Introduction
Definitions and Notations
Reference Point Methods
and Approximation in Multicriteria Optimization
C. Büsing, Kai-Simon Goetzmann, J. Matuschke and S. Stiller
OR Hannover, September 05, 2012
Kai-Simon Goetzmann
Reference Point Methods
Approximation
Introduction
Definitions and Notations
Multicriteria Optimization
Kai-Simon Goetzmann
Reference Point Methods
Approximation
Introduction
Definitions and Notations
Multicriteria Optimization
Kai-Simon Goetzmann
Reference Point Methods
Approximation
Introduction
Definitions and Notations
Multicriteria Optimization
Kai-Simon Goetzmann
Reference Point Methods
Approximation
Introduction
Definitions and Notations
Multicriteria Optimization
Kai-Simon Goetzmann
Reference Point Methods
Approximation
Introduction
Definitions and Notations
Multicriteria Optimization
min{y ∶ y ∈ Y}
Kai-Simon Goetzmann
where Y ⊆ Zk
Reference Point Methods
Approximation
Introduction
Definitions and Notations
Approximation
Pareto Optimality
Definition
A solution y ∈ Y of miny∈Y y is Pareto optimal if there is no
y ′ ∈ Y ∖ {y} with y ′ ≤ y.
y2
Y
y1
Kai-Simon Goetzmann
Reference Point Methods
Introduction
Definitions and Notations
Approximation
Pareto Optimality
Definition
A solution y ∈ Y of miny∈Y y is Pareto optimal if there is no
y ′ ∈ Y ∖ {y} with y ′ ≤ y.
y2
y1
Kai-Simon Goetzmann
Reference Point Methods
Introduction
Definitions and Notations
Approximation
Pareto Optimality
Definition
A solution y ∈ Y of miny∈Y y is Pareto optimal if there is no
y ′ ∈ Y ∖ {y} with y ′ ≤ y.
y2
YP
y1
Kai-Simon Goetzmann
Reference Point Methods
Introduction
Definitions and Notations
Reference Point Solutions
Motivation:
▸ identify a single, Pareto optimal, balanced solution
▸ reference point methods:
part of many state-of-the-art MCDM tools,
little theoretical background
▸ A powerful concept:
all Pareto optimal solutions can be RPS,
approximation of RPS yields approximate Pareto set
Kai-Simon Goetzmann
Reference Point Methods
Approximation
Introduction
Definitions and Notations
1
Introduction
2
Definitions and Notations
3
Approximation
Kai-Simon Goetzmann
Reference Point Methods
Approximation
Introduction
Definitions and Notations
1
Introduction
2
Definitions and Notations
3
Approximation
Kai-Simon Goetzmann
Reference Point Methods
Approximation
Introduction
Definitions and Notations
Approximation
Definition (Ideal Point)
Given a multicriteria optimization problem miny∈Y y,
the ideal point y id = (y1id , . . . , ykid ) is defined by
yiid = min yi
y∈Y
∀ i.
y2
Y
feasible
reference
points
y id
y1
Kai-Simon Goetzmann
Reference Point Methods
Introduction
Definitions and Notations
Approximation
Definition (Compromise Solution, Yu 1973)
Given a multicriteria optimization problem miny∈Y y with the ideal
point y id ∈ Qk , and a norm ∥⋅∥ on Rk , the compromise solution
w.r.t. ∥⋅∥ is
y cs = min ∥y − y id ∥ .
y∈Y
y2
y id
y1
Kai-Simon Goetzmann
Reference Point Methods
Introduction
Definitions and Notations
Approximation
Definition (Compromise Solution, Yu 1973)
Given a multicriteria optimization problem miny∈Y y with the ideal
point y id ∈ Qk , and a norm ∥⋅∥ on Rk , the compromise solution
w.r.t. ∥⋅∥ is
y cs = min ∥y − y id ∥ .
y∈Y
y2
feasible
reference
points
y id
y1
Kai-Simon Goetzmann
Reference Point Methods
Introduction
Definitions and Notations
Approximation
Definition (Reference Point Solution)
Given a multicriteria optimization problem miny∈Y y, a feasible
reference point y rp ∈ Qk , and a norm ∥⋅∥ on Rk , the reference
point solution w.r.t. ∥⋅∥ is
y rps = min ∥y − y rp ∥ .
y∈Y
y2
y rp
y1
Kai-Simon Goetzmann
Reference Point Methods
Introduction
Definitions and Notations
Approximation
Definition (Reference Point Solution)
Given a multicriteria optimization problem miny∈Y y, a feasible
reference point y rp ∈ Qk , and a norm ∥⋅∥ on Rk , the reference
point solution w.r.t. ∥⋅∥ is
y rps = min ∥y − y rp ∥ .
