A Comparison of Constraint Handling Techniques on NSGA-II

Abstract

Almost all real-world and engineering problems involve multi-objective optimization of some sort that is often constrained. To solve these constrained multi-objective optimization problems, constrained multi-objective optimization evolutionary algorithms (CMOEAs) are enlisted. These CMOEAs require specific constraint handling techniques. This study aims to address and test the most successful constraint handling techniques, seven different penalty constraint techniques, as applied to the Non-dominated Sorting Genetic Algorithm II (NSGA-II). In this paper, NSGA-II is chosen because of its high popularity amongst evolutionary algorithms. Inverted Generational Distance and Hypervolume are the main metrics that are discussed to compare the constraint handling techniques. NSGA-II is applied on 13 constrained multi-objective problems known as CF1-CF10, C1-DTLZ1, C2-DTLZ2, and C3-DTLZ4. The result of IGD and HV values are compared and the feasibility proportions of each combination on each problem are shown. The results of simulation present interesting findings that have been presented at the end of paper as discussion and conclusion.

Appendix

CF1:

$$ {\text{Min}}\quad f_{1\,\,} (x) = x_{1} + \frac{2}{{\left| {J_{1} } \right|}}\sum\limits_{{j \in J_{1} }} {\left( {x_{j} - x_{1}^{{0.5\left( {1.0 + \frac{3(j - 2)}{n - 2}} \right)}} } \right)} $$

$$ {\text{Min}}\quad f_{2\,\,} (x) = 1 - x_{1} + \frac{2}{{\left| {J_{2} } \right|}}\sum\limits_{{j \in J_{2} }} {\left( {x_{j} - x_{1}^{{0.5(1.0 + \frac{3(j - 2)}{n - 2})}} } \right)^{2} } $$

$$ {\text{s}}.{\text{t}}.\quad f_{1} + f_{2} - a\left| {\sin \left[ {N\pi (f_{1} - f_{2} + 1)} \right]} \right| - 1 \ge 0 $$

where $J_{1} = \left\{ {j\left| {j\,{\text{is}}\,{\text{odd}}\,{\text{and}}\,2 \le j \le n} \right.} \right\}$ and $J_{1} = \left\{ {j\left| {j\,{\text{is}}\,{\text{even}}\,{\text{and}}\quad 2 \le j \le n} \right.} \right\}$. N is an integer and $a \ge \frac{1}{2N}$.

CF2:

$$ {\text{Min}}\quad f_{1\,\,} (x) = x_{1} + \frac{2}{{\left| {J_{1} } \right|}}\sum\limits_{{j \in J_{1} }} {\left( {x_{j} - \sin \left( {6\pi x_{1} + \frac{j\pi }{n}} \right)} \right)^{2} } $$

$$ {\text{Min}}\quad f_{2\,\,} (x) = 1 - \sqrt {x_{1} } + \frac{2}{{\left| {J_{2} } \right|}}\sum\limits_{{j \in J_{2} }} {\left( {x_{j} - \cos \left( {6\pi x_{1} + \frac{j\pi }{n}} \right)} \right)^{2} } $$

S.t.

$$ \frac{t}{{1 + e^{4\left| t \right|} }} \ge 0 $$

where $J_{1} = \left\{ {j\left| {j\,{\text{is}}\,{\text{odd}}\,{\text{and}}\quad 2 \le j \le n} \right.} \right\}$ and $J_{2} = \left\{ {j\left| {j\,{\text{is}}\,{\text{even}}\,{\text{and}}\quad 2 \le j \le n} \right.} \right\}$.

