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J Glob Optim (2016) 64:17–32
DOI 10.1007/s10898-015-0270-y
Multi objective optimization of computationally
expensive multi-modal functions with RBF surrogates
and multi-rule selection
Taimoor Akhtar ·Christine A. Shoemaker
Received: 31 October 2013 / Accepted: 15 January 2015 / Published online: 8 February 2015
© The Author(s) 2015. This article is published with open access at Springerlink.com
Abstract GOMORS is a parallel response surface-assisted evolutionary algorithm approach
to multi-objective optimization that is designed to obtain good non-dominated solutions
to black box problems with relatively few objective function evaluations. GOMORS uses
Radial Basic Functions to iteratively compute surrogate response surfaces as an approxi-
mation of the computationally expensive objective function. A multi objective search uti-
lizing evolution, local search, multi method search and non-dominated sorting is done on
the surrogate radial basis function surface because it is inexpensive to compute. A balance
between exploration, exploitation and diversification is obtained through a novel procedure
that simultaneously selects evaluation points within an algorithm iteration through different
metrics including Approximate Hypervolume Improvement, Maximizing minimum domain
distance, Maximizing minimum objective space distance, and surrogate-assisted local search,
which can be computed in parallel. The results are compared to ParEGO (a kriging surro-
gate method solving many weighted single objective optimizations) and the widely used
NSGA-II. The results indicate that GOMORS outperforms ParEGO and NSGA-II on prob-
lems tested. For example, on a groundwater PDE problem, GOMORS outperforms ParEGO
with 100, 200 and 400 evaluations for a 6 dimensional problem, a 12 dimensional problem
and a 24 dimensional problem. For a fixed number of evaluations, the differences in perfor-
mance between GOMORS and ParEGO become larger as the number of dimensions increase.
As the number of evaluations increase, the differences between GOMORS and ParEGO
become smaller. Both surrogate-based methods are much better than NSGA-II for all cases
considered.
Electronic supplementary material The online version of this article (doi:10.1007/s10898-015-0270-y)
contains supplementary material, which is available to authorized users.
T. Akhtar (B)·C. A. Shoemaker
School of Civil and Environmental Engineering and School of Operations Research and Information
Engineering, Cornell University,
Holister Hall, Ithaca, NY, USA
e-mail: ta223@cornell.edu
C. A. Shoemaker
e-mail: cas12@cornell.edu
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18 J Glob Optim (2016) 64:17–32
Keywords Function approximation ·Multi objective optimization ·Radial basis
functions ·Expensive optimization ·Global optimization ·Evolutionary optimization ·
Parallel ·Metamodel
1 Introduction
Multi-objective optimization (MO) approaches involve a large number of function evalu-
ations, which make it difficult to use MO in simulation–optimization problems where the
optimization is multi-objective and the nonlinear simulation is computationally expensive
and has multiple local minima (multi modal). Many applied engineering optimization prob-
lems involve multiple objectives and the computational cost of evaluating each objective is
high (e.g minutes to days per objective function evaluation by simulation) [12,32]. We focus
on special algorithms that aim to produce reasonably good results within a limited number
of expensive objective function evaluations.
Many authors (for instance, Deb et al. [5]andZhangetal.[52]) have successfully
employed evolutionary strategies for solving multi-objective optimization problems. Even
with the improvement over traditional methods, these algorithms require, typically, many
objective function evaluations which can be infeasible for computationally expensive prob-
lems. Added challenges to multi-objective optimization of expensive functions arise with
increase in dimensionality of the decision variables and objectives.
The use of iterative response surface modeling or function approximation techniques
inside an optimization algorithm can be highly beneficial in reducing time for computing
objectives for multi-objective optimization of such problems. Since the aim of efficient multi-
objective optimization is to identify good solutions within a limited number of expensive
function evaluations, approximating techniques can be incorporated into the optimization
process to reduce computational costs. Gutmann [15] introduced the idea of using radial
basis functions (RBF) [3] for addressing single objective optimization of computationally
expensive problems. Jin et al. [18] appears to be the first journal paper to combine a non
quadratic response surface with a single objective evolutionary algorithm by using neural net
approximations. Regis and Shoemaker [36] were the first to use Radial Basis Functions (not
a neural net) to improve the efficiency of an evolutionary algorithm with limited numbers
of evaluations. Later they introduced a non-evolutionary algorithm Stochastic-RBF [37],
which is a very effective radial basis function-based method for single objective optimization
of expensive global optimization problems. These methods have been extended to include
parallelism [40]; high-dimensional problems [42]; constraints [38]; local optimization [49,
50]; integer problems [30,31] and other extensions [39,41]. Kriging-based methods have
also been explored for addressing single objective optimization problems [9,16,19]. Jones
et al. [19] introduced Efficient Global Optimization (EGO), which is an algorithm for single
objective global optimization within a limited budget of evaluations.
Response surface methods have also become popular for multi-objective optimization
problems, with kriging-surrogate techniques being the most popular. Various authors have
used kriging-based methods by extending the EGO framework for multi-objective optimiza-
tion of computationally expensive problems. For instance, Knowles [21] combined EGO with
Tchebycheff weighting to convert an MO problem into numerous single objective problems.
