A New Hybrid Evolutionary Algorithm for the Treatment of Equality Constrained MOPs
Abstract
:1. Introduction
2. Background and Related Work
2.1. Multi-Objective Optimization Problem (MOP)
2.2. Related Work
- 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].
2.3. Test Suites for Constrained MOPs
2.4. Pareto Tracer (PT)
3. Proposed Test Problems
3.1. Eq1-ZDT1
- For the first objective we have that .
- For the second objective we have thatThen,Finally,
3.2. Eq2-ZDT1
- (a)
- (b)
- As second step, we need to remove all the points in that do not satisfy the box constraints. In particular, as , we focus on and . For , we have that and , i.e., some values of do not satisfy the lower bound.We can express as follows:After removing the non-feasible points from we have a gap in Pareto set/front. Now, notice that, some points (that is, ), could be within the gap. That is, we have to find the values of such that .For this we consider:Notice that and is a continuous function, then for the intermediate value theorem such that and , respectively.For , such values are and . Then , with , as and .
3.3. Eq-Quad
4. Proposed Algorithm (-NSGA-IIPT)
4.1. First Stage: Rough Approximation via -NSGA-II
- –
- the number of solutions in the roughly approximated set is small, in order to reduce the computational burden of the local search (PT). Indeed, in case of a completely connected front, one single approximated solution might allow us to build the entire Pareto front.
- –
- the MOEA should promote diversity, since the rough approximation produced should cover all the extent of the Pareto front and identify all the different components, in case of a disconnected front.
- –
- the MOEA must be able to handle equality constraints. As mentioned before, a severely constrained problem might cause diversity issues that should be overcome by the MOEA.
4.2. Second Stage: Refinement via PT
- Let be the desired minimal distance between two solutions in objective space. In this first step, go over P and eliminate elements that are too close to each other (if needed). This leads to the new archive .
- Apply the Newton method (A11) to all elements of . Remove all dominated solutions, and elements that are too close to each other as in the first step. This leads to the archive .
- To obtain a “global picture” of the part of the Pareto front that will be computed by PT, construct a partition of a potentially interesting subset S of the image space into a set of hyper-cubes (or k-dimensional boxes) with radius . This partition can easily be constructed via using a binary tree whose root represents S (see [45] for details, where, however, the partition is used in decision variable space). S is a box that is constructed out of as follows: denote by and the minimal and maximal value of the i-th objective value of all elements in , respectively. Then the i-th element of the center of S is set to and its i-th element of the radius to . In the computations, we will only allow a storing one candidate solution within each of these boxes in the archive A to guarantee a spread of the solutions.
Algorithm 1:-NGSA-II |
Evaluate each individual to obtain and |
Compute and set |
for to do |
▹ Parent selection through tournament and -constraint |
, |
Fill P with , using crowding distance if necessary |
if then |
Complete P with |
end if |
Update through Equation (21) |
end for |
▹ return 20 solutions |
Return P and |
- if is dominated by any element of A, then PT the current application of PT is stopped.
- else, it is checked if the unique box that contains is already contained in C. If this is not the case, add this box to C and add to A. Else, decline and proceed with .
5. Numerical Results
5.1. Performance Assessment
5.2. Solving Equality Constrained MOPs with Mathematical Programming Techniques
5.3. Solving Equality Constrained MOPs with MOEAs
- NSGA-II. The popular non-dominated sorting genetic algorithm II [50] was adopted in our comparative study. NSGA-II employs a binary tournament-based on feasibility in the mating selection procedure. In order to determine the next generation, the crowding comparison operator considers the feasibility of solutions. In our study, NSGA-II was performed using the standard parameters given by its authors, i.e., .
- GDE3. The third evolution step of generalized differential evolution [51] was also adopted in our experimental analysis. GDE3 introduces the concept of constraint-domination explained before to discriminate solutions. GDE3 was employed using and .
- cMOEA/D-DE. We also adopted the first version of the multi-objective evolutionary algorithm based on decomposition for constraint multi-objective optimization [52]. cMOEA/D-DE utilizes a penalty function in order to satisfy the constraint of the problem. The penalty function is straightforward added to the scalarizing function employed by MOEA/D-DE [53] to approximate the PF of a constrained MOP. cMOEA/D-DE was employed using , and .
- eMOEA/D-DE. A version of MOEA/D-DE based on the -constraint method for constrained optimization [54] is also adopted in our experimental study. eMOEA/D-DE employs the -constraint method to satisfy the constraints of the problem by obtaining information about feasible solutions in the neighborhood of MOEA/D-DE. Thus, the neighboring solutions are used to defined the -constraint value which is dynamically adapted during the search process of eMOEA/D-DE. eMOEA/D-DE was performed using the standard parameters suggested by its authors, i.e., , and .
