European Journal of Operational Research 260 (2017) 841–855

Contents lists available at ScienceDirect

European Journal of Operational Research

journal homepage: www.elsevier.com/locate/ejor

Efficient computation of the search region in multi-objective

optimization

Kerstin Dächert a , Kathrin Klamroth

b , Renaud Lacour b , ∗, Daniel Vanderpooten

c

a University of Duisburg-Essen, Chair for Management Science and Energy Economics, Universitätsstr. 12, 45117 Essen, Germany b Department of Mathematics and Computer Science, University of Wuppertal, Germany c Université Paris-Dauphine, PSL Research University, CNRS, UMR [7243], LAMSADE, 75016 Paris, France

a r t i c l e i n f o

Article history:

Received 1 September 2015

Accepted 13 May 2016

Available online 20 May 2016

Keywords:

Multi-objective optimization

Nondominated set

Search region

Local upper bounds

Scalarization

a b s t r a c t

Multi-objective optimization procedures usually proceed by iteratively producing new solutions. For this

purpose, a key issue is to determine and efficiently update the search region, which corresponds to the

part of the objective space where new nondominated points could lie. In this paper we elaborate a spe-

cific neighborhood structure among local upper bounds. Thanks to this structure, the update of the search

region with respect to a new point can be performed more efficiently compared to existing approaches.

Moreover, the neighborhood structure provides new insight into the search region and the location of

nondominated points.

© 2016 Published by Elsevier B.V.

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. Introduction

Practical optimization problems often involve more than one

bjective function. Recent progress in the development of algo-

ithms that (i) compute the complete nondominated set, or (ii)

n approximation of it with provable quality (given an appropriate

uality indicator) have boosted the applicability of multi-objective

ptimization methods in many areas, including continuous and

ombinatorial optimization. Many of these methods, particularly in

he very active field of multi-objective combinatorial optimization

MOCO), rely on a concise decomposition of the search region, i.e.,

hat part of the objective space that may still contain nondom-

nated points, and involve repeated computations of bounds on

he nondominated set. Prominent examples are two-phase meth-

ds ( Dhaenens, Lemesre, & Talbi, 2010; Przybylski, Gandibleux, &

hrgott, 2010; Tenfelde-Podehl, 2003; Ulungu & Teghem, 1995 )

nd branch and bound algorithms ( Sourd & Spanjaard, 2008; Vil-

arreal & Karwan, 1981 ). But also methods that iteratively solve

calarized subproblems (e.g., ε-constraint or weighted Tchebychev

calarizations) rely on repeated updates of the search region in or-

er to define subsequent subproblems. While this is an easy task

n the biobjective case ( Aneja & Nair, 1979; Chalmet, Lemonidis, &

lzinga, 1986; Ralphs, Saltzman, & Wiecek, 2006 ), it is not obvi-

∗ Corresponding author. Tel.: +33 662204647.

E-mail addresses: kerstin.daechert@uni-due.de (K. Dächert), klamroth@math.uni-

uppertal.de (K. Klamroth), lacour@math.uni-wuppertal.de (R. Lacour),

dp@lamsade.dauphine.fr (D. Vanderpooten).

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ttp://dx.doi.org/10.1016/j.ejor.2016.05.029

377-2217/© 2016 Published by Elsevier B.V.

us in multi-objective problems involving three and more objec-

ives. Corresponding methods have been proposed, e.g., in Klein

nd Hannan (1982) , Sylva and Crema (2004) , Laumanns, Thiele, and

itzler (2006) , Sylva and Crema (2008) , Ozlen, Burton, and MacRae

2014) , Kirlik and Sayın (2014) and Dächert and Klamroth (2015) .

The search region can be described as a union of axis-parallel

yperrectangles of R

p called search zones . The literature in com-

utational geometry provides some insight into this topic. From

oissonnat, Sharir, Tagansky, and Yvinec (1998) and Bringmann

2013) , it can be derived that the worst-case number of search

ones that describe the search region associated to a set of n

oints is O(n � p/ 2 � ) for p ≥ 2. Kaplan, Rubin, Sharir, and Verbin

2008) provide the same complexity as well as an algorithm

o identify these search zones when the nondominated points

re computed in non-decreasing order of some component value.

owever, this assumption is too restrictive in the multi-objective

ontext, in which points are generated in any order, in gen-

ral, and thus an incremental update of the search region is

equired.

The goal of this paper is to provide new theoretical insights

nto structural properties of the search region in multi-objective

ptimization, assuming that a finite set of mutually nondominated

oints has already been computed, or is known otherwise. The re-

ults imply a new and highly efficient algorithm for identifying

minimal set of search zones that decompose the search region.

oreover, a neighborhood relation providing information on the

eometric structure is generated and maintained by the proposed

lgorithm. This information may be used, for example, if quick

842 K. Dächert et al. / European Journal of Operational Research 260 (2017) 841–855

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searches in adjacent search zones are required, e.g., in the context

of interactive procedures.

In defining the search region we closely follow ideas from

Przybylski et al. (2010) and, particularly, Klamroth, Lacour, and

Vanderpooten (2015) . In the latter, properties of the search region

and its representation through local upper bounds for the gen-

eral multi-objective case are studied and an incremental algorithm

to generate the search region is proposed. Moreover, we general-

ize the concept of a neighborhood relation that is introduced in

Dächert and Klamroth (2015) for the tri-objective case to any num-

ber of objectives.

The organization of the remainder of the paper is as follows.

Section 2 contains relevant preliminaries including an illustrative

example that is used throughout the paper. In Section 3 we derive

a neighborhood relation among local upper bounds and study its

properties for any number of objectives. In Section 4 we show how

to use the neighborhood relation to efficiently update the search

region. Based on the theoretical results, we propose a new incre-

mental algorithm in Section 5 and perform numerical experiments

to show its efficiency. Concluding remarks are given in Section 6 .

2. Preliminaries

2.1. Multi-objective optimization

In multi-objective optimization (MOO) we consider not only

one but several objective functions which are in conflict, i.e. it is

typically impossible to optimize them simultaneously. Formally, we

consider the following general formulation of an MOO problem:

min f (x ) = ( f 1 (x ) , . . . , f p (x )) s.t. x ∈ X

(1)

where X � = ∅ is the feasible set and f 1 , . . . , f p are p ≥ 2 objective

functions mapping from X to R . We denote by Y = f (X ) the image

set of X in the objective space R

p , consisting of all feasible points

of (1) . We define the following usual binary relations for the com-

parison of points of R

p :

z � z ′ (z weakly dominates z ′ ) ⇔ z i ≤ z ′ i , i ∈ { 1 , . . . , p} ,

z ≤ z ′ (z dominates z ′ ) ⇔ z � z ′ and z � = z ′ , z < z ′ (z strictly dominates z ′ ) ⇔ z i < z ′

i , i ∈ { 1 , . . . , p}

A point z ′ ∈ R

p is called dominated by z ∈ R

p if z ≤ z ′ . If, moreover,

z < z ′ then z ′ is called strictly dominated by z . A subset N of R

p

is stable with respect to the dominance relation ≤ , or simply, stable

if for any z , z ′ ∈ N , z �≤ z ′ . For any subset Q of R

p , Q nd is the set

of all nondominated points of Q , i.e. Q nd = { z ∈ Q : � z ∈ Q with z ≤z} . We refer to Y nd as the nondominated set of (1) , and every point z

∈ Y nd is called nondominated . We recall the definitions of the ideal

and nadir points, z I and z N , respectively:

z I =

(min

z∈ Y nd

z i

)i ∈{ 1 , ... ,p}

, z N =

(max z∈ Y nd

z i

)i ∈{ 1 , ... ,p}

.

For any z ∈ R

p let z −i denote the (p − 1) -dimensional vector

containing all components of z except component i for a given i ∈{ 1 , . . . , p} . Finally, for any z, a ∈ R

p and any i ∈ { 1 , . . . , p} , (z i , a −i )

denotes the vector (a 1 , . . . , a i −1 , z i , a i +1 , . . . , a p ) .

2.2. Search region and local upper bound set

Given a stable set N of known feasible points, the search region

is that part of the objective space (or, more generally, of a region

Z , of interest to the decision maker in the objective space), where

remaining nondominated points must lie. More precisely, we de-

fine the search region associated to N and with respect to the

search interval Z = [ m, M) ⊆ R

p , where m j ∈ R and M j ∈ R ∪ { + ∞}for j ∈ { 1 , . . . , p} so as to allow a search on a restricted part of the

bjective space. In the standard case where we look for the entire

ondominated set we must set m � z I and M > z N . We give several

ormal definitions of the search region as well as some important

roperties.

efinition 2.1 ( Klamroth et al., 2015 ) . Let N be a stable set of fea-

ible points. The search region associated to N and with respect to

search interval Z is the set

(N) = { z ∈ Z : ∀ z ′ ∈ N, z ′ � � z} . The search region can be equivalently defined from a set of aux-

liary points called upper bound set (in the minimization case).

efinition 2.2 ( Klamroth et al., 2015; Przybylski et al., 2010 ) . Let N

e a finite and stable set of points. Then U ( N ) is an upper bound

et with respect to N if and only if U ( N ) consists of all vectors u ofˆ = (m, M] that satisfy the following two properties:

(P 1 ) no point of N strictly dominates u , i.e. there is no z ∈ N such

that z < u ;

(P 2 ) u is maximal for property ( P 1 ), i.e. for all u ′ ≥ u , there exists

z ∈ N such that z < u ′ .

Any vector u that satisfies ( P 1 ) and ( P 2 ) is called a local upper

ound with respect to N .

The interest in upper bound sets is justified by

roposition 2.3 which provides an equivalent definition of the

earch region using upper bound sets.

roposition 2.3 ( Klamroth et al., 2015 ) . Let N be a stable set of

oints and let U ( N ) be the associated upper bound set. Then,

(N) = { z ∈ Z : ∃ u ∈ U(N) , z < u } . Using Proposition 2.3 , one can decompose the search region

nto search zones C(u ) = { z ∈ Z : z < u } induced by each local up-

er bound u ∈ U ( N ), see Fig. 1 for a visualization.