Tsang et al. [53] and Emmerich et al. [10] combined EGO with evolutionary search assis-
tance. Ponweiser et al. [33] and Beume et al. [2] explored the idea of maximizing expected
improvement in hypervolume. Authors have also explored the use of other function approxi-
mation techniques inside an optimization algorithm, including Radial Basis Functions (RBFs)
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J Glob Optim (2016) 64:17–32 19
[13,20,25,45,51], Support Vector Machines [24,43] and Artificial Neural Networks [7]. Evo-
lutionary algorithms are the dominant optimization algorithms used in these methods. Some
papers also highlight the effectiveness of local search in improving performance of response
surface based methods [20,23,45].
Various authors have indicated that RBFs could be more effective than other approximation
methods in multi-objective optimization of computationally expensive problems with more
than 15 decision variables [8,29,43]. Various authors have employed RBFs for surrogate-
assisted multi-objective optimization of expensive functions with focus on problems with
more than 15 decision variables. For instance, Karakasis and Giannakoglou [20]employ
RBFs within an MOEA framework and use an inexact pre-evaluation phase (IPE) to select
a subset of solutions for expensive evaluations in each generation of an MOEA. They also
recommend the use of locally fitted RBFs for surrogate approximation. Georgopoulou and
Giannakoglou [13] build upon [20] by employing gradient based refinements of promising
solutions highlighted by RBF approximation during each MOEA generation. Santana et
al. [45] divide the heuristic search into two phases where the first phase employs a global
surrogate-assisted search within an MOEA framework, and rough set theory is used in the
second phase for local refinements.
This paper focuses on the use of RBFs and evolutionary algorithms for multi-objective
optimization of computationally expensive problems, where the number of function evalua-
tions are limited relative to the problem dimension (e.g. to a few hundred evaluations for the
example problems tested here). An added purpose of the investigation is to be able to solve
MO problems where the number of decision variables varies between 15 and 25.
To this effect we propose a new algorithm, GOMORS, that combines radial basis function
approximation with multi-objective evolutionary optimization, within the general iterative
framework of surrogate-assisted heuristic search algorithms. Our approach is different from
prevalent RBF based MO algorithms that use evolutionary algorithms [13,20,25,45,51]. Most
RBF based evolutionary algorithms employ surrogates in an inexact pre-evaluation phase
(IPE) in order to inexpensively evaluate child populations after every MOEA generation. By
contrast our proposed methodology employs evolutionary optimization within each iteration
of the algorithm framework to identify numerous potential points for expensive evaluations.
Hence, multiple MOEA generations evolve via surrogate-assisted search in each algorithm
iteration in GOMORS.
The novelty of the optimization approach is in the amalgamation of various evaluation
point selection rules in order to ensure that various value functions are incorporated in select-
ing some points for expensive evaluations from the potential points identified by surrogate-
assisted evolutionary search during each algorithm iteration. The combination of multiple
selection rules targets a balanced selection between exploration, exploitation and front diver-
sification. The selection strategy incorporates the power of local search and also ensures that
the algorithm can be used in a parallel setting to further improve its efficiency.
2 Problem description
Let D(domain space) be a unit hypercube and a subset of Rdand xbe a variable in the
domain space, i.e., x∈[0,1]d. Let the number of objectives in the multi-objective problem
equal kand let fi(x)be the ith objective. fiis a function of xand fi:D → Rfor 1≤i≤k.
The framework of the multi-objective optimization problem we aim to solve is as follows:
minimize F(x)=[f1(x),..., fk(x)]T
subject to x∈[0,1]d(1)
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20 J Glob Optim (2016) 64:17–32
Our goal for the multi-objective optimization problem is (within a limited number of
objective function evaluations) to find a set of Pareto-optimal solutions P∗={x∗
i|x∗
i∈
D,1≤i≤n}.P∗is defined via the following definitions:
Domination A solution x1∈Ddominates another solution x2∈Dif and only if fi(x1)≤
fi(x2)for all 1≤i≤k,and fi(x1)< fi(x2)for at least one i∈{1,...,k}.
Non-domination Given a set of solutions S={xi|xi∈D,1≤i≤n}, a solution x∗∈Sis
non-dominated in Sif there does not exist a solution xj∈Swhich dominates x∗.
Pareto optimality A candidate solution x∗∈Swhich is non-dominated in Sis called a
globally Pareto-optimal solution if S=D, i.e., Sis the entire feasible domain space of the
defined problem.
3 Algorithm description
3.1 General framework
The proposed optimization algorithm is called Gap Optimized Multi-objective Optimization
using Response Surfaces (GOMORS). GOMORS employs the generic iterative setting of
surrogate-assisted algorithms in which the response surface model used to approximate the
costly objective functions is updated after each iteration. Multiple points are selected for
expensive function evaluation in each iteration, evaluated in parallel, and subsequently used
to update the response surface model. The algorithm framework is defined below:
Step 0 - Define Algorithm Inputs:
M- Maximum number of expensive function evaluations
r- The Gap radius parameter used in Step 2.3
t- Number of expensive evaluations to be performed after each algorithm iteration
Step 1 - Initial Evaluation Points Selection: Select an initial set of points {x1,...,xm},wherexi∈D,for
1≤i≤m, using an experimental design. Evaluate the objectives F=[f1,..., fk]at the selected mpoints,
via expensive simulations. Let Sm={xi|xi∈D,i=1,...,m}denote the initial set of evaluated points.