y∈Y
y2
y rp
y1
Kai-Simon Goetzmann
Reference Point Methods
Introduction
Definitions and Notations
Approximation
The norms we consider:
k
∥y∥p ∶=
1/p
(∑ yip )
i=1
,
(`p -Norm)
p ∈ [1, ∞)
(Maximum (`∞ -)Norm)
∥y∥∞ ∶= max yi
i=1,...,k
⟨ y⟩⟩p ∶= ∥y∥∞ +
1
∥y∥1 ,
p
p ∈ [1, ∞]
1
1
1
`p -Norm
(Cornered p-Norm)
1
Cornered p-Norm
Degree of balancing controlled by adjusting p.
Kai-Simon Goetzmann
Reference Point Methods
p=1
p=2
p=5
p=∞
Introduction
Definitions and Notations
Approximation
The norms we consider:
1/p
k
∥y∥p ∶=
(∑ yip )
i=1
,
(`p -Norm)
p ∈ [1, ∞)
(Maximum (`∞ -)Norm)
∥y∥∞ ∶= max yi
i=1,...,k
⟨ y⟩⟩p ∶= ∥y∥∞ +
1
∥y∥1 ,
p
p ∈ [1, ∞]
(Cornered p-Norm)
Weighted version: For any norm and λ ∈ Qk , λ ≥ 0, λ ≠ 0 ∶
∥y∥λ = ∥(λ1 y1 , λ2 y2 , . . . , λk yk )∥ .
General properties: Norms we consider are
▸ monotone (if y ≥ y ′ then ∥y∥ ≥ ∥y ′ ∥)
▸ polynomially decidable (∥y∥ ≥ ∥y ′ ∥ can be decided in
polynomial time)
Kai-Simon Goetzmann
Reference Point Methods
Introduction
Definitions and Notations
1
Introduction
2
Definitions and Notations
3
Approximation
Kai-Simon Goetzmann
Reference Point Methods
Approximation
Introduction
Definitions and Notations
Approximation
Approximate Pareto sets
Definition (α-approximate Pareto set)
Let YP be the Pareto set of miny∈Y y, and let α > 1.
Yα ⊆ Y is an α-approximate Pareto set if for all y ∈ YP there is
y ′ ∈ Yα such that
yi′ ≤ αyi ∀i = 1, . . . , k
Kai-Simon Goetzmann
Reference Point Methods
Introduction
Definitions and Notations
How to find approximate Pareto sets
Theorem (Papadimitriou&Yannakakis,2000)
Gap(y, α) tractable for all y ∈ Qk
⇒ α2 -approximation for the Pareto set.
α-approximation for the Pareto set
⇒ Gap(y, α) tractable for all y ∈ Qk .
Kai-Simon Goetzmann
Reference Point Methods
Approximation
Introduction
Definitions and Notations
How to find approximate Pareto sets
Theorem (Papadimitriou&Yannakakis,2000)
Gap(y, α) tractable for all y ∈ Qk
⇒ α2 -approximation for the Pareto set.
α-approximation for the Pareto set
⇒ Gap(y, α) tractable for all y ∈ Qk .
Gap(y, α): Given y ∈ Qk and α > 1.
y2
y
y1
Kai-Simon Goetzmann
Reference Point Methods
Approximation
Introduction
Definitions and Notations
How to find approximate Pareto sets
Theorem (Papadimitriou&Yannakakis,2000)
Gap(y, α) tractable for all y ∈ Qk
⇒ α2 -approximation for the Pareto set.
α-approximation for the Pareto set
⇒ Gap(y, α) tractable for all y ∈ Qk .
Gap(y, α): Given y ∈ Qk and α > 1.
y2
y
y′
y1
Kai-Simon Goetzmann
Reference Point Methods
Approximation
Introduction
Definitions and Notations
Approximation
How to find approximate Pareto sets
Theorem (Papadimitriou&Yannakakis,2000)
Gap(y, α) tractable for all y ∈ Qk
⇒ α2 -approximation for the Pareto set.
α-approximation for the Pareto set
⇒ Gap(y, α) tractable for all y ∈ Qk .
Gap(y, α): Given y ∈ Qk and α > 1.
y2
y2
y
y
no sol’n
y′
y1
y1
Kai-Simon Goetzmann
1
y
α
Reference Point Methods
Introduction
Definitions and Notations
Approximate Pareto sets ⇔ approximate RPS
Kai-Simon Goetzmann
Reference Point Methods
Approximation
Introduction
Definitions and Notations
Approximate Pareto sets ⇔ approximate RPS
Relate objective value to size of the vectors:
min ∥y − y rp ∥ + ∥y rp ∥
y∈Y ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¶
r(y)
We call r(y) the relative distance.
