Compromise Solutions Kai-Simon Goetzmann, TU Berlin (Joint work - - PowerPoint PPT Presentation

▶

Jan 13, 2023 20 likes •386 views

Introduction Definitions and Notations Approximation Compromise Solutions Kai-Simon Goetzmann, TU Berlin (Joint work with Christina B using, Jannik Matuschke and Sebastian Stiller) SCOR 2012 Kai-Simon Goetzmann, TU Berlin Compromise

SLIDE 1

Introduction Definitions and Notations Approximation

Compromise Solutions

Kai-Simon Goetzmann, TU Berlin

(Joint work with Christina B¨ using, Jannik Matuschke and Sebastian Stiller)

SCOR 2012

Kai-Simon Goetzmann, TU Berlin Compromise Solutions

SLIDE 2

Introduction Definitions and Notations Approximation

Multicriteria Optimization

Kai-Simon Goetzmann, TU Berlin Compromise Solutions

SLIDE 3

Introduction Definitions and Notations Approximation

Multicriteria Optimization

Kai-Simon Goetzmann, TU Berlin Compromise Solutions

SLIDE 4

Introduction Definitions and Notations Approximation

Multicriteria Optimization

Kai-Simon Goetzmann, TU Berlin Compromise Solutions

SLIDE 5

Introduction Definitions and Notations Approximation

Multicriteria Optimization

Kai-Simon Goetzmann, TU Berlin Compromise Solutions

SLIDE 6

Introduction Definitions and Notations Approximation

Multicriteria Optimization

min{y ∶ y ∈ Y} where Y ⊆ ❩k

Kai-Simon Goetzmann, TU Berlin Compromise Solutions

SLIDE 7

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. Y y1 y2

Kai-Simon Goetzmann, TU Berlin Compromise Solutions

SLIDE 8

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. y1 y2

Kai-Simon Goetzmann, TU Berlin Compromise Solutions

SLIDE 9

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. YP y1 y2

Kai-Simon Goetzmann, TU Berlin Compromise Solutions

SLIDE 10

Introduction Definitions and Notations Approximation

Compromise Solutions

Motivation: ▸ identify a single, Pareto optimal, balanced solution ▸ reference point methods: part of many state-of-the-art MCDM tools, little theoretical background

Kai-Simon Goetzmann, TU Berlin Compromise Solutions

SLIDE 11

Introduction Definitions and Notations Approximation

1 Introduction 2 Definitions and Notations 3 Approximation

Kai-Simon Goetzmann, TU Berlin Compromise Solutions

SLIDE 12

Introduction Definitions and Notations Approximation

1 Introduction 2 Definitions and Notations 3 Approximation

Kai-Simon Goetzmann, TU Berlin Compromise Solutions

SLIDE 13

Introduction Definitions and Notations Approximation

Definition (Ideal Point) Given a multicriteria optimization problem miny∈Y y, the ideal point y∗ = (y∗

1,...,y∗ k) is defined by

y∗

i = min y∈Y yi

∀ i. y1 y2 Y y∗

Kai-Simon Goetzmann, TU Berlin Compromise Solutions

SLIDE 14

Introduction Definitions and Notations Approximation

Definition (Compromise Solution, Yu 1973) Given a multicriteria optimization problem miny∈Y y with the ideal point y∗ ∈ ◗k, the compromise solution w.r.t. the norm ∥⋅∥ on ❘k is ycs = min

y∈Y ∥y − y∗∥.

y1 y2 y∗

Kai-Simon Goetzmann, TU Berlin Compromise Solutions

SLIDE 15

Introduction Definitions and Notations Approximation

Definition (Compromise Solution, Yu 1973) Given a multicriteria optimization problem miny∈Y y with the ideal point y∗ ∈ ◗k, the compromise solution w.r.t. the norm ∥⋅∥ on ❘k is ycs = min

y∈Y ∥y − y∗∥.

y1 y2 y∗

Kai-Simon Goetzmann, TU Berlin Compromise Solutions

SLIDE 16

Introduction Definitions and Notations Approximation

The norms we consider: ∥y∥p ∶= (

∑

i=1

yp

i )

1/p

, p ∈ [1,∞) (ℓp-Norm) ∥y∥∞ ∶= max

i=1,...,k yi

(Maximum (ℓ∞-)Norm) ⟨⟨y⟩⟩p ∶= ∥y∥∞ + 1 p ∥y∥1 , p ∈ [1,∞] (Cornered p-Norm) ℓp-Norm

1 1

Cornered p-Norm

1 1

p = 1 p = 2 p = 5 p = ∞ Degree of balancing controlled by adjusting p.

Kai-Simon Goetzmann, TU Berlin Compromise Solutions

SLIDE 17

Introduction Definitions and Notations Approximation

The norms we consider: ∥y∥p ∶= (

∑

i=1

yp

i )

1/p

, p ∈ [1,∞) (ℓp-Norm) ∥y∥∞ ∶= max

i=1,...,k yi

(Maximum (ℓ∞-)Norm) ⟨⟨y⟩⟩p ∶= ∥y∥∞ + 1 p ∥y∥1 , p ∈ [1,∞] (Cornered p-Norm) Weighted version: For any norm and λ ∈ ◗k,λ ≥ 0,λ ≠ 0 ∶ ∥y∥λ = ∥(λ1y1,λ2y2,...,λkyk)∥.