Recall from the introduction that it is well known that U ( N ) is

finite set whose cardinality depends polynomially on the car-

inality of N . In particular, | U ( N )| can be bounded by | N| + 1 in

he bi-objective case ( Chalmet et al., 1986 ) and 2 | N| + 1 in the tri-

bjective case ( Dächert & Klamroth, 2015 ).

The concept of search zones provides an intuitive interpretation

f properties ( P 1 ) and ( P 2 ) for the associated local upper bound. If

vector u does not satisfy ( P 1 ), then the associated C ( u ) contains

ome known point of N . If u does not satisfy ( P 2 ), then C ( u ) is con-

ained in C ( u ′ ), for some u ′ ∈ U ( N ). Therefore, in the latter case, u

s redundant .

For any, possibly empty, stable set of points N of Z , we de-

ote by ˆ N the augmented set N ∪ { z 1 , . . . , z p } where ˆ z 1 , . . . , z p are

ummy points such that ˆ z i = (M i , m −i ) , i = 1 , . . . , p.

roposition 2.4 ( Klamroth et al., 2015 ) . For any u ∈ U ( N ), the fol-

owing property holds:

(P 3 ) there exist p points of ˆ N , denoted by z 1 (u ) , . . . , z p (u ) , such

that {z i

i (u ) = u i

z i −i (u ) < u −i

, i = 1 , . . . , p

For the sake of completeness and because ( P 3 ) is used exten-

ively in what follows, we recall the proof given in Klamroth et al.

2015) .

roof. If u i = M i , then the dummy point ˆ z i ∈

ˆ N satisfies the re-

uired conditions. Otherwise and since ˆ N is a finite set, there ex-

sts an ε > 0 sufficiently small such that no point of ˆ N has its i th

omponent value in the interval (u i , u i + ε) . Let u ′ = (u i + ε, u −i ) .

ccording to Definition 2.2 , since u ′ ∈

ˆ Z and u ′ ≥ u , there exists

K. Dächert et al. / European Journal of Operational Research 260 (2017) 841–855 843

Fig. 1. Illustration of the concepts of local upper bound and search zone in the bi-objective case.

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∈ N such that (i) z < u ′ and (ii) z � < u . It follows from (i) that

e have z −i < u −i , which imposes z i ≥ u i from (ii). Together with

i < u i + ε from (i) and the choice of ε, we get z i = u i . �

Any such z i ( u ) is a defining point for component i of u or sim-

ly a defining point . Proposition 2.4 allows us to derive remarkable

roperties of defining points.

orollary 2.5. The following statements hold for any local upper

ound u:

(i) The defining points of u are all distinct.

(ii) If z i ( u ) is a dummy point, it can only be ˆ z i .

roof.

(i) For any j, k ∈ { 1 , . . . , p} such that j � = k , z j (u ) = z k (u ) would

violate ( P 3 ).

(ii) If z i (u ) = ˆ z k , for some k ∈ { 1 , . . . , p} , k � = i , then z i k (u ) =

M k ≥ u k since u � M holds for any local upper bound u ∈U ( N ). This also violates ( P 3 ). �

Moreover, we have the following Corollary of Proposition 2.4 ,

hich provides an equivalent characterization of local upper

ounds involving defining points:

orollary 2.6. Any vector u ∈ R

p satisfying (P 1 ) and (P 3 ) for a finite

table set N ⊆ R

p is a local upper bound with respect to N.

roof. If u satisfies ( P 3 ), then for any i ∈ { 1 , . . . , p} and ε > 0 we

ave z i (u ) < (u i + ε, u −i ) . Since z i (u ) ∈

ˆ N , this ensures that u satis-

es ( P 2 ). �

If N � = ∅ , the associated local upper bound set U ( N ) contains the

ollowing p particular elements.

emma 2.7 (Extreme local upper bounds) . Let N � = ∅ and z ∗i

=in z∈ N z i be the minimal value reached on objective i over N , i = , . . . , p. For all i ∈ { 1 , . . . , p} , the vector u i = (z ∗

i , M −i ) is a local up-

er bound of U ( N ) and is referred to as an extreme local upper bound .

roof. For all i ∈ { 1 , . . . , p} , u i clearly satisfies ( P 1 ) and ( P 3 ). �

When deriving the neighborhood relation, the following as-

umption will be useful.

ssumption GP. N is a nonempty and stable set of points in gen-

ral position , i.e. for all z , z ′ ∈ N , z � = z ′ , we have z i � = z ′ i

for all

= 1 , . . . , p.

As a consequence of this assumption, each component value of

local upper bound is defined by a unique point of ˆ N . Even if this

ssumption is common in computational geometry, it is quite re-

trictive in multi-objective optimization. We will show how to re-

ax it in Section 4.3 .

.3. Illustrative example

From Przybylski et al. (2010) we know that we can construct

he upper bound set U ( N ) incrementally by adding (nondominated)

oints one by one. Starting from U ( N ) and adding z ∈ R

p to N ,

e obtain U(N ∪ { z } ) by replacing all u ∈ U ( N ) that would not

atisfy ( P 1 ) with respect to N ∪ { z } , i.e. for which z < u holds, by

i = ( z i , u −i ) for all i ∈ { 1 , . . . , p} . While it is shown that these new

ectors contain all new local upper bounds, some may not sat-

sfy ( P 2 ) with respect to N ∪ { z } and are thus redundant. A filter-

ng step is performed to eliminate those that are redundant. In the

ollowing example we apply a slightly different version of the fil-

ering from Klamroth et al. (2015) where, unlike Przybylski et al.

2010) , only new candidate local upper bounds u i = ( z i , u −i ) with

espect to the same index i are compared to detect redundancies.

hile a more efficient approach will be elaborated in this paper,

e apply the filtering approach to construct the upper bound set

f the following example, which will be used throughout this pa-

er for illustration.

xample 2.8. Let the three points z 1 = (4 , 0 , 4) , z 2 = (3 , 3 , 1) and

3 = (2 , 2 , 2) be given. We insert these three points one by one

nto the initial search region and indicate the corresponding upper

ound sets.

We use the following notation. The last digit of the upper index

f a local upper bound denotes the component which is defined

y the new point. All other defining points are inherited from the

ame local upper bound of the previous iteration, which is referred

o by the leading digits of the upper index. For example, u 32 has

he same defining points as u 3 apart from component 2.

(0) N = ∅ : U(N) = { u 0 } . By construction, u 0 = (M 1 , M 2 , M 3 ) de-

scribes a search region corresponding to the whole search

interval Z . It is defined by the three dummy points ˆ z 1 =(M 1 , m 2 , m 3 ) , ˆ z 2 = (m 1 , M 2 , m 3 ) and ˆ z 3 = (m 1 , m 2 , M 3 ) , i.e.

z i (u 0 ) = ˆ z i for all i = 1 , 2 , 3 .

u u 1 u 2 u 3

u

0 M 1 M 2 M 3

u z 1 (u ) z 2 (u ) z 3 (u ) u

0 ˆ z 1 ˆ z 2 ˆ z 3

(1) N = { z 1 } : Since z 1 < u 0 , we replace u 0 by u 0 ,i = (z 1 i , u 0 −i

)

for all i ∈ {1, 2, 3}. For simplicity, we set u 0 i = u i for all

i = 1 , 2 , 3 . Hence, U(N) = { u 1 , u 2 , u 3 } with

u u 1 u 2 u 3

u

1 4 M 2 M 3

u

2 M 1 0 M 3

u

3 M 1 M 2 4

u z 1 (u ) z 2 (u ) z 3 (u ) u

1 z 1 ˆ z 2 ˆ z 3

u

2 ˆ z 1 z 1 ˆ z 3

u

3 ˆ z 1 ˆ z 2 z 1

The extreme local upper bounds are u i = u i for all i ∈ {1, 2,

3}.

844 K. Dächert et al. / European Journal of Operational Research 260 (2017) 841–855

Fig. 2. Visualization of u ′ = νk (u ) and, conversely, u = ν j (u ′ ) .

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(2) N = { z 1 , z 2 } : Since z 2 < u 1 , z 2 � < u 2 and z 2 < u 3 , we re-

place u 1 and u 3 by u 1 ,i = (z 2 i , u 1 −i

) and u 3 ,i = (z 2 i , u 3 −i

) for all

i ∈ {1, 2, 3}, respectively. As u 13 = (4 , M 2 , 1) � (M 1 , M 2 , 1) =u 33 and u 31 = (3 , M 2 , 4) � (3 , M 2 , M 3 ) = u 11 , u 13 and u 31

are redundant. Thus, we obtain U(N) = { u 11 , u 12 , u 2 , u 32 , u 33 }with

u u 1 u 2 u 3

u

11 3 M 2 M 3

u

12 4 3 M 3

u

2 M 1 0 M 3

u

32 M 1 3 4

u

33 M 1 M 2 1

u z 1 (u ) z 2 (u ) z 3 (u ) u

11 z 2 ˆ z 2 ˆ z 3

u

12 z 1 z 2 ˆ z 3

u

2 ˆ z 1 z 1 ˆ z 3

u

32 ˆ z 1 z 2 z 1

u

33 ˆ z 1 ˆ z 2 z 2

The extreme local upper bounds are u 1 = u 11 , u 2 = u 2 and

u 3 = u 33 .

(3) N = { z 1 , z 2 , z 3 } : We obtain U(N) = { u 111 , u 113 , u 122 , u 2 , u 322 ,

u 323 , u 33 } with

u u 1 u 2 u 3

u

111 2 M 2 M 3

u

113 3 M 2 2

u

122 4 2 M 3

u

2 M 1 0 M 3

u

322 M 1 2 4

u

323 M 1 3 2

u

33 M 1 M 2 1

u z 1 (u ) z 2 (u ) z 3 (u ) u

111 z 3 ˆ z 2 ˆ z 3

u

113 z 2 ˆ z 2 z 3

u

122 z 1 z 3 ˆ z 3

u

2 ˆ z 1 z 1 ˆ z 3

u

322 ˆ z 1 z 3 z 1

u

323 ˆ z 1 z 2 z 3

u

33 ˆ z 1 ˆ z 2 z 2

The extreme local upper bounds are u 1 = u 111 , u 2 = u 2 and

u 3 = u 33 .

