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Optimization over Integers with Robustness in Cost and Few - - PowerPoint PPT Presentation

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Introduction Cost Robustness Applications Optimization over Integers with Robustness in Cost and Few Constraints Kai-Simon Goetzmann (Joint work with Sebastian Stiller and Claudio Telha) WAOA 2011 Sept 09 Kai-Simon Goetzmann Robust

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SLIDE 1

Introduction Cost Robustness Applications

Optimization over Integers with Robustness in Cost and Few Constraints

Kai-Simon Goetzmann

(Joint work with Sebastian Stiller and Claudio Telha)

WAOA 2011 – Sept 09

Kai-Simon Goetzmann Robust Optimization over Integers

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SLIDE 2

Introduction Cost Robustness Applications

Classical Optimization over Integers. x1 x2 max e.g. totally unimodular IPs, Unbounded Knapsack Problem, IPs with two variables per inequality, . . .

Kai-Simon Goetzmann Robust Optimization over Integers

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SLIDE 3

Introduction Cost Robustness Applications

Classical Optimization over Integers. But what if. . . ? x1 x2 e.g. totally unimodular IPs, Unbounded Knapsack Problem, IPs with two variables per inequality, . . .

Kai-Simon Goetzmann Robust Optimization over Integers

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SLIDE 4

Introduction Cost Robustness Applications

Cost Robust Counterpart

Given set of feasible solutions X ⊆ ❩n cost vector c ∈ ◗n ◆ find x ∈ X minimizing the cost cTx

Kai-Simon Goetzmann Robust Optimization over Integers

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SLIDE 5

Introduction Cost Robustness Applications

Cost Robust Counterpart

Given set of feasible solutions X ⊆ ❩n cost vector c ∈ ◗n vector of possible cost changes d ∈ ◆n maximal number of changes Γ ∈ [n] find x ∈ X minimizing the cost cTx

Kai-Simon Goetzmann Robust Optimization over Integers

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SLIDE 6

Introduction Cost Robustness Applications

Cost Robust Counterpart

Given set of feasible solutions X ⊆ ❩n cost vector c ∈ ◗n vector of possible cost changes d ∈ ◆n maximal number of changes Γ ∈ [n] find x ∈ X minimizing the worst case cost cTx + max

S⊆[n] |S|≤Γ

  • i∈S

|dixi|

Kai-Simon Goetzmann Robust Optimization over Integers

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SLIDE 7

Introduction Cost Robustness Applications

Cost Robust Counterpart

Given set of feasible solutions X ⊆ ❩n cost vector c ∈ ◗n vector of possible cost changes d ∈ ◆n maximal number of changes Γ ∈ [n] find x ∈ X minimizing the worst case cost cTx + max

S⊆[n] |S|≤Γ

  • i∈S

|dixi| This defines the (d, Γ)-CRC of P = (c, X).

Kai-Simon Goetzmann Robust Optimization over Integers

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SLIDE 8

Introduction Cost Robustness Applications

1

Introduction

2

General Result for Cost Robust IPs

3

Applications of the General Result

Kai-Simon Goetzmann Robust Optimization over Integers

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SLIDE 9

Introduction Cost Robustness Applications

1

Introduction

2

General Result for Cost Robust IPs

3

Applications of the General Result

Kai-Simon Goetzmann Robust Optimization over Integers

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SLIDE 10

Introduction Cost Robustness Applications

To solve CRC of P: Solve the (α, c′)-MMin of P. min

x∈X n

  • j=1
  • cj(xj),
  • cj(xj) = cj+max{c′

jxj−α, 0}+max{−c′ jxj−α, 0}

xj

  • cj(xj)

α/c′

j

−α/c′

j Kai-Simon Goetzmann Robust Optimization over Integers

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SLIDE 11

Introduction Cost Robustness Applications

Theorem Let X ⊆ ❩n, c ∈ ◗n, d ∈ ◆n, Γ ∈ [n]. If there is a ρ-approximation algorithm for the (α, d)-MMin of P for all α ≥ 0 bounds uj on the absolute value of xj can be computed in polynomial time, then there is a pseudopolynomial ρ-approximation algorithm for the (d, Γ)-CRC of P = (c, X).

