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A Crash Course in Robust Optimization

Arie M.C.A. [email protected]

INRIA – Project COATI – 12 February 2013

Lehrstuhl II für Mathematik

Outline

1 Motivation2 Chance-Constrained Programming3 Robust Optimization4 Γ-Robust Optimization5 Recoverable Robustness6 Robust Network Design with Affine Recourse7 Multi-Band Robustness8 Conclusions

Arie Koster – RWTH Aachen University 2 / 48

How robust is a network design?

Demand uncertainties Traffic fluctuations in the US abilene Internet2network in time intervals of 5 minutes during one week:

Traffic fluctuates heavily between node-pairs

Load of links will fluctuate alikeTo avoid congestion, demand is overestimated by, e.g., ≥ 300%Can we do better?




Traffic fluctuates heavily between node-pairsLoad of links will fluctuate alike

To avoid congestion, demand is overestimated by, e.g., ≥ 300%Can we do better?




Traffic fluctuates heavily between node-pairsLoad of links will fluctuate alikeTo avoid congestion, demand is overestimated by, e.g., ≥ 300%Can we do better?


Alternative Approaches

Lower overestimationThe network is designed such that capacities are as small as possible; trafficfluctuations might result in high network congestion

Stochastic ProgrammingNetwork design has to be computed for many scenarios; high computationaleffort

Multi-period Network DesignMany traffic matrices have to be considered simultaneously; highcomputational effort


Outline



Chance-Constrained Programming

Chance-constrained Optimization Problem (COP)Find among all solutions that satisfy all constraints with high probability asolution with optimal objective value.

How to solve a COP?Stochastic OptimizationI Modelling with random variablesI Quite challenging to solve resulting problemsI Probability distribution have to be determined

Robust Optimization

I Uncertainty comes from a known set, the uncertainty setI No information on probability distribution neededI Seeks for solution with best worst-case objective guarantee




How to solve a COP?Stochastic OptimizationI Modelling with random variablesI Quite challenging to solve resulting problemsI Probability distribution have to be determinedRobust OptimizationI Uncertainty comes from a known set, the uncertainty set

I No information on probability distribution neededI Seeks for solution with best worst-case objective guarantee




How to solve a COP?Stochastic OptimizationI Modelling with random variablesI Quite challenging to solve resulting problemsI Probability distribution have to be determinedRobust OptimizationI Uncertainty comes from a known set, the uncertainty setI No information on probability distribution needed

I Seeks for solution with best worst-case objective guarantee




How to solve a COP?Stochastic OptimizationI Modelling with random variablesI Quite challenging to solve resulting problemsI Probability distribution have to be determinedRobust OptimizationI Uncertainty comes from a known set, the uncertainty setI No information on probability distribution neededI Seeks for solution with best worst-case objective guarantee



Chance-Constrained Linear Programming

min cT x

s.t. Ax ≤ b

x ≥ 0

with Entries of A, b and/or c are not constant but random variables



Chance-Constrained Linear Programming with joint constraints

min cT x

s.t. P (Ax ≤ b) ≥ 1− εx ≥ 0




Chance-Constrained Linear Programming with individual constraints

min cT x

s.t. P (Aix ≤ bi ) ≥ 1− εi ∀i = 1, . . . ,mx ≥ 0



Example

Chance-Constrained Knapsack:Knapsack with n Items, profits ci , uncertain weights ai , and capacity b

maxn∑

i=1

cixi

s.t. P

(n∑

i=1

aixi ≤ b

)≥ 1− ε

x ∈ {0, 1}n

How to solve this problem?Assumption: Weights are independently and normally distributed withexpectation mi and standard deviation σi .


Example


maxn∑

i=1

cixi

s.t. P

(n∑

i=1

aixi ≤ b

)≥ 1− ε

x ∈ {0, 1}n

How to solve this problem?

Assumption: Weights are independently and normally distributed withexpectation mi and standard deviation σi .


Example


maxn∑

i=1

cixi

s.t. P

(n∑

i=1

aixi ≤ b

)≥ 1− ε

x ∈ {0, 1}n

How to solve this problem?Assumption: Weights are independently and normally distributed withexpectation mi and standard deviation σi .


Example (cont.)Assumption: Weights are independently and normally distributed withexpectation mi and standard deviation σi .

P

(n∑

i=1

aixi ≤ b

)= P

∑ni=1 (aixi −mixi )√∑n

i=1 σ2i x

2i

≤b −

∑ni=1mixi√∑n

i=1 σ2i x

2i

=

P

Z ≤b −

∑ni=1mixi√∑n

i=1 σ2i x

2i

≥ 1− ε

with Z =∑n

i=1(ai xi−mi xi )√∑ni=1 σ

2i x2i

Let Φ(.) be the cumulative distribution function of the standard normaldistribution. Then,

b −∑n

i=1mixi√∑ni=1 σ

2i x

2i

≥ Φ−1(1− ε)


Example (cont.)Assumption: Weights are independently and normally distributed withexpectation mi and standard deviation σi .