Kai-Simon Goetzmann
Reference Point Methods
Approximation
Introduction
Definitions and Notations
Approximate Pareto sets ⇔ approximate RPS
Relate objective value to size of the vectors:
min ∥y − y rp ∥ + ∥y rp ∥
y∈Y ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¶
r(y)
We call r(y) the relative distance.
Theorem
α-approximation of the Pareto set
⇒ α-approximation for miny∈Y r(y).
Kai-Simon Goetzmann
Reference Point Methods
Approximation
Introduction
Definitions and Notations
Approximate Pareto sets ⇔ approximate RPS
Relate objective value to size of the vectors:
min ∥y − y rp ∥ + ∥y rp ∥
y∈Y ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¶
r(y)
We call r(y) the relative distance.
Theorem
α-approximation of the Pareto set
⇒ α-approximation for miny∈Y r(y).
Theorem
α-approximation for miny∈Y r(y)
⇒ Gap(y, β) tractable for all y ∈ Qk , β ∈ Θ(α).
⇒ β 2 -approximation for the Pareto set.
Kai-Simon Goetzmann
Reference Point Methods
Approximation
Introduction
Definitions and Notations
Approximation
Equivalences of approximability
RPS(∥⋅∥),
∥⋅∥ monotone
& poly decidable
Pareto set
RPS(∥⋅∥p ) and
RPS(⟨⟨⋅⟩⟩p ), p ≥ 1
P&Y
Gap(y, α)
∀y∈ k
Kai-Simon Goetzmann
Q
Reference Point Methods
RPS(∥⋅∥∞ )
Introduction
Definitions and Notations
Approximation
Equivalences of approximability
RPS(∥⋅∥),
∥⋅∥ monotone
& poly decidable
Pareto set
RPS(∥⋅∥p ) and
RPS(⟨⟨⋅⟩⟩p ), p ≥ 1
P&Y
Gap(y, α)
∀y∈ k
Kai-Simon Goetzmann
Q
Reference Point Methods
RPS(∥⋅∥∞ )
Introduction
Definitions and Notations
Approximation
Equivalences of approximability
RPS(∥⋅∥),
∥⋅∥ monotone
& poly decidable
Pareto set
CS(∥⋅∥),
∥⋅∥ monotone
& poly decidable
RPS(∥⋅∥p ) and
RPS(⟨⟨⋅⟩⟩p ), p ≥ 1
P&Y
RPS(∥⋅∥∞ )
Gap(y, α)
∀y∈ k
Q
CS(∥⋅∥p ) and
CS(⟨⟨⋅⟩⟩p ), p ≥ 1
Kai-Simon Goetzmann
Reference Point Methods
CS(∥⋅∥∞ )
Introduction
Definitions and Notations
Approximation
Equivalences of approximability
RPS(∥⋅∥),
∥⋅∥ monotone
& poly decidable
RPS(∥⋅∥p ) and
RPS(⟨⟨⋅⟩⟩p ), p ≥ 1
RPS(∥⋅∥∞ )
add. factor k
Pareto set
P&Y
Gap(y, α)
∀y∈ k
Q
bin. search
miny λT y
(weighted sum)
add. factor k
CS(∥⋅∥),
∥⋅∥ monotone
& poly decidable
CS(∥⋅∥p ) and
CS(⟨⟨⋅⟩⟩p ), p ≥ 1
Kai-Simon Goetzmann
Reference Point Methods
CS(∥⋅∥∞ )
Introduction
Definitions and Notations
Approximation through LP-rounding
Extend approximation algorithms based on LP-rounding
(e.g. Set Cover, Scheduling) to compromise programming
Kai-Simon Goetzmann
Reference Point Methods
Approximation
Introduction
Definitions and Notations
Approximation through LP-rounding
Extend approximation algorithms based on LP-rounding
(e.g. Set Cover, Scheduling) to compromise programming
▸ LP-relaxation:
min cT x
s.t.
Ax ≤ b
▸ rounding procedure R ∶ Qn≥0 → Nn with
cT R(x) ≤ αcT x
Kai-Simon Goetzmann
Reference Point Methods
Approximation
Introduction
Definitions and Notations
Approximation
Approximation through LP-rounding
Extend approximation algorithms based on LP-rounding
(e.g. Set Cover, Scheduling) to compromise programming
▸ LP-relaxation:
min cT x
s.t.
min ∆ +
Ax ≤ b
s.t.
