Investigation on Stochastic Local Search for Decentralized Asymmetric

Multi-objective Constraint Optimization Considering Worst Case

Toshihiro Matsui

Nagoya Institute of Technology, Gokiso-cho Showa-ku Nagoya Aichi 466-8555, Japan

Keywords:

Asymmetric Distributed Constraint Optimization, Multi-Objective, Stochastic Local Search, Multiagent.

Abstract:

The Distributed Constraint Optimization Problem (DCOP) has been studied as a fundamental problem in

multiagent cooperation. With the DCOP approach, various cooperation problems including resource allocation

and collaboration among agents are represented and solved in a decentralized manner. Asymmetric Multi-

Objective DCOP (AMODCOP) is an extended class of DCOPs that formalizes the situations where agents

have individual objectives to be simultaneously optimized. In particular, the optimization of the worst case

objective value among agents is important in practical problems. Existing works address complete solution

methods including extensions with approximation. However, for large-scale and dense problems, such solution

methods are insufﬁcient. Although the existing studies also address a few simple deterministic local search

methods, there are opportunities to introduce stochastic local search methods. As the basis for applying

stochastic local search methods to AMODCOPs for the preferences of agents, we introduce stochastic local

search methods with several optimization criteria. We experimentally analyze the inﬂuence of the optimization

criteria on perturbation in the exploration process of search methods and investigate additional information

propagation that extends the knowledge of the agents who are performing the local search.

1 INTRODUCTION

The Distributed Constraint Optimization Problem

(DCOP) has been studied as a fundamental prob-

lem in multiagent cooperation (Fioretto et al., 2018).

With the DCOP approach, various cooperation prob-

lems including resource allocation and collaboration

among agents are represented and solved in a decen-

tralized manner. Asymmetric Multi-Objective DCOP

(AMODCOP) is an extended class of DCOPs that

formalizes the situations where agents have individ-

ual objectives to be simultaneously optimized (Mat-

sui et al., 2018a). In particular, the optimization of

the worst case objective value among agents is im-

portant in practical problems. Existing works address

complete solution methods including extensions with

approximation. However, for large-scale and dense

problems, such solution methods are insufﬁcient. Al-

though the current studies also address a few sim-

ple deterministic local search methods (Matsui et al.,

2018b), there are opportunities to introduce stochas-

tic local search methods. As the basis for applying

stochastic local search methods to AMODCOPs for

the preferences of agents, we introduce stochastic lo-

cal search methods with several optimization criteria.

We experimentally analyze the inﬂuence of the op-

timization criteria on perturbation in the exploration

process of the search methods. We also investigate ad-

ditional information propagation, which extends the

knowledge of the agents that are performing the local

search.

The contribution of this study is as follows. 1) We

apply several variants of a fundamental stochastic lo-

cal search method to a class of Asymmetric Multi-

Objective DCOPs where the worst case cost value

among agents is improved. 2) The effect of the

stochastic local search with different optimization cri-

teria is experimentally investigated, and we show the

cases where a leximin based criterion is effective with

the stochastic local search.

The rest of our paper is organized as follows. In

the next section, we present our preliminary study

that includes standard DCOPs, solution methods for

DCOPs, Asymmetric Multi-Objective DCOPs that

consider the worst case, social welfare, and the scal-

ability issues of solution methods for Asymmetric

Multi-Objective DCOPs. In Section 3, we pro-

pose decentralized stochastic local search methods for

AMODCOPs considering the worst case cost value

among agents. We apply fundamental stochastic lo-

462

Matsui, T.

Investigation on Stochastic Local Search for Decentralized Asymmetric Multi-objective Constraint Optimization Considering Worst Case.

DOI: 10.5220/0010395504620469

In Proceedings of the 13th International Conference on Agents and Artiﬁcial Intelligence (ICAART 2021) - Volume 1, pages 462-469

ISBN: 978-989-758-484-8

Copyright

c

2021 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

cal search methods with several optimization criteria

to this class of problems. We experimentally investi-

gate the proposed approach in Section 4. We brieﬂy

discuss our work in Section 5 and conclude in Sec-

tion 6.

2 PRELIMINARY

We address a class of DCOPs where agents asymmet-

rically evaluate their local objectives, and multiple ob-

jectives of the agents are simultaneously optimized.