P

(n∑

i=1

aixi ≤ b

)= P

∑ni=1 (aixi −mixi )√∑n

i=1 σ2i x

2i

≤b −

∑ni=1mixi√∑n

i=1 σ2i x

2i

= P

Z ≤b −

∑ni=1mixi√∑n

i=1 σ2i x

2i

≥ 1− ε

with Z =∑n

i=1(ai xi−mi xi )√∑ni=1 σ

2i x2i

Let Φ(.) be the cumulative distribution function of the standard normaldistribution. Then,

b −∑n


2i x

2i

≥ Φ−1(1− ε)


Example (cont.)

b −∑n


2i x

2i

≥ Φ−1(1− ε)

If 1− ε > 0.5, Φ−1(1− ε) > 0

and the chance constrained knapsack can bereformulated as

minn∑

i=1

cixi

s.t. Φ−1(1− ε)

√√√√ n∑i=1

σ2i x2i +

n∑i=1

aixi ≤ b

x ∈ {0, 1}n

After relaxing the integrality of x , a second order cone problem remains,which can be solved in polynomial time.


Example (cont.)

b −∑n


2i x

2i

≥ Φ−1(1− ε)

If 1− ε > 0.5, Φ−1(1− ε) > 0 and the chance constrained knapsack can bereformulated as

minn∑

i=1

cixi

s.t. Φ−1(1− ε)

√√√√ n∑i=1

σ2i x2i +

n∑i=1

aixi ≤ b

x ∈ {0, 1}n

After relaxing the integrality of x , a second order cone problem remains,which can be solved in polynomial time.


Example (cont.)

b −∑n


2i x

2i

≥ Φ−1(1− ε)

If 1− ε > 0.5, Φ−1(1− ε) > 0 and the chance constrained knapsack can bereformulated as

minn∑

i=1

cixi

s.t. Φ−1(1− ε)

√√√√ n∑i=1

σ2i x2i +

n∑i=1

aixi ≤ b

x ∈ {0, 1}n

After relaxing the integrality of x , a second order cone problem remains,which can be solved in polynomial time.Arie Koster – RWTH Aachen University 10 / 48

Outline



Uncertain LPs

ObservationIn the example, normal distribution of the weights was assumed. What if,the weights are distributed differently, or unknown?

Uncertain Linear ProgramAn Uncertain Linear Optimization problem (ULO) is a collection of linearoptimization problems (instances){

min{cT x : Ax ≤ b}}

(c,A,b)∈U

where all input data stems from an uncertainty set U ⊂ Rn × Rm×n × Rm.


Robust CounterpartULO

{min{cT x : Ax ≤ b}

}(c,A,b)∈U

Robust feasible solutionA vector x ∈ Rn is robust feasible for ULO if

Ax ≤ b ∀(c ,A, b) ∈ U

Robust solution valueGiven a vector x ∈ Rn, the robust solution value c(x) is defined as

c(x) := sup(c,A,b)∈U

cT x

Robust CounterpartThe robust counterpart of an ULO is the optimization problem

min {c(x) : x is robust feasible}


Example

Let{min{cT x : Ax ≤ b, x ≥ 0}

}(c,A,b)∈U

be an ULO with uncertain

right-hand-side

b ∈ [b, b + b]

uncertain matrix A,

aij ∈ [aij , aij + aij ]

but certain objective vector c .

The robust counterpart can be written as

min{cT x : (A + A)x ≤ b, x ≥ 0}


Example


}(c,A,b)∈U


right-hand-sideb ∈ [b, b + b]

uncertain matrix A,

aij ∈ [aij , aij + aij ]



min{cT x : (A + A)x ≤ b, x ≥ 0}


Example


}(c,A,b)∈U


right-hand-sideb ∈ [b, b + b]

uncertain matrix A,aij ∈ [aij , aij + aij ]



min{cT x : (A + A)x ≤ b, x ≥ 0}


Robust Counterpart

ObservationIf the objective is certain, the robust counterpart can be constructedrow-wise, i.e.,

keep the objectivereplace every constraint aT

i x ≤ bi by its robust counterpart

aTi x ≤ bi ∀(ai , bi ) ∈ Ui

where

Ui :={

(ai , bi ) ∈ Rn+1 : ∃(A, b) ∈ U with Ai . = ai , bi = bi

}

Note: the robust counterpart does not change if U = U1 × U2 × . . .× Um

instead of U is used.


Robust Counterpart

ObservationIf the objective is certain, the robust counterpart can be constructedrow-wise, i.e.,

keep the objectivereplace every constraint aT

i x ≤ bi by its robust counterpart

aTi x ≤ bi ∀(ai , bi ) ∈ Ui

where

Ui :={

(ai , bi ) ∈ Rn+1 : ∃(A, b) ∈ U with Ai . = ai , bi = bi

}Note: the robust counterpart does not change if U = U1 × U2 × . . .× Um

instead of U is used.