CF3:

$$ {\text{Min}}\quad f_{1} = x_{1} + \frac{2}{{\left| {J_{1} } \right|}}\left( {4\sum\limits_{{j \in J_{1} }} {y_{j}^{2} } - 2\mathop \prod \limits_{{j \in J_{1} }} \cos \left( {\frac{{20y_{j} \pi }}{\sqrt j }} \right) + 2} \right) $$

$$ {\text{Min}}\quad f_{2} = 1 - x_{1}^{2} + \frac{2}{{\left| {J_{2} } \right|}}\left( {4\sum\limits_{j \in J2} {y_{j}^{2} } - 2\mathop \prod \limits_{{j \in J_{2} }} \cos \left( {\frac{{20y_{j} \pi }}{\sqrt j }} \right) + 2} \right) $$

S.t.

$$ f_{2} + f_{1}^{2} - a\sin \left[ {N\pi (f_{1}^{2} - f_{2} + 1)} \right] - 1 \ge 0 $$

where $y_{j} = x_{j} - \sin \left( {6\pi x_{1} + \frac{j\pi }{n}} \right),\quad j = 2, \ldots ,n$, and $J_{1} ,J_{2}$ are same as mentioned above equations.

CF4:

$$ {\text{Min}}\quad f_{1} = x_{1} + \sum\limits_{{j \in J_{1} }} {h_{j} (y_{j} )} $$

$$ {\text{Min}}\quad f_{2} = 1 - x_{1} + \sum\limits_{{j \in J_{2} }} {h_{j} (y_{j} )} $$

S.t.

$$ \frac{t}{{1 + e^{4\left| t \right|} }} \ge 0 $$

where $y_{j} = x_{j} - \sin \left( {6\pi x_{1} + \frac{j\pi }{n}} \right),\quad j = 2, \ldots ,n$, $h_{2} (t) = \left\{ {\begin{array}{*{20}l} {\left| t \right|} \hfill & {{\text{if}}\;t < \frac{3}{2}\left( {1 - \frac{\sqrt 2 }{2}} \right)} \hfill \\ {0.125 + (t - 1)^{2} } \hfill & {\text{otherwise}} \hfill \\ \end{array} } \right.$, and $t = x_{2} - \sin \left( {6\pi x_{1} + \frac{2\pi }{n}} \right) - 0.5x_{1} + 0.25$$J_{1} ,J_{2}$ are same as mentioned above equations.

CF5:

$$ {\text{Min}}\quad f_{1} = x_{1} + \sum\limits_{{j \in J_{1} }} {h_{j} (y_{j} )} $$

$$ {\text{Min}}\quad f_{2} = 1 - x_{1} + \sum\limits_{{j \in J_{2} }} {h_{j} (y_{j} )} $$

S.t.

$$ x_{2} - 0.8x_{1} \sin (6\pi x_{1} + \frac{2\pi }{n}) - 0.5x_{1} + 0.25 \ge 0 $$

where $y_{j} = \left\{ {\begin{array}{*{20}l} {x_{j} - 0.8x_{1} \cos \left( {6\pi x_{1} + \frac{j\pi }{n}} \right)\quad {\text{if}}\;j \in J_{1} } \hfill \\ {x_{j} - 0.8x_{1} \sin \left( {6\pi x_{1} + \frac{j\pi }{n}} \right)\quad {\text{if}}\;j \in J_{2} } \hfill \\ \end{array} } \right.$ and $h_{2} (t) = \left\{ {\begin{array}{*{20}l} {\left| t \right|} \hfill & {{\text{if}}\;t < \frac{3}{2}\left( {1 - \frac{\sqrt 2 }{2}} \right)} \hfill \\ {0.125 + (t - 1)^{2} } \hfill & {\text{otherwise}} \hfill \\ \end{array} } \right.$$h_{j} (t) = 2t^{2} - \cos (4\pi t) + 1$ for j = 3,4,…,n

CF6:

$$ {\text{Min}}\quad f_{1} = x_{1} + \sum\limits_{{j \in J_{1} }} {y_{j}^{2} } $$

$$ {\text{Min}}\quad f_{2} = (1 - x_{1} )^{2} + \sum\limits_{{j \in J_{2} }} {y_{j} }^{2} $$

