The Power of Compromise Approximation in Multicriteria Optimization - PowerPoint PPT Presentation

Introduction Definitions and Notations Approximation The Power of Compromise Approximation in Multicriteria Optimization C. B¨ using, Kai-Simon Goetzmann , J. Matuschke and S. Stiller FRICO, August 15, 2012 Kai-Simon Goetzmann The Power of Compromise

Introduction Definitions and Notations Approximation Multicriteria Optimization Kai-Simon Goetzmann The Power of Compromise

Introduction Definitions and Notations Approximation Multicriteria Optimization Kai-Simon Goetzmann The Power of Compromise

Introduction Definitions and Notations Approximation Multicriteria Optimization Kai-Simon Goetzmann The Power of Compromise

Introduction Definitions and Notations Approximation Multicriteria Optimization Kai-Simon Goetzmann The Power of Compromise

Introduction Definitions and Notations Approximation Multicriteria Optimization Y ⊆ ❩ k min { y ∶ y ∈ Y} where Kai-Simon Goetzmann The Power of Compromise

Introduction Definitions and Notations Approximation Pareto Optimality Definition A solution y ∈ Y of min y ∈Y y is Pareto optimal if there is no y ′ ∈ Y ∖ { y } with y ′ ≤ y . y 2 Y y 1 Kai-Simon Goetzmann The Power of Compromise

Introduction Definitions and Notations Approximation Pareto Optimality Definition A solution y ∈ Y of min y ∈Y y is Pareto optimal if there is no y ′ ∈ Y ∖ { y } with y ′ ≤ y . y 2 y 1 Kai-Simon Goetzmann The Power of Compromise

Introduction Definitions and Notations Approximation Pareto Optimality Definition A solution y ∈ Y of min y ∈Y y is Pareto optimal if there is no y ′ ∈ Y ∖ { y } with y ′ ≤ y . y 2 Y P y 1 Kai-Simon Goetzmann The Power of Compromise

Introduction Definitions and Notations Approximation 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 ▸ The Power of Compromise : all Pareto optimal solutions can be CS, approximation of CS yields approximate Pareto set Kai-Simon Goetzmann The Power of Compromise

Introduction Definitions and Notations Approximation 1 Introduction 2 Definitions and Notations 3 Approximation Kai-Simon Goetzmann The Power of Compromise

Introduction Definitions and Notations Approximation 1 Introduction 2 Definitions and Notations 3 Approximation Kai-Simon Goetzmann The Power of Compromise

Introduction Definitions and Notations Approximation Definition (Ideal Point) Given a multicriteria optimization problem min y ∈Y y , the ideal point y id = ( y id 1 ,...,y id k ) is defined by y id i = min ∀ i. y ∈Y y i y 2 Y feasible reference y id points y 1 Kai-Simon Goetzmann The Power of Compromise

Introduction Definitions and Notations Approximation Definition (Compromise Solution, Yu 1973) Given a multicriteria optimization problem min y ∈Y y with the ideal point y id ∈ ◗ k , and a norm ∥⋅∥ on ❘ k , the compromise solution w.r.t. ∥⋅∥ is y cs = min y ∈Y ∥ y − y id ∥ . y 2 y id y 1 Kai-Simon Goetzmann The Power of Compromise

Introduction Definitions and Notations Approximation Definition (Compromise Solution, Yu 1973) Given a multicriteria optimization problem min y ∈Y y with the ideal point y id ∈ ◗ k , and a norm ∥⋅∥ on ❘ k , the compromise solution w.r.t. ∥⋅∥ is y cs = min y ∈Y ∥ y − y id ∥ . y 2 feasible reference y id points y 1 Kai-Simon Goetzmann The Power of Compromise

Introduction Definitions and Notations Approximation Definition (Reference Point Solution) Given a multicriteria optimization problem min y ∈Y y , a feasible reference point y rp ∈ ◗ k , and a norm ∥⋅∥ on ❘ k , the reference point solution w.r.t. ∥⋅∥ is y rps = min y ∈Y ∥ y − y rp ∥ . y 2 y rp y 1 Kai-Simon Goetzmann The Power of Compromise

Introduction Definitions and Notations Approximation Definition (Reference Point Solution) Given a multicriteria optimization problem min y ∈Y y , a feasible reference point y rp ∈ ◗ k , and a norm ∥⋅∥ on ❘ k , the reference point solution w.r.t. ∥⋅∥ is y rps = min y ∈Y ∥ y − y rp ∥ . y 2 y rp y 1 Kai-Simon Goetzmann The Power of Compromise

