A Theoretical Analysis of Curvature Based Preference Models Pradyumn - - PowerPoint PPT Presentation

A Theoretical Analysis of Curvature Based Preference Models

Pradyumn Shukla1 Michael Emmerich2 André Deutz 2

1Karlsruhe Institute of Technology – Institute AIFB 2Leiden University – LIACS

Multi-objective Optimization Problem

Given m, n ∈ N such that m ≥ 2, a multi-objective optimization problem (MOP) is a 3-tuple (F, X, C) such that F(x) := (F1(x), F2(x), . . . , Fm(x)) is a vector valued objective function X ⊆ Rn a feasible set C is a cone (set) that induces a partial ordering on Rm

Pradyumn Shukla, Michael Emmerich, André Deutz A Theoretical Analysis of Curvature Based Preference Models

An Optimality Notion

Definition

A point ˆ x ∈ X is C-optimal if ({F(ˆ x)} − C) ∩ F(X) = {F(ˆ x)}. F2(x) F1(x) A B F(ˆ x) −C Image of C-optimal points of a bicritetia problem

Pradyumn Shukla, Michael Emmerich, André Deutz A Theoretical Analysis of Curvature Based Preference Models

Preference Models

Let Xp(F, X, C) be the set of C-optimal optimal points.

Definition

A preferred solution set, denoted by XP(F, X), is a proper subset of Xp(F, X, Rm

+). The set XP(F, X) is said to be induced by a preference

model P.

Pradyumn Shukla, Michael Emmerich, André Deutz A Theoretical Analysis of Curvature Based Preference Models

Characteristics of Preference Models

Domination transformation: A (convex) cone C ⊃ Rm

+ exists such that

XP(F, X) = Xp(F, X, C) Objective transformation: A function T : Rm → Rk exists such that XP(F, X) = Xp(T ◦ F, X, Rm

+)

Lemma

If T := A, where A is a m by k matrix, then the above two transformations are equivalent and, C is the polyhedral cone {d ∈ Rm|Ad ≥ 0}. In general, these do not imply each other.

Pradyumn Shukla, Michael Emmerich, André Deutz A Theoretical Analysis of Curvature Based Preference Models

A Polyhedral Model

−0.5 0.5 1 1.5 −2 −1 1 0.2 0.4 0.6 0.8 1 1.2

A polyhedral domination cone. The k can be much larger than m. Some real world applications use k = m(m − 1).

Pradyumn Shukla, Michael Emmerich, André Deutz A Theoretical Analysis of Curvature Based Preference Models

A Piecewise Polyhedral Model

• ÷

÷ ø ö ç ç è æ = 1 1 1

1

A ÷ ÷ ø ö ç ç è æ = 1 1 1

2

A

Piecewise polyhedral transformations of the objectives in the case of equitable efficiency. There are m! polyhedral transformation for an m-objective problem.

Pradyumn Shukla, Michael Emmerich, André Deutz A Theoretical Analysis of Curvature Based Preference Models

An Algorithm

Algorithm: General cone-based hypervolume computation Input: An m by k matrix A, points S ⊂ Rm, and a reference point r

1

Let r′ = Ar.

2

For all i = 1, . . . , |S|, let Q = {q(1), . . . , q|S|}, with q(i) = As(i).

3

Compute the standard hypervolume HI(Q, r′).

4

Return CHI(S) = (1/ det(A⊤A)) · HI(Q, r′).

Pradyumn Shukla, Michael Emmerich, André Deutz A Theoretical Analysis of Curvature Based Preference Models

An Application in Equitable Efficiency

0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4 0.42 0.38 0.4 0.42 0.44 0.46 0.48 0.5 0.52 F1 F2 Distribution of 30 points in original space, for problem ZDT1

Distribution of points in the original ZDT1 objective space

Pradyumn Shukla, Michael Emmerich, André Deutz A Theoretical Analysis of Curvature Based Preference Models

An Application in Equitable Efficiency

0.38 0.4 0.42 0.44 0.46 0.48 0.5 0.52 0.75 0.755 0.76 0.765 T1(F) T2(F) Distribution of 30 points in transformed space, for problem ZDT1

Distribution of points in the transformed ZDT1 objective space

Pradyumn Shukla, Michael Emmerich, André Deutz A Theoretical Analysis of Curvature Based Preference Models

Summary

Analyzed various preference models Presented new theoretical results

Pradyumn Shukla, Michael Emmerich, André Deutz A Theoretical Analysis of Curvature Based Preference Models

Summary

Analyzed various preference models Presented new theoretical results Proposed an algorithm to compute general cone based hypervolume Applied on a piecewise polyhedral preference model

Pradyumn Shukla, Michael Emmerich, André Deutz A Theoretical Analysis of Curvature Based Preference Models

Summary

Analyzed various preference models Presented new theoretical results Proposed an algorithm to compute general cone based hypervolume Applied on a piecewise polyhedral preference model Many other details in the paper

Pradyumn Shukla, Michael Emmerich, André Deutz A Theoretical Analysis of Curvature Based Preference Models