S.t:

$$ x_{2} - 0.8x_{1} \sin \left( {6\pi x_{1} + \frac{2\pi }{n}} \right) - sign(0.5(1 - x_{1} ) - (1 - x_{1} )^{2} )\sqrt {\left| {0.5(1 - x_{1} ) - (1 - x_{1} )^{2} } \right|} \ge 0 $$

$$ x_{4} - 0.8x_{1} \sin \left( {6\pi x_{1} + \frac{4\pi }{n}} \right) - sign(0.25(1 - x_{1} ) - 0.5(1 - x_{1} )^{2} )\sqrt {\left| {0.25(1 - x_{1} ) - 0.5(1 - x_{1} )^{{}} } \right|} \ge 0 $$

$$ y_{j} = \left\{ {\begin{array}{*{20}l} {x_{j} - 0.8x_{1} \cos \left( {6\pi x_{1} + \frac{j\pi }{n}} \right)\quad {\text{if}}\;j \in J_{1} } \hfill \\ {x_{j} - 0.8x_{1} \sin \left( {6\pi x_{1} + \frac{j\pi }{n}} \right)\quad {\text{if}}\;j \in J_{2} } \hfill \\ \end{array} } \right. $$

CF7:

$$ {\text{Min}}\quad f_{1} = x_{1} + \sum\limits_{{j \in J_{1} }} {h_{j} (y_{j} )} $$

$$ {\text{Min}}\quad f_{2} = 1 - x_{1} + \sum\limits_{{j \in J_{2} }} {h_{j} (y_{j} )} $$

$$ y_{j} = \left\{ {\begin{array}{*{20}l} {x_{j} - 0.8x_{1} \cos \left( {6\pi x_{1} + \frac{j\pi }{n}} \right)\quad {\text{if}}\;j \in J_{1} } \hfill \\ {x_{j} - 0.8x_{1} \sin \left( {6\pi x_{1} + \frac{j\pi }{n}} \right)\quad {\text{if}}\;j \in J_{2} } \hfill \\ \end{array} } \right. $$

S.t.

$$ x_{2} - \sin (6\pi x_{1} + \frac{2\pi }{n}) - sign(0.5(1 - x_{1} ) - (1 - x_{1} )^{2} )\sqrt {\left| {0.5(1 - x_{1} ) - (1 - x_{1} )^{2} } \right|} \ge 0 $$

$$ x_{4} - \sin (6\pi x_{1} + \frac{4\pi }{n}) - sign(0.25\sqrt {(1 - x_{1} )} - 0.5(1 - x_{1} )^{{}} )\sqrt {\left| {0.5(1 - x_{1} ) - (1 - x_{1} )^{2} } \right|} \ge 0 $$

where $h_{2} (t) = h_{4} (t) = t^{2}$ and $h_{j} (t) = 2t^{2} - \cos (4\pi t) + 1$ for j = 3,5,6,…,n.

CF8:

$$ \begin{aligned} Min\,f_{1} = \cos (0.5x_{1} \pi )\cos (0.5x_{2} \pi ) + \frac{2}{{\left| {J_{1} } \right|}}\sum\limits_{{j \in J_{1} }} {\left( {x_{j} - 2x_{2} \sin \left( {2\pi x_{1} + \frac{j\pi }{n}} \right)} \right)^{2} } \hfill \\ Min\,f_{2} = \cos (0.5x_{1} \pi )\sin (0.5x_{2} \pi ) + \frac{2}{{\left| {J_{2} } \right|}}\sum\limits_{{j \in J_{2} }} {\left( {x_{j} - 2x_{2} \sin \left( {2\pi x_{1} + \frac{j\pi }{n}} \right)} \right)^{2} } \hfill \\ Min\,f_{3} = \sin (0.5x_{1} \pi ) + \frac{2}{{\left| {J_{3} } \right|}}\sum\limits_{{j \in J_{3} }} {\left( {x_{j} - 2x_{2} \sin \left( {2\pi x_{1} + \frac{j\pi }{n}} \right)} \right)^{2} } \hfill \\ \end{aligned} $$

S.t.

