Multi Objective Optimization Problems

Most real-world optimization problems involve multiple conflicting objectives that should be optimized (minimized or maximized) simultaneously. This simultaneous optimization could provide better quality solutions and better insights to the problem which in turn will help making better decisions. Such problems are known as Multi-objective Optimization Problems (MOPs). In MOP, there is no longer a single optimum solution, but rather a set of trade-off solutions known as the Pareto optimal set or Pareto optimal solutions from which the decision maker can select one according to his/her preference. These solutions are optimal in a sense that no other solution is superior when all objectives are considered [31][3].

For example, consider the problem of determining the most efficient choice for purcha- sing a mobile phone. Assume we use two criteria to be optimized: the width of the mobile screen (to be maximized) and the price or the cost of the mobile(to be minimized). Due

to the conflicting relationship between these two objectives, many trade-off solutions can be found. For instance, the buyer can choose one of the trade off solutions (A, B, C) that

are shown in Figure2.2. If the buyer is concerned about the cost, then he/she can choose

solution A. If the buyer is interested in a wide screen mobile, then he/she can choose solution C.

MOP Formulation

Three basic elements define any MOP, namely: a set of decision variables, a set of ob- jectives and a set of constraints. Basically, a MOP consists of exploring the search space of the decision variables in order to find the vector of variables that optimizes the set of objectives while satisfying the set of constraints. More formally, a MOP is defined as

follows [31][15]:

A D-dimensional multi-objective optimization problem consists of finding a decision vector

~

x∗ _{= (x}∗

1, x∗2, . . . , x∗D)T that optimizes (minimizes or maximizes) a vector of M objectives

~

F (~x) = (f1(~x), f2(~x), ...., fM(~x)) subject to:

• Inequality and equality constraints that delimit the feasible region containing po- tential solutions

gj(~x) < 0 j = 1, 2, . . . , J hk(~x) = 0 k = 1, 2, . . . , K

• Domain constraints are specified by the lower bound x(L)i and upper bound x

(U )

i of

each decision variable xi . They delimit a subspace of the D-dimensional space RD

called the search space which includes the feasible region [15].

Figur 2.2: Simple example of a MOP inspired by [31] .

Dominance and Pareto Optimality

The notion of optimality does not apply directly to the multi-objective context as in the single objective optimization domain. In the context of single objective optimization problems, the solutions are easily compared against each other according to their objective values. The question is how to handle the comparison among solutions in the multi- objective context? Pareto Dominance is the concept used for this purpose. For instance,

we say that solution x1dominates solution x2if it is better than x2in at least one objective

and not worse than x2 in all other objectives. In this case x1 is better than x2 and we

dominates all other potential solutions to the problem. Moreover, if x1 does not dominate

x2 and vice versa, then both x1 and x2 are considered as non-dominated solutions [31].

• Pareto Dominance

Definition [22]: A vector u = (u1, . . . , uk) is said to dominate vector v = (v1, . . . , vk)

denoted by (u  v) if and only if (in the minimization case):

∀i ∈ {1, 2, . . . , k}, ui 6vi ∧ ∃i ∈ {1, 2, . . . , k} : ui < vi

Therefore, the best solutions with respect to the Pareto dominance relation consti- tute the Pareto optimal set and are called non-dominated solutions.

• Pareto Optimal Set

Definition [3][22]: A solution is said to be Pareto optimal if and only if it is not

dominated by any other solution in the search space Ω. The set of all Pareto optimal

solutions is called Pareto optimal set P∗ _{and is defined as:}

P∗ = {~x ∈ Ω|¬∃~x′ _{∈ Ω ~f (~x}′_{)  ~f (~x)}}

• Pareto Front

Definition [22]: Each solution vector in the Pareto optimal set corresponds to a

vector of the related objectives values. Therefore, a Pareto front represents the set

of Pareto optimal solutions in the objective space [22][31]. The Pareto front is defined

as:

Given a MOP ~F (x) and Pareto optimal set P∗_{, the Pareto front P F}∗ _{is defined as :}

Figure2.3shows an example of a Pareto front of a MOP with two objective functions and some other solutions dominated by the Pareto front solutions.

Figur 2.3: Example of a Pareto front and dominated solutions in the objective space for

a minimization problem. Taken from [31].

SOP versus MOP

Figure 2.4 illustrates the difference between the single objective and the multi-objective

optimization problems. We can see from Figure 2.4 (a) that we need to find the best

solution which is shown as the global optimum point in the figure. Figure 2.4 (b) shows

Figur 2.4: Difference between SOP and MOP inspired by [31].

With SOP, we deal with only one search space which is the decision space. While in MOP, we deal with two search spaces, namely the decision space and the objective space. The objective space is defined as the space in which the objective vector belongs. The decision space is the feasible search space of the problem in which the decision variables

belong [31]. Table 2.1 summarizes the differences between single objective and multi-

objective optimization problems.

SOP MOP

One objective function Multiple objective functions

One search space (decision search space) Two search spaces(decision and objective search spaces) Interested in one optimal solution Interested in a set of optimal solutions

Requires search only Requires search and decision making Optimality is related to objective fitness value Optimality is related to dominance concept

Tabel 2.1: SOP vs. MOP

blem is that in the case of multi-objective optimization, a multi-dimensional space called the objective space is constituted by the objective functions. Therefore, each solution in

the decision space maps to a point in the objective space. Figure 2.5 shows the mapping

process from decision space to objective space.

Figur 2.5: Mapping between decision space and objective space taken from [31].

The challenge in a multi-objective optimization problem is to find the Pareto optimal solutions that are as close as possible to the true Pareto optimal front and to maintain

diversity in this developed Pareto optimal set [31].

Multi-objective optimization problems have been solved in different ways by several

methods as will be seen in the review of related work given in section 2.3. Some of these

methods produce only one solution at each run. Therefore to obtain a set of compromise solutions, several runs are required. This issue has raised interest toward population based metaheuristics as they can produce several solutions at each run and they exhibit good

search capabilities in the search space [31][25]. In our study, we are interested in swarm

based metaheuristics for which a presentation is given in section 2.2.