Let Pm={xi∈Sm|xiis non-dominated in Sm}be the set of non-dominated points from Sm.
Step 2 - Iterative Improvement: Run algorithm iteratively until termination condition is satisfied:
While m≤M
Step 2.1 - Fit/Update Response Surface Models: Fit/update response surface models for each objective
based on the set of already evaluated simulation points Sm.Let
Fm(x)=[
fm,1,...,
fm,k]be the
inexpensive approximate objective functions obtained by fitting a response surface model on Sm.
Step 2.2 - Surrogate Assisted Global Search: Use a multi-objective evolutionary algorithm (MOEA)
to solve the following optimization problem:
minimize:
Fm(x)=[
fm,1(x),...,
fm,k(x)]T(2)
Let
PA
m={x1,...,xnA}denote the solutions obtained by solving Equation 2, i.e, utilizing an MOEA
for searching on the response surfaces.
Step 2.3a - Identify Least Crowded Solution:. Using the crowding distance calculation procedure
proposed by Deb et al. [5], identify the least crowded element of the expensively evaluated non-dominated
set, Pm.Letxcr owdbe the least crowded element of Pm(see definition in Table 1).
Step 2.3b - Local Search:. Use a multi-objective evolutionary algorithm in a small neighborhood of
xcro wdto solve the following optimization problem:
minimize:
Fm(x)={
fm,1(x),...,
fm,k(x)}
subject to: (xcro wd−r)≤x≤(xcrowd+r)(3)
The problem solved in Equation 3 we will call the “Gap Optimization Problem”.
Let
PB
m={x1,..., xnB}denote the solutions obtained by solving Equation (3) ,i.e, utilizing an MOEA
for local search on the response surfaces, within a radius, r, around the least crowded solution, xcrowd.
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J Glob Optim (2016) 64:17–32 21
Step 2.4 - Select Points for Expensive Function Evaluations: Our evaluation points are then selected
from the sets
PA
mand
PB
m(from Equations 2 and 3, and also called candidate points) based on the rules
described in Sect. 3.4. Let Scur r =x1,...,xtbe the set of tpoints selected for expensive evaluations in
current algorithm iteration. (Discussed in detail in Sect. 3.4)
Step 2.5 - Do expensive function evaluations and update Non-dominated solution set: Evaluate costly
functions, F, on the selected evaluation points, Scurr . Update Sm, i.e., add the new expensively evaluated
points to the set of already evaluated points. Consequently, m=m+t,Sm={Sm}∪{Scur r }. Update
Pm, i.e., compute Pm={xi∈Sm|xiis non-dominated in Sm}.
End Loop
Step 3 - Return Best Approximated Front: Return PM={xi∈SM|xiis non-dominated in SM}as an
approximation to the globally optimal solution set, i.e, P∗={x∗
i|x∗
iis non-dominated in D}.
The algorithm initiates with selection of an initial set of points for costly function evalua-
tions of the objective function set, F. Latin hypercube sampling, a space-filling experimental
design scheme proposed by McKay et al. [26] is used for selection of the initial evaluation
set. The iterative framework of GOMORS (Step 2) follows, and consists of three major seg-
ments: (1) Building a Response Surface Model for approximating expensive objectives (Step
2.1), (2) Applying an Evolutionary algorithm for solving multi-objective response surface
problems (Steps 2.2 and 2.3) and (3) Selecting multiple points for expensive evaluations
from the solutions of the multi-objective response surface problems, and evaluating them in
parallel (Step 2.4). The algorithm terminates after Mexpensive evaluations and the output is
a non-dominated set of solutions PM, which is an approximation to the true Pareto solution of
the problem i.e P∗. Sections 3.2–3.4 and the online supplement discuss details of the major
steps of the algorithm.
