# Detection of Hidden Objectives and Interactive Objective Reduction

Seyed Mahdi Shavarani  Alliance Manchester Business School, University of
Manchester, Manchester, M13 9PL, UK
Email: {seyedmahdi.shavarani,manuel.lopez-ibanez,
<EMAIL_ADDRESS>
This work has been submitted to the IEEE for possible publication. Copyright
may be transferred without notice, after which this version may no longer be
accessible.  Manuel López-Ibáñez  Alliance Manchester Business School,
University of Manchester, Manchester, M13 9PL, UK
Email: {seyedmahdi.shavarani,manuel.lopez-ibanez,
<EMAIL_ADDRESS>
This work has been submitted to the IEEE for possible publication. Copyright
may be transferred without notice, after which this version may no longer be
accessible.  Richard Allmendinger  Alliance Manchester Business School,
University of Manchester, Manchester, M13 9PL, UK
Email: {seyedmahdi.shavarani,manuel.lopez-ibanez,
<EMAIL_ADDRESS>
This work has been submitted to the IEEE for possible publication. Copyright
may be transferred without notice, after which this version may no longer be
accessible.

###### Abstract

In multi-objective optimization problems, there might exist _hidden_
objectives that are important to the decision maker but are not being
optimized. On the other hand, there might also exist _irrelevant_ objectives
that are being optimized but are of less importance to the DM. The question
that arises here is whether it is possible to detect and reduce irrelevant
objectives without deteriorating the quality of the final results? In fact,
when dealing with multi-objective problems, each objective implies a
significant cost best avoided if possible. However, existing methods that
pertain to the reduction of objectives are computationally intensive and
ignore the preferences of the decision maker. In this paper, we propose an
approach to exploit the capabilities of interactive evolutionary multi-
objective optimization algorithms (EMOAs) and the preference information
provided by the decision maker, to detect and eliminate the irrelevant
objectives during the optimization process and replace them with hidden ones,
if any. The proposed method, which is based on uni-variate feature selection,
is computationally effective and can be integrated into any ranking-based
interactive EMOA. Using synthetic problems developed in this study, we
motivate different scenarios in the experiments and prove the effectiveness of
the proposed method in improving the computational cost, the total number of
objective evaluations and the quality of the final solutions.

###### Index Terms:

Interactive Multi-Objective Optimization, Hidden Objective Functions,
Redundant Objectives, Irrelevant Objectives, Machine Learning, Dimension
Reduction, Feature Selection

##  I Introduction

Interactive Evolutionary Multi-Objective Algorithms (iEMOAs) mostly assume
that the objectives being optimized match the ones that interest all potential
Decision Makers (DMs), however, this may not be the case. As instance, the DM
may not be consulted during the modelling phase or his desires may not be well
satisfied during the group decision-making on final objectives to be
optimized. As the result, there may exist objectives that are optimized but
the DM is not really interested in and on the other hand, there are objectives
that the DM cares for but are not being optimized. These inconsistencies may
cause the elicited preference information appear non-rational, e.g. when a
solution that is dominated with respect to the modelled objectives is
preferred by the DM over a non-dominated one because the former is better than
the latter with respect to features that are not modelled as objectives by the
algorithm. A real-world example would be the case discussed by Ramos-Pérez et
al. [1], where a problem is defined to optimize the food served to school
children in terms of not only cost, but also a number of metrics of
nutritional value and food variety. The variations in food types is optimized
by penalizing repetitions of each food group. Repetitions in each food group
may be considered as an independent objective. However, in that study they are
merged “a priori” into a single objective. A DM that is interacting with the
system may consider a given non-optimal menu (not) appealing but may not be
able to justify this decision. Is it because there is a hidden tendency to
optimize a certain type of food type? When making decisions, does the DM
really considers all nutritious and food variety metrics as assumed by the
model? Does the system really need to optimize all of the objectives? Or are
there any extra objectives that the DM considers but are not reflected in the
model? Another example would be optimizing the behavior of robots [2]. There
might exist a set of potential objectives (e.g., average speed, morphology,
accuracy, etc.) for how well the robot behaves but, at the end, a human
decides which behaviors are better by observing the robot perform tasks. Thus,
there might be a disconnection between the set of objectives being optimized
and the set of (relevant) objectives that the DM uses in evaluating
alternative behaviors.

Under such circumstances, it is desirable to have an algorithm that identifies
relevant and irrelevant objectives and updates the optimization model in such
a way that only the relevant objectives take part in the optimization process.
Removing irrelevant objectives is specifically important in many-objective
optimization problems. In some scenarios there is a large number of solution
features that DMs may wish to optimise and there is no way but to model the
problem as a many-objective optimization problem. However, if not all
objectives are relevant to a particular DM, removing those irrelevant
objectives will be beneficial to solving many-objective problems where the
non-dominance strategy loses its selection pressure as most of the solutions
become non-dominated with respect to the current population [3, 4, 5].Even in
the case of many-objective algorithms that overcome problems associated with
the dominance-based EMOAs [6, 7], objective reduction techniques are still
beneficial because, first, the running time increases exponentially with the
number of objectives [8] and, second, the number of non-dominated solutions
required for covering the whole Pareto front increases exponentially with the
number of objectives [9, 10], hence, complicating the decision-making phase
[11].

Given the benefits of reducing the number of objectives, previous research has
considered removing objectives that are highly correlated to other objectives
[11, 12], or their elimination does not have much of an effect on dominance
relations [13, 5].Here, we consider an approach for objective reduction that
is different and complementary to previous ones.

Here, for the first time in the literature, we consider the preference
information elicited in interactive EMOAs as an opportunity to detect
irrelevant objectives during the optimization and update the model in a way
that only relevant objectives, i.e. important for the DM, are optimized, while
the rest of the objectives are made inactive. The relevance of objectives may
change from a DM to another. The proposed approach gives the interactive EMOA
the ability to adapt to different DMs and update objectives that are relevant
to the DM who is interacting with the EMOA. The main contributions of this
study can be summarized as follows:

  * •

A motivation and formal definitions of the problem of identifying relevant
objectives amongst a set of potential objectives in interactive multi-
objective optimization.

  * •

Proposal of an approach to detect relevant and hidden objectives and to update
the set of objectives during the optimization to optimize only the relevant
ones. The approach draws on feature selection methods and can be applied to
any ranking-based interactive EMOA.

  * •

An approach is proposed to simulate hidden and irrelevant objectives which can
be applied to any multi-objective problem and utility function. This approach
is demonstrated for DTLZ problems [14] and multi-objective NK-landscapes
$\rho$MNKs [15].

  * •

A validation of the proposed detection method is performed which includes:

    1. i.

Problems of varying dimensionality, complexity, and Pareto front structure.

    2. ii.

Different utility functions to represent different DMs.

    3. iii.

Different feature selection methods to detect relevant objectives, and

    4. iv.

A sensitivity analysis to understand the performance impact of key parameters
of the proposed approach.