Robust Counterpart

CorollaryIf only the right hand side b is uncertain, the robust counter part reads

Ax ≤ b

with bi = min{bi : (A, b, c) ∈ U}.

Max-Flow with uncertain capacities:Take minimum capacity on every arc, and solve the max flow problem.

Min-Cut with uncertain capacities:Objective vector c is uncertain! Requires solving of a new problem.

Corollary: Robust Max-Flow 6= Robust Min-Cut


Robust Counterparts – Limitations


Challenge: Find another way to handle right-hand-side uncertainty.

Minoux [16] considers

maxb∈U

min cT x(b) : Ax(b) ≤ b, x ≥ 0}

instead ofmin cT x : Ax ≤ b ∀b ∈ U , x ≥ 0}

Problem is NP-hard for commonly used uncertainty sets [17, 18]Intermediate solutions required!




Challenge: Find another way to handle right-hand-side uncertainty.Minoux [16] considers

maxb∈U

min cT x(b) : Ax(b) ≤ b, x ≥ 0}







maxb∈U

min cT x(b) : Ax(b) ≤ b, x ≥ 0}


Problem is NP-hard for commonly used uncertainty sets [17, 18]

Intermediate solutions required!





maxb∈U

min cT x(b) : Ax(b) ≤ b, x ≥ 0}




Uncertainty Sets

How to define the uncertainty sets?

Uncertainty set is an ellipsoid [4], e.g.,

Ui ={

(a, b) ∈ Rn+1 : ‖(a, b)− (a, b)‖ < κ}

Uncertainty set is an polyhedron, e.g.,

Ui ={

(a, b) ∈ Rn+1 : D · (a, b) ≤ d}

with D ∈ Rk×n, d ∈ Rk for some k ∈ N [1].

equivalent: set of discrete scenarios (extreme points of polyhedron)special case: Γ-Robustness


Uncertainty Sets



Ui ={

(a, b) ∈ Rn+1 : ‖(a, b)− (a, b)‖ < κ}


Ui ={

(a, b) ∈ Rn+1 : D · (a, b) ≤ d}

with D ∈ Rk×n, d ∈ Rk for some k ∈ N [1].equivalent: set of discrete scenarios (extreme points of polyhedron)

special case: Γ-Robustness


Uncertainty Sets



Ui ={

(a, b) ∈ Rn+1 : ‖(a, b)− (a, b)‖ < κ}


Ui ={

(a, b) ∈ Rn+1 : D · (a, b) ≤ d}

with D ∈ Rk×n, d ∈ Rk for some k ∈ N [1].equivalent: set of discrete scenarios (extreme points of polyhedron)special case: Γ-Robustness


Outline



Robust Optimization ApproachSimplifying assumption: b and c are certain

Uncertainty Set by Bertsimas & Sim [5, 6]: Let aij ∈ R, aij ≥ 0 be given,and Γ ∈ R+ a parameter.

Ui (Γ) = {ai ∈ Rn : aij = aij + aijzij ∀j = 1, . . . , n, zi ∈ Zi (Γ)}

with

Zi (Γ) =

zi ∈ Rn : |zij | ≤ 1 ∀j = 1, . . . , n,n∑

j=1

|zij | ≤ Γ

Stated otherwise:

nominal values aij and deviations aij

aij ∈ [aij − aij , aij + aij ]

Sum of relative deviations from the nominal values is bounded by Γ

At most Γ many entries might deviate from their nominal value


Robust Optimization ApproachSimplifying assumption: b and c are certainUncertainty Set by Bertsimas & Sim [5, 6]: Let aij ∈ R, aij ≥ 0 be given,and Γ ∈ R+ a parameter.


with

Zi (Γ) =

zi ∈ Rn : |zij | ≤ 1 ∀j = 1, . . . , n,n∑

j=1

|zij | ≤ Γ

Stated otherwise:nominal values aij and deviations aij







with

Zi (Γ) =

zi ∈ Rn : |zij | ≤ 1 ∀j = 1, . . . , n,n∑

j=1

|zij | ≤ Γ

Stated otherwise:








with

Zi (Γ) =

zi ∈ Rn : |zij | ≤ 1 ∀j = 1, . . . , n,n∑

j=1

|zij | ≤ Γ

Stated otherwise:



Sum of relative deviations from the nominal values is bounded by ΓAt most Γ many entries might deviate from their nominal value


Example

a =

255

, a =

121

, Γ = 2

1 1.5 2 2.5 34

64

5

6

a1

a2

a 3

Provided by Manuel Kutschka


Γ-Robust CounterpartRobust Counterpart

minn∑

i=1

cixi

s.t.n∑

j=1

aijxj + maxzi∈Zi (Γ)

n∑j=1

aijzijxj

≤ bi i = 1, . . . ,m

x ≥ 0

ObservationSince Zi defines a (bounded) polyhedron, only the extreme points have tobe treated.

