#### 2.2. Related Work

The development of solution tools for the constrained MOP described in the previous subsection has generated a recent interest, mainly within the evolutionary computation community. This subsection provides a review of previous works related to this field, focusing on pure evolutionary techniques and on hybrid strategies as well.

Regarding MOEAs, two constraint handling mechanisms, classically used in the single-objective optimization framework, are adapted in [

13] to MOEA/D [

14]. The first one is stochastic ranking (SR, [

15]), which accounts for the need of directing the search according to both feasibility and objective value. Therefore, when comparing two solutions and at least one of them is infeasible, the comparison criterion is the objective function value with probability

${p}_{f}$, while, with probability

$1-{p}_{f}$, individuals are distinguished through the overall constraint violation

$\varphi $ (where for any solution

x,

$\varphi \left(x\right)={\sum}_{i=1}^{p}\left|{h}_{i}\left(x\right)\right|+{\sum}_{j=1}^{p}max\{0,{g}_{j}\left(x\right)\}$). On the other hand, the constraint-domination principle (CDP) is a multi-objective extension [

16] of Deb’s feasibility rules, where feasible solutions are compared according to dominance. The computational experiments, performed on the CTP-series [

17] and CF-series [

18], demonstrate that MOEA/D-CDP consistently outperforms MOEA/D-SR concerning hypervolume, inverted generational distance (IGD) and set coverage. It also performs reasonably well when compared with IDEA [

19] and DMOEA-DD [

20].

Besides, the Infeasibility Driven Evolutionary Algorithm (IDEA) [

19] is a modification of the classical CDP-based NSGA-II [

16] that enforces the participation of infeasible solutions through a parameter

$\alpha $, which represents the fraction of the current population allocated for those solutions. During the selection step, the pool combining parent and offspring populations is first divided into two (feasible and infeasible) sets. Then, non-dominated ranking is applied to both these sets, considering a function of the constraint violations as an additional objective for the infeasible set. Then, if

N is the population size, the

$\alpha \xb7N$ infeasible solutions with the best-ranking values, as well as the

$\left(1-\alpha \right)\xb7N$ feasible solutions with the best ranks, are included into the next population. Note that the constraint violation function used as an additional objective for infeasible solutions is computed as the sum of the ranking of the solutions, sorted in increasing order of the magnitude of constraint violation for each constraint (instead of the number of violated constraints formerly used [

21]). The experimental results conducted over some test functions of the CTP-series [

17] demonstrate that IDEA consistently outperforms CDP-NSGA-II.

Another adaptation of MOEA/D for handling constrained MOPs is introduced in [

22] where, for each solution, a modified function for the overall constraint violation accounts for the number of active constraints, in addition to the simple constraint violation. The mean value of this modified function over the population is weighted by the ratio of feasible individuals in the population, in order to produce a threshold on the allowed amount of constraint violation. Solutions that are within this threshold are considered as feasible and compared in terms of their objective values. Furthermore, a gradient-based local search is periodically invoked in order to repair infeasible solutions. The resulting algorithm, tested on some of the CTP-series test functions [

17], performs similarly or better than NSGA-II with CDP.

In [

23], the

$\u03f5$-constraint technique originally developed for single-objective optimization is extended for the solutions of MOPs. This strategy, proposed in [

24], consists in relaxing the tolerance level on constraints up to a value

$\u03f5$. Thus, when two solutions have an overall constraint violation lower than

$\u03f5$, they are both considered as feasible and compared in terms of their objective value. In [

24], the value of

$\u03f5$ is monotonically decreased according to a polynomial function, until some generation

${T}_{c}$. From then on,

$\u03f5$ is set to 0 in order to narrow the search on the feasible space. In [

23], the authors allow increasing the

$\u03f5$ level when the ratio of feasible solutions is greater than a threshold value, to promote exploration. This strategy, embedded in MOEA/D, is compared with MOEA/D-CDP [

13] and with the original

$\u03f5$-constraint mechanism (decreasing

$\u03f5$ pattern), over a set of nine constrained MOPs introduced earlier by the same authors [

25]. Furthermore, this strategy is also compared with classical MOEAs (either dominance or decomposition-based) in a later work [

26], over the CTP [

17] and CF series [

18], using IGD as a performance indicator. In any case, the MOEA/D-IEpsilon algorithm outperforms all its contenders, except IDEA [

19], which obtains similar performance levels.