3. A neighborhood relation among local upper bounds

Updating efficiently the search region when a new point is

added to N is a key issue. We define a neighborhood relation

among local upper bounds that allows an efficient update of the

search region in Section 3.1 and study its properties in Section 3.2 .

3.1. Definitions

Definition 3.1. Two local upper bounds u and u ′ are said to share q

defining points if, for some I , I ′ ⊆ { 1 , . . . , p} with | I | = | I ′ | = q, there

is a one-to-one mapping σ : I → I ′ such that z i (u ) = z σ (i ) (u ′ ) , for

all i ∈ I .

Example 3.2. Consider U ( N ) of Example 2.8 (2): u 11 and u 12 share

two defining points, since z 1 (u 11 ) = z 2 (u 12 ) = z 2 and z 3 (u 11 ) =z 3 (u 12 ) = ˆ z 3 , i.e. I = { 1 , 3 } and I ′ = { 2 , 3 } . u 12 and u 33 share one

defining point, since z 2 (u 12 ) = z 3 (u 33 ) = z 2 , i.e. I = { 2 } and I ′ ={ 3 } . Lemma 3.3. Let u , u ′ ∈ U ( N ) be local upper bounds sharing q ≥ 1

defining points. Then, z i (u ) = z i (u ′ ) holds for all i ∈ I ∩ I ′ , where I , I ′ ⊆{ 1 , . . . , p} and | I| = | I ′ | = q .

Proof. Assume, to the contrary, that z i ( u ) � = z i ( u ′ ), for some i ∈I ∩ I ′ . Then there must exist j ∈ I and k ∈ I ′ such that z j (u ) = z i (u ′ )and z k (u ′ ) = z i (u ) . Due to ( P 3 ), we have z

j i (u ) < z i

i (u ) and z k

i (u ′ ) <

z i i (u ′ ) . This implies that z i

i (u ′ ) = z

j i (u ) < z i

i (u ) and z i

i (u ) = z k

i (u ′ ) <

z i i (u ′ ) , a contradiction. �

Example 3.4. Consider the two local upper bounds u = u 11 and

u ′ = u 12 from Example 3.2 which share q = 2 defining points. We

have I ∩ I ′ = { 3 } and, indeed, z 3 (u 11 ) = z 3 (u 12 ) = ˆ z 3 .

Corollary 3.5. Let u , u ′ ∈ U ( N ) . If u and u ′ share

1. p defining points, then u = u ′ holds.

2. p − 1 defining points, then u and u ′ are equal in p − 2 compo-

nents .

roof. The first assertion immediately follows from Lemma 3.3 .

or the second assertion, let I and I ′ denote the indices of

he shared defining points of u and u ′ , respectively. Since

, I ′ ⊆ { 1 , . . . , p} and | I| = | I ′ | = p − 1 , we have | I ∩ I ′ | ≥ p − 2 . On

he other hand, | I ∩ I ′ | ≤ p − 2 , since, otherwise, the stability of

( N ) would be violated. Hence, | I ∩ I ′ | = p − 2 holds. Applying

emma 3.3 yields the assertion. �

Local upper bounds that share p − 1 defining points are of par-

icular interest in the following. We define them to be neighbors.

efinition 3.6 (Neighbors) . Two local upper bounds u and u ′ ∈ ( N ) are neighbors if they share p − 1 defining points, that is,

j (u ) = z k (u ′ ) for some j , k with j � = k and z i (u ) = z i (u ′ ) for all

∈ { 1 , . . . , p}\{ j, k } . We then say that u is a neighbor of u ′ , de-

oted by u = ν(u ′ ) , or symmetrically u ′ is a neighbor of u , denoted

y u ′ = ν(u ) . More precisely, when the indices of u with different

efining points are to be emphasized, we say that u is a j -neighbor

f u ′ , denoted by u = ν j (u ′ ) , and that u ′ is a k -neighbor of u , de-

oted by u ′ = νk (u ) .

A visualization of Definition 3.6 is given in Fig. 2 .

xample 3.7. The local upper bounds u 11 and u 12 from

xample 3.2 are neighbors, since they share p − 1 = 2

efining points. Moreover, as z 1 (u 11 ) = z 2 (u 12 ) = z 2 and

3 (u 11 ) = z 3 (u 12 ) = ˆ z 3 , u 11 is a 1-neighbor of u 12 and, conversely,

12 is a 2-neighbor of u 11 , i.e. ν1 (u 12 ) = u 11 and ν2 (u 11 ) = u 12 .

It is convenient to represent neighborhood relations ν andk using graphs. Since relation ν is symmetric, it can be rep-

esented by an undirected graph G = (U(N) , ν) where each edge

u , u ′ ) is such that u ′ = ν(u ) (and u = ν(u ′ ) ). An example is de-

icted in Fig. 3 . Relation νk is represented by a directed graph

k = (U(N) , νk ) where each arc ( u , u ′ ) is such that u ′ = νk (u ) ,

= 1 , . . . , p. Fig. 4 shows an example.

.2. Properties of k -neighbors

In this section, we show the existence and uniqueness of k -

eighbors as well as the connectedness of the neighborhood re-

ation. To this end, we first provide a straightforward result on the

elative component values of two neighbor local upper bounds.

emma 3.8. Let u , u ′ ∈ U ( N ) be local upper bounds such that u ′ is a

-neighbor of u , and u is a j-neighbor of u ′ . Then, we have u j < u ′ j ,

k > u ′ k

and u i = u ′ i

for i � = j , k.

roof. Equalities directly follow from Definition 3.6 . Consider now

he two inequalities. We have u j = z j j (u ) = z k

j (u ′ ) < u ′

j where the

ast inequality results from ( P 3 ). Similarly, using ( P 3 ), we get u ′ k

=

k k (u ′ ) = z

j

k (u ) < u k . �

xample 3.9. Recall from Example 3.7 that u 11 is a 1-neighbor of

12 and u 12 is a 2-neighbor of u 11 . Since u 11 = (3 , M 2 , M 3 ) and

12 = (4 , 3 , M 3 ) , we have u 11 1 < u 12

1 , u 11 2 > u 12

2 and u 11 3 = u 12

3 .

The next result shows, under Assumption GP , that all but one

ocal upper bound admit a unique k -neighbor.

K. Dächert et al. / European Journal of Operational Research 260 (2017) 841–855 845

Fig. 3. The undirected graph G that is associated to the local upper bounds from Example 2.8 (2).

Fig. 4. The directed graphs G 1 , G 2 and G 3 that are associated to the local upper bounds from Example 2.8 (2).

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roposition 3.10. Under Assumption GP , any local upper bound u ∈ ( N ) has a unique k-neighbor, except u k which has no k-neighbor, for

= 1 , . . . , p.

roof. We first show that u k has no k -neighbor. From Lemma 3.8 a

-neighbor u ′ of u k would be such that u ′ k

< u k k

= z ∗k , where z ∗

k =

in { z k : z ∈ N} , see Lemma 2.7 , which is impossible.

Consider now any local upper bound u ∈ U(N) \ { u k } . Claim 1: u admits at most one k-neighbor u ′ . Let

j ∗ = arg max j � = k

z j k (u ) . (2)

e show that if u ′ is a k -neighbor of u , then u must be a j ∗-

eighbor of u ′ . Suppose by contradiction that u is a j -neighbor

f u ′ with j � = j ∗. Then we would have u ′ k

= z k k (u ′ ) = z

j

k (u ) <

j ∗k

(u ) = z j ∗k

(u ′ ) , contradicting Property ( P 3 ). Observe first that, un-

er Assumption GP , j ∗ is uniquely defined. To complete the proof

f Claim 1, we show that there cannot exist a second local upper

ound u ′ ′ � = u ′ such that u ′ ′ is a k -neighbor of u and u is also a

∗-neighbor of u ′ ′ . Indeed, in this case we would have u ′ i = u ′′

i , for

ll i � = j ∗, which implies either u ′ ≤ u ′ ′ or u ′ ′ ≤ u ′ , violating the

tability of U ( N ) in each case.

Claim 2: u admits at least one k-neighbor u ′ . For this purpose,

e define u ′ by specifying its p defining points in N and show

hat it satisfies properties ( P 1 ) and ( P 3 ), which, by Corollary 2.6 ,

stablishes that u ′ is indeed a local upper bound. For i � = j ∗, k , we

ave z i (u ′ ) = z i (u ) and we just showed that z k (u ′ ) = z j ∗(u ) . Thus,

e just need to specify z j ∗(u ′ ) . Consider the set Q = { z ∈

ˆ N : z i <

′ i , i � = j ∗} , which is non-empty since ˆ z j

∗ ∈ Q and select as z j ∗(u ′ )

he point in Q with the smallest value on component j ∗ ( z j ∗(u ′ ) is

niquely defined under Assumption GP ).

( P 1 ): Since u ′ j ∗ = z

j ∗j ∗ (u ′ ) = min { z j ∗ : z ∈ Q} , there is no point in

ˆ that strictly dominates u ′ , which establishes ( P 1 ) for u ′ .

( P 3 ): Many of the inequalities z i −i (u ′ ) < u ′ −i

, i = 1 , . . . , p to be

roved are already satisfied since u satisfies ( P 3 ). In order to show

he remaining cases, we need u j ∗ < u ′ j ∗ . This is shown as Claim 3

elow. Hence, assume now that u j ∗ < u ′ j ∗ is valid. The remaining

ases to be proved to show ( P 3 ) for u ′ are:

• when i = j ∗, we have indeed z j ∗− j ∗ (u ′ ) < u ′ − j ∗ since z j

∗(u ′ ) ∈ Q .

• when i = k, we have z k −k (u ′ ) = z

j ∗−k

(u ) . Then, owing to ( P 3 ) for

u , we have z j ∗− j ∗ (u ) < u − j ∗ and z

j ∗j ∗ (u ) = u j ∗ . Then, since u i = u ′

i

for i � = j ∗, k and u j ∗ < u ′ j ∗ we get z k −k

(u ′ ) < u ′ −k .

• when i � = j ∗, k , the two inequalities to be proved are:

- z i j ∗ (u ′ ) < u ′

j ∗ , which results from z i j ∗ (u ′ ) = z i

j ∗ (u ) < u j ∗ <

u ′ j ∗ , where the first inequality comes from ( P 3 ) for u .