Kai-Simon Goetzmann Robust Optimization over Integers

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SLIDE 12

Introduction Cost Robustness Applications

Proof (Sketch): min

x∈X

  • cTx + max

S⊆[n] |S|≤Γ

  • j∈S

|djxj|

  • Kai-Simon Goetzmann

Robust Optimization over Integers

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SLIDE 13

Introduction Cost Robustness Applications

Proof (Sketch): min

x∈X

  • cTx + max

S⊆[n] |S|≤Γ

  • j∈S

|djxj|

  • ⇓ duality

min

x∈X ϑ≥0

  • cTx + Γϑ +

n

  • j=1
  • max{djxj − ϑ, 0} + max{−djxj − ϑ, 0}
  • Kai-Simon Goetzmann

Robust Optimization over Integers

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SLIDE 14

Introduction Cost Robustness Applications

Proof (Sketch): min

x∈X

  • cTx + max

S⊆[n] |S|≤Γ

  • j∈S

|djxj|

  • ⇓ duality

min

x∈X ϑ≥0

  • cTx + Γϑ +

n

  • j=1
  • max{djxj − ϑ, 0} + max{−djxj − ϑ, 0}
  • Γ

|djxj|

Kai-Simon Goetzmann Robust Optimization over Integers

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SLIDE 15

Introduction Cost Robustness Applications

Proof (Sketch): min

x∈X

  • cTx + max

S⊆[n] |S|≤Γ

  • j∈S

|djxj|

  • ⇓ duality

min

x∈X ϑ≥0

  • cTx + Γϑ +

n

  • j=1
  • max{djxj − ϑ, 0} + max{−djxj − ϑ, 0}
  • Γ

|djxj| Γ ϑ

Kai-Simon Goetzmann Robust Optimization over Integers

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SLIDE 16

Introduction Cost Robustness Applications

Proof (Sketch): min

x∈X

  • cTx + max

S⊆[n] |S|≤Γ

  • j∈S

|djxj|

  • ⇓ duality

min

x∈X ϑ≥0

  • cTx + Γϑ +

n

  • j=1
  • max{djxj − ϑ, 0} + max{−djxj − ϑ, 0}
  • = min

ϑ≥0

  • Γϑ+min

x∈X

  • n
  • j=1
  • cj(xj)
  • xj
  • cj(xj)

ϑ/dj

−ϑ/dj

Kai-Simon Goetzmann Robust Optimization over Integers

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SLIDE 17

Introduction Cost Robustness Applications

Finding the optimal ϑ min

x∈X ϑ≥0

  • cTx + Γϑ +

n

  • j=1
  • max{djxj − ϑ, 0} + max{−djxj − ϑ, 0}
  • ❩

ϑ

Kai-Simon Goetzmann Robust Optimization over Integers

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SLIDE 18

Introduction Cost Robustness Applications

Finding the optimal ϑ min

x∈X ϑ≥0

  • cTx+Γϑ+

n

  • j=1
  • max{djxj − ϑ, 0} + max{−djxj − ϑ, 0}
  • =0 for ϑ≥djuj
  • ϑ∗ ≤ maxj{djuj}

❩

maxj{djuj}

ϑ

Kai-Simon Goetzmann Robust Optimization over Integers

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SLIDE 19

Introduction Cost Robustness Applications

Finding the optimal ϑ min

x∈X ϑ≥0

  • cTx + Γϑ +

n

  • j=1
  • max{djxj
  • ∈❩

−ϑ, 0} + max{−djxj

∈❩

−ϑ, 0}

  • ϑ∗ ≤ maxj{djuj}

ϑ∗ ∈ ❩ ϑ

maxj{djuj}

Kai-Simon Goetzmann Robust Optimization over Integers

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SLIDE 20

Introduction Cost Robustness Applications

Finding the optimal ϑ min

x∈X ϑ≥0

  • cTx + Γϑ +

n

  • j=1
  • max{djxj − ϑ, 0} + max{−djxj − ϑ, 0}
  • ϑ∗ ≤ maxj{djuj}

ϑ∗ ∈ ❩ ϑ

maxj{djuj}

⇒ only maxj{djuj} + 1 possible values for ϑ∗, can enumerate in pseudopolynomial time.