The results of the experiments show that our proposed method is able to
quickly replace irrelevant objectives with relevant ones and significantly
improve the quality of final solutions. Moreover, it is observed that, even in
cases where the utility of final solution is not favored by the proposed
method, the computational effort is highly reduced. The rest of the paper is
organized as follows. The problem is defined and formulated in Section II. A
summarized background on previous efforts towards objective reduction is given
in Section III. In Section IV the proposed solution method are elaborated in
detail. The experimental setup is laid out in Section V. The results of the
experiments are discussed in Section VI. Finally Section VII provides
conclusions and future research directions.

##  II Definitions and formulations

Let us consider a many-objective optimization problem in which we could
potentially optimize simultaneously a set $\\{f_{1},\dotsc,f_{m}\\}\in F$ of
$m$ objectives:

$\begin{split}\text{Minimize}\quad&\bigl{(}f_{1}(\mathbf{x}),\dotsc,f_{m}(\mathbf{x})\bigr{)}\\\
\text{subject to}\quad&\mathbf{x}\in\mathcal{X}\\\ \end{split}$ (1)

where each objective $f_{i}\colon\mathcal{X}\to\mathbb{R}$ depends on a
solution vector $\mathbf{x}=(x_{1},\dotsc,x_{n})$ of $n$ design (or decision)
variables, and $\mathcal{X}$ is the feasible decision space.

######  Definition II.1 (Domination).

A solution $\mathbf{x}\in\mathcal{X}$ dominates solution
$\mathbf{y}\in\mathcal{X}$ if $\mathbf{f}(\mathbf{x})$ is not worse than
$\mathbf{f}(\mathbf{y})$ in any objective and $\mathbf{f}(\mathbf{x})$ is
strictly better than $\mathbf{f}(\mathbf{y})$ in at least one objective.

######  Definition II.2 (Non-domination).

A feasible solution $\mathbf{x}\in\mathcal{X}$ is said to be non-dominated if
there is no solution $\mathbf{y}\in\mathcal{X}$ that dominates it. Non-
dominated solutions are also known as Pareto optimal solutions.

######  Definition II.3 (Pareto front).

The projection of the Pareto solutions on the objective space is known as the
Pareto front (PF).

######  Definition II.4 (Potential objectives).

All $m$ objectives in $F$ are called _potential_ objectives.

######  Definition II.5 (Redundant objectives).

An objective is called _redundant_ if it can be eliminated without changing
the set of Pareto optimal solutions [16, 17]. Saxena et al. [18] extend this
definition to include objectives that are not conflicting with a non-redundant
objective.

The above definitions are independent of the preferences of a DM interacting
with the EMOA. Interactive methods can be classified into ad-hoc and non-ad-
hoc methods [19]. Non-ad-hoc methods assume there exists a utility function
(UF) which derive the DM’s decisions but are unknown to the EMOA [20]. Due to
the popularity of UFs in modeling preference models, vast majority of iEMOAs
are non-ad-hoc methods thus we focus on those in the remainder of the paper.

Without loss of generality, here we assume an unknown UF
$U\colon\mathbb{R}^{m_{\textsc{dm}}}\to[0,1]$ the value of which depends
mainly on $F_{\textsc{dm}}\subseteq F$ objectives ($m_{\textsc{dm}}=\lvert
F_{\textsc{dm}}\rvert\leq m=\lvert F\rvert$). An instance will be a linear
additive utility function where the weights associated to irrelevant
objectives are close to zero.

######  Definition II.6 (Irrelevant objectives).

An objective is “ _irrelevant_ ” to the DM if it does not appear in the DM’s
utility function. In other words, irrelevant objectives ($F\setminus
F_{\textsc{dm}}$) do not contribute to the utility of solutions.

Interactive methods try to somehow estimate the DM’s UF by exploiting the
preference information that is elicited from the DM in the interactions that
are performed during the optimization process. The estimated UF is then used
in the next generations of the interactive EMOA to guide the search towards
that part of the Pareto front that is of interest to the DM. Let us assume
that, at some moment of its execution, a given EMOA only optimizes a subset
$\hat{F}\subseteq F$ ($\hat{m}=\lvert\hat{F}\rvert\leq m$) of the potential
objectives.

######  Definition II.7 (Active objective).

An objective is _active_ if it is optimized by the EMOA. The set of active
objectives is represented by $\hat{F}$. Inactive objectives
($F\setminus\hat{F}$) are either evaluated but ignored by the EMOA or not
evaluated at all, unless required by the DM during an interaction.

######  Definition II.8 (Objective evaluations & computational efficiency).

An objective evaluation is defined as a single evaluation of any of the
objectives corresponding to a solution ($f_{i}(\mathbf{x})$) and is different
than solution evaluation ($\mathbf{f}$($\mathbf{x}$)) which corresponds to
evaluation of all of the objectives of a given solution. In line with our
previous definitions, each solution evaluation is equivalent to $\hat{m}$
objective evaluations. On the other hand, computational cost can be defined in
a broader scale inclusive of objective evaluations and other aspects of the
solution procedure such as the cost of non-domination sorting, which are again
affected by the number of active objectives which participate in sorting and
optimization. Thus, we use the number of active objectives and objective
evaluations as an indicator of computational efficiency; i.e. reducing the
number of active objectives reduces computational costs.

######  Definition II.9 (Hidden objectives).

An objective $f_{i}\in F$, is said to be _hidden_ if $f_{i}\in
F_{\textsc{dm}}\land f_{i}\notin\hat{F}$. That is, an objective that appears
in the DM’s UF (relevant), but is not (currently) active. If
$F_{\textsc{dm}}\subset\hat{F}$, then no hidden objectives exist, but the EMOA
is still optimizing some irrelevant objectives. Similarly, if
$F_{\textsc{dm}}=\hat{F}$, then neither hidden nor irrelevant objectives
exist, and the EMOA is optimizing precisely the objectives that the DM cares
about.

Being able to reveal hidden objectives is beneficial, since otherwise there is
a disconnection between what the algorithm is optimizing and what the DM cares
about. Moreover, the interactive algorithm may become confused if the
interaction with the DM is consistent with the dominance criterion for the
objectives in $F_{\textsc{dm}}$ but appears to contradict it for the
objectives in $\hat{F}$.

Without any prior information about $F_{\textsc{dm}}$, most EMOAs would
optimize all potential objectives $\hat{F}=F$ leading to a many-objective
optimization problem, which is inherently challenging for EMOAs as discussed
above.

From these definitions, it can be concluded that while _redundant_ objectives
are determined based on the structure of the problem, _irrelevant_ and
_hidden_ objectives are defined from the DM’s perspective. While there are
studies on detection and elimination of redundant objectives, which we review
in the next section, there is no prior research on the identification of
irrelevant and hidden objectives to the best of our knowledge. Our focus here
is to fill this gap and we propose a method to tackle it in Section IV.