1
p
⋅ 1T Cx + ∥y rp ∥∞
Ax ≤ b
Cx − y rp ≤ ∆ ⋅ 1
∆≥0
▸ rounding procedure R ∶ Qn≥0 → Nn with
cT R(x) ≤ αcT x
Kai-Simon Goetzmann
Reference Point Methods
Introduction
Definitions and Notations
Approximation
Approximation through LP-rounding
Extend approximation algorithms based on LP-rounding
(e.g. Set Cover, Scheduling) to compromise programming
▸ LP-relaxation:
min cT x
s.t.
min ∆ +
Ax ≤ b
s.t.
1
p
⋅ 1T Cx + ∥y rp ∥∞
Ax ≤ b
Cx − y rp ≤ ∆ ⋅ 1
∆≥0
▸ rounding procedure R ∶ Qn≥0 → Nn with
cT R(x) ≤ αcT x
⇒
r(CR(x)) ≤ α ⋅ r(Cx)
Kai-Simon Goetzmann
Reference Point Methods
Introduction
Definitions and Notations
Approximation
Approximation through dynamic programming
Extend result from min-max-regret robustness (Aissi et al. 2006)
(e.g. Shortest Path, MST) to compromise programming
Kai-Simon Goetzmann
Reference Point Methods
Introduction
Definitions and Notations
Approximation
Approximation through dynamic programming
Extend result from min-max-regret robustness (Aissi et al. 2006)
(e.g. Shortest Path, MST) to compromise programming
▸ exact solution by dynamic programming
in pseudopolynomial time
▸ upper and lower bounds on optimal value
▸ scale and round the instance
to achieve polynomial running time
▸ x∗ , x∗ compromise solutions for original and rounded instance.
∥Cx∗ − y id ∥∞ ≤ (1 + ε) ⋅ ∥Cx∗ − y id ∥∞
Kai-Simon Goetzmann
Reference Point Methods
Introduction
Definitions and Notations
Approximation
Approximation through dynamic programming
Extend result from min-max-regret robustness (Aissi et al. 2006)
(e.g. Shortest Path, MST) to compromise programming
▸ exact solution by dynamic programming
in pseudopolynomial time
▸ upper and lower bounds on optimal value
▸ scale and round the instance
to achieve polynomial running time
▸ x∗ , x∗ compromise solutions for original and rounded instance.
∥Cx∗ − y id ∥∞ ≤ (1 + ε) ⋅ ∥Cx∗ − y id ∥∞
⟨ Cx∗ − y rp ⟩ p + ⟨ y rp ⟩ p ≤ (1 + ε) ⋅ (⟨⟨Cx∗ − y rp ⟩ p + ⟨ y rp ⟩ p )
Kai-Simon Goetzmann
Reference Point Methods
Introduction
Definitions and Notations
Approximation
Summary
RPS(∥⋅∥),
∥⋅∥ monotone
& poly decidable
RPS(∥⋅∥p ) and
RPS(⟨⟨⋅⟩⟩p ), p ≥ 1
RPS(∥⋅∥∞ )
Gap(y, α)
∀y∈ k
miny λT y
(weighted sum)
add. factor k
Pareto set
P&Y
Q
bin. search
add. factor k
CS(∥⋅∥),
∥⋅∥ monotone
& poly decidable
CS(∥⋅∥p ) and
CS(⟨⟨⋅⟩⟩p ), p ≥ 1
CS(∥⋅∥∞ )
Approximation Algorithms:
▸ Approximations based on LP-relaxations
▸ FPTAS based on dynamic programming
Kai-Simon Goetzmann
Reference Point Methods
Introduction
Definitions and Notations
Approximation
Summary
RPS(∥⋅∥),
∥⋅∥ monotone
& poly decidable
RPS(∥⋅∥p ) and
RPS(⟨⟨⋅⟩⟩p ), p ≥ 1
RPS(∥⋅∥∞ )
Gap(y, α)
∀y∈ k
miny λT y
(weighted sum)
add. factor k
Pareto set
P&Y
Q
bin. search
add. factor k
CS(∥⋅∥),
∥⋅∥ monotone
& poly decidable
CS(∥⋅∥p ) and
CS(⟨⟨⋅⟩⟩p ), p ≥ 1
CS(∥⋅∥∞ )
Approximation Algorithms:
▸ Approximations based on LP-relaxations
▸ FPTAS based on dynamic programming
Thank you for your attention.
Kai-Simon Goetzmann
Reference Point Methods