Tab l e 1 Definitions of sets and variables
Item Description
F(x)=[f1(x),..., fk(x)]Expensive objectives for the multiple-objective optimization problem
P∗={x∗
1,...,x∗
n}Set of Pareto-optimal solutions given the objectives F
Sm={x1,...,xm}Set of points which have been evaluated via costly simulation until
iteration mof the optimization algorithm
Fm(x)=[
fm,1(x),...,
fm,k(x)]
fm,iis the inexpensive surrogate approximation of the ith objective,
fi, generated using the set of points, Sm
Pm={x∈Sm}The non-dominated solutions in Smbased on expensive evaluations
xcro wd=arg max
x∈Pm
[d(x)]The least crowded element in Pm, according to the crowding definition,
d(.),whered(xj)indicates the function by [6] to compute crowding
distance for xjgiven (xj,F(xj)) for all xj∈Pm
PA
m={x1,..., xnA}Candidate points obtained by Surrogate-Assisted Global Search on the
approximate objective function set
Fm(see Equation 2)
PB
m={x1,..., xnB}Candidate points obtained by Gap Optimization of the approximate
objectives,
Fm(see Equation 3)
HV(P)Hypervolume [1] of the objective space dominated by the objective
vectors corresponding to the set P. The hyperspace is such that
y∈Rkand yis bounded by a reference vector b(see Fig. 1for
illustration)
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22 J Glob Optim (2016) 64:17–32
500 1000 1500 2000 2500
1000
1200
1400
1600
1800
2000
2200
2400
F1(x)
F2(x)
Reference
Point
(A) - Hypervolume
500 1000 1500 2000 2500
1000
1200
1400
1600
1800
2000
2200
2400
F1(x)
F2(x)
Hypervolume Improvement:
Objective SpaceObjective Space
(B) - Hypervolume Imrovement
HYPERVOLUME:
HV(P
m)
Non-Dominated
Solutions:
P
m
Ideal Solution(s):
UNCOVERED
HYPERVOLUME:
Reference
Point
Non-Dominated
Solutions:
P
m
Fig. 1 Visualization of ahypervolume and uncovered hypervolume and bhypervolume improvement
employed in Rule 1 and Rule 4 of Step 2.4 (discussed in Sect. 3.4). Maximization of hypervolume improve-
ment implies selection of a point for function evaluation that predicts maximum improvement in hypervolume
coverage of a non-dominated set as per RBF approximation
3.2 Response surface modeling
The first major component of the iterative framework is the procedure used for approximating
the costly functions, F, by the inexpensive surrogates,
Fm(x)=[
fm,1,...,
fm,k]. A response
surface model based on the evaluated points is generated in Step 2.1 of the algorithm and
subsequently updated after each iteration. For example artificial neural networks [35], Support
Vector Machines (SVM) [47], kriging [44], and radial basis functions (RBFs) [3,34] could
be employed to generate the response surface model.
While kriging has been used in MO [21,33,53], the number of parameters to be esti-
mated for kriging meta-models increases quickly with an increase in the number of decision
variables [16,17], and hence, re-evaluating kriging surrogates in each iteration of a kriging
based algorithm may itself become a computationally expensive step. Various authors [29,42]
including Manriquez et al. [8] have reported that kriging-based surrogate optimization is not
effective for high dimensional problems (approximately defined as problems with more than
15 decision variables). In [8] they demonstrate the relative effectiveness of RBF approxi-
mation in tackling such high dimensional problems. The GOMORS algorithm proposed in
this paper hence makes use of RBFs as the surrogate model for approximating the costly
functions, although other surrogate models could be used in the GOMORS strategy.
3.3 Evolutionary optimization
The surrogate optimization problems defined in Steps 2.2 (Surrogate-assisted global search)
and 2.3 (Gap Optimization) aim to find near optimal fronts given that the expensive functions,
F(x), are replaced by the corresponding response surface model approximations, i.e.,
Fm(x)
(see Table 1). In Steps 2.2 and 2.3, two different multi-objective optimization problems are
solved on the surrogate surface.
Steps 2.2 and 2.3 solve the MO problems depicted in Equations 2 and 3 respectively. Since
the objective functions of these problems are derived from the surrogate model, solving the
MO problems is a relatively inexpensive step. However, the MO problems of Equations 2 and
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J Glob Optim (2016) 64:17–32 23
3 are not trivial and could potentially have non-linear and multi-modal objectives. Hence, we
employ evolutionary algorithms for MO optimization of surrogates in Steps 2.2 and 2.3.
Three algorithm were considered as potential alternatives for optimization of surrogates
in Steps 2.2 and 2.3, namely, NSGA-II, [5], MOEA/D [52] and AMALGAM [48]. NSGA-
II handles the evolutionary search optimization process by ranking and archiving parent
and child populations according to a non-domination sorting. MOEA/D [52] uses aggregate
functions, and simultaneously solves many single-objective Tchebycheff decompositions
of multi-objective problems in an evolutionary generation. AMALGAM [48] is a multi-
method evolutionary algorithm, which incorporates search mechanics of various algorithms.
Extensive computer experiments on test problems were performed on GOMORS with either
of NSGA-II, MOEA/D and AMALGAM as embedded evolutionary schemes (see Section
C of the online supplement for a detailed discussion on the experiments) and AMALGAM
was identified as the best performing evolutionary algorithm (although the differences were
small) embedded in GOMORS.
3.4 Expensive evaluation point selection: Step 2.4
Step 2.4 of the GOMORS algorithm determines which of the candidate points are evaluated
by the expensive objective functions, F(x), within an algorithm iteration. The candidate
points are obtained from Steps 2.2 and 2.3, and are denoted as
PA
mand
PB
mrespectively.
As mentioned earlier, selection of points for expensive evaluation is a critical step in the
algorithm because this is usually by far the most computationally expensive step.
A balance between exploration [19], exploitation and diversification [6] is crucial for
selecting points for expensive evaluations from candidate points,
PA
mand
PB
m.Exploration
of the decision space aims at selecting points in unexplored regions of the decision space.