##  III Background

A brief review of the literature on the reduction of _redundant_ objectives is
presented here. These methods are applied either “a priori” to facilitate the
optimization process or “a posteriori” to assist the DM in decision making.
Presented studies either consider correlations among objectives or their
importance in preserving the dominance relations among solutions. We have
categorized existing methods accordingly in such a way that makes it easy to
compare similar studies and also discuss their limitations. We acknowledge
there are reduction methods that attend to the dimensions of the decision
space, such as those discussed in [21, 22, 23], as well as those that focus on
constraint reduction [24, 25]. However, they fall out of the scope of this
research as our focus is on the objective space. Many of these studies use the
term _dimension_ reduction for the same concept. However, to avoid confusion
with methods that reduce the number of decision variables [21], we use the
term “objective reduction”.

####  III-1 Subset selection & preserving Pareto front solutions

Gal and Leberling [16] proposed one of the earliest approaches that considered
preserving the PF. Although the method was later extended by Agrell [17], they
both make exhaustive assumptions about the problem structure that are
impossible to validate for real-life problems. Brockhoff and Zitzler [13, 5]
consider objective reduction while preserving the PF. They introduce two
minimum objective subset selection (MOSS) methods: $\delta$MOSS finds a
minimal set of objectives with maximum error of $\delta$ and $k$-EMOSS finds a
subset with fixed size $k$ having the minimum error. Error here is a measure
of change in the set of PF solutions. The performance of these methods is
compared with PCA-based methods in [26] and the results indicated the
competitiveness of the methods. In [27], the authors use their minimal set
approach online in simple indicator-based evolutionary algorithm (SIBEA) [28].
As MOSS problems are computationally expensive, Guillén-Gosálbez [29] proposed
a linear programming model for $k$-EMOSS problem, which was demonstrated to
have acceptable performance.

López Jaimes et al. [30] use an unsupervised feature selection technique [31]
for preserving the most conflicting objectives and identifying the subset of
non-redundant objectives. In their approach, neighborhoods of objective
functions are formed based on correlation of some non-dominated solutions.
Then an objective is selected from each compact neighborhood while the rest
are dropped. Later, this technique was incorporated into NSGA-II [32] to
construct an online reduction method [33].

Singh et al. [10] focus on the Pareto corner points to identify which
objectives are sufficient to reproduce the PF. Some drawbacks of their
approach are: not capturing the whole PF, losing objectives that are
contributing somewhere on the PF away from corner points and possible
overestimation of dimensions.

####  III-2 Recombining non-conflicting objectives

Freitas et al. [34] used the harmonic level, introduced in [35, 36, 13, 37],
to identify objectives that can be merged into a new compound scalarized
objective with minimal effect on the PF. As opposed to conflict, there is
“harmony” among objectives when improvement of either of them does not lead to
deterioration of the other [36, 35, 37]. When there is harmony between two
objectives, they can be merged without any loss in quality of the results.
Similar to other objective reduction methods, the DM preferences are not
considered in the reduction process in [34]. However, the DM can decide on the
final number of objectives “a priori”. Similarly, de Freitas et al. [38] use
aggregation trees to identify harmonious objectives that could be aggregated
into single scalarized objectives with minimal loss.

####  III-3 PCA & Correlation among objectives

One of the first methods that relied on linear PCA to spot redundant
objectives was by Deb and Saxena [39] that was implemented within the NSGA-II
framework. Linear PCA relies on correlation among objectives to detect
conflicting objectives, thus it does not guarantee that the dominance relation
is preserved [11].Furthermore, linear PCA does not provide a good performance
for non-linear data, is limited by the type of the data it can handle and it
might fail identify all non-conflicting objectives properly, but in local
regions where solutions are close it may still be used for determination of
conflicts and harmony between objectives [40]. To address these issues, two
more methods were introduced in [41], which used correntropy PCA [42] and
maximum variance unfolding as was previously used in [43, 44]. PCA was also
used in [45], however the method was validated on a problem with $4$
objectives only. PCA was also used inline with NSGA-II in [46, 47] among
others, to handle redundant objectives in non-linear contexts. PCA-based
methods are applied on non-dominated solutions and thus their performance
relies heavily on proper approximation of the PF [10].

In their study, Costa and Oliveira [48] have demonstrated that objectives that
are deemed redundant by PCA may be “informative”, i.e., contain trade-off
information that would be lost if omitted. They suggest using PCA coupled with
BIPLOT [49] representation to confirm if a PCA-identified redundant objective
is not “informative”. The applicability of the method is limited to the
problems with only one independent objective.

####  III-4 Visualization

Some studies suggest identification of redundant objectives through
visualization [50, 51, 52, 53]. However, most of these studies aim to
facilitate intuitive visualization of the solutions rather than the
optimization process itself and are limited in the number of objectives they
can handle. Thus, they use objective reduction methods such as PCA to enable
the visualization and hence bring new insights to the DM.

##  IV Methods

As laid out, existing approaches for objective reduction suffer from
dependency on non-dominated solutions, expensive computations, elimination of
informative objectives, offline reduction, and not exploiting the DM’s
preferences. In this section, we illustrate a method that is able to use the
DM’s preferences, elicited during optimization by an interactive method, to
identify irrelevant objectives as well as hidden ones dynamically.The
experimental study will show that the proposed method is able to efficiently
reduce the number of objectives being optimized by the optimizer, hence
dedicating more computational resources to the optimization of relevant
objectives only.

In a nutshell, our proposal works as follows. At some point during the run of
the EMOA, the method interacts with the DM by showing the value of all
potential objectives of a selected subset of solutions and asking the DM to
rank the solutions. Feature selection applied to the rankings and the
objective values identifies which objectives have the most significant effect
on the ranking. The method uses this information to possibly activate
currently inactive objectives and/or deactivate currently active ones. The
EMOA then continues the search using the new set of active objectives. In what
follows the method is described in detail.

###  IV-A Feature Selection

We explore two feature selection methods in this study: Uni-variate feature
selection and Recursive Feature Elimination (RFE). This gives us the
opportunity to investigate and compare the results with respect to different
feature selection methods. Hereafter, feature and objective are used
interchangeably in this context.

####  IV-A1 Uni-variate Feature Selection

We propose the application of F-test uni-variate feature selection for
identifying the most relevant features. In uni-variate methods, each feature
is considered independently and any correlation between features is ignored
[54]. The uni-variate detection method applied here estimates the degree of
linear dependency between two random variables. Let $T$ be the set of
solutions presented to the DM at an interaction, where $\mathbf{f}_{j}\in T$
is the $j^{\text{th}}$ vector of objective values, and $f_{ji}$ denotes the
value of its $i^{\text{th}}$ objective out of the $m$ potential objectives.
The DM ranks the solutions according to her own preference of the objective
values. The vector of rankings is given by $\mathbf{r}$, where $r_{j}$ is the
rank corresponding to $\mathbf{f}_{j}\in T$. There is no restriction on the
rankings and two solutions may have the same rank.