For Γ ∈ Z+:

n∑j=1

aijxj + maxS⊆{1,...,n}:|S |≤Γ

∑j∈S

aijxj

≤ bi (1)


Γ-Robustness – Theory

Theorem 1 (Bertsimas & Sim [6])Let x? be an optimal solution of the Γ-robust counterpart. If aij ,j = 1, . . . , n, are independent and symmetric distributed random variables in[aij − aij , aij + aij ], then

P

(n∑

i=1

aix?i > b

)≤ B(n, Γ)

with

limn→∞

B(n, Γ) = 1− Φ

(Γ√n

)where Φ(.) is the CDF of the standard normal distribution.

Instead of the limit: B(n, Γ) ≈ 1− Φ(

Γ−1√n

)


Γ-Robustness – Theory

Theorem 1 (Bertsimas & Sim [6])Let x? be an optimal solution of the Γ-robust counterpart. If aij ,j = 1, . . . , n, are independent and symmetric distributed random variables in[aij − aij , aij + aij ], then

P

(n∑

i=1

aix?i > b

)≤ B(n, Γ)

with

limn→∞

B(n, Γ) = 1− Φ

(Γ√n

)where Φ(.) is the CDF of the standard normal distribution.

Instead of the limit: B(n, Γ) ≈ 1− Φ(

Γ−1√n

)Arie Koster – RWTH Aachen University 23 / 48

Choice of Γ

Choice of Γ as a function of n so that theprobability of constraint violation is less than p%:

Γn p = 5 p = 2 p = 1 p = 0.5 p = 0.15 4.7 5.0 5.0 5.0 5.0

10 6.2 7.5 8.4 9.1 10.020 8.4 10.2 11.4 12.5 14.850 12.6 15.5 17.4 19.2 22.9

100 17.4 21.5 24.3 26.8 31.9200 24.3 30.0 33.9 37.4 44.7

1,000 53.0 65.9 74.6 82.5 98.72,000 74.6 92.8 105.0 116.2 139.2

Note: Result is independent of actual distribution of random variables aij ,only symmetry and independence are required.


Solving the robust counterpart

n∑j=1


∑j∈S

aijxj

≤ bi

Observations:

Inequality (1) can be linearized by∑j 6∈S

aijxj +∑j∈S

(aij + aij )xj ≤ bi ∀S ⊆ {1, . . . , n}, |S | ≤ Γ (2)

This number of inequalities is exponential if Γ = O(n)

Separation can be done in polynomial timeAlternatively, a compact formulation can be obtained via dualization



n∑j=1


∑j∈S

aijxj

≤ bi

Observations:Inequality (1) can be linearized by∑

j 6∈S

aijxj +∑j∈S

(aij + aij )xj ≤ bi ∀S ⊆ {1, . . . , n}, |S | ≤ Γ (2)





n∑j=1


∑j∈S

aijxj

≤ bi


j 6∈S

aijxj +∑j∈S

(aij + aij )xj ≤ bi ∀S ⊆ {1, . . . , n}, |S | ≤ Γ (2)


Separation can be done in polynomial time

Alternatively, a compact formulation can be obtained via dualization



n∑j=1


∑j∈S

aijxj

≤ bi


j 6∈S

aijxj +∑j∈S

(aij + aij )xj ≤ bi ∀S ⊆ {1, . . . , n}, |S | ≤ Γ (2)




Separation of robust inequalities

Given x?, find a subset S ⊆ {1, . . . , n} with |S | ≤ Γ such that∑j 6∈S

aijx?j +

∑j∈S

(aij + aij )x?j > bi

Separation problem:

ZSEP = maxn∑

j=1

aijx?j zj

s.t.n∑

j=1

zj ≤ Γ

If ZSEP > bi , add robust inequality (2) for S = {j : zj = 1}.Optimization = Separation implies polynomial solvability of LP



Given x?, find a subset S ⊆ {1, . . . , n} with |S | ≤ Γ such that

n∑j=1

aijx?j +

∑j∈S

aijx?j > bi

Separation problem:

ZSEP = maxn∑

j=1

aijx?j zj

s.t.n∑

j=1

zj ≤ Γ

If ZSEP > bi , add robust inequality (2) for S = {j : zj = 1}.Optimization = Separation implies polynomial solvability of LP




∑j∈S

aijx?j > bi −

n∑j=1

aijx?j

Separation problem:

ZSEP = maxn∑

j=1

aijx?j zj

s.t.n∑

j=1

zj ≤ Γ

If ZSEP > bi −∑n

j=1 aijx?j , add robust inequality (2) for S = {j : zj = 1}.