$$ \frac{{f_{1}^{2} + f_{2}^{2} }}{{1 - f_{3}^{2} }} - a\left| {\sin \left[ {N\pi \left( {\frac{{f_{1}^{2} + f_{2}^{2} }}{{1 - f_{3}^{2} }} + 1} \right)} \right]} \right| - 1 \ge 0 $$

where

$$ \begin{aligned} J_{1} = \left\{ {\left. j \right|3 \le j \le n,and\,j - 1\,is\,a\,multiplication\,of\,3} \right\} \hfill \\ J_{2} = \left\{ {\left. j \right|3 \le j \le n,and\,j - 2\,is\,a\,multiplication\,of\,3} \right\} \hfill \\ J_{3} = \left\{ {\left. j \right|3 \le j \le n,and\,j\,is\,a\,multiplication\,of\,3} \right\} \hfill \\ \end{aligned} $$

CF9:

$$ \begin{aligned} Min\,f_{1} = \cos (0.5x_{1} \pi )\cos (0.5x_{2} \pi ) + \frac{2}{{\left| {J_{1} } \right|}}\sum\limits_{{j \in J_{1} }} {\left( {x_{j} - 2x_{2} \sin \left( {2\pi x_{1} + \frac{j\pi }{n}} \right)} \right)^{2} } \hfill \\ Min\,f_{2} = \cos (0.5x_{1} \pi )\sin (0.5x_{2} \pi ) + \frac{2}{{\left| {J_{2} } \right|}}\sum\limits_{{j \in J_{2} }} {\left( {x_{j} - 2x_{2} \sin \left( {2\pi x_{1} + \frac{j\pi }{n}} \right)} \right)^{2} } \hfill \\ Min\,f_{3} = \sin (0.5x_{1} \pi ) + \frac{2}{{\left| {J_{3} } \right|}}\sum\limits_{{j \in J_{3} }} {\left( {x_{j} - 2x_{2} \sin \left( {2\pi x_{1} + \frac{j\pi }{n}} \right)} \right)^{2} } \hfill \\ \end{aligned} $$

S.t.

$$ \frac{{f_{1}^{2} + f_{2}^{2} }}{{1 - f_{3}^{2} }} - a\sin \left[ {N\pi \left( {\frac{{f_{1}^{2} + f_{2}^{2} }}{{1 - f_{3}^{2} }} + 1} \right)} \right] - 1 \ge 0 $$

where

$$ \begin{aligned} J_{1} = \left\{ {\left. j \right|3 \le j \le n,and\,j - 1\,is\,a\,multiplication\,of\,3} \right\} \hfill \\ J_{2} = \left\{ {\left. j \right|3 \le j \le n,and\,j - 2\,is\,a\,multiplication\,of\,3} \right\} \hfill \\ J_{3} = \left\{ {\left. j \right|3 \le j \le n,and\,j\,is\,a\,multiplication\,of\,3} \right\} \hfill \\ \end{aligned} $$

CF10:

$$ \begin{aligned} Min\,f_{1} = \cos (0.5x_{1} \pi )\cos (0.5x_{2} \pi ) + \frac{2}{{\left| {J_{1} } \right|}}\sum\limits_{{j \in J_{1} }} {\left[ {4y_{j}^{2} - \cos (8\pi y_{j} ) + 1} \right]} \hfill \\ Min\,f_{2} = \cos (0.5x_{1} \pi )\sin (0.5x_{2} \pi ) + \frac{2}{{\left| {J_{2} } \right|}}\sum\limits_{{j \in J_{2} }} {\left[ {4y_{j}^{2} - \cos (8\pi y_{j} ) + 1} \right]} \hfill \\ Min\,f_{3} = \sin (0.5x_{1} \pi ) + \frac{2}{{\left| {J_{3} } \right|}}\sum\limits_{{j \in J_{3} }} {\left[ {4y_{j}^{2} - \cos (8\pi y_{j} ) + 1} \right]} \hfill \\ \end{aligned} $$

S.t.