Our problem is based on the deﬁnition in (Matsui

et al., 2018a), and we mainly concentrate on the prob-

lems where the worst case cost among agents is min-

imized.

2.1 Distributed Constraint

Optimization Problems

A Distributed Constraint Optimization Problem

(DCOP) is deﬁned by hA, X, D, Fi where A is a set of

agents, X is a set of variables, D is a set of the domains

of the variables, and F is a set of objective functions.

Variable x

i

∈ X represents a state of agent i ∈ A. Do-

main D

i

∈ D is a discrete ﬁnite set of values for x

i

.

Objective function f

i, j

(x

i

, x

j

) ∈ F deﬁnes a utility ex-

tracted for each pair of assignments to x

i

and x

j

. The

objective value of assignment {(x

i

, d

i

), (x

j

, d

j

)} is de-

ﬁned by binary function f

i, j

: D

i

× D

j

→ N

0

. For as-

signment A of the variables, global objective function

F(A ) is deﬁned as F(A ) =

∑

f

i, j

∈F

f

i, j

(A

↓x

i

, A

↓x

j

),

where A

↓x

i

is the projection of assignment A on x

i

.

The value of x

i

is controlled by agent i, which lo-

cally knows the objective functions that are related to

x

i

in the initial state. The goal is to ﬁnd global opti-

mal assignment A

∗

that minimizes the global objec-

tive value in a decentralized manner. For simplicity,

we focus on the fundamental case where the scope of

constraints/functions is limited to two variables, and

each agent controls a single variable.

2.2 Solution Methods for DCOPs

The solution methods for DCOPs are categorized into

exact and inexact solution methods (Fioretto et al.,

2018). The former are based on tree search and dy-

namic programming (Fioretto et al., 2018). However,

the time/space complexity of such exact methods

is generally exponential for one of size parameters

of problems. Therefore, applying the exact meth-

ods to large-scale and densely constrained problems

is difﬁcult. On the other hand, inexact solution meth-

ods consist of a number of approaches (Fioretto et al.,

2018) including hill-climbing local search, stochastic

random sampling and belief propagation.

In this study, we focus on Distributed Stochas-

tic search Algorithm (DSA) (Zhang et al., 2005; Zi-

van, 2008), which is a baseline algorithm of inexact

methods. DSA is the simplest stochastic local search

method for DCOPs where agents locally determine

the assignment to their own variables by exchanging

their assignments with neighborhood agents that are

related by constraints/functions. For each agent i, an

iteration of the algorithm is synchronously performed

as follows.

1. Set the initial assignment to variable x

i

of agent i.

2. Collect the current assignment of the variables of

the neighborhood agents in Nbr

i

. With the col-

lected assignment and functions related to agent i,

locally evaluate all the assignments of variable x

i

.

Then select a new assignment to x

i

with a stochas-

tic strategy and the evaluation for x

i

.

3. Update the assignment to x

i

. Repeat from step 2.

Each agent i has a view of assignments to the vari-

ables of the neighborhood agents Nbr

i

that is denoted

by A

i

. Agent i’s local cost is deﬁned as f

i

(A

i

) =

∑

j∈Nbr

i

f

i, j

(A

i↓x

i

, A

i↓x

j

). Here we assume that A

i

also

contains i’s own assignment. The agent only interests

to improve its local cost value and does not receive

local cost values of other agents.

Although there are a few versions of stochas-

tic search strategies, most of them randomly select

1) hill-climb, 2) stay as is, or 3) switch to one of other

assignments to its own variable whose local objective

value equals the current local objective value. With

preliminary experimental investigation, we prefer the

following settings in this work. 1) If there are other

assignments to the agent’s own variable that improves

the local objective, the agent performs a hill-climb

with probability P

a

and randomly selects one of the

assignments related to the best improvement with uni-

form distribution. 2) Otherwise, the agent randomly

selects one of other assignments with probability P

b

.

With a relatively large P

a

and a relatively smaller P

b

,

the hill-climb and escape from local optimal solutions

are stochastically performed.

Although there are a number of sophisticated so-

lution methods for DCOPs, DSA is considered a base-

line method. Therefore, there are opportunities to

investigate the performance of the stochastic local

search for the problems based on min-max problems.