Optimization = Separation implies polynomial solvability of LP


Separation of robust inequalitiesGiven x?, find a subset S ⊆ {1, . . . , n} with |S | ≤ Γ such that

∑j∈S

aijx?j > bi −

n∑j=1

aijx?j

Separation problem:

ZSEP = maxn∑

j=1

aijx?j zj

s.t.n∑

j=1

zj ≤ Γ

zj ∈ {0, 1}







∑j∈S

aijx?j > bi −

n∑j=1

aijx?j

Separation problem:

ZSEP = maxn∑

j=1

aijx?j zj

s.t.n∑

j=1

zj ≤ Γ

0 ≤ zj ≤ 1







∑j∈S

aijx?j > bi −

n∑j=1

aijx?j

Separation problem:

ZSEP = maxn∑

j=1

aijx?j zj

s.t.n∑

j=1

zj ≤ Γ

0 ≤ zj ≤ 1



Optimization = Separation implies polynomial solvability of LPArie Koster – RWTH Aachen University 26 / 48

Compact formulation

Let βi (x , Γ) = maxS⊆{1,...,n}:|S|≤Γ

∑j∈S

aijxj

, and hence (1) reads

n∑j=1

aijxj + βi (x , Γ) ≤ bi

Given x?, βi (x?, Γ) is the optimization problem

βi (x?, Γ) = max

n∑j=1

aijx?j zj

= minΓπi +n∑

j=1

ρij

s.t.n∑

j=1

zj ≤ Γ

s.t.πi + ρij ≥ aijx?j ∀j = 1, . . . , n

0 ≤ zj ≤ 1

πi , ρij ≥ 0


Compact formulation

Let βi (x , Γ) = maxS⊆{1,...,n}:|S|≤Γ

∑j∈S

aijxj


n∑j=1



βi (x?, Γ) = max

n∑j=1

aijx?j zj = minΓπi +

n∑j=1

ρij

s.t.n∑

j=1

zj ≤ Γ

s.t.πi + ρij ≥ aijx?j ∀j = 1, . . . , n

0 ≤ zj ≤ 1

πi , ρij ≥ 0


Compact formulation

Let βi (x , Γ) = maxS⊆{1,...,n}:|S|≤Γ

∑j∈S

aijxj


n∑j=1



βi (x?, Γ) = max

n∑j=1

aijx?j zj = minΓπi +

n∑j=1

ρij

s.t.n∑

j=1

zj ≤ Γ s.t.πi + ρij ≥ aijx?j ∀j = 1, . . . , n

0 ≤ zj ≤ 1 πi , ρij ≥ 0


Compact Reformulation

Thus, (1) now reads

n∑j=1

aijxj + min

Γπi +n∑

j=1

ρij : πi + ρij ≥ aijxj ∀j , πi ≥ 0, ρij ≥ 0

≤ bi

or equivalently

n∑j=1

aijxj + Γπi +n∑

j=1

ρij ≤ bi

πi + ρij ≥ aijxj ∀j = 1, . . . , nπi ≥ 0, ρij ≥ 0


Further results

Theorem 2 (Bertsimas & Sim [6])If only the objective is uncertain and x ∈ {0, 1}n, then the robustcounterpart can be solved by solving n + 1 nominal problems of the sametype.

Corollary 3The knapsack problem with uncertain objective can be solved in O(n2B).

Theorem 4 (Pferschy et al., 2012)The knapsack problem with uncertain weights can be solved in O(nΓB).


Robust Optimization Revisited

Advantages Robust Optimization:Only marginal complexity increase (compared to deterministic case)

Trade-off between level of robustness and cost of solution by parameter Γ

Optimizes result in the worst-case (in advance)No information on probability distribution needed

Disadvantages Robust Optimization:

Right-hand-side uncertainty not satisfyingSingle solution without any flexibity!The almost always optimal solution might be infeasible

⇒ Two-Stage Robustness Concepts

Very inprecise description of uncertainty (only two values)

⇒ More detailed description of uncertainty



Advantages Robust Optimization:Only marginal complexity increase (compared to deterministic case)Trade-off between level of robustness and cost of solution by parameter Γ


Disadvantages Robust Optimization:

Right-hand-side uncertainty not satisfyingSingle solution without any flexibity!The almost always optimal solution might be infeasible







Optimizes result in the worst-case (in advance)

No information on probability distribution needed

Disadvantages Robust Optimization:Right-hand-side uncertainty not satisfying

Single solution without any flexibity!The almost always optimal solution might be infeasible









Disadvantages Robust Optimization:Right-hand-side uncertainty not satisfyingSingle solution without any flexibity!The almost always optimal solution might be infeasible

⇒ Two-Stage Robustness ConceptsVery inprecise description of uncertainty (only two values)







Disadvantages Robust Optimization:Right-hand-side uncertainty not satisfyingSingle solution without any flexibity!The almost always optimal solution might be infeasible⇒ Two-Stage Robustness Concepts







Disadvantages Robust Optimization:Right-hand-side uncertainty not satisfyingSingle solution without any flexibity!The almost always optimal solution might be infeasible⇒ Two-Stage Robustness ConceptsVery inprecise description of uncertainty (only two values)