The procedure for F-test uni-variate feature selection can be described as
follows (for reference on F-test feature selection see [55]):

Step 1:

The correlation $\rho_{i}$ between each objective (feature) $i$ and
$\mathbf{r}$ is computed as:

$\rho_{i}=\sum_{j=1}^{|T|}\frac{(f_{ji}-\bar{f_{\cdot
i}})\cdot(r_{j}-\bar{\mathbf{r}})}{\operatorname{Var}(f_{\cdot
i})\operatorname{Var}(\mathbf{r})}$ (2)

where $\bar{f_{\cdot i}}$ and $\operatorname{Var}(f_{\cdot i})$ are,
respectively, the mean and variance of the $i^{\text{th}}$ objective over all
solutions in $T$; and $\bar{\mathbf{r}}$ and $\operatorname{Var}(\mathbf{r})$
are the same for the vector of rankings.

Step 2:

The F-statistic for each objective is computed as:

$\mathcal{F}_{i}=\dfrac{\rho_{i}}{1-\rho_{i}}\cdot(\lvert T\rvert-2)$ (3)

Here, the F-statistic is a notion of how well an objective can explain the
rankings provided by the DM.

Step 3:

The $p$-values corresponding to each F-statistic is calculated by any
statistical software.

Step 4:

Features with lower $p$-value are selected. Number of selected features can be
either fixed ($k=\hat{m}$) or variable. In the latter case, objectives with
$p$-values less than a predetermined threshold $\tau$ are selected. These two
variants are explained in Section IV-B.

The lower the $p$-value, the better is the corresponding objective function in
explaining the DM’s rankings. The pseudo-code of uni-variate feature selection
is illustrated in Algorithm 1.

Input:

$F$ : Set of all potential objectives

$T$ : Set of ranked objective vectors

$\mathbf{r}$ : vector of ranks

Either

  * •

$k<\lvert F\rvert$ (for fixed number of objectives) or

  * •

$\tau$ (for variable number of objectives)

Output: $\hat{F}$: Selected objectives

1 for  _$i\leftarrow 1$ to $\lvert F\rvert$_ do

2 Step 1: Calculate $\rho_{i}$ using Eq. (2)

3 Step 2: Calculate $\mathcal{F}_{i}$ using Eq. (3)

4 Step 3: Calculate $p_{i}$ ($p$-value) from $\mathcal{F}_{i}$

5if  _Fixed number of objectives_ then

6 $\hat{F}$ $\leftarrow$$k$ objectives with lowest $p$-value

7else

8 $\hat{F}\leftarrow\\{f_{i}\in F\mid p_{i}<\tau\\}$

return _$\hat{F}$_

Algorithm 1 Uni-Variate Feature Selection

####  IV-A2 Recursive Feature Elimination

RFE is different than uni-variate feature selection in that, here, first
logistic regression is used to build a model based on all the features to
predict the rankings. In the next step, the feature with the minimum
contribution to the constructed model is excluded from the selected subset
[56].

The two steps are repeated until a preset number of objectives $k$ is selected
(fixed number of objectives) or the minimum contribution of the remaining
objectives is greater than a threshold $\tau$ (variable number of objectives).
There are several ways to measure contribution to the model ($\phi$) such as
the coefficient of the objective in the model or the importance of the
objective. Here, we use the latter to be consistent with uni-variate feature
selection. The importance of each feature is calculated based on the drop in
the accuracy of the model when that feature is eliminated. The algorithm for
RFE is depicted in Algorithm 2.

Input:

$F$: Set of all potential objectives

$T$: Set of ranked solutions

$\mathbf{r}$: vector of ranks

Either

  * •

$k<\lvert F\rvert$ (for fixed number of objectives) or

  * •

$\tau$ (for variable number of objectives)

Output: $\hat{F}$: Selected objectives

1 $\hat{F}\leftarrow F$

2 while  _True_ do

3 Step 1: $M$ $\leftarrow$Build_Model($T$, $\mathbf{r}$, $\hat{F}$)

4 Step 2:
$f_{j}\leftarrow\operatorname*{arg\,min}_{f_{i}\in\hat{F}}\phi(f_{i})$

5 if  _$\phi(f_{j}) >\tau$ or $|\hat{F}|=k$ or $|\hat{F}|=2$_ then

6 break

7 Step 3: $\hat{F}\leftarrow\hat{F}\setminus f_{j}$

return _$\hat{F}$_

Algorithm 2 Recursive Feature Elimination

###  IV-B Fixed versus Variable Number of Active Objectives

The number of features (active objectives) selected can be defined in
different ways. Here we explore the following two alternatives:

####  IV-B1 Fixed number of objectives ($k$)

The optimization starts with $k$ active objectives and this number is kept
constant throughout the optimization process such that activating an inactive
objective implies deactivating an active one.

####  IV-B2 Variable number of objectives

We select the subset of objectives which meet a predetermined threshold
$\tau$. In case of uni-variate feature-selection, the lower the value of
$\tau$, the lower would be the number of objectives with acceptable
$p$-values. If there is only one objective with a $p$-value lower than $\tau$,
the two objectives with lowest $p$-values are selected instead. For RFE, in
each step an objective is selected only when its contribution to the model is
greater than $\tau$. Thus, higher values of $\tau$ corresponds to less number
of active objectives. Having two feature selection methods and two approaches
with fixed and variable number of objectives as explained above, we have four
total variations defined as follows:

  1. 1.

$k$-_HD_ : Uni-variate feature selection with fixed number of objectives

  2. 2.

$\tau$-_HD_ : Uni-variate feature selection with variable number of objectives

  3. 3.

$k$-_HDR_ : RFE with fixed number of objectives

  4. 4.

$\tau$-_HDR_ : RFE with variable number of objectives

The proposed methods can be applied to any ranking-based interactive EMOA for
objective reduction and/or detection of hidden objectives in order to find the
objectives that are relevant to the DM. Here, we will focus on extending
BCEMOA [57] with our proposed method to show how the method can be integrated
with any ranking based algorithm. In what follows, the modified BCEMOA here
called BCEMOA-HD, is explained in detail.

###  IV-C BCEMOA-HD

As discussed, our proposed HD/HDR method can be applied to any ranking based
interactive method. In this study, BCEMOA is selected and equipped with
detection of hidden objectives. BCEMOA [57] is an interactive EMOA based on
NSGA-II. The algorithm starts with a population of randomly generated
solutions (_pop_). The population is evolved with NSGA-II for $\text{gen}_{1}$
generations. Next, at each interaction step, the best
$N^{\text{exa}}$solutions are selected from the evolved population and ranked
by the DM. The data along their ranks are then used to train a Support Vector
Machine (SVM) model to learn a utility function ($U_{\textsc{svm}}$). The
evaluations of the learned UF replaces the crowding distance in the next
generations. Further interactions with the DM provide additional samples to
re-train the SVM model and improve the predictions of the learned utility.