- z i k (u ′ ) < u ′

k , which results from z i

k (u ′ ) = z i

k (u ) < z

j ∗k

(u ) =z k

k (u ′ ) = u ′

k , where the inequality comes from the definition

of index j ∗.

This finishes the proof of Claim 2. It remains to show that u j ∗ <

′ j ∗ .

Claim 3: u j ∗ < u ′ j ∗ . By construction of u ′ , we have u ′

i = u i for

� = j ∗, k . Moreover, we have u ′ k

< u k , since using ( P 3 ) for u

e get u k > z j ∗k

(u ) = z k k (u ′ ) = u ′

k . Considering that z j

∗(u ′ ) ∈ Q, we

ave z j ∗i

(u ′ ) < u ′ i

for all i � = j ∗. Thus, we get z j ∗i

(u ′ ) < u i for all

� = j ∗. It follows that u j ∗ ≤ u ′ j ∗ , since otherwise we would get

′ j ∗ = z

j ∗j ∗ (u ′ ) < u j ∗ and thus z j

∗(u ′ ) < u, contradicting ( P 1 ) for u .

o show that the inequality u j ∗ ≤ u ′ j ∗ is actually a strict inequal-

ty, we observe that u j ∗ = u ′ j ∗ would imply z j

∗(u ) = z j

∗(u ′ ) owing

o Assumption GP . Then, since z j ∗(u ′ ) ∈ Q and k � = j ∗, we would

et u ′ k

> z j ∗k

(u ′ ) = z j ∗k

(u ) = z k k (u ′ ) = u ′

k . �

xample 3.11. Consider u 11 and all other local upper bounds from

xample 2.8 (2). For k = 1 , since u 11 is the extreme local upper

ound u 1 , it has no 1-neighbor. For all other indices, i.e. for k = , 3 , a unique k -neighbor exists. Let u ′ denote the 2-neighbor of

11 . According to (2) , we have

j ∗ = arg max j � =2

z j 2 (u

11 ) = arg max { z 1 2 (u

11 ) , z 3 2 (u

11 ) } = arg max { z 2 2 , z 3 2 } = arg max { 3 , m 2 } = 1 ,

.e. u 11 is a 1-neighbor of u ′ , which implies z 2 (u ′ ) = z 1 (u 11 ) =

2 and z 3 (u ′ ) = z 3 (u 11 ) = ˆ z 3 . The unique local upper bound from

( N ) \ { u 11 } which satisfies these conditions is u 12 .

Neighborhood relations νk have a remarkable structure which

an be conveniently expressed on their associated graphs G

k , k = , . . . , p.

emma 3.12. Graph G

k is a directed tree rooted at u k , k = 1 , . . . , p.

846 K. Dächert et al. / European Journal of Operational Research 260 (2017) 841–855

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p z z

Proof. From Proposition 3.10 , any local upper bound has exactly

one successor, except u k which has none. Consider any directed

path in G

k . By Lemma 3.8 , consecutive vertices along this path cor-

respond to local upper bounds with strictly decreasing value on

component k . It follows that G

k has no cycle, which completes the

proof. �

The graphs G

1 , G

2 , and G

3 depicted in Fig. 4 are directed trees

with root nodes u 1 = u 11 , u 2 = u 2 and u 3 = u 33 , respectively.

As a consequence of the previous result, we can now prove the

connectedness of the global neighborhood relation ν .

Proposition 3.13. The neighborhood relation ν is connected.

Proof. For any k ∈ { 1 , . . . , p} , each arc in G

k gives rise to an edge

in G . By Lemma 3.12 , G

k is connected and therefore G is connected,

too. �

4. Update of the search region based on the neighborhood

relation

In this section, we exploit the neighborhood relation estab-

lished in Section 3 to derive a new update procedure of the search

region. In Section 4.1 , we explain how it can be used to identify all

search zones that contain a candidate feasible point and avoid gen-

erating redundant search zones. Besides, the neighborhood relation

needs to be updated at the same time. We provide properties to

achieve this efficiently in Section 4.2 . Finally, in Section 4.3 we ex-

plain how to relax Assumption GP .

4.1. Update of the search region

When a new point z is inserted into N , we have to update the

search region. In particular, we update all u ∈ U ( N ) for which z < u

holds, since ( P 1 ) would be violated for them. As stated in Section 2 ,

all candidates for new local upper bounds are of the form u i =( z i , u −i ) , i ∈ { 1 , . . . , p} . However, not all of them satisfy ( P 2 ) with

respect to N ∪ { z } . For convenience, if u i satisfies ( P 2 ) we call u i

the i -child of u and u the parent of u i .

We recall that Przybylski et al. (2010) use a filtering step

to avoid redundancies, i.e. candidate local upper bounds that

do not satisfy ( P 2 ). Nevertheless, criteria to avoid generat-

ing redundant candidates without filtering, i.e. without com-

parisons among local upper bounds, have been elaborated in

Dächert and Klamroth (2015) for the tri-objective case and in

Klamroth et al. (2015) for the multi-objective case, respectively.

Under Assumption GP , the criterion proposed in the latter con-

sists in comparing z component-wise to z max k

(u ) = max l � = k z l k (u ) ,

which, using (2) , equals u ′ k

with u ′ being k -neighbor of u , since

max l � = k z l k (u ) = z j ∗k

(u ) = z k k (u ′ ) = u ′

k .

In the next result we reformulate the criterion of Klamroth

et al. (2015) for the neighborhood relation. For convenience, we

define νk ( u k ) = m for all k ∈ { 1 , . . . , p} in the following, which en-

ables us to treat the extreme local upper bound u k as a regular

local upper bound.

Proposition 4.1. Let N ⊆ Z and z ∈ Z such that N ∪ { z } is a finite and

stable set of points satisfying Assumption GP . Consider a local upper

bound u ∈ U ( N ) .

For any k ∈ { 1 , . . . , p} , u k = ( z k , u −k ) is a local upper bound with

respect to N ∪ { z } if and only if the two following conditions are sat-

isfied:

(C 1 ) z < u

(C 2 ) z k > u ′ k

where u ′ = νk (u ) is the k-neighbor of u.

roof.

( ⇒ ) Assume ( C 1 ) does not hold, i.e., z i ≥ u i for some i ∈{ 1 , . . . , p} , or, using Assumption GP , z i > u i . If i = k, this im-

plies z k > u k = z k k (u ) . Hence, using that ( P 3 ) holds for u , we

have z k ( u ) < u k . Since z k (u ) ∈ N ∪ { z } , this contradicts ( P 1 )

for u k . If i � = k , we have z −k � < u −k = u k −k . Since, under

Assumption GP , z is the defining point for component k of

u k , this violates ( P 3 ) for u k .

Assume ( C 2 ) does not hold, i.e. z k ≤ u ′ k . Since u ′ is a k -

neighbor of u , z k (u ′ ) = z j (u ) for some j � = k . Moreover, since

j � = k , z j (u ) = z j (u k ) . Thus, from z k = u k k , we have u k

k ≤ z

j

k (u k )

which implies z j − j

(u k ) � < u k − j , violating ( P 3 ) for u k .

( ⇐ ) From ( C 1 ), we have u k ≤ u . Thus, since u satisfies ( P 1 ), the

only point that could strictly dominate u k is z , but since u k k

=z k , we have z � < u k Therefore u k satisfies ( P 1 ).

To show that u k satisfies ( P 3 ), we consider two cases with

respect to the index i of a defining point of u k , and finally

show z i −i (u k ) < u k −i

for all i ∈ { 1 , . . . , p} . i = k : From ( C 1 ) we have z k −k

(u k ) = z −k < u −k = u k −k ;

i � = k : We have in this case z i (u ) = z i (u k ) . Since u satis-

fies ( P 3 ), we have z i −i (u ) < u −i . Moreover, since z i ( u )

is also a defining point of u ′ and u ′ satisfies ( P 3 ), we

have in particular z i k (u ) ≤ u ′

k . Since, from ( C 2 ), u ′

k <

u k k

= z k , we obtain z i k (u ) < u k

k , and, finally, z i −i

(u k ) <

u k −i . �

xample 4.2. Consider the upper bound set U(N) = { u 1 , u 2 , u 3 }f Example 2.8 (1), where u 1 = (4 , M 2 , M 3 ) , u 2 = (M 1 , 0 , M 3 ) and

3 = (M 1 , M 2 , 4) , and its update when adding z = z 2 = (3 , 3 , 1) .

ondition ( C 1 ) is satisfied for u 1 and u 3 . Since ν1 (u 1 ) = m,2 (u 1 ) = u 2 and ν3 (u 1 ) = u 3 , Condition ( C 2 ) for u 1 is only satis-

ed for k = 1 and k = 2 . Hence, u 11 = (z 2 1 , u 1 −1 ) and u 12 = (z 2 2 , u

1 −2 )

re local upper bounds with respect to N ∪ { z 2 }. Analogously, from

3 , we obtain u 32 and u 33 as new local upper bounds.

The next result establishes that either a local upper bound u

hat is dominated by the current point z is decomposed with re-

pect to component k , i.e. the resulting point u k = ( z k , u −k ) is an

lement of U(N ∪ { z } ) , or the k -neighbor of u is dominated by z .

roposition 4.3. Let u ∈ U z = { u ′ ∈ U(N) : z < u ′ } , i.e. ( C 1 ) from

roposition 4.1 holds. Then, for every k = 1 , . . . , p, one of the follow-

ng two assertions holds exclusively:

1. u k = ( z k , u −k ) ∈ U(N ∪ { z } ) 2. νk (u ) ∈ U z .

roof. Let u ′ = νk (u ) . We consider the two cases z k > u ′ k

and z k <

′ k . The case z k = u ′

k does not occur due to Assumption GP . If z k >

′ k , u k = ( z k , u −k ) ∈ U(N ∪ { z } ) due to Proposition 4.1 . If z k < u ′

k , we

btain z < u ′ , i.e. u ′ = νk (u ) ∈ U z , since, according to the definition

f the k -neighbor of u , u j ∗ < u ′ j ∗ with j ∗ as defined in (2) and u i =

′ i

for all i � = j ∗, k , thus, z −k < u −k ≤ u ′ −k . �

xample 4.4. From Example 4.2 , in which we add z = z 2 to N = z 1 } , we know that u 1 satisfies ( C 1 ). Since ( C 2 ) is only satisfied

or k = 1 , 2 , we have u 11 , u 12 ∈ U(N ∪ { z } ) , but ν1 (u 1 ) = m / ∈ U z and2 (u 1 ) = u 2 / ∈ U z . As ( C 2 ) is not satisfied for k = 3 , we have u 13 /∈(N ∪ { z } ) , but ν3 (u 1 ) = u 3 ∈ U z .