Disadvantages Robust Optimization:Right-hand-side uncertainty not satisfyingSingle solution without any flexibity!The almost always optimal solution might be infeasible⇒ Two-Stage Robustness ConceptsVery inprecise description of uncertainty (only two values)⇒ More detailed description of uncertainty


Outline



Recoverable Robustness

Recoverable robustness [14, 7]uncertainty as two-stage process:

1st stage: a-priori decision2nd stage: recovery:

limited change of first-stage decisionafter realization of uncertainty is known

optimize worst-case w. r. t. recovery

Example:Recoverable Robust Knapsack problem (RRKP) with

Discrete Scenarios [9]Γ Scenarios [8]

Recoverable Robust Network Topology Design (discrete scenarios) [2]


(k , `)-RRKP with Discrete Scenarios

Given items N = {1, . . . , n},first stage: profits p0, weight w0, capacity c0,

scenarios S ∈ SD with profits pS , weight wS , capacity cS ,recovery set X (X ): delete ≤ k items, add ≤ ` items

Find subset X ⊆ NSuch that w0(X ) ≤ c0,

for all S ∈ SD there exists X S ∈ X (X ) with wS (X S ) ≤ cS ,

total profit

pT (X ) = p0(X )

+ minS∈SD

maxX S

pS (X S )

is maximized.

First



Given items N = {1, . . . , n},first stage: profits p0, weight w0, capacity c0,scenarios S ∈ SD with profits pS , weight wS , capacity cS ,

recovery set X (X ): delete ≤ k items, add ≤ ` items


for all S ∈ SD there exists X S ∈ X (X ) with wS (X S ) ≤ cS ,total profit

pT (X ) = p0(X ) + minS∈SD

maxX S

pS (X S )

is maximized.

S_1 S_2First



Given items N = {1, . . . , n},first stage: profits p0, weight w0, capacity c0,scenarios S ∈ SD with profits pS , weight wS , capacity cS ,recovery set X (X ): delete ≤ k items, add ≤ ` items


for all S ∈ SD there exists X S ∈ X (X ) with wS (X S ) ≤ cS ,total profit

pT (X ) = p0(X ) + minS∈SD

maxX S

pS (X S )

is maximized.

S_1 S_2First


k-RRKP with Γ Scenarios

Given Items N = {1, . . . , n},first stage: profits p0, weights w0, capacity c0,

Γ-scenarios: weights [w , w + w ], capacity c , Γ ∈ N,recovery set X (X ): delete ≤ k items from X ⊆ N

Find subset X ⊆ N,Such that w0(X ) ≤ c0,

for all S ∈ SΓ there exists X S ∈ X (X ) with wS (X S ) ≤ c ,

total profit p0(X ) is maximized

S_1SecondFirst


k-RRKP with Γ Scenarios

Given Items N = {1, . . . , n},first stage: profits p0, weights w0, capacity c0,Γ-scenarios: weights [w , w + w ], capacity c , Γ ∈ N,recovery set X (X ): delete ≤ k items from X ⊆ N

Find subset X ⊆ N,Such that w0(X ) ≤ c0,

for all S ∈ SΓ there exists X S ∈ X (X ) with wS (X S ) ≤ c ,total profit p0(X ) is maximized

S_1SecondFirst


RRKP with Γ scenarios - MP

Mathematical Programming formulation:

max∑i∈N

p0i xi

s. t.∑i∈N

w0i xi ≤c0

∑i∈N

wixi + maxX⊆N|X |≤Γ

∑i∈X

wixi − maxY⊆N|Y |≤k

(∑i∈Y

wixi +∑

i∈X∩Y

wixi

) ≤cxi ∈ {0, 1}

Question: Compact Linear reformulation?Answer: LP duality and enumeration of solution values!




max∑i∈N

p0i xi

s. t.∑i∈N

w0i xi ≤c0

∑i∈N


∑i∈X


(∑i∈Y

wixi +∑

i∈X∩Y

wixi

) ≤cxi ∈ {0, 1}

Question: Compact Linear reformulation?

Answer: LP duality and enumeration of solution values!




max∑i∈N

p0i xi

s. t.∑i∈N

w0i xi ≤c0

∑i∈N


∑i∈X


(∑i∈Y

wixi +∑

i∈X∩Y

wixi

) ≤cxi ∈ {0, 1}

Question: Compact Linear reformulation?Answer: LP duality and enumeration of solution values!


Outline



Static vs. Dynamic Routing

Static Routing:Capacities have to be installed in integer amountsRouting templates fixes percentual distribution of traffic volume alongpaths

Dynamic Routing:

Capacities have to be installed in integer amountsRouting can be adapted to actual traffic volumes (realization fromuncertainty set)


Static vs. Dynamic Routing

Static Routing:Capacities have to be installed in integer amountsRouting templates fixes percentual distribution of traffic volume alongpaths

Dynamic Routing:Capacities have to be installed in integer amountsRouting can be adapted to actual traffic volumes (realization fromuncertainty set)


Robust Network Design with DynamicRouting

ykij (d) = fraction of demand k ∈ K routed along arc (i, j) ∈ A for realization d ∈ D.

xe = number of link capacity modules to be installed on link e ∈ E .