Similar to the original BCEOMA, the BCEMOA-HD algorithm, proposed here, starts
with a set of active objectives $\hat{F}$. All inactive objectives not in
$\hat{F}$ do not need to be evaluated during the optimization and do not
participate in dominance ranking and evolution of the population. However,
during the interactions, all objectives in $F$ are evaluated for the solutions
that are presented to the DM. Immediately after each interaction, the feature
selection method described in Section IV-A is applied to the objective vectors
and their rankings to identify relevant objectives and update $\hat{F}$.
Consequently, the population should also be updated to be evaluated for
$f_{i}\in\hat{F}$. SVM is also used to learn $U_{\textsc{svm}}$based on active
objectives in updated $\hat{F}$ and their rankings. An overview of BCEMOA-HD
is shown in Algorithm 3.

As described above, compared to the original BCEMOA we have modified the
algorithm in lines 3 and 3, where feature selection is deployed and the set of
active objectives is updated. Another modification was applied to the original
BCEMOA in the selection of the best solutions presented to the DM. When there
is no variance in the values of some objective, for example, because its
values are near-optimal, then their correlation with the rankings provided by
the DM is almost zero, which would result in the feature selection considering
that objective as irrelevant and replacing it with a different one. To address
this issue and to preserve the elitism, the solution that was ranked best by
the DM in the last interaction is always included in the next set of solutions
presented to the DM by BCEMOA-HD.

Input:

$N^{\textup{int}}\colon$ Total number of interactions

$N^{\text{exa}}\colon$ Number of training examples per interaction

$\textit{pop}\colon$ population of solutions

$\textit{gen}_{1}$: Generations before first interaction

$\textit{gen}_{i}$: Generations between two interactions

$F$: Set of potential objectives

$\hat{F}$: Set of active objectives

Either

  * •

$k<\lvert F\rvert$ (for fixed number of objectives) or

  * •

$\tau$ (for variable number of objectives)

Output: The most preferred solution

1

2$\textit{T}\leftarrow\\{\\}$

3 $\textit{r}\leftarrow\\{\\}$

4 pop $\leftarrow$NSGA-II for $\textit{gen}_{1}$ generations (pop)

5 for  _$1$ to $N^{\textup{int}}$_ do

6
$\textit{T}_{i}\leftarrow\texttt{select}\,N^{\text{exa}}\,\texttt{solutions}$

7 Evaluate solutions in $\textit{T}_{i}$ for all objectives in $F$

8 $\textit{r}_{i}$ $\leftarrow$DM_ranks($\textit{T}_{i}$)

9 $\textit{T}\leftarrow\textit{T}_{i}$ $\cup$ T

10 r $\leftarrow$$\textit{r}_{i}$ $\cup$ r

11 $\hat{F}$ $\leftarrow$feature_selection(T, r, $F$, $\tau$ / $k$)

12 Evaluate pop for $f_{i}\in\hat{F}$

13 $U_{\textsc{svm}}$$\leftarrow$train_SVM($T$ with objectives $\in\hat{F}$,
$\mathbf{r}$)

14 Crowding_Distance $\leftarrow$$U_{\textsc{svm}}$

15 pop $\leftarrow$NSGA-II for $\textit{gen}_{i}$ generations, only
considering $\hat{F}$

return _Best (first) solution in the pop ranked by non-domination sorting and
$U_{\textsc{svm}}$_

Algorithm 3 BCEMOA-HD

##  V Experimental Setup

To evaluate the effectiveness of our proposed method, we design a set of
experiments that can comprehensively cover different aspects of the problem of
identifying hidden and irrelevant objectives. To do so, we propose a possible
way of simulating hidden and irrelevant objectives. Two sets of well-known
benchmark problems are selected from the literature. Each problem features
various difficulties to different parts of the solution method including
convergence towards most preferred point and performance in highly rugged
landscapes. In what follows, a detailed description of the design of the
experiments is laid out.

In the experiments with variable number of objectives we seek to investigate
how the method performs for objective reduction purposes. Thus, for these set
of experiments all objectives are active from the start of the run
($\hat{F}=F$), while for fixed number of objectives only designated objectives
are active ($\hat{F}\subset F$) at the beginning of the optimization.

###  V-A Synthetic problems with hidden objectives

We create synthetic problems that feature irrelevant and hidden objectives by
extending existing benchmark problems as follows. Given a problem with
$m=\lvert F\rvert$ potential objectives, we extend it with a binary vector
$\mathbf{d}\in\\{0,1\\}^{m}$ that specifies which objectives are considered by
the optimizer, i.e., $d_{i}=1$ iff $f_{i}\in\hat{F}\subset F$, where $f_{i}$
indicates the $i^{\text{th}}$ objective function. That is, given a solution
$\mathbf{x}$, whose objective vector is
$\mathbf{f}(\mathbf{x})=(f_{1}(\mathbf{x}),\dotsc,f_{m}(\mathbf{x}))$, the
optimizer only considers
$\mathbf{\hat{f}}(\mathbf{x})=\mathbf{f}(\mathbf{x})\odot\mathbf{d}$, where
$\odot$ denotes the element-wise product of two vectors. The optimizer is able
to change the set of active objectives by changing the vector $\mathbf{d}$.

On the other hand, feature selection methods and the DM have access to
$\mathbf{f}(\mathbf{x})$, that is, the unknown UF used by the DM when asked to
compare pairs of objective vectors actually evaluates
$U(\mathbf{f}(\mathbf{x}))$.

###  V-B Benchmark Problems

We consider two well-known numerical and binary benchmark problems, namely,
multi-objective NK landscape problems with correlation between objectives
($\rho$MNK) [15] and DTLZ problems [14] with $m\in\\{4,10,20\\}$ objectives.
Problems with $m=4$ help us to better understand and investigate the dynamics
of the proposed methods, while larger number of objectives allows us to
evaluate the efficiency of the feature selection with variable number of
objectives in many-objective problems.

$\rho$MNK problems are used to analyse the effects of correlation among
objectives and smoothness of the landscape on the performance of the proposed
method. We consider $\rho$MNK instances with different values of correlation
among objectives $\rho\in\\{-0.25,0,0.25,0.5,0.75,0.9\\}$, taking into account
the restriction that $\rho\geq-1/(m-1)$ [15] and different values of parameter
$K$, which controls the smoothness of the landscape, namely,
$K\in\\{1,4,6,8\\}$ for problems with 4 objectives and $K\in\\{1,5,10,15\\}$
for many objective problems, considering the constraint $K<n$. The greater the
value of $K$, the more rugged is the fitness landscape. The value of $n$ is
kept fixed at 10 for problems with $m=4$, 20 for problems with $m=10$ and 30
for problems with $m=20$ for $\rho$MNK problems.

We use DTLZ1, DTLZ2, DTLZ7 from DTLZ test suit, which were also used in the
experiments on BCEMOA by its authors [57] and also in [27] for objective
reduction. DTLZ1 contains $11^{k}-1$ local Pareto-optimal fronts, and each of
them can attract the evolutionary algorithm. Thus, it can be used to test the
ability of the algorithm to deal with multiple local attractors. DTLZ2
investigates the performance of the algorithm when dealing with many objective
algorithms and the quality of the solutions in regard with proximity to true
PF. Finally, DTLZ7 has $2^{m-1}$ disconnected Pareto-optimal regions in the
objective space and is used to check the diversity of the solutions and the
performance of the algorithm in disconnected feasible space.