The next result shows that the set U z is connected for the

eighborhood relation ν , which implies that once we know one

¯ ∈ U z , we can find all u ∈ U z \{ u } with the help of Proposition 4.3 .

o prove the connectedness of U z , we construct a stable set of

oints N from N with respect to which U is an upper bound set.

K. Dächert et al. / European Journal of Operational Research 260 (2017) 841–855 847

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o this end, we transform every point z of N by projecting it onto

he dominance cone D z = { z ′ ∈ Z : z � z ′ } in the following way:

max z : z �→ ( max { z 1 , z 1 } , . . . , max { z p , z p } ) . ote that points of N that are (weakly) dominated by z remain un-

hanged by the transformation. Moreover, due to the GP assump-

ion, every point z � � z is projected to an individual point on the

oundary of cone D z . Also, only nondominated points among trans-

ormed points are considered for inclusion in N z . To summarize, we

efine N z as follows:

z = { pmax z (z) : z ∈ N } nd .

xample 4.5. Consider the set N = { z 1 , z 2 , z 3 } from Example 2.8 ,

here z 1 = (4 , 0 , 4) , z 2 = (3 , 3 , 1) and z 3 = (2 , 2 , 2) . Let z =(1 , 4 , 3) be a new point according to which N z is to be defined. We

ave pmax z (z 1 ) = (4 , 4 , 4) , pmax z (z 2 ) = (3 , 4 , 3) , and pmax z (z 3 ) =(2 , 4 , 3) . Thus N z = { (4 , 4 , 4) , (3 , 4 , 3) , (2 , 4 , 3) } nd = { (2 , 4 , 3) } .

emma 4.6. Let z ∈ S(N) . The set U z is an upper bound set with re-

pect to N z and restricted to the search interval Z z = [ z , M) .

roof. Denote by U(N z ) the upper bound set with respect to N z

nd restricted to the search interval Z z . We show that U z = U(N z ) ,

.e. U z contains exactly all the vectors of ˆ Z z = ( z , M] that satisfy ( P 1 )

nd ( P 2 ) with respect to N z .

• U z ⊆ U(N z ) : Since U z ⊆ U(N) , the vectors of U z satisfy ( P 1 )

and ( P 2 ) for N . We show that the properties are also satisfied

with respect to N z .

- ( P 1 ). For all u ∈ U z and z ∈ N , we have z � < u, which implies

pmax z (z) � < u, i.e. for all z ′ ∈ N z , z ′ � < u . This ensures ( P 1 ) for

all u ∈ U z with respect to N z .

- ( P 2 ). For all u ∈ U z and u ′ ≥ u , there exists ˆ z ∈ N such that

ˆ z < u ′ . Since by definition of U z , z < u, this implies that

pmax z ( z ) < u ′ . Either pmax z ( z ) ∈ N z or it is dominated by

some z ′ ∈ N z . In both cases, ( P 2 ) is satisfied for all u ∈ U z

with respect to N z . • U(N z ) ⊆ U z : For all u ∈ U(N z ) , it holds that z < u, since U(N z )

is restricted to ˆ Z z . We show that since u ∈ U(N z ) satisfies ( P 1 )

and ( P 2 ) with respect to N z , the properties are also satisfied

with respect to N .

- ( P 1 ). For all u ∈ U(N z ) and z ′ ∈ N z , it holds z ′ � < u . We have

z ′ = pmax z (z) for some z ∈ N . Since z < u, this implies that

z � < u, which ensures ( P 1 ) for u with respect to N .

- ( P 2 ). For all u ∈ U(N z ) and u ′ ≥ u , there exists z ′ ∈ N z such

that z ′ < u ′ . Since z ′ = pmax z (z) for some z ∈ N , we have z

< u ′ , which ensures ( P 2 ) for u with respect to N . �

We are now able to show that the subset of local upper bounds

ominated by some point z ∈ S(N) is connected.

roposition 4.7. The neighborhood relation νz restricted to U z , i.e.

z = { (u, u ′ ) ∈ ν : u, u ′ ∈ U z } is connected under Assumption GP .

roof. From Lemma 4.6 , U z is an upper bound set with re-

pect to a particular set of points N z . Therefore, according to

roposition 3.13 and since the vectors of U z are in general position,

he neighborhood relation defined on U z with respect to the defin-

ng points of N z is connected. Each point z ′ = pmax z (z) of N z has at

east one common component with z . Thus under Assumption GP ,

ach element of N z has a unique preimage through pmax z in N .

his implies that two local upper bounds that are neighbors with

espect to N z are also neighbors with respect to N . �

.2. Update of the neighborhood relation

Now, we discuss how to update the neighborhood relation for

(N ∪ { z } ) in an efficient way. In particular, we make use of the

urrent neighborhood in U ( N ) to establish the update. Therefore,

e first introduce the notion of the n th k -neighbor of a local upper

ound and define associated reverse indices.

efinition 4.8 ( n th k -neighbor ) . Consider u ∈ U ( N ) and the k -

eighborhood relation νk represented by graph G

k . The n th k -

eighbor of u , if it exists, is denoted by νk , n ( u ) and is the tail of

he unique path of length n starting at u , when this path exists.

e also define νk, 0 (u ) = u .

efinition 4.9 (Reverse indices) . For each arc ( u , u ′ ) of a path be-

ween two local upper bounds there exists a backward arc ( u ′ , u ) inxactly one of the graphs G

j for all j ∈ { 1 , . . . , p}\{ k } , respectively.

e call the corresponding index j = π(u ′ , u ) the reverse index and

enote the reverse indices of a path of length n by r 1 , . . . , r n .

xample 4.10. Consider the neighborhood structure depicted in

ig. 5 . We have ν1 , 0 (u 32 ) = u 32 , ν1 , 1 (u 32 ) = u 12 and ν1 , 2 (u 32 ) =

11 . The corresponding reverse indices are r 1 = 3 and r 2 = 2 .

roposition 4.11 (Update of the neighborhood relation) . Let N be a

table set of points satisfying Assumption GP and z ∈ S(N) . Consider a

ocal upper bound u ∈ U z and denote by I(u ) ⊆ { 1 , . . . , p} the subset

f indices for which u has an i-child u i , that is u i = ( z i , u −i ) ∈ U(N ∪ z } ) . Then, for all u i , i ∈ I ( u ), the j-neighbors ν j (u i ) , j = 1 , . . . , p, are

etermined as follows:

1. If j = i, then we have ν i (u i ) ∈ U(N) \ U z , i.e. ν i ( u i ) remains un-

changed by the insertion of z . More precisely, ν i (u i ) = ν i (u ) , i.e.

the i-neighbor of u i is the former i-neighbor of u. Moreover, the re-

verse index on this arc does not change, i.e. ν j (ν i (u )) = u i where

j is such that ν j (ν i (u )) = u .

2. If j � = i , then we have

(a) ν j (u i ) ∈ U (N ∪ { z } ) \ U (N) , i.e., ν j ( u i ) is a newly generated

local upper bound, and ν j ( u i ) is either the i- or the j-child

of some ˆ u ∈ U z .

(b) Moreover, ˆ u is linked to u by a path that consists of n 1 j-

neighbors, starting from u , followed by (n − n 1 ) i-neighbors,

for some 0 ≤ n 1 ≤ n. More precisely,

ν j (u

i ) =

{ˆ u

j i f r 0 = r 1 = · · · = r n 1 = i

ˆ u

i i f r 0 = r 1 = · · · = r n 1 −1 = i, r n 1 � = i (3)

where ˆ u = ν i,n −n 1 (ν j,n 1 (u )) , r 0 = i, and r 1 , . . . , r n denote

the reverse indices that correspond to the arcs in the path

from u to ˆ u . For n = 0 we obtain

ν j (u

i ) = u

j for all j ∈ I(u ) \{ i } (4)

as a special case of (3) , i.e., ˆ u = u .

roof. From Proposition 3.10 we know that a unique j -neighbor of

i exists. Hence, as soon as we find a suitable candidate that sat-

sfies Definition 3.6 , we have identified the correct j -neighbor of

i .

1. j = i : Since i ∈ I ( u ), ν i (u ) ∈ U(N ∪ { z } ) according to

Proposition 4.3 . Moreover, due to Lemma 3.3 and Definition 3.6 ,

z i ( u ) is not a defining point of ν i ( u ), but all z k ( u ), k � = i , are.

Since z i ( u i ) is the only defining point of u i that is different

from the defining points of u , ν i (u i ) = ν i (u ) holds. With the

same argumentation, for the index j such that ν j (ν i (u )) = u, u i

becomes the new j -neighbor of ν i ( u ), i.e. ν j (ν i (u )) = u i .

2. j � = i :

(a) We first show that ν j (u i ) ∈ U (N ∪ { z } ) \ U (N) . Indeed,

since any u ′ ∈ U ( N ) \ { u } and u share at most p − 1 defin-

ing points, and since z i (u i ) = z , u i and u ′ share p − 1

defining points if and only if u ′ = ν i (u ) , which corre-

sponds to case 1. Hence, ν j ( u i ) can not equal any of the

unchanged local upper bounds u ′ ∈ U ( N ) �{ u }, but must

be one of the newly generated local upper bounds.

848 K. Dächert et al. / European Journal of Operational Research 260 (2017) 841–855

Fig. 5. Neighborhood structure in Graph G related to the upper bound set U ( N ) of Example 2.8 (2).

Fig. 6. Visualization of the update of the neighborhood structure in part 2b of the proof of Proposition 4.11 : The nodes in each row represent components or defining points

of the local upper bounds which are denoted in the first column. A edge between two nodes indicates that the corresponding defining points are identical. A gray node

represents the component the corresponding local upper bound that will be defined by the new point z .