Integer Linear Programming formulation:

min∑e∈E

κe xe

s.t.∑

j∈N(i)

(ykij (d)− yk

ji (d)) =

dk (d) i = s(k)

−dk (d) i = t(k)

0 else

, ∀d ∈ D, i ∈ V , k ∈ K

∑k∈K

yke ≤ Cxe , ∀d ∈ D, e ∈ E

y(d) ≥ 0, x ∈ Z|E|+

Theorem (Mattia [15])

The vector x ∈ Px if and only if for all length functions` : E → R+ holds

∑e∈E

`(e)xe ≥ maxd∈D

{∑k∈K

dk(d)`(sk , tk )

}


Robust Network Design with DynamicRouting

ykij (d) = fraction of demand k ∈ K routed along arc (i, j) ∈ A for realization d ∈ D.

xe = number of link capacity modules to be installed on link e ∈ E .Integer Linear Programming formulation:

min∑e∈E

κe xe

s.t.∑

j∈N(i)

(ykij (d)− yk

ji (d)) =

dk (d) i = s(k)

−dk (d) i = t(k)

0 else

, ∀d ∈ D, i ∈ V , k ∈ K

∑k∈K

yke ≤ Cxe , ∀d ∈ D, e ∈ E

y(d) ≥ 0, x ∈ Z|E|+

Theorem (Mattia [15])

The vector x ∈ Px if and only if for all length functions` : E → R+ holds

∑e∈E

`(e)xe ≥ maxd∈D

{∑k∈K

dk (d)`(sk , tk )

}Arie Koster – RWTH Aachen University 38 / 48

Affine Routing Function

Robust Network Design with Affine Routing:Capacities have to be installed in integer amountsRouting follows a linear function of all traffic values

ykij (d) := hk0

ij +∑k∈K

hkkij d k

where hk0ij , h

kkij ∈ R for all ij ∈ A, k , k ∈ K .

Theorem (Poss & Raack [19])Let D be an arbitrary demand uncertainty set. Then

OPTdyn(D) ≤ OPTaff (D) ≤ OPTstat(D)


Outline



Multi-band robustnessIdea: Refinement of Γ-robustness approach

valuevalue

prob prob

da d1 d2 d3da d3 a0

Γ-robustness

� dk ≥ 0

� dk ≥ 0

� [dk , dk + dk ]

� Γ ∈ N

Multi-band robustness

� dk ≥ 0

� 0 = dk0 ≤ dk

1 ≤ . . . ≤ dk|B| = dk

� [dk + dkb−1, d

k + dkb ] forall b ∈ B

� u0, u1, . . . , u|B| ∈ N

Γ-robustness ≡ multi-band robustness with B = {1}, u0 = |K |, u1 = Γ

work by Büsing and D’Andreagiovanni [10]; based on Bienstock


Multi-band RND

Compact ILP formulation:

min∑e∈E

cexe

s. t.∑

j∈V :ij∈E

f kij −

∑j∈V :ji∈E

f kji =

1 , i = sk

−1 , i = tk

0 , otherwise∀i , k

∑k∈K

dk f ke +

∑b∈B

ubwe,b +∑k∈K

zke ≤ Cxe ∀e

we,b + zke ≥ dk

b fk

e ∀b, kxe ∈ Z+, f

kij ∈ [0, 1], we,b ≥ 0, zk

e ≥ 0 ∀e, ij , b, k


Outline



Concluding Remarks

Robust optimization is an emerging field in mathematical optimizationbut requires an additional effort to solve robust problems

Correct modelling of uncertainties in network applications requiredIntegration of other (real) networking aspects (survivability, multi-layer)⇒ Real applications [3, 12, 11]⇒ Robustness & 1+1 Protection [13]Recoverable Robustness allows a two-stage approach⇒ What recovery action is possible?⇒ Algorithmic implications?Quality of robust approach has to be evaluated⇒ Which value of Γ is enough to obtain robust designs?


Concluding Remarks

Robust optimization is an emerging field in mathematical optimizationbut requires an additional effort to solve robust problemsCorrect modelling of uncertainties in network applications requiredIntegration of other (real) networking aspects (survivability, multi-layer)⇒ Real applications [3, 12, 11]⇒ Robustness & 1+1 Protection [13]

Recoverable Robustness allows a two-stage approach⇒ What recovery action is possible?⇒ Algorithmic implications?Quality of robust approach has to be evaluated⇒ Which value of Γ is enough to obtain robust designs?