Exploitation aims at exploiting the inexpensive response surface approximations of Step 2.1
to assist in choosing appropriate points for expensive evaluations. Diversification strives to
ensure that the non-dominated evaluated points are spread out in the objective space.
GOMORS employs a multi-rule based strategy for selection of a small subset of candidate
points (
PA
mand
PB
m) for actual function evaluations (computing Fvia simulation). The
various “rules” employed in the strategy target a balance between exploration,exploitation
and diversification. A detailed framework of Step 2.4 of the algorithm is given below which
gives an overview of the selection rules (The ith rule is referred as Rule i)(ReferTable1for
definitions of sets and symbols):
Details of Step 2.4 (in Section 3.1:)
Inputs from previous steps: (m,Sm,Pm,
PA
m,
PB
m)
Evaluation Points selection: Select t=4
i=0tipoints for expensive evaluations via rules 0–4. Let Scur r be
the selected points. So there are tipoints generated using Rule i, where the Rules are listed below.
Apply Rule 0 - Random Sampling: Pick t0points for expensive function evaluation via random sampling
from a uniform distribution. Ensures exploration of decision space.
Apply Rule 1- Hypervolume Improvement: Use hypervolume improvement to choose t1points from the
candidate points
PA
m.Let HV(Pm)denote the hypervolume [1] of the objective space dominated by the vectors
in the set Pm(see Figure 1a). A new point for expensive function evaluation is selected as follows (see Table 1
for definition of sets and variables):
x∗=arg max
xj∈
PA
mHV(Pm∪xj)−HV(Pm)(4)
Equation 4 exploits the RBF approximation and chooses a point for function evaluation which maximizes the
improvement in hypervolume in the objective space as per the RBF approximations of the candidate points,
as illustrated in Fig. 1b.
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Apply Rule 2- Maximize Minimum Domain Euclidean Distance: Let xi,A∈PA
mand xj∈Sm. Choose
t2points for expensive evaluation such that the maximum of minimum distances of each point from already
evaluated points is maximized, i.e.:
x∗=arg max
xi,A∈
PA
mmin
xj∈Sm
xi,A−xj(5)
Apply Rule 3- Maximize Minimum Objective Space Euclidean Distance: Choose t3points such that the
maximum of minimum distances of each point from already evaluated points, as per the objective function
space, is maximized, i.e.:
x∗=arg max
xi,A∈
PA
mmin
xj∈Sm
Fm(xi,A)−F(xj)(6)
Rule 2 and Rule 3 are a hybrid of exploration and exploitation.
Apply Rule 4 - Hypervolume Improvement in “Gap Optimization” candidates: Use “Rule 1” to select
t4points from the candidate points obtained via “Gap Optimization”, i.e,
PB
m. Select points for expensive
function evaluations as follows (See Table 1 for definition of sets and variables):
x∗=arg max
xj∈
PB
mHV(Pm∪xj)−HV(Pm)(7)
The difference between (4) and (7) is that
PA
min (4) comes from Step 2.2 and
PB
min (7) comes from Step 2.3.
The number of points to be selected for expensive evaluation via each rule i.e, timay
vary. However, we performed all our computer experiments with ti=1, for Rules 1–4. A
point is selected via Rule 0 in each algorithm iteration with a probability of 0.1. Hence, either
four or five points are selected for expensive function evaluations in each iteration of the
algorithm. The expensive evaluations of points are performed in parallel to further speed up
the algorithm.
In order to assess the individual effectiveness of the rules, we performed computer exper-
iments on eleven test problems with the exclusive use of all rules (i.e, one point is selected
from one rule, in all algorithm iterations). Furthermore, we also performed experiments with
the idea of cycling between all rules in subsequent iterations of GOMORS framework. These
different selection strategies were compared against the multiple rule selection strategy for
simultaneous selection of evaluation points (described above). Results of the computer exper-
iments indicated that if the value of parallelization is considered, the multiple rule selection
strategy outperforms the other strategies employed on our analysis (See Section D of the
online supplement for details.). However, if GOMORS is used in a serial setting, i,e, one
point is evaluated in each algorithm iteration, cycling between rules, and use of Rule 3 are
most beneficial (see section D of online supplement).
4 Test problems
In order to test the performance of GOMORS, computational experiments were performed
on various test functions and a groundwater remediation problem. Certain characteristics
of various problems can lead to inadequate convergence or poor distribution of points
on the Pareto front. These characteristics include high dimensionality, non-convexity and
non-uniformity of the Pareto front [6], the existence of multiple locally optimal Pareto
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J Glob Optim (2016) 64:17–32 25
Tab l e 2 Two MO variants, GWDA and GWDM, of the groundwater optimization problem (Sect. 4.1)
GWDA GWDM
minimize: f1(X)=T
t=1Ct(Xt)minimize: f1(X)=T
t=1Ct(Xt)
minimize: f2(X)=M
m=1sm,T(X)
Mminimize: f2(X)=maxM
m=1sm,T(X)
subject to: X∈[0,1]N×Tsubject to: X∈[0,1]N×T
Mis total number of finite element grid nodes, Nis the total number of wells, Tis the total number of
management periods, Xtis the pumping decision in management period t,sm,tis the contaminant concentration
at grid mand at the end of the last time period T,andCt(Xt)is the cost of cleanup for the pumping decision
Xt
fronts [4], low density of solutions close to the Pareto front, and existence of compli-
cated Pareto sets (this implies that the Pareto solutions are defined by a complicated curve
in the decision space) [22]. We have employed eleven test problems in our experimental
analysis which incorporate the optimization challenges mentioned above. Five test prob-
lems are part of the ZDT test suite [4], while six were derived from the work done by
Li and Zhang [22]. Mathematical formulations and optimization challenges of the test
problems are discussed in detail in Section B.1 of the online supplement to this docu-
ment. These problems are collectively referred as synthetic test problems in subsequent
discussions.