Several random-walk solution methods re-

quire/employ snapshot algorithms to capture global

objective values and related global solutions. A fun-

Investigation on Stochastic Local Search for Decentralized Asymmetric Multi-objective Constraint Optimization Considering Worst Case

463

variable of agent a

1

asymmetric constraint

x

1

x

2

x

3

x

4

local valuation of a

1

x

1

x

2

f

1,2

f

2,1

0 0 a b

0 1 c d

1 0 e f

1 1 g h

Figure 1: AMODCOP.

damental approach is based on a spanning tree on a

constraint graph and the implicit/explicit timestamps

of locally evaluated objective values to aggregate

them in a synchronized manner (Mahmud et al.,

2020). With such methods, global objective values

can be aggregated with the number of communication

cycles that equals the depth of the spanning tree.

A similar additional snapshot algorithm has been

applied to DSA (Zivan, 2008). Note that it is possible

to employ similar mechanism so that the agents

symmetrically and simultaneously collect snapshots

using the same single spanning tree, where each agent

performs as a root node of the spanning tree. Here

we concentrate on a solution method and employ

snapshots on a simulator for simplicity.

2.3 Asymmetric Multi-objective

Distributed Constraint

Optimization Problem

An Asymmetric Multiple Objective DCOP on the

preferences of agents (AMODCOP) is deﬁned by

hA, X, D, Fi, where A, X and D are similarly deﬁned

for the DCOP in Section 2.1. Agent i ∈ A has its local

problem deﬁned for X

i

⊆ X . For neighboring agents

i and j, X

i

∩ X

j

6=

/

0. F is a set of objective functions

f

i

(X

i

). Function f

i

(X

i

) : D

i

1

× ··· × D

i

k

→ N

0

repre-

sents the objective value for agent i based on the vari-

ables in X

i

= {x

i

1

, ··· , x

i

k

}. For simplicity, we con-

centrate on the case where each agent has a single

variable and relates to its neighborhood agents with

binary functions. The functions are asymmetrically

deﬁned and locally aggregated. Variable x

i

of agent

i is related to other variables by objective functions.

When x

i

is related to x

j

, agent i evaluates objective

function f

i, j

(x

i

, x

j

). On the other hand, j evaluates

another function f

j,i

(x

j

, x

i

). Each agent i has function

f

i

(X

i

) that represents the local problem of i that ag-

gregates f

i, j

(x

i

, x

j

). We deﬁne the local evaluation of

agent i as summation f

i

(X

i

) =

∑

j∈Nbr

i

f

i, j

(x

i

, x

j

) for

neighborhood agents j ∈ Nbr

i

related to i by objec-

tive functions.

Global objective function F(A ) is deﬁned as

[ f

1

(A

1

), ··· , f

|A|

(A

|A|

)] for assignment A to all the

variables. Here A

i

denotes the projection of assign-

ment A on X

i

. The goal is to ﬁnd assignment A

∗

that

minimizes the global objective based on a set of ag-

gregation and evaluation structures. Figure 1 shows

an example of AMODCOP.

2.4 Multiple Objectives and Social

Welfare

Since multiple objective problems among individual

agents cannot be simultaneously optimized in gen-

eral cases, several criteria such as Pareto optimality

are considered. However, there are generally a huge

number of candidates of optimal solutions based on

such criteria. Therefore, several social welfare and

scalarization functions are employed. With aggrega-

tion and comparison operators ⊕ and ≺, the mini-

mization of the objectives is represented as follows:

A

∗

= argmin

≺

A

⊕

i∈A

f

i

(A) .

Several types of social welfare (Sen, 1997) and

scalarization methods (Marler and Arora, 2004) are

employed to handle objectives. In addition to the

summation and comparison of scalar objective val-

ues, we consider several criteria based on the worst

case objective values (Matsui et al., 2018a). Although

some operators and criteria are designed for the maxi-

mization problems of utilities, we employ similar cri-

teria for minimization problems.

Summation

∑

a

i

∈A

f

i

(X

i

) only considers the total

utilities. Min-max criterion min max

a

i

∈A

f

i

(X

i

) im-

proves the worst case cost value. This criterion is

called the Tchebycheff function. Although it does not

consider global cost values and is not Pareto optimal,

the criterion to improve the worst case is practically

important in situations without trades of cost (util-

ity) values. To improve the global cost values, ties

of min-max are broken by comparing the summation

values, and the criterion is Pareto optimal. We em-

ploy the lexicographic augmented Tchebycheff func-

tion that independently compares maximum and sum-

mation values (Marler and Arora, 2004).

