z ≤ Bxe) ≥ β can be approximated by

ve +1

1− β(π1(−ve1, ye1) + π2(−ve2, ye2)) ≤ Bxe

ve = ve1 + ve2

ye = ye1 + ye2


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The Abilene case

Outline

.. .1 Introduction




.. .5 Conclusion


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The Abilene case

Abilene case study

11 nodes, 15 edges, 110 demands.

Demand for each 5 minute period over 7 days starting 22ndDecember 03.

Data available for download fromhttp://math.bu.edu/people/kolaczyk/datasets.html


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The Abilene case

Abilene case study

Only data for the first day is used to approximate the distribution of z.

Let K = C and let the demand for each commodity c equal zc

Let W = [z, z], W = [µ, µ] In other words:

F = P : z = EP[z] ∈ [µ, µ],P(z ∈ [z, z]) = 1,EP[(z− z)(z− z)′] = Σ


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The Abilene case

minimize∑e∈E

κexe

subject to

∑j :(i,j)∈E

f ckij −∑

j :(j,i)∈E

f ckji =

1 if i = o(c) and c = k−1 if i = d(c) and c = k0 otherwise

∀c ∈ C, ∀i ∈ N,k = 0, 1, ...K

− z′γe1 + z′γe2 ≥ 0 ∀e ∈ E

− γe1 + γe2 = fcij ∀(i, j) ∈ E

ve +1

1 − β(te1 + te2) ≤ Bxe ∀e ∈ E

ve1 + ve2 = ve ∀e ∈ E

ye1 + ye2 =∑c∈C


µ′γe3 − µ

′γe4 + ve1 + z′γe5 − z′γe6 ≤ te1 ∀e ∈ E

µ′γe3 − µ

′γe4 + z′γe7 − z′γe8 ≤ te1 ∀e ∈ E

γe3 − γe4 = s ∀e ∈ E

γe5 − γe6 = ye1 − s ∀e ∈ E

γe7 − γe8 = −s ∀e ∈ E

1

2u +

1

2

√u2 + ye2

′Σye2 ≤ te2 ∀e ∈ E

ve2 + µ′γe9 − µ

′γe10 ≤ u ∀e ∈ E

γe9 − γe10 ∀e ∈ E

γe1, γe2, γe3, γe4, γe5,γe6, γe7, γe8, γe9, γe10 ≥ 0 ∀e ∈ E

x ∈ Z|E|+


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The Abilene case

Some preliminary results

Out of sample testing for the remaining 6 days.

β = 0.9 β = 0.95 β = 0.97


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Conclusion

Outline

.. .1 Introduction




.. .5 Conclusion


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Conclusion

Conclusion

Distributionally robust optimization provides a tractableapproximation to the network design problem under uncertainty.

The preliminary results are very promising.


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Conclusion

Thank you very much!


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Telecommunication Network Design under Uncertainty: A ... · Viet Anh Nguyen Telecommunication Network Design under Uncertainty: A Distributionally Robust Optimization Approach30th