4.1 Groundwater remediation design problem
Optimization problems pertaining to groundwater remediation models usually require solving
complex and computationally expensive PDE systems to find objective function values for a
particular input [27]. The groundwater remediation problem used in our analysis is based on
a PDE system which describes the movement and purification of contaminated groundwater
given a set of bioremediation and pumping design decisions [28]. Detailed description of the
problem is provided in Section B.2 of the online supplement. The decision variables of the
problem are the injection rates of remediation agent at 3 different well locations during each of
mmanagement periods. The input dimension size ranges between 6 and 36 variables, depend-
ing upon the number of management periods incorporated in the numerical computation
model.
The remediation optimization problem can be formulated as two separate multi-objective
optimization problems, called GWDA and GWDM (Table 2). In the first formulation,
GWDM, the aim is to minimize the maximum contaminant concentration at the end of the
pumping time, and to minimize the cost of remediation. The second formulation, GWDA,
aims to minimize the average contaminant concentration, along with cost. The mathematical
formulation of GWDA and GWDM are depicted in Table 2.
5 Results and analysis
5.1 Experimental setup
We tested our algorithm on the test functions and groundwater problems discussed in the
previous section. The performance of GOMORS was compared against the Non-Dominated
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26 J Glob Optim (2016) 64:17–32
Sorting Algorithm-II (NSGA-II) proposed by Deb et al. [5] (discussed in Sect. 3.3)and
the kriging-based ParEGO algorithm proposed by Knowles [21]. All three algorithms are
quite different. ParEGO is a multi-objective version of EGO [19] where the multi-objective
problem is converted into many single objective optimization problems through Tchebycheff
weighting [46]. One single objective problem is chosen at random from the predefined set of
decomposed problems and EGO is applied to it to for selection of one point for evaluation
per algorithm iteration. ParEGO is not designed for high dimensional problems (more than
15 variables). GOMORS on the other hand embeds RBFs within an evolutionary framework,
and selects multiple points (from various rules defined in Sect. 3.4) for simultaneous (parallel)
evaluations in each iteration and is designed for low and higher (15–25 decision variables)
dimensional problems.
Since the objective of GOMORS is to find good solutions to MO problems within a limited
function evaluation budget, our experiments were restricted to 400 function evaluations.
Since all algorithms compared are stochastic, ten optimization experiments were performed
for each algorithm, on each test problem, and results were compared via visual analysis of
fronts and a performance metric based analysis. A detailed description of the experimental
setup, including parameter settings for all algorithms and source code references for ParEGO
and NSGA-II, is provided in Section E of the online supplement.
The uncovered hypervolume metric [1] was used to compare the performance of various
algorithms. Uncovered hypervolume is the difference between the total feasible objective
space (defined by the reference and ideal points in Fig. 1a) and the objective space dominated
by estimate of the Pareto front obtained by an algorithm. A lower value of the metric indicates
a better solution and the ideal value is zero.
Results from the synthetic test problems were analyzed in combination through the
metric. Experiments were performed for each test problem with 8 decision variables,
16 decision variables and 24 decision variables to highlight performance differences of
GOMORS, ParEGO and NSGA-II with varying problem dimensions. Results of all syn-
thetic test problems were compiled for analysis by summing the metric values of each
of the ten optimization experiments performed on each test problem. Since the uncov-
ered hypervolume metric values obtained for each individual test problem and algorithm
combination could be considered as independent random variables, a sum of them across
a single algorithm is another random variable which is a convolution of the indepen-
dent random variables. This convolution based metric summarizes overall performance
of an algorithm on all synthetic test problems and is used as basis of our analysis
methodology.
Results from the two variants of the groundwater remediation problem (GWDA and
GWDM) were also analyzed through the uncovered hypervolume metric in a similar
manner. Experiments were performed with 6 decision variables, 12 decision variables
and 24 decision variables for each groundwater problem. Results from individual sub-
sequent experiments performed on each problem were summed to obtain a convoluted
metric analysis of the groundwater problems. Performance of GOMORS and ParEGO on
the GWDM test problem was further assessed through visual analysis of non-dominated
solutions obtained from each algorithm (median solution). The numerical comparisons
are based on the number of objective function evaluations and hence do not incorpo-
rate parallel speedup. ParEGO does not have a parallel implementation so its wall-clock
time is much longer than GOMORS. However, the comparisons here focus on the num-
ber of function evaluations and evaluate an estimate of total CPU time. These com-
parisons do not consider the additional advantage of GOMORS (parallel) in wall-clock
time.