Since z defines one of the components for all u ′ ∈ U(N ∪{ z } ) \ U(N) and z = z i (u i ) holds, the sought j -neighbor of

u i must be either the i - or the j -child of some ˆ u ∈ U z . For

all other cases Definition 3.6 is not satisfied.

(b) Consider first the special case n = 0 which holds if,

and only if, u j = ( z j , u − j ) ∈ U(N ∪ { z } ) , i.e. j ∈ I ( u ) \ { i }.

Since z l (u i ) = z l (u j ) for all l � = i , j and z i (u i ) = z j (u j ) =z , u i and u j share p − 1 defining points. According to

Definition 3.6 , ν j (u i ) = u j (and ν i (u j ) = u i ).

Now we deal with the case n ≥ 1, i.e., j �∈ I ( u ). Consider

ν j, 1 (u ) , ν j, 2 (u ) , . . . until either ν j,n 1 (u ) has a j -child or

r n 1 � = i, for some n 1 ≥ 1. The first situation is illustrated

in Fig. 6 a while the latter is depicted in Fig. 6 b.

In the first case, let ˆ u = ν j,n 1 (u ) and ˆ u j be the j -child

of ˆ u . From r 1 = · · · = r n 1 = i, z l (u ) = z l ( u ) for all l � = i ,

j . Thus, additionally to z , u i and ˆ u j have p − 2 common

defining points, which, according to Definition 3.6 , im-

plies that ˆ u j is the j -neighbor of u i (and u i is the i -

neighbor of ˆ u j ).

In the second case, wew have r n 1 � = i, j. Let r n 1 = k .

We proceed in two steps: (i) We show that, if ˆ u =ν i,n −n 1 (ν j,n 1 (u )) exists and has an i -child for some n ≥ n 1 with r n 1 = · · · = r n = k, then it is the sought j -neighbor of

u i . (ii) It remains to show that the above path leading to

ˆ u does exist and is finite. This will conclude the proof

since in all cases, the existence of ˆ u with the required

properties is shown.

(i) Assume that ˆ u = ν i,n −n 1 (ν j,n 1 (u )) exists and has an i -

child for some n ≥ n 1 with r n 1 = · · · = r n = k . Since u and

ν j,n 1 −1 (u ) have identical defining points for all compo-

nents different from i , j , z l (u ) = z l ( u ) for l � = i , j , k and

z k (u ) = z j ( u ) . From the above and z i (u i ) = z i ( u i ) = z , ˆ u i

is, according to Definition 3.6 , the j -neighbor of u i (and

u i is the k -neighbor of ˆ u i ).

(ii) The existence of ˆ u i , i.e. in particular the fact that

r n = · · · = r n = k can be seen as follows.

1

We start with the case n = n 1 : If ν j,n 1 (u ) has an i -child

this local upper bound is the j -neighbor of u i according

to (i). Otherwise, i.e. if no i -child of ν j,n 1 (u ) exists and

hence n > n 1 , we know from Proposition 4.3 that the

search zone defined by the i -neighbor of ν j,n 1 (u ) con-

tains the current point z , i.e. that ν i (ν j,n 1 (u )) ∈ U z or, in

other words,

z < ν i (ν j,n 1 (u )) . (5)

We show in the following that (5) holds if and only

if r n 1 +1 = k = r n 1 . Let u ′ = ν j,n 1 (u ) and u ′′ = ν i (u ′ ) =ν i (ν j,n 1 (u )) . First, since u ′ ′ is i -neighbor of u ′ , r n 1 +1 � =i . Moreover, z i (u ′′ ) = z

r n 1 +1 (u ′ ) by definition of the i -

neighbor and the reverse index. Now suppose that

the above claim is not true, i.e., that r n 1 +1 � = k . Then

the defining point z i ( u ′ ′ ) equals one of the defining

points of u i different from z i ( u i ) (i.e., that of compo-

nent i of u i ), since if r n 1 +1 = j, then z i (u ′′ ) = z r n 1 +1 (u ′ ) =

z j (u ′ ) = z k (u ) = z k (u i ) , and if r n 1 +1 � = i, j, k, then z i (u ′′ ) =z

r n 1 +1 (u ′ ) = z r n 1 +1 (u ) = z

r n 1 +1 (u i ) . With Proposition 2.4 ,

which applied to u i states that z l i (u i ) < z i

i (u i ) for all

l � = i , and the fact that z i ( u ′ ′ ) equals one of the defin-

ing points of u i different from that of component i

we obtain u ′′ i

< z i i (u i ) = z i . This, however, is a contra-

diction to (5) , which therefore holds if and only if

r n 1 +1 = k .

In order to show that r n 1 = · · · = r n = k we proceed

in an iterative way: If the i -child of u ′ ′ exists it is

the j -neighbor of u i . Otherwise, from Proposition 4.3 ,

ν i, 2 (ν j,n 1 (u )) ∈ U z and, with exactly the same arguments

as above, r n 1 +2 = · · · = r n 1 +1 = r n 1 = k must hold.

The fact that U z is finite and that no local upper bound

is repeated in this construction implies that the i -child

of some ˆ u = ν i,n −n 1 (ν j,n 1 (u ))) exists, which finishes the

proof. �

K. Dächert et al. / European Journal of Operational Research 260 (2017) 841–855 849

Fig. 7. Update of the neighborhood structure.

E

u

{

b

t

ν

h

a

p

{

r

a

u

n

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p

v

3

u

P

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4

xample 4.12. In Fig. 7 we illustrate Proposition 4.11 for the local

pper bounds from Example 2.8 . In Fig. 7 a, we start with U(N) = u 0 } . Since u 0 ∈ U z 1 and I(u 0 ) = { 1 , 2 , 3 } , three new local upper

ounds u i , i = 1 , 2 , 3 are generated and the neighborhood among

hese local upper bounds is determined by Proposition 4.11 (2) , i.e.j (u i ) = u j for all i, j = 1 , 2 , 3 , j � = i . Moreover, ν i (u i ) = ν i (u 0 ) = m

olds for all i = 1 , 2 , 3 according to Proposition 4.11 (1) , i.e. all u i

re extreme local upper bounds.

In the second iteration, which is depicted in Fig. 7 b, local up-

er bounds u 1 , u 3 ∈ U z 2 holds and since I(u 1 ) = { 1 , 2 } and I(u 2 ) = 2 , 3 } , the children u 11 and u 12 as well as u 32 and u 33 are created,

espectively. The neighborhood between u 12 and u 2 as well as u 32

nd u 2 is determined by Proposition 4.11 (1) . Moreover, u 11 and

33 become new extreme local upper bounds. For the remaining

eighbors, Proposition 4.11 (2) , Case n ≥ 1 is used. Consider, e.g.,

he 3-neighbor of u 11 ( j = 3 , i = 1 ). We consider u 1 which is the

arent node of u 11 and consider ν3 (u 1 ) = u 3 . The associated re-

erse index is r = 1 = i . Hence, according to Case 1 for n = 1 , the

1

p

-child of u 3 is the 3-neighbor of u 11 , i.e. ν3 (u 11 ) = u 33 . Conversely,

11 is r 1 -neighbor of u 33 , i.e. ν1 (u 33 ) = u 11 .

In the third iteration, represented in Fig. 7 c, again all cases of

roposition 4.11 are used to obtain the new neighbors. In partic-

lar, consider u 323 and determine its 1-neighbor, i.e. j = 1 , i = 3 .

ince the parent node u 32 has no 1-child, we investigate its 1-

eighbor u 12 . The corresponding reverse index is r 1 = 3 = i . Hence,

e check whether a 1-child of u 12 exists. Since this is not the case,

e continue with the 1-neighbor of u 12 , which is u 11 . The reverse

ndex is r 2 = 2 � = i . Thus, we check whether a 3-child of u 11 exists.

ince the check is positive, u 113 is the 1-neighbor of u 323 .

Algorithm 1 summarizes an implementation of the results of

his section, i.e. the update of the search region and the neighbor-

ood relation with respect to a new point z .

.3. The non-general position case

Most of the previous results rely on Assumption GP . However,

articularly for discrete multi-objective optimization problems, this

850 K. Dächert et al. / European Journal of Operational Research 260 (2017) 841–855

Algorithm 1: Update procedure of an upper bound set based

on Proposition 4.11 .

input : z , u – Assumption: z < u

output : updated neighborhood relation

1 O ← { u } – List of LUBs to visit, all u ∈ O satisfy z < u

2 B ← ∅ – List of LUBs created at the current iteration, eventually

U(N ∪ { z } ) \ U(N)

3 while O � = ∅ do

4 Select u ∈ O ; O ← O \ { u } – u is not in U(N ∪ { z } ) 5 visited( u ) ← true

6 I ← { i = 1 , . . . , p : z i ≥ ν i i (u ) }

7 for i ∈ I do

– Let u i = ( z i , u −i )

8 B ← B ∪ { u i } – u i is in U(N ∪ { z } ) 9 for j ∈ I : j < i do

10 ν j (u i ) ← u j ; ν i (u j ) ← u i – Apply Proposition 4.11 (2)

11 ν i (u i ) ← ν i (u ) – Apply Proposition 4.11 (1)

12 νr (ν i (u )) ← u i with r = π(ν i (u ) , u )

13 for j ∈ J = { 1 , . . . , p} \ I do

14 neighborFound( u i , j) ← false

15 if not visited( ν j (u ) ) then

16 O ← O ∪ { ν j (u ) }

– Apply Proposition 4.11 (2) to link remaining neighbors

17 for u i ∈ B do

18 for j ∈ { 1 , . . . , p} do

19 if not neighborFound( u i , j) then

20 ˆ u ← ν j (u ) – Consider the j-neighbor of the parent

of u i

21 r ← π( u , u ) – Reverse index, i.e. νr ( u ) = u

22 t = j

23 while not neighborFound( u i , j) do

24 if r � = i then

25 t = i

26 if ∃ u t then

27 ν j (u i ) ← ˆ u t ; νr ( u t ) ← u i

28 neighborFound( u i , j) ← true

29 else

30 r ← π(νt ( u ) , u )

31 ˆ u ← νt ( u )

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assumption is not satisfied, in general. In order to be able to ap-

ply these results, and, in particular, Algorithm 1 in any situation,

we must discriminate among equal values on each objective. This

can be achieved by defining p total orders ≺i , compatible with < ,

on the values reached by the points in N for each objective i =1 , . . . , p. Assuming the points are assigned distinct indices, we de-

fine a lexicographic order ≺i for all i ∈ { 1 , . . . , p} . Let

z r i �i z s i ⇔ z r i < z s i or (z r i = z s i and r ≤ s ) . (6)

We then define ≺i as the asymmetric part of �i .