Lexmin for maximization problems is an exten-

sion of the max-min that is the maximization ver-

sion of min-max. With this criterion, utility values

are represented as a vector whose values are sorted

in ascending order, and the comparison of two vec-

tors is based on the dictionary order of the values in

the vectors. Maximization with leximin is Pareto op-

timal and relatively improves the fairness among ob-

jectives. We address ‘leximax’, which is an inverted

lexmin for minimization problems, where objective

values are sorted in descending order. See the liter-

ature for details (Sen, 1997; Marler and Arora, 2004;

Matsui et al., 2018a).

ICAART 2021 - 13th International Conference on Agents and Artiﬁcial Intelligence

464

In this study, we mainly concentrate on improving

the worst case value, since it is a challenge for decen-

tralized local search methods.

To evaluate experimental results, we also consider

the Theil index T , which is a measurement of unfair-

ness: T =

1

|A|

∑

|A|

i=1

f

i

(X

i

)

f

ln

f

i

(X

i

)

f

, where f denotes

the average value for all f

i

(X

i

). T takes zero if all

f

i

(X

i

) are identical.

2.5 Scalability of Exact Solution

Methods and Stochastic Local

Search

Several exact solution methods based on tree search

and dynamic programming for AMODCOPs with

preferences for individual agents have been pro-

posed (Matsui et al., 2018a). However, such meth-

ods cannot be applied to large-scale and complex

problems with dense constraints/functions where the

tree-width of constraint graphs, which is related to

the number of combinations of assignments in partial

problems, is intractable. Although several approxi-

mations have been proposed for those types of solu-

tion methods (Matsui et al., 2018b), the accuracy of

the solutions decreases when a number of constraints

are eliminated from densely constrained problems.

Therefore, opportunities can be found for employ-

ing local search methods. Although several local

search methods were addressed in earlier studies for

such problems (Matsui et al., 2018b), stochastic local

search methods have not been adequately addressed.

Since such solution methods are an important class

of baseline methods, we focus on the stochastic local

search for AMODCOPs with variants of a min-max

criterion for the worst case cost value among agents.

3 DECENTRALIZED

STOCHASTIC LOCAL SEARCH

FOR AMODCOPS WITH

WORST CASE CRITERIA

3.1 Applying Stochastic Local Search to

AMODCOPs

To handle multiple objectives, the objective values

of other agents must be evaluated in addition to the

objective value of each agent even in a local search

with a narrow view. For Asymmetric Multi-Objective

problems, we adjusted the original DSA so that each

agent collects the local objective values of the neigh-

borhood agents. Each agent i repeats the following

steps in each iteration.

1. Set the initial assignment to variable x

i

of agent i.

2. Collect the current assignment of the variables of

the neighborhood agents in Nbr

i

. With the assign-

ment and functions related to each agent, evaluate

local objective value f

i

(A

i

) to the current assign-

ment.

3. Collect the current objective values f

j

of the

neighborhood agents j ∈ Nbr

i

. With the objective

values of the neighborhood agents, the current as-

signment and functions related to each agent, lo-

cally evaluate all the assignments to x

i

. Then se-

lect a new assignment with a stochastic strategy

and the evaluation of assignments of x

i

.

4. Update the assignment to variable x

i

of agent i.

Repeat from step 2.

Here we insert an additional step 3. Each agent up-

dates its current objective value based on a received

partial assignment to the variables of neighborhood

agents and then receives the current objective values

of the neighborhood agents. This step is necessary

to improve the consistency of the current assignment

and the related objective values.

The local evaluation is also modiﬁed to consider

the current cost values f

j

for neighborhood agents

j ∈ Nbr

i

. Although each agent can employ several so-

cial welfares and scalarization functions, their scope

is limited to the objective values of neighborhood

agents. In this work, it is assumed that all the agents

employ the same criterion in the optimization process

for a common goal.

3.2 Adjusting for the Criteria

With the modiﬁcation of the collected information,

each agent has a view of the current objective values

of the neighborhood agents. In a local search, each

agent only evaluates the multiple objectives within the

view and the evaluation partially overlaps thoughout

the whole system.