$$ \frac{{f_{1}^{2} + f_{2}^{2} }}{{1 - f_{3}^{2} }} - a\sin \left[ {N\pi (\frac{{f_{1}^{2} + f_{2}^{2} }}{{1 - f_{3}^{2} }} + 1)} \right] - 1 \ge 0 $$

where

$$ \begin{aligned} J_{1} = \left\{ {\left. j \right|3 \le j \le n,and\,j - 1\,is\,a\,multiplication\,of\,3} \right\} \hfill \\ J_{2} = \left\{ {\left. j \right|3 \le j \le n,and\,j - 2\,is\,a\,multiplication\,of\,3} \right\} \hfill \\ J_{3} = \left\{ {\left. j \right|3 \le j \le n,and\,j\,is\,a\,multiplication\,of\,3} \right\} \hfill \\ \end{aligned} $$

and $y_{j} = x_{j} - 2x_{2} \sin \left( {2\pi x_{1} + \frac{j\pi }{n}} \right)$

C1_DTLZ1:

Regarding to [11]:DTLZ1 test problem is as follows:

$$ \begin{aligned} Min\,f_{1} (x) = \frac{1}{2}x_{1} x_{2} x_{M - 1} (1 + g(x_{M} )) \hfill \\ Min\,f_{2} (x) = \frac{1}{2}x_{1} x_{2} (1 - x_{M - 1} )(1 + g(x_{M} )) \hfill \\ \end{aligned} $$

$$ \begin{aligned} Min\,f_{M - 1} (x) = \frac{1}{2}x_{1} (1 - x_{2} )(1 + g(x_{M} )) \hfill \\ Min\,f_{M} (x) = \frac{1}{2}(1 - x_{1} )(1 + g(x_{M} )) \hfill \\ 0 \le x_{i} \le 1 \hfill \\ \end{aligned} $$

where $g(x_{M} ) = 100(\left| {x_{M} } \right| + \sum\limits_{{x_{i} \in x_{M} }} {(x_{i} - 0.5)^{2} - \cos (20\pi (x_{i} - 0.5)))}$. Objective functions of C1-DTLZ1 is same as DTLZ1, only one constraint is added as follows:

$$ c(x) = 1 - \frac{{f_{M} (x)}}{0.6} - \sum\nolimits_{i = 1}^{M - 1} {\frac{{f_{i} (x)}}{0.5}} \ge 0 $$

C2_DTLZ2:

$$ \begin{aligned} Min\,f_{1} (x) = (1 + g(x_{M} ))\cos (x_{1} \pi /2) \ldots \cos (x_{M - 1} \pi /2) \hfill \\ Min\,f_{2} (x) = (1 + g(x_{M} ))\cos (x_{1} \pi /2) \ldots \sin (x_{M - 1} \pi /2) \hfill \\ \end{aligned} $$

$$ Min\,f_{M(x)} = (1 + g(x_{M} ))\sin (x_{1} \pi /2) $$

$$ 0 \le x_{i} \le 1,\;{\text{for}}\;i = 1, \ldots ,n $$

$$ g(x_{M} ) = \sum\limits_{{x_{i} \in X_{M} }} {(x_{i} - 0.5)^{2} } $$

Objective functions of C2-DTLZ2 is same as DTLZ2, only one constraint is added as follows:

$$ c(x) = \hbox{max} \left\{ {\max_{i = 1}^{M} } \right.[(f_{i} (x) - 1)^{2} + \sum\limits_{j = 1,j \ne i}^{M} {f_{j}^{2} } - r^{2} \,\left. {} \right],[\sum\limits_{i = 1}^{M} {(f_{i}^{{}} } (x) - 1/\sqrt M )^{2} - r^{2} ]\left. {} \right\} $$

where r = 0.4

C3_DTLZ4:

In [11], DTLZ2 has been modified by mapping a meta-variable:$x_{i} \to x_{i}^{\alpha }$ where $\alpha = 100$.

$$ c(x) = \frac{{f_{j}^{2} }}{4} + \sum\limits_{i = 1,i \ne j}^{M} {f_{i}^{2} (x) - 1 \ge 0,\quad \forall j = 1, \ldots ,M} $$