The only place where component values are compared in

Algorithm 1 is at Step 6, i.e. in the application of Proposition 4.1 .

Therefore these total orders are only explicitly required for this

step. Moreover, the comparison always occurs between the current

point and the component value of some point considered before.

By assigning to the current point an index that is strictly greater

than the points considered before, the comparisons are always of

the form z s i �i z

r i

with s > r , which is z s i ≥ z r

i . That is why we di-

ectly formulated Step 6 using ≥ in order to cover also the non-

eneral position case.

Proceeding this way with instances in non-general position en-

ures that the computed upper bound set U

′ ( N ) is stable with re-

pect to � defined as follows:

� u

′ ⇔ z i i (u ) �i z i i (u

′ ) for all i ∈ { 1 , . . . , p} and there exists i such that z i i (u ) ≺i z

i i (u

′ ) . t may, however, not be stable with respect to the dominance re-

ation ≤, i.e. it may induce redundant search zones. Nevertheless,

rom the stability with respect to �, u ≤ u ′ for some u , u ′ ∈ U

′ ( N )

mplies that there exists i ∈ { 1 , . . . , p} such that u i = u ′ i , i.e. the case

< u ′ is not possible. The set U

′ ( N ) is referred to as a quasi-upper

ound set and the search zone C ( u ) associated to such a vector u is

ualified as quasi-nonredundant . If not too many points share the

ame component value in a given dimension, such cases of quasi-

onredundant search zones C ( u ) are limited.

In the case of Algorithm 1 , removing the corresponding local

pper bounds may disconnect the neighborhood relation. There-

ore, we maintain these quasi-nonredundant local upper bounds in

lgorithm 1 , but avoid searching for new points in them.

xample 4.13. Let z 1 = (4 , 0 , 4) z 2 = (4 , 3 , 1) and z 3 = (2 , 3 , 2) .

hile these three points are pairwise nondominated, they do not

atisfy Assumption GP , since z 2 1

= z 1 1

and z 3 2

= z 2 2

.

Using ≺i as defined in Eq. (6) , we have z 2 1 ≺1 z 1 1 and z 3

2 ≺2 z

2 2 .

he corresponding quasi-upper bound set is:

u u 1 u 2 u 3

u

11 2 M 2 M 3

u

12 4 3 M 3

u

2 M 1 0 M 3

u

312 4 3 4

u

313 4 M 2 2

u

32 M 1 3 4

u

33 M 1 M 2 1

u z 1 (u ) z 2 (u ) z 3 (u ) u

11 z 3 ˆ z 2 ˆ z 3

u

12 z 1 z 3 ˆ z 3

u

2 ˆ z 1 z 1 ˆ z 3

u

312 z 2 z 3 z 1

u

312 z 2 ˆ z 2 z 3

u

32 ˆ z 1 z 2 z 1

u

33 ˆ z 1 ˆ z 2 z 2

e observe that u 312 is quasi-nonredundant since u 312 ≤ u 12 and

312 ≤ u 32 . Note that here one other case of quasi-nonredundant

ocal upper bounds is nevertheless avoided by the application of

C 2 ): u 13 = (4 , M 2 , 2) .

. Complexity and computational experiments

In this section, we analyze the efficiency of our approach

ompared to two procedures that have been proposed recently,

amely the “redundancy elimination” (RE) approach of Klamroth

t al. (2015) , which slightly enhances a procedure proposed by

rzybylski et al. (2010) , and the “redundancy avoidance” (RA) ap-

roach of Klamroth et al. (2015) .

In Section 5.1 , we analyze the computational complexity of

lgorithm 1 as compared to the RE and RA algorithms. Next, we

emonstrate the practical efficiency of Algorithm 1 in computa-

ional tests. We describe the general setting in Section 5.2 , the

nstances used in Section 5.3 and finally provide and discuss our

esults in Section 5.4 .

.1. Complexity analysis

Algorithm 1 is specifically designed for the case where non-

ominated points are generated iteratively in selected search

ones. This is a common strategy, for example, in scalarization

ased methods. For this reason, we assume in the following anal-

sis that, together with a new point z , one search zone C( u ) con-

aining z is known. Note that we do not assume that all search

ones in the set U z , i.e., all search zones containing the new point

¯ , are known. We make the usual assumption that the number p

K. Dächert et al. / European Journal of Operational Research 260 (2017) 841–855 851

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f objectives is fixed, and that Assumption GP is satisfied. Know-

ng one search zone containing the new point z and using the fact

hat all local upper bounds in U z are connected ( Proposition 4.7 ),

e can upper bound the complexity of Algorithm 1 in terms of

U z | . roposition 5.1. Algorithm 1 has time complexity O(| U z | ) . roof. The time needed for the first loop (Steps 3–16) is O(| U z | )ince every local upper bound of U z is visited once.

For the second part of Algorithm 1 (Steps 17–31), O(p| U z | ) =(| U z | ) newly generated local upper bounds in the set B have

o be integrated into the neighborhood structure between lo-

al upper bounds. We show below that each local upper bound

f U z cannot be visited more than O(1) times. In this part of

lgorithm 1 , the neighborhood relation is updated using part 2(b)

f Proposition 4.11 . Any u ′ ∈ U z that is visited in this construction

elongs to a path of j - or i -neighbors that is traversed to find the

-neighbor of some new local upper bound, which will be the j -

r the i -child, respectively, of some ˆ u ∈ U z . The endpoint ˆ u of such

path always exists, i.e., a neighbor is found at the end of the

ath, and it only depends on the existence of the j - or the i -child

hich, in turn, only depends on the component values of z , which

re fixed. Since these children, like any local upper bound, may

ave at most p neighbors ( Proposition 3.10 ) and may thus be the

eighbor of at most p new local upper bounds, any of these paths

ields at most p visits of u ′ . Given that there are p paths of each of

he two types (“j ” and “i ”), u ′ may be visited at most 2 p 2 = O(1)

imes. �

The following list compares this worst case complexity with the

ounds for the RE and RA algorithms, respectively (see Klamroth

t al., 2015 ).

• Algorithm 1 : O(| U z | ) • Algorithm RE: O(| U(N) | + | U z | 2 ) • Algorithm RA: O(| U(N) | + | U z | )

The term | U ( N )| in Algorithms RE and RA comes from the de-

ermination of the set U z . If efficient data structures like, for exam-

le, dynamic p -dimensional range trees are used to store the set

( N ), this can be reduced to O( log p | U(N) | + | U z | ) , see Klamroth

t al. (2015) for further details and references. Note that | U(N) | =(| N| � p/ 2 � ) in the worst case. Even though O(| U z | ) = O(| U(N) | ) in

he worst case, this only occurs in pathological instances and only

or a minority of the generated points z . In practice, | U z | is usu-

lly very small as compared to U ( N ) and we observe substantial

avings with respect to the computation time of Algorithm 1 com-

ared to the RE and RA algorithms, particularly when N increases

see Section 5.4 below).

If we drop the assumption that a search zone containing the

ew point z is initially known, Algorithm 1 would have to search

he complete set U ( N ) in the worst case so as to find a first search

one containing the new point z , which overrides its advantage

ver Algorithms RE and RA to some extent. However, preliminary

ests indicate that, even if no search zone containing the new point

¯ is initially known, such a zone is often found quickly during the

rocedure. From that time on, Algorithm 1 can stop searching the

et U ( N ) and utilize the neighborhood structure instead, while Al-

orithms RE and RA have to continue searching the complete set

( N ) for potential further search zones containing the new point z .

.2. Setting

The compared algorithms are all embedded into a general

cheme which simulates a solution strategy based on a scalariza-

ion technique to compute the nondominated set. For our tests, we

se as instance a set of points instead of actually generating non-

ominated points of a MOCO problem on the fly with the help of

ome solver.

The procedure is summarized in Algorithm 2 and described

n what follows. Starting with an initial search region described

y a unique search zone u 0 = M, a local upper bound u is se-

ected at each iteration and it is checked whether the associated

earch zone C ( u ) contains a point from the instance. If so, the cur-

ent upper bound set is updated using either our new approach

Algorithm 1 ), the “redundancy elimination” (RE) approach or the

redundancy avoidance” (RA) approach of Klamroth et al. (2015) .

therwise, when u does not contain further nondominated points,

t is moved to a list E of empty zones. We do not remove empty

earch zones for two reasons. First, when the algorithm terminates,

ist E corresponds to U ( N ) (or a superset of it for Algorithm 1 in the

GP-case). Knowing U ( N ) is useful when additional points are gen-

rated after having inserted the given ones. Secondly, in the case of

lgorithm 1 , the set E is required to apply criterion ( C 2 ) correctly.

imilarly, in the case of Algorithm (RE), the set E is needed to filter

ut redundant zones in the NGP-case.

Algorithm 2: Simulation of a solution strategy that relies on

a scalarization technique.

input : N, M

1 O ← { M} ; – List of search zones that have not been explored

yet

2 E ← ∅ ; – List of search zones proven to be empty

3 while O � = ∅ do

4 Select u ∈ O

5 if there exists z ∈ N such that z < u then

6 Update O using either Algorithm 1 , Algorithm (RE) or

Algorithm (RA)

7 else

8 E ← E ∪ { u } 9 O ← O \ { u }

If Assumption GP is not satisfied, we modify

lgorithm 2 slightly in order to avoid that redundant local

pper bounds are selected, since this would imply the solution of

nnecessary scalarizations, which are costly in terms of computa-

ion time. Firstly, we make sure that in Step 4 of Algorithm 2 we

lways select an element from the set of local upper bounds which

s not redundant. Recall that redundancy might only occur in the

GP-case. We can check quickly with the help of the neighborhood

elation whether a local upper bound is redundant. If so, we test

ts respective neighbor (with respect to which it is redundant)

ext and go on until we find a non-redundant local upper bound.

s a second modification with respect to Algorithm 2 , we move

edundant local upper bounds to E as soon as a local upper bound

ontaining them turns out to be empty and is moved to E in Step

of Algorithm 2 .