An issue is that local evaluation with a criterion

of social welfare is inexact, and the choice of assign-

ments to variables might unexpectedly inﬂuence the

perturbation in the search process. For example, min-

imization of the maximum objective value might im-

prove perturbation more than the minimization of the

summation for the best solution based on a differ-

ent criterion. Therefore, we simultaneously employ

several different criteria to monitor and capture snap-

shots of the best solution. We capture the best solu-

tions based on different multiple criteria, although the

Investigation on Stochastic Local Search for Decentralized Asymmetric Multi-objective Constraint Optimization Considering Worst Case

465

search process is performed based on a single crite-

rion. As mentioned in Section 2.2, we capture snap-

shots of the best solutions by a simulator.

3.3 Considering Opposite Constraints

In the original problem setting, each agent locally ag-

gregates its own asymmetric constraints related to the

agent. However, when each agent locally searches

for the assignment to its variable, each agent cannot

evaluate the inﬂuence of the new assignment on the

neighborhood agents. By revealing the opposite part

of an asymmetric constraint to neighborhood agents,

the change of the objective values of the neighbor-

hood agents can also be evaluated. This might be too

optimistic, since each agent independently estimates

the improvement of the cost values.

3.4 Local Agreement

Since each agent locally determines the assignment to

its own variable, the selection usually causes a mis-

match among neighborhood agents. To reduce such

cases, we employ an agreement mechanism based on

a local leader election that resembles MGM (Fioretto

et al., 2018), which is a local search method for

DCOPs. For a leader election, an additional commu-

nication step is introduced as follows.

1. Set the initial assignment to variable x

i

of agent i.

2. Collect the current assignment of the variables of

the neighborhood agents in Nbr

i

. With the assign-

ment and functions related to each agent, evaluate

local objective value f

i

(A

i

) to the current assign-

ment.

3. Collect the current objective values f

j

of the

neighborhood agents j ∈ Nbr

i

. With the objective

values of the neighborhood agents, the current as-

signment and functions related to each agent, lo-

cally evaluate all the assignments to x

i

. Then se-

lect a new assignment with a stochastic strategy

and the evaluation of the assignment to x

i

. Com-

pute the improvement g

i

of the objective value by

the new assignment.

4. Collect the improvement g

j

of objective values of

the neighborhood agents j ∈ Nbr

i

. Evaluate the

best improvement value g

∗

k

among the neighbor-

hood agents with a tie-breaker based on the iden-

tiﬁers of the agents.

5. Update the assignment to a variable of agent i if

its improvement is the best value g

∗

i

. Repeat from

step 2.

In Step 4, agents compute the improvement g

i

of

the local evaluations and exchange information about

the improvement with the neighborhood agents. The

information of the improvement is the same type as

the information for the criterion of optimization. For

example, a sorted objective vector is exchanged for

leximax criterion.

3.5 Employing Global Information

Although each agent locally searches for the assign-

ment to its local value, there are opportunities to ac-

cess some global information. We employ global

objective values by assuming the snapshot methods

mentioned in Section 2.2. Here we simply employ

snapshots in a simulator by inserting a sufﬁcient de-

lay of iterations that equals the number of agents so

that the agents do not access the global objective val-

ues too early.

While the global summation of objective val-

ues cannot be compared with the local evaluation of

each agent, the worst case objective value among the

agents can directly be a bound for each agent. There-

fore, we limit the locally worst case objective value

in each agent’s view by the globally worst case value.

The locally maximum cost value f

max

k

in a view of

agent i is limited by globally maximum cost value

F

max

as f

max

k

← min( f

max

k

, F

max

). In the case of lex-

imax, the globally maximum value is employed in-

stead of the ﬁrst value in the sorted objective vector

of the ‘maximum’ cost values. We do not employ

this information when the optimization criterion is the

summation.

4 EVALUATION

4.1 Settings

We experimentally evaluated our proposed meth-

ods. The benchmark problems consist of n

agents/variables and c asymmetric constraints, which

are a pair of binary objective ﬁctions for two agents.

Each variable takes a value in its domain that contains

d discrete values. We employed the following types

of cost functions for minimization problems. 1) rnd:

The cost values are randomly set to integer values

in [1, 100] based on uniform distribution. 2) gmm:

The cost values are random integer values in [1, 100]

that are rounded down from random values based on

gamma distribution with α = 9 and β = 2.