As suggested [14], the decision space dimension ($n$) is $m+4$ for DTLZ1,
$m+9$ for DTLZ2 and $m+19$ for DTLZ7. In DTLZ problems optimizing a subset of
objectives will optimize the rest of the objectives as well. To make the
problem more challenging and also to avoid collapsing the PF to one point when
projected to $k<m$ objectives, we follow [27] and map $x_{i}$ to
$x_{i}/2+0.25$, $i=1,\dotsc,n$, for DTLZ2 and bound $x_{i}$ within
$[0.25,0.75]$ for DTLZ1, which is also suggested by the authors of BCEMOA
[57]. Please note that this modification is not needed for DTLZ7 because it
does not collapse to a single point.

###  V-C Machine Decision Maker (MDM)

We adhere to the MDM framework introduced in [58] and simulate the DM’s
preference with a UF that implicitly expresses which objectives are relevant.
We define $\mathbf{c}$ as the ordered index set of relevant objectives such
that $i<j\to c_{i}<c_{j}$, and consider the following quadratic UFs that were
proposed in experiments on the original BCEMOA [57]:

$\displaystyle\text{UF1}(\mathbf{f})$
$\displaystyle=0.28f_{c_{1}}^{2}+0.38f_{c_{2}}^{2}+0.29f_{c_{1}}f_{c_{2}}+0.05f_{c_{1}}$
(4) $\displaystyle\text{UF2}(\mathbf{f})$
$\displaystyle=0.6f_{c_{1}}^{2}+0.05f_{c_{1}}f_{c_{2}}+0.23f_{c_{1}}+0.38f_{c_{2}}$
(5) $\displaystyle\text{UF3}(\mathbf{f})$
$\displaystyle=0.44f_{c_{1}}^{2}+0.14f_{c_{2}}^{2}+0.09f_{c_{1}}f_{c_{2}}+0.33f_{c_{1}}$
(6)

In addition, we consider the following Tchebychef UF:

$U_{\text{tch}}(\mathbf{f})=\max_{i\in\mathbf{c}}w_{i}|f_{i}-f_{i}^{*}|$ (7)

with $\mathbf{0}$ as ideal point $f^{*}$. The weights $w_{i}$ of irrelevant
objectives ($i\notin\mathbf{c}$) are set to zero while the weights of relevant
objectives are selected by trial and error in a way that the most preferred
solution is away from the corner points as far as possible. As shown in the
utilities above, in all cases, the DM only considers
$F_{\textsc{dm}}=\\{f_{i}|i\in\mathbf{c}\\}$, while other objectives are
irrelevant. We explain how we selected relevant objectives for each problem in
the next section.

Since the DTLZ objectives are to be minimized and to preserve consistency, we
assume minimization for utility values as well; i.e. solutions with lower
utility value are preferred by the DM.

###  V-D Selecting relevant objectives

Projection of the PF on lower dimensions might make it collapse to a single
point, for some problems. This is true for DTLZ problems even when the problem
is bounded [27]. Thus, the extended problem and its PF should be given a
careful examination before proceeding to experiments. For instance, the PF of
DTLZ7 would collapse to a point if the first two objectives are active and the
rest are inactive. After a careful examination, here the first and fourth
objectives are selected as relevant for DTLZ problems
($\mathbf{c}=\\{1,4\\}$). In the case of $\rho$MNK problems the first two
objectives are selected ($\mathbf{c}=\\{1,2\\}$). For $\rho$MNK problem with
four objectives, having $F_{\textsc{dm}}=\\{f_{1},f_{2}\\}$ and given an
initial $\mathbf{d}=(0,1,0,1)$, we can see that $f_{1}$ is a hidden objective
(relevant but not optimized), $f_{2}$ is both relevant and optimized, $f_{3}$
is irrelevant and not optimized, and $f_{4}$ is irrelevant and optimized. The
projection of the PF and the UF contours are depicted in Figure 1 for DTLZ
problems and selected relevant objectives. The location of the most preferred
solution is indicated by a triangle and the worst solution with a rectangle.
The graphs for UF1 and UF3 are identical to UF2 on DTLZ1 and hence omitted to
save space. The same is true for DTLZ2.

[width=0.48]graphs/Hidden_pfs_reduced.jpg

Figure 1: PF of the selected problems with first and fourth objectives being
active depicted over the contour lines of the different UFs. The position of
the worst and best solutions is marked respectively with a rectangle and a
triangle.

###  V-E Evaluation of the results

The experiments are performed in three different modes to enable the
assessment of the algorithms:

  1. 1.

Golden mode: No interaction is done in this mode and the algorithm directly
accesses the true UF of the DM instead of learning a UF. Moreover, only
relevant objectives are optimised from the start to end. This is the ideal
scenario.

  2. 2.

Only Learning mode: This mode corresponds to the original BCEMOA without any
detection of hidden objectives. The algorithm does not have access to the DM’s
UF and instead a UF is learned from pairwise comparisons provided by the MDM
at each interaction; i.e. at each interaction the MDM uses true UF to rank
solutions. Predictions from the learned utility are used to rank non-dominated
solutions, replacing the crowding distance in NSGA-II. The algorithm still
uses non-dominated sorting as the first criteria to rank solutions. Both non-
dominated sorting and the learned utility only consider the set of active
objectives $\hat{\mathbf{f}}(x)$. The set of active objectives never changes,
that is, $\mathbf{d}$ remains constant throughout the run.

  3. 3.

Learning $+$ detection mode: This is our proposed BCEMOA-HD that performs
detection of hidden objectives and is able to modify the set of active
objectives. Within this mode, we test $4$ variants of the HD method: $k$-_HD_
, $\tau$-_HD_ , $k$-_HDR_ , $\tau$-_HDR_. Similar to the Only Learning mode,
the optimization algorithm relies on non-dominated sorting and an UF that is
learned based on $\hat{\mathbf{f}}(x)$, and not the DM’s true utility.
However, in this mode, $\mathbf{d}$ and subsequently $\mathbf{\hat{f}}$ may
change after each DM interaction with the ultimate goal of converging to the
objectives that are actually relevant for the DM ($F_{\textsc{dm}}$).

Having these three modes makes it possible to evaluate the performance of the
proposed method compared to the original BCEMOA and to the best solution that
is achieved under an ideal scenario in Golden mode. The criteria for
evaluation of the performance is the true utility value of the final solution
returned by the algorithm in different modes. We also make record of the
active objectives after each interaction for Learning $+$ detection mode to
investigate how good the proposed method performs in detecting and diverging
towards the true set of objectives.

####  V-E1 Parameter settings

In all evaluation modes, BCEMOA uses the parameter settings proposed in the
original paper [57], including the parameters of the SVM learning model. In
particular, it uses a population size of $100$ and creates $100$ new solutions
at each generation, The total number of generations is 500 and
$N^{\text{exa}}=5$ solutions are shown to the DM at each interaction.