More recently, the same authors developed a two-stage (Push and Pull Search, PPS) procedure, which first solves the unconstrained MOP (solutions are pushed towards the unconstrained Pareto front) and, in a second stage, include constraints to modify the first (unconstrained) approximation and identify the constrained Pareto front (solutions are pulled from infeasible regions towards the feasible space). The switching criterion between both phases is based on no-evolution of the identified ideal and Nadir points. During the “push stage”, the canonical MOEA/D (with the Tchebycheff scalarizing function) is employed, while a modified

$\u03f5$-constraint technique is applied in the second one (still with MOEA/D) to find feasible solutions. A specific decreasing scheme for the

$\u03f5$ level is adopted, where exponential or polynomial (as in [

24]) decrease can be used, depending on the feasibility ratio. Tested over a 14 functions benchmark earlier proposed by the same authors [

25], the resulting PPS-MOEA/D outperforms some classical MOEAs quoted in this section but requires tuning many parameters.

As the last example of MOEA-based solution procedures, an innovative idea is introduced in [

27], where MOEA/D is modified in such a way that two solutions are assigned to each weight vector. The aim is having one individual on each side of the feasibility boundary (one feasible and one infeasible), in order to focus the search on this region where the Pareto-optimal solutions might lie. The consequences of this working mode are: (i) the doubled size of the neighborhood of each weight vector for offspring generation (since each neighboring weight vector has two associated solutions); (ii) for solution replacement, the created offspring is now compared to two individuals, in terms of both the scalarizing function used within MOEA/D and the overall constraint violation. In this bi-objective space, dominance is used to select two surviving individuals among the three contenders. If the three are non-dominated, that with the larger constraint violation is discarded, while if one solution dominates the two others, the former is the only one to survive. The algorithm is successfully compared with MOEA/D-CDP over several functions of the C-DTLZ test suite [

28]. However, for CTP and CF series [

17,

18], this Dual-grid MOEA/D is outperformed by IDEA [

19] or MOEA/D-IEpsilon [

26].

Besides, as for single-objective optimization, hybrid strategies have been adapted for solving MOPs in recent years. In general, these so-called memetic algorithms combine a global search engine (a MOEA) with some local search technique based on exact algorithms. In this framework, proposals differ one from another according to:

The kind of local search technique used, which may be gradient-based (quasi-Newton in [

29,

30], or sequential quadratic programming in [

31]) or direct search for nonlinear problems (Nelder and Mead’s algorithm in [

32]).

The problem reformulation on which local search is applied, which may be based on

$\u03f5$-constraint [

33] or a scalarization of the MOP [

31].

The hybridization scheme, which can consists on seeding the initial population of the MOEA [

34], interleaving global and local search steps by applying local search to some selected individuals of the population [

33] or periodically (every

t generations) [

35], or using the non-dominated solutions obtained by the exact algorithm to reconstruct the whole Pareto front [

36].

However, to the best of our knowledge, most of these hybrid strategies were applied to unconstrained MOPs through classical test suites (ZDT, DTLZ, WFG) and there is almost no proposal for dealing with constraints, particularly equality constraints.

There already exist some strategies for deal with equality constraints: for instance, if all equalities are linear, one can use orthogonal projections in order to obtain feasible points near to a given candidate solution ([

8]). For vehicle routing problems, several repair mechanisms can be found in [

37,

38,

39]), and in [

4,

40] such mechanisms can be found for portfolio selection problems. Finally, [

41] proposes a repair mechanism that makes use of first-order Taylor approximations of the constraints.