In a stochastic local search, one of the following

criteria is employed. 1) sum: The summation of the

objective values in the view of each agent. 2) max:

ICAART 2021 - 13th International Conference on Agents and Artiﬁcial Intelligence

466

300

350

400

450

500

550

600

650

0 50 100 150 200

max. cost

second

sum sumo

maxg maxo

msg mso

lxmg lxgoa

Figure 2: Example of anytime curve (rnd, n = 100, d = 3,

c = 250, (P

a

, P

b

) = (0.9, 0.1), max. cost).

18000

20000

22000

24000

26000

0 50 100 150

sum. cost

second

sum sumo

maxg maxgoa

msg msa

lxmg lxma

Figure 3: Example of anytime curve (rnd, n = 100, d = 3,

c = 250, (P

a

, P

b

) = (0.9, 0.1), sum. cost).

The maximum of the objective values in the view of

each agent. 3) ms: An extension of ‘max’ where ties

are broken by ‘sum’. 4) lxm: The leximax criteria for

cost values that is similar to leximin for utility values.

These criteria are also employed to capture snapshots

of the best solutions in each iteration. We captured

four different best solutions based on different crite-

ria, including the one for optimization. We also eval-

uated the Theil index values for the solutions.

For stochastic searches, we set probabilities

(P

a

, P

b

) = (1, 0) and (0.9, 0.1) by preliminary exper-

iments. The former setting is a deterministic local

search method.

We denote the additional mechanisms in Section 3

as follows. 1) (none): base method. 2) o: evaluation

of part of the opposite objective functions. 3) a: local

agreement with neighborhood agents. 4) g: modiﬁca-

tion of local evaluation by globally worst cost value.

These are added to the names of optimization criteria

(e.g. ‘maxgoa’).

We compared the best solutions that are captured

with several criteria at the cutoff iteration 200000.

The results are averaged over ten trials with differ-

ent initial solutions on ten problem instances for each

setting.

4.2 Results

Table 1 shows the solution quality for ‘rnd’, n = 50,

d = 3, c = 150 and (P

a

, P

b

) = (1, 0)). For each op-

timization criterion, we selected the variants of so-

lution that result the best maximum and summation

cost values. In this result, the solution quality of the

variants of ‘max’ was relatively worse than the oth-

ers. A possible reason is that the search can easily be

captured by locally optimal solutions, since the local

search process is deterministic in these settings. The

results of ‘ms’ and ‘lxm’ that are assisted by addi-

tional characteristics of the criteria are relatively bet-

ter. In particular, ‘lxm’ with ‘oa’ signiﬁcantly out-

performed the other methods for the maximum cost

value. It can be considered that the effect of leximax

to improve fairness by comparing more than secondly

worst cases might also improved the local search. In

addition, several variants of ‘lxm’ relatively improved

the Theil index. It reveals the possibility that the prop-

erty of leximax to improve fairness is effective in sev-

eral settings of local search.

Tables 2 and 3 show the solution quality in the

case of ‘rnd’ and (P

a

, P

b

) = (0.9, 0.1). In these cases,

the solution quality of variants of ‘max’ was relatively

better in comparison to the deterministic search. It is

considered the effect of the stochastic escape form lo-

cally optimal solutions. With additional information,

the solution quality was improved in several cases.

Although ‘o’ that referred opposite cost functions was

too optimistic without different additional informa-

tion in several cases, it was effective with some addi-

tional information. We found that ‘g’ that employed

the global min-max values looks improved the vari-

ants of ‘max’ in a few cases. However, the effect

decreased by employing other types of additional in-

formation. As mentioned above, several variants of

‘lxm’ were relatively better, while its computational

overhead to maintain sorted objective vectors is sig-

niﬁcantly higher than ‘max’.

Table 4 shows the solution quality in the case of

gmm, n = 50, d = 3, c = 150 and (P

a

, P

b

) = (0.9, 0.1).

Here, the effectiveness of the solution methods was

different from the previous cases. It can be consid-

ered that the non-uniform distribution of cost values

prevented some cooperative evaluation among neigh-

borhood agents.