Within BCEMOA, NSGA-II runs for $gen_{1}=200$ generations before the first
interaction and there are $gen_{i}=30$ generations between subsequent
interactions. The total number of generations after the last interaction is
calculated by $500-gen_{1}-gen_{i}(N_{\text{int}}-1)$. Thus, changing the
number of interactions ($N_{\text{int}}$) would not alter total number of
generations. We run experiments with $1$, $3$ and $6$ interactions for DTLZ
problems. For $\rho$MNK problems, we only consider $6$ interactions and,
instead, we investigate the effect of different levels of correlation ($\rho$)
and ruggedness($K$). Each test was repeated 40 times with different random
seed.

####  V-E2 Implementations

The algorithms, machine DM and $\rho$MNK problems are implemented in Python
3.7.6. The implementations of NSGA-II within BCEMOA and DTLZ benchmarks are
provided by the Pygmo library 2.16.0 [59], the uni-variate feature-selection
and RFE implementations are based on Scikit-learn 0.23.1 (http://scikit-
learn.org/).

##  VI Experimental Results & Discussion

We have considered $3$ DTLZ benchmark problems and each of them are considered
with $3$ different values for number of interactions. The $\rho$MNK problems
are considered with $6$ values of $\rho$ and $4$ values of $K$. Considering
$3$ different number of objectives, in total we have investigated $99$
problems, each of which are considered with $4$ utility functions. Each
experiment is repeated $40$ times with different random seeds and the average
values are reported. Comparison of different modes is done with regard to the
utility value of the final solution returned by the algorithm averaged over
$40$ runs. In this section we focus on the most important findings. Figures
that do not include key findings are not presented to save space. However, the
complete set of results and figures can be found in the Appendix.

###  VI-A DTLZ problems with fixed number of objectives

The results of experiments on DTLZ problems with fixed number of active
objectives is illustrated in Fig. 2. When using online detection of hidden
objectives with DTLZ1 problem with $m=4$ or $m=20$ and fixed number of active
objectives, almost no improvement is observed in terms of the utility value
compared to Only learning. However, when $m=10$ the performance of $k$-_HD_
and $k$-_HDR_ is significantly better than Only learning and almost as good as
Golden mode. The only exception is when UF2 is used.

For DTLZ2, improvements in the performance can be seen when detection of
hidden objectives is active in the case of $m=10$ and $m=20$ when utility
functions UF1 and UF2 are used. Another important observation is better
performance of $k$-_HDR_ with more interactions, although it fails to get as
good as $k$-_HD_.

For DTLZ7, there are slight improvements when detection methods ($k$-_HD_ ,
$k$-_HDR_) are used. Complete list of figures can be found in the Appendix. In
general, it is observed that the proposed method can significantly improve the
utility value of the final solutions. Although in some cases it fails to do
so, the utility value is not deteriorated by the method.

[width=0.48]graphs/Combined_DTLZ_Fixed.jpg

Figure 2: Comparison of the performance of different modes for DTLZ problems.
Vertical axis indicated the utility value. Number of active objectives are
fixed. Horizontal axis indicates number of interactions. Lower utility values
are preferred.

###  VI-B DTLZ problems with variable number of objectives

In this set of experiments, the effectiveness of the proposed detection method
is investigated with regard to objective reduction capabilities and thus the
number of active objectives is not fixed. Furthermore, the experiments start
with all the objectives being active. The key results of these experiments are
illustrated in Figure 3. For DTLZ1 with $m=4$, the $\tau$-_HD_ and
$\tau$-_HDR_ perform better than the only learning mode on Tchebychef UF,
while for other UFs they have almost the same performance. With $m=10$ and
$m=20$, $\tau$-_HD_ and $\tau$-_HDR_ performs as good the Golden mode while
$\tau$ is slightly outperformed by $\tau$-_HD_. Results for DTLZ2, are
identical to those of DTLZ1; i.e. $\tau$-_HD_ and $\tau$-_HDR_ still doing as
good as Golden mode and outperform Only learning mode with $m=10$ and $m=20$.
For $m=4$ the performance of $\tau$-_HD_ and $\tau$-_HDR_ outperforms only
learning when UF3 is used, but cannot perform as good as Golden mode.

For DTLZ7 with $m=4$ all algorithms perform as good as Golden mode which
suggests the problem may be too easy for the algorithms and optimizing any
subset of objectives may lead to optimization of other objectives as well
[27]. However, with higher number of objectives, they fail to perform as good
as the Golden mode and $\tau$-_HD_ and $\tau$-_HDR_ outperform Only learning.

[width=0.48]graphs/Combined_DTLZ_Reduction.jpg

Figure 3: Comparison of the performance of different modes for DTLZ problems.
The number of active objectives is not fixed and detection mode is used as a
mean of objective reduction technique. Vertical axis indicated the utility
value. Horizontal axis indicates number of interactions. Lower utility values
are preferred.

###  VI-C $\rho$MNK problems

The results of experiments on $\rho$MNK problems with fixed number of
objectives and variable number of objectives are depicted in Figures 4 and 5,
respectively. In all of these experiments, detection method fails to make
significant improvements in terms of the utility value compared to Only
learning mode. However, there are significant savings in terms of
computational cost when $\tau$-_HD_ and $\tau$-_HDR_ are used. In the next
Section, this aspect will be discussed in full.

[width=0.48]graphs/Combined_rMNK_Fixed.png

Figure 4: Comparison of the performance of different modes for $\rho$MNK
problems. Number of active objectives is fixed. Vertical axis indicated the
utility value. Horizontal axis indicates the $\rho$ values. Lower utility
values are preferred.

[width=0.48]graphs/Combined_rMNK_Reduction.jpeg

Figure 5: Comparison of the performance of different modes for $\rho$MNK
problems. The number of active objectives is not fixed and detection mode is
used as an objective reduction technique. Vertical axis indicated the utility
value. Horizontal axis indicates the $\rho$ values. Lower utility values are
preferred.

###  VI-D Sensitivity analysis and Discussions

####  VI-D1 Power of Detection of True Set of Objectives

In terms of the power of the detection, a heatmap plot is provided in Figure
6. The plot illustrates the frequency of the times the true set of objectives
are activated by the detection of hidden objectives across all experiments on
$\rho$MNK problems with $10$ objectives when $\tau$-_HD_ is on. The x-axis
indicates the number of interaction within a single run and the y-axis (rows)
pertain to different objectives numbered from $1$ to $10$. Interaction $0$
refers to the state of the algorithm before the first interaction, when all
objectives are active. At the first interaction all objectives are active.
Then immediately after the first interaction most of the objectives are
dropped and first and second objectives that are relevant and included in the
utility functions, are kept active. It can be easily verified that the
$\tau$-_HD_ converges fast towards the true set of objectives. Another
surprising observation is that after the $6^{th}$ interaction, almost all
objectives become active, although to a lesser degree comparing with the
relevant ones. This observation can be justified as follows: When detection of
hidden objectives is enabled, the selected objectives might be optimized to
near-optimal values and then become fixed in the next generations. The uni-
variate correlation of these objectives having fixed values with ranking
vector would diminish to almost $0$. As a result, the objectives would be
identified as irrelevant and substituted with new ones whether or not they are
important to the DM.