Kai-Simon Goetzmann, TU Berlin Compromise Solutions

SLIDE 18

Introduction Definitions and Notations Approximation

Known Properties

Gearhardt 1979: ▸ for p < ∞ all compromise solutions are Pareto optimal ▸ all Pareto optimal solution are a compromise solution, for p big enough y1 y2 YP

Kai-Simon Goetzmann, TU Berlin Compromise Solutions

SLIDE 19

Introduction Definitions and Notations Approximation

1 Introduction 2 Definitions and Notations 3 Approximation

Kai-Simon Goetzmann, TU Berlin Compromise Solutions

SLIDE 20

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 y′

i ≤ αyi

∀i = 1,...,k

Kai-Simon Goetzmann, TU Berlin Compromise Solutions

SLIDE 21

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 y′

i ≤ αyi

∀i = 1,...,k

Kai-Simon Goetzmann, TU Berlin Compromise Solutions

SLIDE 22

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 y′

i ≤ αyi

∀i = 1,...,k

Kai-Simon Goetzmann, TU Berlin Compromise Solutions

SLIDE 23

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 y′

i ≤ αyi

∀i = 1,...,k

Kai-Simon Goetzmann, TU Berlin Compromise Solutions

SLIDE 24

Introduction Definitions and Notations Approximation

How to find approximate Pareto sets

Theorem (Papadimitriou&Yannakakis,2000) Gap(y,α) tractable for all y ∈ ◗k ⇒ α2-approximation for the Pareto set. α-approximation for the Pareto set ⇒ Gap(y,α) tractable for all y ∈ ◗k. Gap(y,α): Given y ∈ ◗k and α ≥ 1.

Kai-Simon Goetzmann, TU Berlin Compromise Solutions

SLIDE 25

Introduction Definitions and Notations Approximation

How to find approximate Pareto sets

Theorem (Papadimitriou&Yannakakis,2000) Gap(y,α) tractable for all y ∈ ◗k ⇒ α2-approximation for the Pareto set. α-approximation for the Pareto set ⇒ Gap(y,α) tractable for all y ∈ ◗k. Gap(y,α): Given y ∈ ◗k and α ≥ 1.

y1 y2 y

Kai-Simon Goetzmann, TU Berlin Compromise Solutions

SLIDE 26

Introduction Definitions and Notations Approximation

How to find approximate Pareto sets

Theorem (Papadimitriou&Yannakakis,2000) Gap(y,α) tractable for all y ∈ ◗k ⇒ α2-approximation for the Pareto set. α-approximation for the Pareto set ⇒ Gap(y,α) tractable for all y ∈ ◗k. Gap(y,α): Given y ∈ ◗k and α ≥ 1.

y′ y1 y2 y

Kai-Simon Goetzmann, TU Berlin Compromise Solutions

SLIDE 27

Introduction Definitions and Notations Approximation

How to find approximate Pareto sets

Theorem (Papadimitriou&Yannakakis,2000) Gap(y,α) tractable for all y ∈ ◗k ⇒ α2-approximation for the Pareto set. α-approximation for the Pareto set ⇒ Gap(y,α) tractable for all y ∈ ◗k. Gap(y,α): Given y ∈ ◗k and α ≥ 1.

y′ y1 y2 y

1 α y

no sol’n y y1 y2

Kai-Simon Goetzmann, TU Berlin Compromise Solutions

SLIDE 28

Introduction Definitions and Notations Approximation

Approximate Pareto sets ⇔ approximate CS

◗

Kai-Simon Goetzmann, TU Berlin Compromise Solutions

SLIDE 29

Introduction Definitions and Notations Approximation

Approximate Pareto sets ⇔ approximate CS

Relate objective value to size of the vectors: min

y∈Y ∥y − y∗∥ + ∥y∗∥

ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ

r(y)

We call r(y) the relative distance. ◗

Kai-Simon Goetzmann, TU Berlin Compromise Solutions

SLIDE 30

Introduction Definitions and Notations Approximation

Approximate Pareto sets ⇔ approximate CS

Relate objective value to size of the vectors: min

y∈Y ∥y − y∗∥ + ∥y∗∥

ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ

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, TU Berlin Compromise Solutions

SLIDE 31

Introduction Definitions and Notations Approximation

Approximate Pareto sets ⇔ approximate CS

Relate objective value to size of the vectors: min

y∈Y ∥y − y∗∥ + ∥y∗∥

ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ

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 ∈ ◗k, β ∈ Θ(α). ⇒ β2-approximation for the Pareto set.

Kai-Simon Goetzmann, TU Berlin Compromise Solutions

SLIDE 32

Introduction Definitions and Notations Approximation

Summary

approximate Pareto set ⇓ ⇑ approximate compromise solutions

Kai-Simon Goetzmann, TU Berlin Compromise Solutions

SLIDE 33

Introduction Definitions and Notations Approximation

Summary

approximate Pareto set ⇓ ⇑ approximate compromise solutions Approximation Algorithms: ▸ Approximations based on LP-relaxations (e.g. Set Cover, Scheduling)

Kai-Simon Goetzmann, TU Berlin Compromise Solutions

SLIDE 34

Introduction Definitions and Notations Approximation

Summary

approximate Pareto set ⇓ ⇑ approximate compromise solutions Approximation Algorithms: ▸ Approximations based on LP-relaxations (e.g. Set Cover, Scheduling) ▸ FPTAS based on dynamic programming, scaling and rounding (e.g. Shortest Path, Min Spanning Tree)

Kai-Simon Goetzmann, TU Berlin Compromise Solutions

SLIDE 35

Introduction Definitions and Notations Approximation

Summary

approximate Pareto set ⇓ ⇑ approximate compromise solutions Approximation Algorithms: ▸ Approximations based on LP-relaxations (e.g. Set Cover, Scheduling) ▸ FPTAS based on dynamic programming, scaling and rounding (e.g. Shortest Path, Min Spanning Tree) Thank you for your attention.

Kai-Simon Goetzmann, TU Berlin Compromise Solutions

SLIDE 36

AUGUST 19-24 www.ismp2012.org