5.4. Analysis
6. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Appendix A. Pareto Tracer (PT)
Appendix B. Graphical Results for the NBI
Appendix C. Graphical results for the ϵ-NSGAII/PT
Appendix D. Values of γ and η
n | ||
---|---|---|
16 | 0.954380 | 0.863336 |
17 | 0.957029 | 0.848048 |
18 | 0.959445 | 0.832853 |
19 | 0.961656 | 0.817805 |
20 | 0.963686 | 0.802946 |
21 | 0.965554 | 0.788312 |
22 | 0.967278 | 0.773932 |
23 | 0.968874 | 0.759830 |
24 | 0.970353 | 0.746025 |
25 | 0.971727 | 0.732530 |
26 | 0.973006 | 0.719359 |
27 | 0.974199 | 0.706518 |
28 | 0.975314 | 0.694012 |
29 | 0.976357 | 0.681847 |
30 | 0.977336 | 0.670021 |
31 | 0.978253 | 0.658536 |
32 | 0.979116 | 0.647389 |
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Function | Feval | Feval | Feval | Feval | ||
---|---|---|---|---|---|---|
CZDT1 | 0.0010 | 25,759 | 314 | 314 | - | |
CZDT2 | 0.0008 | 30,221 | 314 | 693 | - | |
CZDT3 | 0.2094 | 4587 | 638 | 830 | - | |
CZDT4 | 0.0011 | 22,543 | 186 | 186 | - | |
CZDT6 | 0.0325 | 6902 | 106 | 240 | - | |
D&D | 0.1005 | 84,531 | 1430 | 498 | - | |
Eq1-ZDT1 | 2.7541 | 89,838 | 405 | 4528 | - | |
Eq2-ZDT1 | 2.6567 | 87,539 | 373 | 5181 | - | |
Eq1-Quad | 0.9019 | 22,679 | 357 | 231 | 649 | |
Eq2-Quad | 3.5263 | 10,449 | 147 | 144 | 4739 |
Method | Feval | Method | F. Ratio | ||||||
---|---|---|---|---|---|---|---|---|---|
CZDT1 | -NSGA-IIPT | 0.0038 | 1.0000 | 15,981 | D&D | -NSGA-IIPT | 0.3442 | 1.0000 | 17,175.5 |
(std.dev) | (0.0002) | (std.dev) | (0.6553) | ||||||
-NSGA-IINBI↑ | 0.0015 | 0.9881 | 26,317 | -NSGA-IINBI↑ | 1.2701 | 0.9887 | 66,414 | ||
(std.dev) | (0.0014) | (std.dev) | (1.8398) | ||||||
c-MOEAD | – | 0.0000 | 20,000 | c-MOEAD↑ | 4.5168 | 0.0270 | 20,000 | ||
(std.dev) | (–) | (std.dev) | (2.1485) | ||||||
e-MOEAD | – | 0.0000 | 20,000 | e-MOEAD | – | 0.0000 | 20,000 | ||
(std.dev) | (–) | (std.dev) | (–) | ||||||
GDE3 | – | 0.0000 | 20,000 | GDE3 | – | 0.0000 | 20,000 | ||
(std.dev) | (–) | (std.dev) | (–) | ||||||
NSGA-II | – | 0.0000 | 20,000 | NSGA-II | – | 0.0000 | 20,000 | ||
(std.dev) | (–) | (std.dev) | (–) | ||||||
CZDT2 | -NSGA-IIPT | 0.0038 | 1.0000 | 15,700 | Eq1-ZDT1 | -NSGA-IIPT | 0.0158 | 1.0000 | 15,763.2 |
(std.dev) | (0.0002) | (std.dev) | (0.0015) | ||||||
-NSGA-IINBI↑ | 0.0008 | 0.9980 | 29,298 | -NSGA-IINBI↑ | 0.7960 | 1.0000 | 161,525 | ||
(std.dev) | (0.0001) | (std.dev) | (0.4744) | ||||||
c-MOEAD | – | 0.0000 | 20,000 | c-MOEAD ↑ | 0.4088 | 0.5060 | 20,000 | ||
(std.dev) | (–) | (std.dev) | (0.2504) | ||||||
e-MOEAD | – | 0.0000 | 20,000 | e-MOEAD ↑ | 0.1683 | 0.3787 | 20,000 | ||
(std.dev) | (–) | (std.dev) | (0.0488) | ||||||
GDE3 | – | 0.0000 | 20,000 | GDE3 ↑ | 3.0997 | 0.6653 | 20,000 | ||
(std.dev) | (–) | (std.dev) | (0.5521) | ||||||
NSGA-II | – | 0.0000 | 20,000 | NSGA-II ↑ | – | 0.0013 | 20,000 | ||
(std.dev) | (–) | (std.dev) | (–) | ||||||
CZDT3 | -NSGA-IIPT | 0.0156 | 1.0000 | 16,235.1 | Eq2-ZDT1 | -NSGA-IIPT | 0.1251 | 1.0000 | 16,285.3 |