  • Kai-Simon Goetzmann

Robust Optimization over Integers

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Introduction Cost Robustness Applications

Theorem (Extension 1) Let X ⊆ ❩n, c ∈ ◗n, d ∈ ◆n, Γ ∈ [n]. If there is a ρ-approximation algorithm for the (α, d)-MMin of P for all α ≥ 0 bounds uj on the absolute value of xj can be computed in polynomial time, then there is a pseudopolynomial ρ-approximation algorithm for the (d, Γ)-CRC of P = (c, X).

Kai-Simon Goetzmann Robust Optimization over Integers

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SLIDE 22

Introduction Cost Robustness Applications

Theorem (Extension 1) Let X ⊆ ❩n, c ∈ ◗n, d ∈ ◆n, Γ ∈ [n]. If there is a ρ-approximation algorithm for the (α, d)-MMin of P for all α ≥ 0 bounds uj on the absolute value of xj can be computed in polynomial time, then there is a pseudopolynomial ρ-approximation algorithm for the (d, Γ)-CRC of P = (c, X). If ρ = 1, and if the optimal values of the (α, d)-MMin of P are convex in α, then there is an exact polynomial time algorithm for the (d, Γ)-CRC of P.

Kai-Simon Goetzmann Robust Optimization over Integers

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SLIDE 23

Introduction Cost Robustness Applications

Theorem (Extension 1) Let X ⊆ ❩n, c ∈ ◗n, d ∈ ◆n, Γ ∈ [n]. If there is a ρ-approximation algorithm for the (α, d)-MMin of P for all α ≥ 0 bounds uj on the absolute value of xj can be computed in polynomial time, then there is a pseudopolynomial ρ-approximation algorithm for the (d, Γ)-CRC of P = (c, X). If ρ = 1, and if the optimal values of the (α, d)-MMin of P are convex in α, then there is an exact polynomial time algorithm for the (d, Γ)-CRC of P. Proof (Idea): Binary search for ϑ∗.

Kai-Simon Goetzmann Robust Optimization over Integers

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SLIDE 24

Introduction Cost Robustness Applications

Theorem (Extensions 2 and 3) Let X ⊆ ❩n, c ∈ ◗n, d ∈ ◆n, Γ ∈ [n]. If there is a ρ-approximation algorithm for the (α, d)-MMin of P for all α ≥ 0 bounds uj on the absolute value of xj can be computed in polynomial time, then there is a pseudopolynomial ρ-approximation algorithm for the (d, Γ)-CRC of P = (c, X). ◆

Kai-Simon Goetzmann Robust Optimization over Integers

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SLIDE 25

Introduction Cost Robustness Applications

Theorem (Extensions 2 and 3) Let X ⊆ ❩n, c ∈ ◗n, d ∈ ◆n, Γ ∈ [n]. If there is a ρ-approximation algorithm for the (α, d)-MMin of P for all α ≥ 0 bounds uj on the absolute value of xj can be computed in polynomial time, then there is a pseudopolynomial ρ-approximation algorithm for the (d, Γ)-CRC of P = (c, X). If X ⊆ ◆n and c ≥ 0, then for all ε > 0 there is a polynomial time ρ(1 + ε)-approximation algorithm for the (d, Γ)-CRC of P for minimization problems.

Kai-Simon Goetzmann Robust Optimization over Integers

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SLIDE 26

Introduction Cost Robustness Applications

Theorem (Extensions 2 and 3) Let X ⊆ ❩n, c ∈ ◗n, d ∈ ◆n, Γ ∈ [n]. If there is a ρ-approximation algorithm for the (α, d)-MMax of P for all α ≥ 0 bounds uj on the absolute value of xj can be computed in polynomial time, then there is a pseudopolynomial ρ-approximation algorithm for the (d, Γ)-CRC of P = (c, X). If X ⊆ ◆n and c ≥ 0, and dj

cj ≤ β < 1,

then there is a polynomial time 2ρ-approximation algorithm for the (d, Γ)-CRC of P for maximization problems.