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J Glob Optim (2016) 64:17–32 27
400
Uncovered Hypervolume
GOMORS ParEGO NSGA2
Algorithm
GOMORS ParEGO NSGA2 GOMORS ParEGO NSGA2
Expensive Function Evaluations
Decision Variables
200
16 Variables 24 Variables8 Variables
1
2
3
1
2
3
Fig. 2 Synthetic test problems: box plots of uncovered hypervolume metric values summed over the eleven
test problems. Axis scales are identical across all sub-plots, which describe problems ranging from 8 to 24
decisions and from 200 to 400 function evaluations. Upper row is for 400 evaluations and lower row is for
200 evaluations. In all cases lower values of uncovered hypervolume are best
5.2 Results: synthetic test problems
The metric analysis performed on the synthetic test problems is depicted via box plots in
Fig. 2. The vertical axis in Fig. 2is the convoluted uncovered hypervolume metric described
in Sect. 5.1. The lower the uncovered hypervolume, the better is the quality of the Non-
dominated Front that the algorithm has found.
Figure 2aims to (1) visualize speed of convergence of all algorithms (by comparing results
of 200 and 400 objective function evaluations) and (2) understand the effect of increasing
decision space dimensions on performance of algorithms (by comparing problems with 8,
16 and 24 decision variables). The box plot visualization of each algorithm within each
sub-plot of Fig. 2corresponds to the uncovered hypervolume metric values summed over
the eleven test problems (see Sect. 5.1). Lower values of the metric signify superiority of
performance and a lower spread within a box plot depicts robustness of performance. Each
sub-plot within Fig. 2compares performance of all three algorithms for a specified number of
decision variables (number of decision variables vary between 8 and 24) and a fixed number
of function evaluations (either 200 or 400). Traversing from bottom to top, one can visualize
the change in performance of algorithms as function evaluations increase (from 200 to 400),
and a left to right traversal can help in visualizing performance differences with an increase
in decision space dimensions (8, 16 and 24 decision variables).
Figure 2clearly illustrates that GOMORS outperforms ParEGO for the 24 variable ver-
sions of the synthetic problems, both in terms of speed of convergence (at 200 evaluations)
and at algorithm termination after 400 function evaluations (see top right sub-plot of Fig. 2).
GOMORS’ convergence to a good solution is faster than ParEGO for the 8 and 16 variable
versions of the synthetic test problems, but the difference is not as distinguishable (from Fig.
2) as in the case of the 24 variable versions. The Wilcoxon rank-sum test [14] (at 5 percent
significance) confirms that performance of GOMORS is better than ParEGO for the 8, 16
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28 J Glob Optim (2016) 64:17–32
400
Uncovered Hypervolume
GOMORS ParEGO NSGA2
Algorithm
GOMORS ParEGO NSGA2 GOMORS ParEGO NSGA2
Expensive Function Evaluations
Decision Variables
12 Variables 24 Variables6 Variables
0.2
0.3
0.4
0.5
0.6
0.7
0.2
0.3
0.4
0.5
0.6
0.7
200
Fig. 3 Groundwater application problems: box plots of uncovered hypervolume metric values summed over
the two groundwater remediation design optimization problems, i.e GWDA and GWDM. See Fig. 2caption
for figure explanation
and 24 variable versions of the test problems after 100, 200 and 400 function evaluations.
Figure 2also indicates that both GOMORS and ParEGO significantly outperform NSGA-II
with reference to the synthetic problems when evaluations are limited to 400.
5.3 Results: groundwater remediation design problem
The box-plot analysis methodology for the groundwater remediation problem is similar to
the one employed for the synthetic test problems and the analysis is summarized in Fig. 3.
Results summarized in Fig. 3are consistent with the findings observed with the synthetic
test functions in Fig. 2. GOMORS and ParEGO both outperform NSGA-II for the two MO
groundwater problems defined in Table 2, GWDA and GWDM, when function evaluations
are limited to 400. Performance of GOMORS (as per Fig. 3) is superior to ParEGO with
application to the 12 and 24 variable versions of GWDA and GWDM, both in terms of
speed of convergence (at 200 evaluations) and upon algorithm termination after 400 function
evaluations. The difference between GOMORS and ParEGO is not visually discernible for
the 6 variable versions of the problems, but the rank-sum test (at 5 percent significance)
confirms that performance of GOMORS is better, which is supported by Fig. 4.
