telecommunication network design under uncertainty: a ... · viet anh nguyen telecommunication...

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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

Viet Anh Nguyen () Telecommunication Network Design under Uncertainty: A Distributionally Robust Optimization Approach30th October 2012 2 / 25

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Introduction

Outline

.. .1 Introduction

.. .2 Linear decision rule approximation

.. .3 Approximations of linear probabilistic constraints

.. .4 The Abilene case

.. .5 Conclusion

Viet Anh Nguyen () Telecommunication Network Design under Uncertainty: A Distributionally Robust Optimization Approach30th October 2012 3 / 25

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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.

Viet Anh Nguyen () Telecommunication Network Design under Uncertainty: A Distributionally Robust Optimization Approach30th October 2012 4 / 25

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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

Viet Anh Nguyen () Telecommunication Network Design under Uncertainty: A Distributionally Robust Optimization Approach30th October 2012 5 / 25

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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

Viet Anh Nguyen () Telecommunication Network Design under Uncertainty: A Distributionally Robust Optimization Approach30th October 2012 6 / 25

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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?

Viet Anh Nguyen () Telecommunication Network Design under Uncertainty: A Distributionally Robust Optimization Approach30th October 2012 7 / 25

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Linear decision rule approximation

Outline

.. .1 Introduction

.. .2 Linear decision rule approximation

.. .3 Approximations of linear probabilistic constraints

.. .4 The Abilene case

.. .5 Conclusion

Viet Anh Nguyen () Telecommunication Network Design under Uncertainty: A Distributionally Robust Optimization Approach30th October 2012 8 / 25

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Linear decision rule approximation

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

Viet Anh Nguyen () Telecommunication Network Design under Uncertainty: A Distributionally Robust Optimization Approach30th October 2012 9 / 25

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Linear decision rule approximation

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.

Viet Anh Nguyen () Telecommunication Network Design under Uncertainty: A Distributionally Robust Optimization Approach30th October 2012 10 / 25

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Linear decision rule approximation

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

Viet Anh Nguyen () Telecommunication Network Design under Uncertainty: A Distributionally Robust Optimization Approach30th October 2012 11 / 25

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Linear decision rule approximation

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.

Viet Anh Nguyen () Telecommunication Network Design under Uncertainty: A Distributionally Robust Optimization Approach30th October 2012 12 / 25

. . . . . .

Linear decision rule approximation

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.

Viet Anh Nguyen () Telecommunication Network Design under Uncertainty: A Distributionally Robust Optimization Approach30th October 2012 12 / 25

. . . . . .

Linear decision rule approximation

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.

Viet Anh Nguyen () Telecommunication Network Design under Uncertainty: A Distributionally Robust Optimization Approach30th October 2012 12 / 25

. . . . . .

Approximations of linear probabilistic constraints

Outline

.. .1 Introduction

.. .2 Linear decision rule approximation

.. .3 Approximations of linear probabilistic constraints

.. .4 The Abilene case

.. .5 Conclusion

Viet Anh Nguyen () Telecommunication Network Design under Uncertainty: A Distributionally Robust Optimization Approach30th October 2012 13 / 25

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Approximations of linear probabilistic constraints

CVaR approximation of linear probabilistic constraints

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

.Theorem..

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. ..

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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

Viet Anh Nguyen () Telecommunication Network Design under Uncertainty: A Distributionally Robust Optimization Approach30th October 2012 14 / 25

. . . . . .

Approximations of linear probabilistic constraints

CVaR approximation of linear probabilistic constraints

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

.Theorem..

.

. ..

.

.

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

Viet Anh Nguyen () Telecommunication Network Design under Uncertainty: A Distributionally Robust Optimization Approach30th October 2012 14 / 25

. . . . . .

Approximations of linear probabilistic constraints

CVaR approximation of linear probabilistic constraints

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

.Theorem..

.

. ..

.

.

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

Viet Anh Nguyen () Telecommunication Network Design under Uncertainty: A Distributionally Robust Optimization Approach30th October 2012 14 / 25

. . . . . .

Approximations of linear probabilistic constraints

CVaR approximation of linear probabilistic constraints

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

.Theorem..

.

. ..

.

.

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

Viet Anh Nguyen () Telecommunication Network Design under Uncertainty: A Distributionally Robust Optimization Approach30th October 2012 14 / 25

. . . . . .

Approximations of linear probabilistic constraints

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).

Viet Anh Nguyen () Telecommunication Network Design under Uncertainty: A Distributionally Robust Optimization Approach30th October 2012 15 / 25

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Approximations of linear probabilistic constraints

Bounds taken from Goh and Sim (2010)

.Theorem..

.

. ..

.

.

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

Viet Anh Nguyen () Telecommunication Network Design under Uncertainty: A Distributionally Robust Optimization Approach30th October 2012 16 / 25

. . . . . .

Approximations of linear probabilistic constraints

Bounds taken from Goh and Sim (2010)

.Theorem..

.

. ..

.

.

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

Viet Anh Nguyen () Telecommunication Network Design under Uncertainty: A Distributionally Robust Optimization Approach30th October 2012 16 / 25

. . . . . .

Approximations of linear probabilistic constraints

Bounds taken from Goh and Sim (2010)

.Theorem..

.

. ..

.

.

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.

Viet Anh Nguyen () Telecommunication Network Design under Uncertainty: A Distributionally Robust Optimization Approach30th October 2012 17 / 25

. . . . . .

Approximations of linear probabilistic constraints

Bounds taken from Goh and Sim (2010)

.Theorem..

.

. ..

.

.

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.

Viet Anh Nguyen () Telecommunication Network Design under Uncertainty: A Distributionally Robust Optimization Approach30th October 2012 17 / 25

. . . . . .

Approximations of linear probabilistic constraints

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

Viet Anh Nguyen () Telecommunication Network Design under Uncertainty: A Distributionally Robust Optimization Approach30th October 2012 18 / 25

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The Abilene case

Outline

.. .1 Introduction

.. .2 Linear decision rule approximation

.. .3 Approximations of linear probabilistic constraints

.. .4 The Abilene case

.. .5 Conclusion

Viet Anh Nguyen () Telecommunication Network Design under Uncertainty: A Distributionally Robust Optimization Approach30th October 2012 19 / 25

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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

Viet Anh Nguyen () Telecommunication Network Design under Uncertainty: A Distributionally Robust Optimization Approach30th October 2012 20 / 25

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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

Viet Anh Nguyen () Telecommunication Network Design under Uncertainty: A Distributionally Robust Optimization Approach30th October 2012 21 / 25

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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

(fcij + fcji) ∀e ∈ E

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|+

Viet Anh Nguyen () Telecommunication Network Design under Uncertainty: A Distributionally Robust Optimization Approach30th October 2012 22 / 25

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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

Viet Anh Nguyen () Telecommunication Network Design under Uncertainty: A Distributionally Robust Optimization Approach30th October 2012 23 / 25

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Conclusion

Outline

.. .1 Introduction

.. .2 Linear decision rule approximation

.. .3 Approximations of linear probabilistic constraints

.. .4 The Abilene case

.. .5 Conclusion

Viet Anh Nguyen () Telecommunication Network Design under Uncertainty: A Distributionally Robust Optimization Approach30th October 2012 24 / 25

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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.

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

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Conclusion

Thank you very much!

Viet Anh Nguyen () Telecommunication Network Design under Uncertainty: A Distributionally Robust Optimization Approach30th October 2012 26 / 25