Figures 2 and 3 show typical anytime curves of

maximum and summation cost values in a case of

‘rnd’. The snapshots of cost values are independently

captured with the globally maximum and summation

values. Since the evaluated optimization criteria need

different computational overheads, we plot the re-

sults considering computational time. This experi-

Investigation on Stochastic Local Search for Decentralized Asymmetric Multi-objective Constraint Optimization Considering Worst Case

467

Table 1: Solution quality (rnd, n = 50, d = 3, c = 150, (P

a

, P

b

) = (1, 0)).

alg. sum max ms lxm

max. sum. theil max. sum. theil max. sum. theil max. sum. theil

sum 537.1 12224.4 0.1070 449.1 12926.0 0.0915 449.1 12902.6 0.0922 449.1 12912.2 0.0917

suma 536.5 12160.2 0.1107 466.2 12572.0 0.0995 466.2 12565.0 0.0996 466.2 12593.1 0.0991

maxo 536.4 13443.6 0.0834 480.2 14053.7 0.0730 480.2 14014.6 0.0725 480.2 14047.3 0.0725

ms 537.1 12224.4 0.1070 449.1 12926.0 0.0915 449.1 12902.6 0.0922 449.1 12912.2 0.0917

msa 536.5 12146.2 0.1148 462.6 12711.9 0.0972 462.6 12678.5 0.0974 462.6 12696.4 0.0977

msg 537.1 12224.4 0.1070 449.1 12926.0 0.0915 449.1 12902.6 0.0922 449.1 12912.2 0.0917

lxma 528.9 12160.3 0.1117 458.8 12766.4 0.0963 458.8 12745.6 0.0971 458.8 12760.3 0.0964

lxmgoa 410.7 13061.3 0.0631 394.1 13308.0 0.0570 394.1 13196.0 0.0581 394.1 13266.9 0.0569

lxmoa 410.7 13061.3 0.0631 394.1 13308.0 0.0570 394.1 13196.0 0.0581 394.1 13266.9 0.0569

Table 2: Solution quality (rnd, n = 50, d = 3, c = 150, (P

a

, P

b

) = (0.9, 0.1)).

alg. sum max ms lxm

max. sum. theil max. sum. theil max. sum. theil max. sum. theil

sumo 486.1 11371.2 0.1075 391.8 12150.1 0.0822 391.8 12115.7 0.0823 391.8 12156.2 0.0820

maxgoa 532.6 12214.8 0.1047 431.9 13033.2 0.0780 431.9 12859.9 0.0822 431.9 13004.5 0.0764

maxo 476.9 12523.4 0.0841 391.1 13277.3 0.0593 391.1 13270.7 0.0591 391.1 13277.4 0.0586

msa 517.7 11907.9 0.1135 426.6 12541.6 0.0929 426.6 12519.7 0.0932 426.6 12534.3 0.0930

mso 477.4 12512.4 0.0844 391.7 13321.3 0.0585 391.7 13316.8 0.0587 391.7 13320.5 0.0587

lxma 508.4 11872.4 0.1109 420.9 12579.0 0.0898 420.9 12548.6 0.0904 420.9 12581.1 0.0896

lxmgoa 391.2 12459.9 0.0641 369.4 12973.6 0.0554 369.4 12748.6 0.0576 369.4 12986.5 0.0524

lxmoa 391.2 12459.9 0.0641 369.4 12973.6 0.0554 369.4 12748.6 0.0576 369.4 12986.5 0.0524

Table 3: Solution quality (rnd, n = 100, d = 3, c = 250, (P

a

, P

b

) = (0.9, 0.1)).

alg. sum max ms lxm

max. sum. theil max. sum. theil max. sum. theil max. sum. theil

sumo 512.5 18690.5 0.1402 389.7 20282.9 0.1140 389.7 20235.7 0.1141 389.7 20269.8 0.1140

maxgoa 531.9 19829.8 0.1169 383.5 20971.4 0.0937 383.5 20811.9 0.0948 383.5 20948.5 0.0923

maxo 477.5 21242.9 0.1010 377.9 22416.4 0.0798 377.9 22393.9 0.0796 377.9 22406.0 0.0795

msa 527.7 19173.4 0.1435 413.3 20228.6 0.1276 413.3 20182.9 0.1283 413.3 20228.4 0.1273

mso 468.9 21209.3 0.0997 379.4 22350.2 0.0807 379.4 22342.2 0.0808 379.4 22354.8 0.0806

lxma 518.5 19199.3 0.1409 412.5 20387.0 0.1254 412.5 20314.6 0.1256 412.5 20376.5 0.1250

lxmgoa 352.0 20376.1 0.0793 334.4 21292.6 0.0706 334.4 20765.7 0.0757 334.4 21161.3 0.0703

lxmoa 352.0 20376.1 0.0793 334.4 21292.6 0.0706 334.4 20765.7 0.0757 334.4 21161.3 0.0703