Figure 4provides a visual comparison of non-dominated trade-offs obtained from
GOMORS and ParEGO with application to the GWDM groundwater problem. The red
line within each sub-plot is an estimate of the Pareto front of GWDM. Since knowledge
of the true front is not known, we obtained our estimate of true Pareto front through a sin-
gle trial of NSGA-II with 50,000 function evaluations. The green dots within each sub-plot
correspond to the non-dominated solutions obtained via application of the algorithm ref-
erenced in the sub-plot. There are two sub-figures within Fig. 4and four sub-plots within
each sub-figure. Figure 4a corresponds to the median (the fronts ranked fifth in ten experi-
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J Glob Optim (2016) 64:17–32 29
(A) 6 Decision Variables
0
1
2
3
F2(x)
02000 4000
0
1
2
3
F
1(x)
F2(x)
GOMORS
200 Evauations
ParEGO
200 Evaluations
0
1
2
3
F2(x)
02000 4000
0
1
2
3
F1(x)
F2(x)
GOMORS
200 Evaluations
ParEGO
200 Evaluations
0
1
2
3
F2(x)
02000 4000
0
1
2
3
F1(x)
F2(x)
GOMORS
100 Evaluations
ParEGO
100 Evalutions
0
1
2
3
F2(x)
02000 4000
0
1
2
3
F1(x)
F2(x)
GOMORS
100 Evaluations
ParEGO
100 Evaluations
(B) 24 Decision Variables
0 2000 4000
F
1
(x)
0 2000 4000
F1(x)
0 2000 4000
F1(x)
0 2000 4000
F1(x)
Fig. 4 Estimated non-dominated front for groundwater problem GWDM: visual comparison of median non-
dominated fronts (as per uncovered hypervolume) obtained from GOMORS and ParEGO for GWDM with 6
(Fig. 4a) and 24 (Fig. 4b) decisions, and after 100 and 200 expensive function evaluations. Red line is result
of 50,000 evaluations with NSGA-II. Green circles are non-dominated solutions from GOMORS or ParEGO
ments of each algorithm) non-dominated fronts obtained from each algorithm for GWDM
with 6 decision variables and Fig. 4b corresponds to the median fronts obtained from each
algorithm for GWDM with 24 decision variables. A clock-wise traversal of sub-plots within
each sub-figure depict results in the following order: (1) GOMORS after 100 evaluations, (2)
GOMORS after 200 evaluations, (3) ParEGO after 200 evaluations, and (4) ParEGO after
100 evaluations.
Figure 4indicates that both algorithms seem to converge to the estimated Pareto front and
also manage to find diverse solutions on the front for the 6 variable version of GWDM. More
diverse solutions are obtained from GOMORS however, and with better speed of convergence
than with ParEGO (depicted by the 6 variable version results after 100 and 200 evaluations).
Performance of GOMORS is significantly better than ParEGO for the 24 variable version of
the problem both in terms of speed of convergence (depicted by 24 variable GWDM version
results after 100 and 200 evaluations) and diversity of solutions. In case of computationally
expensive real-world problems it may not be possible to run multiple MO experiments. Hence,
we also visually analyzed trade-offs of worst case solutions (as per uncovered hypervolume)
obtained from GOMORS and ParEGO for both GWDM and GWDA. The visualizations
are provided in Section F of the online supplement to this paper. After performing a metric
based analysis and a visual analysis, it can be concluded that performance of GOMORS is
superior to ParEGO on the test problems examined here, especially on the relatively higher
dimensional problems.
6Conclusion
GOMORS is a new parallel algorithm for multi-objective optimization of black box functions
that improves efficiency for computationally expensive objective functions by obtaining good
solutions with a relatively small number of evaluations (<500). GOMORS accomplishes
this with the construction in each iteration of a surrogate response surface based on all
the values of the objective functions computed in the current and previous iterations. Then
evolution and non-dominated sorting are applied to the inexpensive function describing this
surrogate surface. It is then possible to evaluate a large number of points on the surrogate
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30 J Glob Optim (2016) 64:17–32
inexpensively so multiple Rules can be used to select a diversity of points for expensive
evaluation on parallel processors. The use of these multiple Rules is the innovative aspect
of GOMORS and contributes to its effectiveness and robustness with limited numbers of
evaluations (<500). GOMORS is very different from ParEGO, which solves multiple single
objective problems.
Our numerical results indicate that GOMORS was significantly more effective than
ParEGO when the number of evaluations was quite limited relative to the difficulty of the
problem (Figs. 2,3,4) for both the test functions and the groundwater partial differential
equation models. Computational demands for nonlinear optimization will grow rapidly as
the dimension increases, so the effectiveness of GOMORS becomes more obvious on higher
dimensional problems, as is evident in Figs. 2,3,4.
There are many real application models with computational times that are so large that
the evaluations will be greatly limited, especially for multi-objective problems. For example,
analysis of a carbon sequestration monitoring problem required global optimization with
seven decisions of a nonlinear, multiphase flow transient PDF model that took over 2 hours
per simulation [11]. So even 100 evaluations for a problem like this is probably more eval-
uations than are feasible. Hence the GOMORS approach to multi-objective optimization
is a contribution in the area of surrogate-assisted multi-objective optimization of objective
functions based on computationally expensive, multimodal, black box models.
Acknowledgments The research was supported by a Fulbright Fellowshipto Akhtar and grants to Shoemaker
from NSF (CISE 1116298) and to Mahowald and Shoemaker from DOE (SciDAC).
Open Access This article is distributed under the terms of the Creative Commons Attribution License which
permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source
are credited.
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