Concluding Remarks

Robust optimization is an emerging field in mathematical optimizationbut requires an additional effort to solve robust problemsCorrect modelling of uncertainties in network applications requiredIntegration of other (real) networking aspects (survivability, multi-layer)⇒ Real applications [3, 12, 11]⇒ Robustness & 1+1 Protection [13]Recoverable Robustness allows a two-stage approach⇒ What recovery action is possible?⇒ Algorithmic implications?

Quality of robust approach has to be evaluated⇒ Which value of Γ is enough to obtain robust designs?


Concluding Remarks

Robust optimization is an emerging field in mathematical optimizationbut requires an additional effort to solve robust problemsCorrect modelling of uncertainties in network applications requiredIntegration of other (real) networking aspects (survivability, multi-layer)⇒ Real applications [3, 12, 11]⇒ Robustness & 1+1 Protection [13]Recoverable Robustness allows a two-stage approach⇒ What recovery action is possible?⇒ Algorithmic implications?Quality of robust approach has to be evaluated⇒ Which value of Γ is enough to obtain robust designs?


A Crash Course in Robust Optimization

Arie M.C.A. [email protected]

INRIA – Project COATI – 12 February 2013

Lehrstuhl II für Mathematik

References I

[1] A. Altin, H. Yaman, and M. C. Pinar.The Network Loading Problem under Hose Demand Uncertainty: Formulation, Polyhedral Analysis, andComputations.INFORMS Journal on Computing, 23:75–89, 2011.

[2] E. Alvarez-Miranda, I. Ljubic, S. Raghavan, and P. Toth.The recoverable robust two-level network design problem.Technical report, Vienna University, 2012.http://homepage.univie.ac.at/ivana.ljubic/publications.html.

[3] P. Belotti, K. Kompella, and L. Noronha.A comparison of OTN and MPLS networks under traffic uncertainty.Working paper, 2011.http://myweb.clemson.edu/~pbelott/papers/\robust-opt-network-design.pdf.

[4] A. Ben-Tal, L. E. Ghaoui, and A. Nemirovski.Robust optimization.Princeton University Press, 2009.

[5] D. Bertsimas and M. Sim.Robust discrete optimization and network flows.Math. Program., Ser. B 98:49–71, 2003.

[6] D. Bertsimas and M. Sim.The Price of Robustness.Operations Research, 52(1):35–53, 2004.

[7] C. Büsing.Recoverable Robustness in Combinatorial Optimization.PhD thesis, Technische Universität Berlin, 2011.


http://myweb.clemson.edu/~ pbelott/papers/\robust-opt-network-design.pdf

References II

[8] C. Büsing, A. M. C. A. Koster, and M. Kutschka.Recoverable robust knapsacks: gamma-scenarios.In Proceedings of INOC 2011, International Network Optimization Conference, volume 6701 of Lecture Noteson Computer Science, pages 583–588, 2011.

[9] C. Büsing, A. M. C. A. Koster, and M. Kutschka.Recoverable robust knapsacks: the discrete scenario case.Optimization Letters, 5(3):379–392, 2011.

[10] F. D. C. Büsing.New results about multi-band uncertainty in robust optimization.Proceedings of SEA 2012, 11th Symposium on Experimental Algorithms, Lecture Notes on Computer Science7276, pages 63–74, 2012.

[11] G. Claßen, A. M. C. A. Koster, and A. Schmeink.A robust optimisation model and cutting planes for the planning of energy-efficient wireless networks.Computers & Operations Research, 40(1):80–90, 2013.

[12] S. Duhovniko, A. M. C. A. Koster, M. Kutschka, F. Rambach, and D. Schupke.Γ-robust network design for mixed-line-rate-planning of optical networks.In Proc. Optical Fiber Communication - National Fiber Optic Engineers Conference (OFC/NFOEC), 2013.

[13] A. M. C. A. Koster and M. Kutschka.An integrated model for survivable network design under demand uncertainty.In Proceedings of 8th International Workshop on the Design of Reliable Communication Networks (DRCN2011), pages 54–61, 2011.

[14] C. Liebchen, M. E. Lübbecke, R. H. Möhring, and S. Stiller.The concept of recoverable robustness, linear programming recovery, and railway applications.In R. Ahuja, R. Möhring, and C. Zaroliagis, editors, Robust and Online Large-Scale Optimization, volume5868 of Lecture Notes on Computer Science, pages 1–27, 2009.


References III

[15] S. Mattia.The robust network loading problem with dynamic routing.Computational Optimization and Applications, August 2012.

[16] M. Minoux.On robust maximum flow with polyhedral uncertainty sets.Optimization Letters, 3:367–376, 2009.

[17] M. Minoux.On 2-stage robust lp with rhs uncertainty: complexity results and applications.Journal of Global Optimization, 49:521–537, 2011.

[18] M. Minoux.Two-stage robust lp with ellipsoidal right-hand side uncertainty is np-hard.Optimization Letters, 6:1463–1475, 2012.

[19] M. Poss and C. Raack.Affine recourse for the robust network design problem: between static and dynamic routing.Networks, to appear, 2013.


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