[width=0.49]graphs/a_heatmap.jpg

Figure 6: Divergence of the algorithm towards true objective set through
interactions on problem $\rho$MNK with $10$ objectives ($c=\\{1,2\\}$). The
x-axis indicates the interaction number and the y-Axis indicates the index of
the objective functions. The darker color, the higher is the number of the
times the objective has been activated. The true set of objectives is found
after the first interaction in the majority of the experiments.

####  VI-D2 Analysis of Performance and Threshold $\tau$

As an important parameter of $\tau$-_HD_ and $\tau$-_HDR_ , $\tau$ has a
direct effect on the number of active objectives and those that are removed.
Thus, careful examination should be given in determining this parameter. To
inspect the effect of parameter $\tau$, which indirectly controls the number
of active objectives, the DTLZ problems with $m=20$ objectives are solved for
different values of $\tau$. The results for other problems are similar and
hence not discussed here. Setting $m=20$ would provide for a better
illustration of the efficiency of the proposed method in reducing the
computational requirements and objective evaluations. When $\tau=1$, all
objectives have $p$-value less than the threshold and, thus, all of them are
active; this means no objective reduction is performed and the mode is
identical to Only learning. The results in Figure 7 show that the performance
of the $\tau$-_HD_ improves with lower values of $\tau$ on DTLZ problems. On
the other hand, reducing the $\tau$ value would reduce the number of active
objectives. As explained earlier, in the case where there are fewer than $2$
objectives that pass the selection criteria, the two objectives with least
$p$-values are selected.

Figure 8 illustrates the change in the utility value of the best solution
gained after each interaction in a single run of the algorithm averaged over
40 runs. All changes in the objectives results in improvement of the utility
value after each interaction. The changes become smaller as the algorithm
converges towards a good solution.

The changes in the number of active objectives after each interaction,
averaged over $40$ runs, are depicted in Figure 9 where the shaded areas show
the 95% confidence interval around the mean. It can clearly be verified that
after the first interaction, the number of active objectives experiences a
steep decrease. As expected, when $\tau=1$, no objective reduction is
performed.

[width=0.98]graphs/threshold_analysis_reduced.jpg

Figure 7: Performance analysis of $\tau$-_HDR_ with different values of $\tau$
for DTLZ problems with $m=20$. Number of interaction in all runs is fixed to
6.

[width=0.49]graphs/DTLZ_trend_Reduced.jpg

Figure 8: Utility of the best-so-far solution within a single run after each
interaction on DTLZ2 problem when $\tau$-_HD_ detection mode is on. Results
are averaged over 40 runs. The results for other test problems are identical.

[width=0.49]graphs/threshold_a_analysis_DTLZ_Reduced.jpg

Figure 9: Number of active objectives within a single run after each
interaction for different values of $\tau$ in $\tau$-_HDR_ mode on DTLZ1 and
DTLZ2 test problems. The x-axis indicates the number of the interaction and
the y-axis illustrates the mean value of the total number of active objectives
averaged over 40 runs. The shaded areas show the 95% confidence interval
around the mean. The results for other test problems are identical.

Lower number of active objectives translates to lower computational costs.
However, another important criteria to consider is the ratio of evaluations of
relevant objectives to those of irrelevant ones.

$\tau$-_HD_ effectively reduces the number of objective evaluations and most
of objective evaluations pertain to relevant objectives. As defined, whenever
we refer to objective evaluations, we measure the total number of individual
objective evaluations. Thus, the number of objective evaluations depend on the
number of active objectives. For instance, when $\tau=1$ (equivalent to Only
learning mode), an experiment on DTLZ1 with UF3 and $\tau=0.2$, uses $600,000$
objective evaluations after the first interaction (before the first
interaction everything is identical, thus no comparison is made) and only
$10\%$ of these evaluations pertain to relevant objectives. However, when
$\tau$-_HD_ is used, only $45,000$ objective evaluations are done of which
$30,000$ $(67\%)$ are dedicated to relevant objectives. In general, objective
evaluations are reduced by up to $80\%$ compared to Only learning when
$\tau$-_HD_ or $\tau$-_HDR_ is used.

These results prove the effectiveness of the $\tau$-_HDR_ in terms of both
performance with regard to the quality of the final results and computational
efficiency.

##  VII Conclusion and future work

This study has considered interactive multi-objective problems where only an
unknown subset of all the defined objectives are of relevance to the DM. In
this context, we provided formal definitions of irrelevant, hidden and active
objectives that complement the definition of redundant objectives already
studied in the literature. Consequently, simulation of irrelevant, hidden, and
active objectives are discussed and an efficient method is proposed to deal
with these objectives.

Two variants of the method with variable and fixed number of active objectives
were studied. The results show that the variant with variable number of
objectives can be considered for dimension reduction purposes, reducing
efficiently the number of active objectives even after the first interaction;
this eliminates unnecessary evaluations of irrelevant objectives and improves
computational time and quality of the final solution returned by the
algorithm. The variant with fixed number of active objectives also manages to
detect and switch to relevant objectives. We also explored the application of
recursive feature selection to investigate if further improvements could be
achieved. However, the results indicate that there is no gain in using this
method over the uni-variate feature selection. Comparing the results achieved
for different test problem suits, it can be observed that improvement in the
utility value of the final solutions are more significant for DTLZ problems.
However, achievements with regard to saving in objective evaluations and
computational costs

is similar for both test problems. The benefits of this achievement is
highlighted where expensive objective evaluations are avoided. The effect of
correlation among objectives was investigated and proved to be insignificant
in tests with $\rho$MNK problems.

A sensitivity analysis was performed to scrutinize different aspects of the
problem, effect of key parameters of the algorithms on final results, and to
provide further insight into the proposed method. We showed experimentally
that the value of $\tau$ affects the number of active objectives and can be
used as a tool to control this aspect. The proposed feature selection method
only considers linear relations between objectives and DM’s rankings.
Considering nonlinear regression in feature selection would be a subject worth
to study, although it adds to the computational effort. In the case of BCEMOA,
one can use the SVM’s support vector in order to identify most important
objectives. However, the intention of this study was to target ranking-based
algorithms in general rather than a specific one. We considered 4 different
UFs to capture MDMs with different preferences. Future study could expand on
this and consider a range of further UFs, such as a Sigmoid UF and others
[60]. For some experiments, we observed that once the relevant objectives are
optimized, the proposed methods try to explore optimizing other inactive
objectives in the hope to find an even better utility. In future studies, it
would be desirable to introduce a stopping mechanism to avoid such excess
search and thus increase computational efficiency/reduce resource usage.

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