Introduction Definitions and Notations Approximation The norms we consider: 1 / p ∥ y ∥ p ∶= ( k i ) p ∈ [ 1 , ∞ ) ∑ y p ( ℓ p -Norm) , i = 1 ∥ y ∥ ∞ ∶= max (Maximum ( ℓ ∞ -)Norm) i = 1 ,...,k y i ⟨ ⟨ y ⟩ ⟩ p ∶= ∥ y ∥ ∞ + 1 p ∥ y ∥ 1 , p ∈ [ 1 , ∞ ] ( Cornered p -Norm) 1 1 p = 1 p = 2 p = 5 p = ∞ 1 1 ℓ p -Norm Cornered p -Norm Degree of balancing controlled by adjusting p . Kai-Simon Goetzmann The Power of Compromise

Introduction Definitions and Notations Approximation The norms we consider: 1 / p ∥ y ∥ p ∶ = ( k i ) p ∈ [ 1 , ∞) ∑ y p ( ℓ p -Norm) , i = 1 ∥ y ∥ ∞ ∶ = max (Maximum ( ℓ ∞ -)Norm) i = 1 ,...,k y i ⟨ ⟨ 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 = ∥( λ 1 y 1 ,λ 2 y 2 ,...,λ k y k )∥ . 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 The Power of Compromise

Introduction Definitions and Notations Approximation 1 Introduction 2 Definitions and Notations 3 Approximation Kai-Simon Goetzmann The Power of Compromise

Introduction Definitions and Notations Approximation Approximate Pareto sets Definition ( α -approximate Pareto set) Let Y P be the Pareto set of min y ∈Y y , and let α > 1 . Y α ⊆ Y is an α -approximate Pareto set if for all y ∈ Y P there is y ′ ∈ Y α such that i ≤ αy i ∀ i = 1 ,...,k y ′ Kai-Simon Goetzmann The Power of Compromise

◗ Introduction Definitions and Notations Approximation How to find approximate Pareto sets Theorem (Papadimitriou&Yannakakis,2000) Gap ( y,α ) tractable for all y ∈ ◗ k ⇒ α 2 -approximation for the Pareto set. α -approximation for the Pareto set ⇒ Gap ( y,α ) tractable for all y ∈ ◗ k . Kai-Simon Goetzmann The Power of Compromise

Introduction Definitions and Notations Approximation How to find approximate Pareto sets Theorem (Papadimitriou&Yannakakis,2000) Gap ( y,α ) tractable for all y ∈ ◗ k ⇒ α 2 -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 2 y y 1 Kai-Simon Goetzmann The Power of Compromise

Introduction Definitions and Notations Approximation How to find approximate Pareto sets Theorem (Papadimitriou&Yannakakis,2000) Gap ( y,α ) tractable for all y ∈ ◗ k ⇒ α 2 -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 2 y y ′ y 1 Kai-Simon Goetzmann The Power of Compromise

Introduction Definitions and Notations Approximation How to find approximate Pareto sets Theorem (Papadimitriou&Yannakakis,2000) Gap ( y,α ) tractable for all y ∈ ◗ k ⇒ α 2 -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 2 y 2 y y y ′ no sol’n 1 α y y 1 y 1 Kai-Simon Goetzmann The Power of Compromise

◗ Introduction Definitions and Notations Approximation Approximate Pareto sets ⇔ approximate CS Kai-Simon Goetzmann The Power of Compromise

◗ Introduction Definitions and Notations Approximation Approximate Pareto sets ⇔ approximate CS Relate objective value to size of the vectors: y ∈Y ∥ y − y id ∥ + ∥ y id ∥ min �ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ�ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ� r ( y ) We call r ( y ) the relative distance . Kai-Simon Goetzmann The Power of Compromise

◗ Introduction Definitions and Notations Approximation Approximate Pareto sets ⇔ approximate CS Relate objective value to size of the vectors: y ∈Y ∥ y − y id ∥ + ∥ y id ∥ min �ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ�ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ� r ( y ) We call r ( y ) the relative distance . Theorem α -approximation of the Pareto set ⇒ α -approximation for min y ∈Y r ( y ) . Kai-Simon Goetzmann The Power of Compromise

Introduction Definitions and Notations Approximation Approximate Pareto sets ⇔ approximate CS Relate objective value to size of the vectors: y ∈Y ∥ y − y id ∥ + ∥ y id ∥ min �ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ�ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ� r ( y ) We call r ( y ) the relative distance . Theorem α -approximation of the Pareto set ⇒ α -approximation for min y ∈Y r ( y ) . Theorem α -approximation for min y ∈Y r ( y ) ⇒ Gap ( y,β ) tractable for all y ∈ ◗ k , β ∈ Θ ( α ) . ⇒ β 2 -approximation for the Pareto set. Kai-Simon Goetzmann The Power of Compromise