Kai-Simon Goetzmann Robust Optimization over Integers

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SLIDE 27

Introduction Cost Robustness Applications

Proof (Idea): Try all ϑ ∈ {0, 1, 1 + ε, (1 + ε)2, (1 + ε)3, . . . , maxj(djuj)}.

ϑ 1 max djuj

Kai-Simon Goetzmann Robust Optimization over Integers

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SLIDE 28

Introduction Cost Robustness Applications

Proof (Idea): Try all ϑ ∈ {0, 1, 1 + ε, (1 + ε)2, (1 + ε)3, . . . , maxj(djuj)}.

ϑ 1 max djuj

⇒ polynomially many tests, missing ϑ∗ by a factor of ≤ 1 + ε.

Kai-Simon Goetzmann Robust Optimization over Integers

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SLIDE 29

Introduction Cost Robustness Applications

Proof (Idea): Try all ϑ ∈ {0, 1, 1 + ε, (1 + ε)2, (1 + ε)3, . . . , maxj(djuj)}.

ϑ 1 max djuj

⇒ polynomially many tests, missing ϑ∗ by a factor of ≤ 1 + ε. Minimization missing optimum by at most ρ(1 + ε).

Kai-Simon Goetzmann Robust Optimization over Integers

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SLIDE 30

Introduction Cost Robustness Applications

Proof (Idea): Try all ϑ ∈ {0, 1, 1 + ε, (1 + ε)2, (1 + ε)3, . . . , maxj(djuj)}.

ϑ 1 max djuj

⇒ polynomially many tests, missing ϑ∗ by a factor of ≤ 1 + ε. Minimization missing optimum by at most ρ(1 + ε). Maximization with dj

cj ≤ β, setting ε := 1−β 2β

missing optimum by at most 2ρ.

  • Kai-Simon Goetzmann

Robust Optimization over Integers

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SLIDE 31

Introduction Cost Robustness Applications

Theorem (Extension 2) Let X ⊆ ❩n, c ∈ ◗n, d ∈ ◆n, Γ ∈ [n]. If there is a ρ-approximation algorithm for the (α, d)-MMin of P for all α ≥ 0 bounds uj on the absolute value of xj can be computed in polynomial time, then there is a pseudopolynomial ρ-approximation algorithm for the (d, Γ)-CRC of P = (c, X). If X ⊆ ◆n and c ≥ 0, then for all ε > 0 there is a polynomial time ρ(1 + ε)-approximation algorithm for the (d, Γ)-CRC of P for minimization problems.

Kai-Simon Goetzmann Robust Optimization over Integers

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SLIDE 32

Introduction Cost Robustness Applications

1

Introduction

2

General Result for Cost Robust IPs

3

Applications of the General Result

Kai-Simon Goetzmann Robust Optimization over Integers

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SLIDE 33

Introduction Cost Robustness Applications

Totally Unimodular Problems

If the minimization problem P = (c, X) is given by a bounded totally unimodular description, i.e. conv(X) ={x ∈ ❘n : Ax ≤ b, x ≤ u}, A TUM, b ∈ ❩m, u ∈ ❩n,

Kai-Simon Goetzmann Robust Optimization over Integers

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SLIDE 34

Introduction Cost Robustness Applications

Totally Unimodular Problems

If the minimization problem P = (c, X) is given by a bounded totally unimodular description, i.e. conv(X) ={x ∈ ❘n : Ax ≤ b, x ≤ u}, A TUM, b ∈ ❩m, u ∈ ❩n, then the (α, c′)-MMin of P has a bounded TUM description, the optimal values of the (α, c′)-MMin of P are convex in α. ⇒ The (d, Γ)-CRC of P is efficiently solvable.

Kai-Simon Goetzmann Robust Optimization over Integers

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SLIDE 35

Introduction Cost Robustness Applications

IPs with two variables per inequality

Bounded IP2: min

x∈❩n{cTx :

aT

i x ≤ bi for i = 1, . . . , m,

ℓ ≤ x ≤ u} with b ∈ ❩m, ℓ, u ∈ ❩n, c ∈ ◗n, ai ∈ ◗n with only two non-zero entries.