.3. Instances

The algorithms are tested on random stable sets of points. We

se the same instance type as in Klamroth et al. (2015) . For each

nstance, points are randomly generated in [0 , K] p ⊂ Z

p accord-

ng to the uniform distribution and inserted to the instance be-

ng constructed if they are not dominated by, nor dominate, any

reviously generated point, until the desired number of points is

btained.

We consider instances in general position (GP), i.e., sat-

sfying Assumption GP , for which K was chosen as a large

852 K. Dächert et al. / European Journal of Operational Research 260 (2017) 841–855

Fig. 8. Computation time (second) as a function of | N | on GP instances.

K. Dächert et al. / European Journal of Operational Research 260 (2017) 841–855 853

Fig. 9. Computation time (second) as a function of | N | on NGP instances.

854 K. Dächert et al. / European Journal of Operational Research 260 (2017) 841–855

Table 1

Number of local upper bounds and computation times.

GP instances NGP instances

Avg. comp. time (second) Avg. comp. time (second)

p | N | Avg. | U ( N )| Algorithm 1 Alg. (RE) Alg. (RA) Avg. | U ( N )| Algorithm 1 Alg. (RE) Alg. (RA)

3 10 0 0 0 20 0 01 .00 1 .98 6 .63 3 .10 13584 .30 1 .32 3 .68 2 .13

20 0 0 0 40 0 01 .00 8 .11 58 .03 20 .57 29334 .90 6 .03 38 .38 10 .79

30 0 0 0 60 0 01 .00 19 .80 152 .52 75 .54 44895 .50 15 .64 120 .65 47 .47

40 0 0 0 80 0 01 .00 46 .89 279 .89 151 .80 61633 .00 37 .71 242 .95 132 .96

50 0 0 0 10 0 0 01 .00 95 .37 481 .39 278 .21 78517 .80 68 .87 383 .00 213 .56

4 40 0 0 31799 .30 0 .86 7 .36 8 .61 19122 .00 0 .60 2 .67 1 .55

80 0 0 62400 .00 3 .30 42 .04 39 .40 43068 .20 2 .71 30 .80 23 .45

120 0 0 92354 .70 7 .44 99 .64 92 .30 66438 .30 6 .43 73 .81 59 .48

160 0 0 121382 .40 14 .01 179 .06 154 .70 91602 .30 12 .40 136 .47 106 .86

20 0 0 0 149933 .40 22 .89 278 .77 245 .68 116069 .60 20 .80 229 .03 177 .69

5 20 0 0 77885 .40 0 .89 14 .42 16 .17 38362 .40 0 .70 6 .63 7 .02

40 0 0 155146 .60 3 .22 59 .73 64 .90 93608 .70 2 .48 37 .31 35 .93

60 0 0 227893 .60 6 .53 136 .25 146 .56 152020 .60 5 .69 91 .58 90 .40

80 0 0 298735 .60 11 .20 238 .85 256 .45 209835 .70 9 .87 171 .49 172 .22

10 0 0 0 364530 .00 17 .97 361 .22 382 .88 273079 .90 16 .07 270 .79 300 .50

6 10 0 0 161508 .10 1 .58 36 .53 19 .24 67889 .40 0 .96 10 .31 8 .16

20 0 0 330098 .20 4 .47 120 .64 80 .20 20 0 0 03 .70 3 .53 65 .36 48 .94

30 0 0 494377 .20 8 .45 261 .69 188 .38 354565 .00 7 .29 182 .84 136 .38

40 0 0 664705 .70 14 .22 447 .64 361 .96 514890 .30 12 .05 360 .99 285 .94

50 0 0 812160 .20 20 .80 685 .73 593 .50 6 8074 9 .70 18 .11 571 .21 432 .60

7 200 62980 .40 0 .48 9 .49 1 .72 11596 .90 0 .24 0 .51 0 .26

400 170082 .60 1 .82 58 .17 9 .27 50433 .80 1 .04 7 .41 2 .98

600 290796 .40 3 .56 150 .59 24 .31 118654 .80 2 .52 34 .45 10 .36

800 427613 .20 5 .97 272 .17 49 .01 201791 .50 4 .31 83 .09 21 .99

10 0 0 552532 .20 8 .10 474 .67 74 .87 301500 .20 6 .61 163 .92 41 .28

8 100 57358 .50 0 .45 16 .58 0 .94 5113 .10 0 .26 0 .18 0 .04

200 202062 .00 2 .23 170 .02 5 .57 34198 .30 1 .37 5 .10 1 .19

300 371875 .10 5 .04 488 .70 16 .81 88255 .40 3 .41 25 .76 4 .26

400 575059 .50 8 .68 976 .62 34 .39 177245 .80 6 .58 91 .60 11 .95

500 797109 .00 11 .82 1526 .14 61 .53 304008 .30 11 .02 201 .15 24 .25

9 100 152105 .40 1 .72 157 .10 2 .54 11957 .80 1 .11 0 .76 0 .22

200 608758 .20 8 .69 – 19 .54 98632 .80 7 .08 41 .08 3 .69

300 1313695 .40 23 .78 – 56 .92 272061 .00 15 .92 368 .46 14 .09

400 2182362 .60 44 .54 – 138 .26 639902 .10 32 .58 1364 .86 46 .95

500 3089809 .60 65 .83 – 245 .83 1062888 .20 55 .33 2514 .22 87 .27

10 50 74222 .00 0 .67 73 .74 0 .77 2892 .60 0 .56 0 .07 0 .02

100 386030 .00 5 .21 – 6 .87 27833 .20 3 .55 4 .82 0 .58

150 961036 .40 15 .10 – 23 .16 104077 .60 11 .89 57 .72 3 .12

200 1679918 .20 26 .44 – 52 .13 255362 .70 24 .60 269 .60 9 .71

250 2531965 .30 46 .05 – 100 .81 517430 .60 33 .83 895 .24 22 .63

“–” indicates that some instances could not be solved within one hour.

p

w

a

w

i

p

t

p

b

o

m

s

A

s

i

i

q

t

n

b

number. We also consider instances in non-general position (NGP),

by choosing K so that K | N| = 10 . For both configurations (GP/NGP),

we generate instances with p = 3 , . . . , 10 and with up to 50 0 0 0

points. For a given dimension p , the maximal number of points

in the generated instances is chosen so as to obtain a running

time of the order of 100 seconds for Algorithm 1 . From the

theoretical results reported in the introduction, we expect that

the number of local upper bounds increases with p and, thus,

the update of the search region becomes more time-consuming.

Therefore, as the number of objectives increases, we generate in-

stances with fewer points. While, e.g., for p = 3 we create in-

stances with 50 0 0 0 points, the instances for p = 10 only com-

prise up to 250 points. Finally, each type of instance is generated

10 times.

5.4. Results

All algorithms are implemented in C, compiled with gcc4.3.4 , and run under SUSE Linux Enterprise Server 11 on work-

stations equipped with four Intel Xeon E7540 CPU at 2.00 giga-

hertz and with 128 gigabyte of RAM.

The computation times are reported in Figs. 8 and 9 , for GP and

NGP instances, respectively. They provide a quick overview on the

erformance of each method. Note that we use logarithmic scales

ith respect to both axes, i.e. the number of given points as well

s the computation time. Precise values of computation times, as

ell as the average final number of local upper bounds for each

nstance type, are reported in Table 1 , for half of the instances de-

icted in Figs. 8 and 9 .

For the GP instances Algorithm 1 clearly outperforms the other

wo algorithms for all problem sizes. Moreover, Algorithm (RA)

erforms better than Algorithm (RE) except in a few cases for

p = 4 and p = 5 . While the gain in computation time of the neigh-

orhood variant is considerable for instances with three to seven

bjectives, it becomes rather small for instances with eight or

ore objectives. Recall that the more objectives the tested in-

tances have, the less points are inserted. While Algorithm 1 and

lgorithm (RA) produce similar computation times for rather small

ets of points, the neighborhood relation can show its advantages

n particular for a high number of nondominated points.

In the NGP case, an additional computational effort is made

n order to handle redundant local upper bounds. As a conse-

uence, Algorithm (RA), and even Algorithm (RE), perform better

han Algorithm 1 for a small number of points. However, as the

umber of points increases, our new algorithm performs much

etter, in general. This is all the more true for large values of

K. Dächert et al. / European Journal of Operational Research 260 (2017) 841–855 855

p

r

6

t

p

t

c

r

t

m

p

i

a

p

a

i

n

b

u

c

p

n

m

s

A

t

(

b

5

n

R

A

B

B

C

D

D

K

K

K

K

L

O

P

R

S

S

S

T

U

V

, where the cardinality of the upper bound set U ( N ) increases

apidly.

. Conclusion and further ideas

The computation of upper bound sets that concisely describe

he search region, given a finite set of nondominated points, is a

revailing subproblem when solving multi-objective objective op-

imization problems. In this paper, a new and efficient method for

omputing and updating upper bound sets (and hence the search

egion) is developed that is based on a neighborhood structure be-

ween local upper bounds. Aside from revealing a beautiful geo-

etrical relationship between nondominated points and local up-

er bounds, irrespective of the dimension, the results provide new

nsights for a variety of different applications. A fundamental ex-

mple are nadir point computations in higher dimensions. More

recisely, by investigating local upper bounds that are defined by

t least one dummy point, the “boundary region” of the nondom-

nated set may be accessed and searched for nadir point compo-

ents quickly. Similarly, in the context of generation algorithms

ased on the repeated solution of scalarized subproblems, local

pper bounds and their interrelation can be used towards effi-

ient implementations where selected scalarizations are solved in

arallel.

Further interesting research questions address the use of the

eighborhood relation for generating points within interactive

ethods or the generation of representations of the nondominated

et with respect to some quality measures.

cknowledgments

This work was partially supported by a fellowship within

he Postdoc-Program of the German Academic Exchange Service

DAAD) and by the bilateral cooperation project MOCO-BBC funded

y the German Academic Exchange Service (DAAD, Project-ID

7212018 ) and by Campus France, PHC PROCOPE 2016 (Project

o. 35476QD).

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