(std.dev) | (0.0164) | (std.dev) | (0.0428) | ||||||
-NSGA-IINBI↑ | 0.2681 | 0.8386 | 201,139 | -NSGA-IINBI↑ | 0.6204 | 1.0000 | 159,702 | ||
(std.dev) | (0.2298) | (std.dev) | (0.4138) | ||||||
c-MOEAD | – | 0.0000 | 20,000 | c-MOEAD ↑ | 0.6624 | 0.4700 | 20,000 | ||
(std.dev) | (–) | (std.dev) | (0.2215) | ||||||
e-MOEAD | – | 0.0000 | 20,000 | e-MOEAD ↑ | 0.7800 | 0.4617 | 20,000 | ||
(std.dev) | (–) | (std.dev) | (0.1235) | ||||||
GDE3 | – | 0.0000 | 20,000 | GDE3 ↑ | 3.6144 | 0.8873 | 20,000 | ||
(std.dev) | (–) | (std.dev) | (0.5234) | ||||||
NSGA-II | – | 0.0000 | 20,000 | NSGA-II ↑ | 2.4662 | 0.0037 | 20,000 | ||
(std.dev) | (–) | (std.dev) | (1.6368) | ||||||
CZDT4 | -NSGA-IIPT | 0.0031 | 1.0000 | 16,265.4 | Eq1-Quad | -NSGA-IIPT | 0.1261 | 1.0000 | 149,826.3 |
(std.dev) | (0.0016) | (std.dev) | (0.0043) | ||||||
-NSGA-IINBI↑ | 0.0073 | 0.9990 | 529,146 | -NSGA-IINBI↑ | 0.1880 | 0.2478 | 40,633 | ||
(std.dev) | (0.0055) | (std.dev) | (0.0439) | ||||||
c-MOEAD | – | 0.0000 | 20,000 | c-MOEAD ↑ | 0.5714 | 0.2533 | 150,000 | ||
(std.dev) | (–) | (std.dev) | (0.0953) | ||||||
e-MOEAD | – | 0.0000 | 20,000 | e-MOEAD ↑ | 3.1760 | 0.0014 | 150,000 | ||
(std.dev) | (–) | (std.dev) | (0.6012) | ||||||
GDE3 | – | 0.0000 | 20,000 | GDE3 ↑ | 0.9133 | 0.2666 | 150,000 | ||
(std.dev) | (–) | (std.dev) | (0.0931) | ||||||
NSGA-II | – | 0.0000 | 20,000 | NSGA-II | – | 0.0001 | 150,000 | ||
(std.dev) | (–) | (std.dev) | (–) | ||||||
CZDT6 | -NSGA-IIPT | 0.0884 | 1.0000 | 15,739.5 | Eq2-Quad | -NSGA-IIPT | 1.9969 | 1.0000 | 149,049.2 |
(std.dev) | (0.0180) | (std.dev) | (1.0378) | ||||||
-NSGA-IINBI↑ | 0.0177 | 0.8545 | 22,362 | -NSGA-IINBI | – | 0.0000 | 63,299 | ||
(std.dev) | (0.0102) | (std.dev) | () | ||||||
c-MOEAD | – | 0.0000 | 20,000 | c-MOEAD ↑ | 0.4737 | 0.1583 | 150,000 | ||
(std.dev) | (–) | (std.dev) | (0.2000) | ||||||
e-MOEAD | – | 0.0000 | 20,000 | e-MOEAD | – | 0.0000 | 150,000 | ||
(std.dev) | (–) | (std.dev) | (–) | ||||||
GDE3 | – | 0.0000 | 20,000 | GDE3 ↑ | 2.8142 | 0.0047 | 150,000 | ||
(std.dev) | (–) | (std.dev) | (1.1008) | ||||||
NSGA-II | – | 0.0000 | 20,000 | NSGA-II | – | 0.0000 | 150,000 | ||
(std.dev) | (–) | (std.dev) | (–) |
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Cuate, O.; Ponsich, A.; Uribe, L.; Zapotecas-Martínez, S.; Lara, A.; Schütze, O. A New Hybrid Evolutionary Algorithm for the Treatment of Equality Constrained MOPs. Mathematics 2020, 8, 7. https://doi.org/10.3390/math8010007
Cuate O, Ponsich A, Uribe L, Zapotecas-Martínez S, Lara A, Schütze O. A New Hybrid Evolutionary Algorithm for the Treatment of Equality Constrained MOPs. Mathematics. 2020; 8(1):7. https://doi.org/10.3390/math8010007
Chicago/Turabian StyleCuate, Oliver, Antonin Ponsich, Lourdes Uribe, Saúl Zapotecas-Martínez, Adriana Lara, and Oliver Schütze. 2020. "A New Hybrid Evolutionary Algorithm for the Treatment of Equality Constrained MOPs" Mathematics 8, no. 1: 7. https://doi.org/10.3390/math8010007