Kai-Simon Goetzmann Robust Optimization over Integers

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SLIDE 36

Introduction Cost Robustness Applications

IPs with two variables per inequality

Bounded IP2: min

x∈❩n{cTx :

aT

i x ≤ bi for i = 1, . . . , m,

ℓ ≤ x ≤ u} with b ∈ ❩m, ℓ, u ∈ ❩n, c ∈ ◗n, ai ∈ ◗n with only two non-zero entries. Pseudopolynomial exact algorithm for MMin of bounded monotone IP2s ⇒ Pseudopolynomial exact algorithm for the CRC.

Kai-Simon Goetzmann Robust Optimization over Integers

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SLIDE 37

Introduction Cost Robustness Applications

IPs with two variables per inequality

Bounded IP2: min

x∈❩n{cTx :

aT

i x ≤ bi for i = 1, . . . , m,

ℓ ≤ x ≤ u} with b ∈ ❩m, ℓ, u ∈ ❩n, c ∈ ◗n, ai ∈ ◗n with only two non-zero entries. Pseudopolynomial exact algorithm for MMin of bounded monotone IP2s ⇒ Pseudopolynomial exact algorithm for the CRC. Pseudopolynomial 2-approximation for MMin of bounded IP2s ⇒ Pseudopolynomial 2-approximation for the CRC.

Kai-Simon Goetzmann Robust Optimization over Integers

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SLIDE 38

Introduction Cost Robustness Applications

Unbounded Knapsack Problem (UKP)

UKP = Knapsack where many copies of each item can be packed.

Kai-Simon Goetzmann Robust Optimization over Integers

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SLIDE 39

Introduction Cost Robustness Applications

Unbounded Knapsack Problem (UKP)

UKP = Knapsack where many copies of each item can be packed. Nonlinear UKP (concave cost functions, convex weight functions) has an FPTAS.

W wj is an upper bound on the number of items of type j.

Kai-Simon Goetzmann Robust Optimization over Integers

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SLIDE 40

Introduction Cost Robustness Applications

Unbounded Knapsack Problem (UKP)

UKP = Knapsack where many copies of each item can be packed. Nonlinear UKP (concave cost functions, convex weight functions) has an FPTAS.

W wj is an upper bound on the number of items of type j.

⇒ For any ε > 0, there is a (2 + ε)-approximation algorithm for the (d, Γ)-CRC of UKP, if dj

cj ≤ β < 1.

Kai-Simon Goetzmann Robust Optimization over Integers

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SLIDE 41

Introduction Cost Robustness Applications

Conclusion

Preconditions CRC of P X ⊆ {0, 1}n, ρ-approx (B&S) ρ-approx for P

Kai-Simon Goetzmann Robust Optimization over Integers

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SLIDE 42

Introduction Cost Robustness Applications

Conclusion

Preconditions CRC of P X ⊆ {0, 1}n, ρ-approx (B&S) ρ-approx for P ρ-approx for MMin/MMax, pseudopoly |xj| ≤ uj ρ-approx

Kai-Simon Goetzmann Robust Optimization over Integers

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SLIDE 43

Introduction Cost Robustness Applications

Conclusion

Preconditions CRC of P X ⊆ {0, 1}n, ρ-approx (B&S) ρ-approx for P ρ-approx for MMin/MMax, pseudopoly |xj| ≤ uj ρ-approx x, c ≥ 0, |xj| ≤ uj ρ(1 + ε)-approx ρ-approx for MMin