Table 4: Solution quality (gmm, n = 50, d = 3, c = 150, (P

a

, P

b

) = (0.9, 0.1)).

alg. sum max ms lxm

max. sum. theil max. sum. theil max. sum. theil max. sum. theil

sumo 177.3 4483.9 0.0794 153.5 4719.7 0.0697 153.5 4697.8 0.0699 153.5 4723.8 0.0691

maxo 166.5 4711.4 0.0677 145.5 4938.3 0.0569 145.5 4899.7 0.0577 145.5 4930.6 0.0568

msa 178.7 4555.6 0.0811 160.8 4706.6 0.0741 160.8 4696.0 0.0743 160.8 4711.3 0.0737

mso 165.9 4699.4 0.0673 145.9 4936.2 0.0572 145.9 4908.9 0.0581 145.9 4937.9 0.0570

lxma 181.2 4557.4 0.0815 160.1 4726.0 0.0726 160.1 4700.4 0.0732 160.1 4720.9 0.0725

lxmo 164.6 4708.6 0.0664 145.8 4948.7 0.0563 145.8 4919.7 0.0564 145.8 4940.4 0.0564

ment is performed on a computer with g++ (GCC)

4.4.7, Linux version 2.6, Intel(R) Core(TM) i7-3770K

CPU @ 3.50GHz and 32GB memory. Note that there

are opportunities to improve our experimental imple-

mentation.

For each optimization criterion, we selected the

fastest variant and the best-solution-quality variant.

While the optimization process with complex crite-

ria and more additional information results relatively

better solutions, the process with ‘light’ criteria has

chance to perform more iterations. However, for min-

max cost values of relatively large-scale problems, the

best variant of ‘lxm’ optimization was relatively bet-

ter. We note that for ‘max’, ‘ms’ and ‘lxm’ variants,

employing snapshots of globally maximum cost val-

ues were slightly faster than the simplest one. This

ICAART 2021 - 13th International Conference on Agents and Artiﬁcial Intelligence

468

was a side-effect where part of evaluation was con-

ditionally skipped due to the threshold based on the

globally maximum cost value.

5 DISCUSSION

There are several decentralized min-max problem set-

tings that require several assumptions including the

convexity and continuity of objective functions. How-

ever, the aim of DCOPs with similar criteria is general

discrete problems with non-convexity. To solve large-

scale and complex problems of these classes, inexact

methods are necessary. As a baseline investigation,

we addressed a class of stochastic hill-climb methods

that is an extension of a fundamental approach for tra-

ditional DCOPs.

An approximation of complete solution methods

for AMODCOPs that are based on tree search and dy-

namic programming has been presented. However,

since the approximation partially eliminates or ig-

nores several constraints, its accuracy decreases when

the induced width (Fioretto et al., 2018) of constraint

graphs is relatively larger. In such situations, there are

opportunities to employ stochastic local search meth-

ods. While deterministic local search methods for this

class of problems have been addressed in early stud-

ies, we investigated stochastic local search methods

that also employ some additional information.

In the original DSA and several variants, the prob-

ability of the algorithms is generally employed to de-

termine whether the agents select the hill-climb or

stay as is. Therefore, the methods are basically local

search that cannot escape from locally optimal solu-

tions. We mainly investigated the practical case where

the agents are basically greedy but stochastically es-

cape from locally optimal solutions.

6 CONCLUSION

We investigated the decentralized stochastic search

methods for Asymmetric Multi-Objective Distributed

Constraint Optimization Problems considering the

worst cases among the agents. The experimental

results show the effect and inﬂuence of different

optimization criteria and additional information in

stochastic local search processes. In particular, the re-

sults show that the minimization of the maximum cost

value is not straightforward with min-max based cri-

teria that are also designed to improve the global ob-

jective value or fairness. Min-max optimization with

several additional bits of information performs better

perturbation. We believe such an investigation of fun-

damental search methods for different optimization

criteria can provide a foundation for more sophisti-

cated sampling based methods for extended classes

of DCOPs. Our future work will investigate other

stochastic and sampling-based algorithms with fair-

ness and Pareto optimality.

ACKNOWLEDGEMENTS

This work was supported in part by JSPS KAKENHI

Grant Number JP19K12117.

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