Kai-Simon Goetzmann Robust Optimization over Integers

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SLIDE 44

Introduction Cost Robustness Applications

Conclusion

Preconditions CRC of P X ⊆ {0, 1}n, ρ-approx (B&S) ρ-approx for P ρ-approx for MMin/MMax, pseudopoly |xj| ≤ uj ρ-approx x, c ≥ 0, |xj| ≤ uj ρ(1 + ε)-approx ρ-approx for MMin x, c ≥ 0, |xj| ≤ uj 2ρ-approx ρ-approx for MMax,

dj cj ≤ β < 1

Kai-Simon Goetzmann Robust Optimization over Integers

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SLIDE 45

Introduction Cost Robustness Applications

Conclusion

Preconditions CRC of P X ⊆ {0, 1}n, ρ-approx (B&S) ρ-approx for P ρ-approx for MMin/MMax, pseudopoly |xj| ≤ uj ρ-approx x, c ≥ 0, |xj| ≤ uj ρ(1 + ε)-approx ρ-approx for MMin x, c ≥ 0, |xj| ≤ uj 2ρ-approx ρ-approx for MMax,

dj cj ≤ β < 1

ρ = 1, |xj| ≤ uj exact alg. C∗(ϑ) convex

Kai-Simon Goetzmann Robust Optimization over Integers

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SLIDE 46

Introduction Cost Robustness Applications

Conclusion

Preconditions CRC of P ConsRC of P X ⊆ {0, 1}n, ρ-approx (B&S) ρ-approx ρ-approx for P ρ-approx for MMin/MMax, pseudopoly |xj| ≤ uj ρ-approx x, c ≥ 0, |xj| ≤ uj ρ(1 + ε)-approx ρ-approx for MMin x, c ≥ 0, |xj| ≤ uj 2ρ-approx ρ-approx for MMax,

dj cj ≤ β < 1

ρ = 1, |xj| ≤ uj exact alg. C∗(ϑ) convex

Kai-Simon Goetzmann Robust Optimization over Integers

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SLIDE 47

Introduction Cost Robustness Applications

Conclusion

Preconditions CRC of P ConsRC of P X ⊆ {0, 1}n, ρ-approx (B&S) ρ-approx ρ-approx for P ρ-approx for MMin/MMax, pseudopoly pseudopoly |xj| ≤ uj ρ-approx ρ-approx x, c ≥ 0, |xj| ≤ uj ρ(1 + ε)-approx

  • ρ-approx for MMin

x, c ≥ 0, |xj| ≤ uj 2ρ-approx

  • ρ-approx for MMax,

dj cj ≤ β < 1

ρ = 1, |xj| ≤ uj exact alg.

  • C∗(ϑ) convex

Kai-Simon Goetzmann Robust Optimization over Integers

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SLIDE 48

Introduction Cost Robustness Applications

Conclusion (2)

Cost robust problems with bounded TUM description: polynomial exact algorithm

Kai-Simon Goetzmann Robust Optimization over Integers

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SLIDE 49

Introduction Cost Robustness Applications

Conclusion (2)

Cost robust problems with bounded TUM description: polynomial exact algorithm Cost robust bounded IPs with two variables per inequality: monotone: pseudopolynomial exact algorithm general: pseudopolynomial 2-approximation

Kai-Simon Goetzmann Robust Optimization over Integers

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SLIDE 50

Introduction Cost Robustness Applications

Conclusion (2)

Cost robust problems with bounded TUM description: polynomial exact algorithm Cost robust bounded IPs with two variables per inequality: monotone: pseudopolynomial exact algorithm general: pseudopolynomial 2-approximation Unbounded Knapsack Problem: cost robust: polynomial (2 + ε)-approximation, exact in O(maxj(dj)n2W 2) weight robust: exact in O(maxj(∆wj)n2W 2) cost & weight robust: exact in O(maxj(dj) max(∆wj)n2W 3)

Kai-Simon Goetzmann Robust Optimization over Integers

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SLIDE 51

Introduction Cost Robustness Applications

Conclusion (2)

Cost robust problems with bounded TUM description: polynomial exact algorithm Cost robust bounded IPs with two variables per inequality: monotone: pseudopolynomial exact algorithm general: pseudopolynomial 2-approximation Unbounded Knapsack Problem: cost robust: polynomial (2 + ε)-approximation, exact in O(maxj(dj)n2W 2) weight robust: exact in O(maxj(∆wj)n2W 2) cost & weight robust: exact in O(maxj(dj) max(∆wj)n2W 3) Thank you for your attention.

Kai-Simon Goetzmann Robust Optimization over Integers

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