Power-law fitness scaling on multi-objective evolutionary algorithms: interpretations of experimental results

Abstract

The effect of power-law fitness scaling method on the convergence and distribution of MOEAs is investigated in a systematic fashion. The proposed method is named as gamma (γ) correction-based fitness scaling (GCFS). What scaling does is that the selection pressure of a population can be efficiently regulated. Hence, fit and unfit individuals may be separated well in fitness-wise before going to the selection mechanism. It is then applied to Strength Pareto Evolutionary Algorithm 2 (SPEA2) and Domination Power of an individual Genetic Algorithm (DOPGA). Firstly, the effectiveness of GCFS is tested by 11 static gamma values (including 0.5, 1, 2, …, 9, 10) on nine well-known benchmarks. Simulated study safely states that SPEA2 and DOPGA may perform generally better with the square (γ = 2) and the cubic (γ = 3) of original fitness value, respectively. Secondly, an adaptive version of GCFS is proposed based on statistical merits (standard deviation and mean of fitness values) and implemented to the selected MOEAs. Generally speaking, fitness scaling significantly improves the convergence properties of MOEAs without extra computational burdens. It is observed that the convergence ability of existing MOEAs with fitness scaling (static or adaptive) can be improved. Simulated results also show that GCFS is only effective when fitness proportional selection methods (such as stochastic universal sampling—SUS) are used. GCFS is not effective when tournament selection is used.

Appendix: test problems, n is the number of decision variables

Problem	n	Variable bounds	Objective functions	Comments
ZDT1	30	[0,1]n	$\begin{aligned} f_{1} (x) = x_{1} ,\quad f_{2} (x) = g(x)\left[ { \, 1 - \sqrt {x_{1} /g(x)} } \right] \hfill \\ g(x) = 1 + 9{{\left( {\sum\limits_{i = 2}^{n} {x_{i} } } \right)} \mathord{\left/ {\vphantom {{\left( {\sum\limits_{i = 2}^{n} {x_{i} } } \right)} {\left( {n - 1} \right)}}} \right. \kern-0pt} {\left( {n - 1} \right)}} \hfill \\ \end{aligned}$	Convex
ZDT2	30	[0,1]n	$\begin{aligned} f_{1} (x) = x_{1} ,\quad \, f_{2} (x) = g(x)\left[ { \, 1 - (x_{1} /g(x))^{2} } \right] \hfill \\ g (x )\;{\text{is}}\;{\text{the}}\;{\text{same}}\;{\text{as}}\;{\text{those}}\;{\text{of}}\;{\text{ZDT1}} \hfill \\ \end{aligned}$	Non-convex
ZDT3	30	[0,1]n	$\begin{aligned} f_{1} (x) = x_{1} ,\quad \, f_{2} (x) = g(x)\left[ {1 - \sqrt {x_{1} /g(x)} - \frac{{x_{1} }}{g(x)}\sin (10\pi x_{1} )} \right] \hfill \\ g (x )\;{\text{is}}\;{\text{the}}\;{\text{same}}\;{\text{as}}\;{\text{those}}\;{\text{of}}\;{\text{ZDT1}} \hfill \\ \end{aligned}$	Convex, disconnected
ZDT4	10	[0,1] × [-5,5]n−1	$\begin{aligned} f_{1} (x) = x_{1} ,\quad f_{2} (x) = g(x)\left[ { \, 1 - \sqrt {x_{1} /g(x)} } \right] \hfill \\ g(x) = 1 + 10(n - 1) + \sum\limits_{i = 2}^{n} {\left[ {x_{i}^{2} - 10\cos (4\pi x_{i} )} \right]} \hfill \\ \end{aligned}$	Convex, multifrontal
ZDT6	10	[0,1]n	$\begin{aligned} f_{1} (x) = 1 - \exp ( - 4x_{1} )\sin^{6} (6\pi x_{1} ) ,\quad f_{2} (x) = g(x)\left[ { \, 1 - (f_{1} (x) /g(x))^{2} } \right] \hfill \\ g(x) = 1 + 9\left[ {{{\left( {\sum\limits_{i = 2}^{n} {x_{i} } } \right)} \mathord{\left/ {\vphantom {{\left( {\sum\limits_{i = 2}^{n} {x_{i} } } \right)} {\left( {n - 1} \right)}}} \right. \kern-0pt} {\left( {n - 1} \right)}}} \right]^{0.25} \hfill \\ \end{aligned}$	Non-convex, non-uniformly spaced
UF1	30	[0,1] × [-1,1]n−1	$\begin{aligned} f_{1} = x_{1} + \frac{2}{{\left| {J_{1} } \right|}}\sum\limits_{{j \in J_{1} }} {\left[ {x_{j} - \sin \left( {6\pi x_{1} + \frac{j\pi }{n}} \right)} \right]}^{2} \hfill \\ f_{2} = 1 - \sqrt {x_{1} } + \frac{2}{{\left| {J_{2} } \right|}}\sum\limits_{{j \in J_{2} }} {\left[ {x_{j} - \sin \left( {6\pi x_{1} + \frac{j\pi }{n}} \right)} \right]}^{2} \hfill \\ {\text{where }}J_{1} = \left\{ {j|j\;{\text{is}}\;{\text{odd}}\;{\text{and}}\;2 \le j \le n} \right\}{\text{ and}} \hfill \\ \, J_{2} = \left\{ {j|j\;{\text{is}}\;{\text{even}}\;{\text{and}}\;2 \le j \le n} \right\} \hfill \\ \end{aligned}$	Convex, the Pareto set (PS) is a nonlinear curve
UF2	30	[0,1] × [-1,1]n−1	$\begin{aligned} f_{1} = x_{1} + \frac{2}{{\left| {J_{1} } \right|}}\sum\limits_{{j \in J_{1} }} {y_{j}^{2} } ,\quad f_{2} = 1 - \sqrt {x_{1} } + \frac{2}{{\left| {J_{2} } \right|}}\sum\limits_{{j \in J_{2} }} {y_{j}^{2} } \, \hfill \\ { (}J_{1} \;{\text{and}}\;J_{2} \;{\text{are}}\;{\text{the}}\;{\text{same}}\;{\text{as}}\;{\text{those}}\;{\text{of}}\;{\text{UF1)}} \hfill \\ y_{j} = \left\{ \begin{aligned} x_{j} - \left[ {0.3x_{1}^{2} \cos \left( {24\pi x_{1} + \frac{4j\pi }{n}} \right) + 0.6x_{1} } \right]\cos \left( {6\pi x_{1} + \frac{j\pi }{n}} \right) \, j \in J_{1} \hfill \\ x_{j} - \left[ {0.3x_{1}^{2} \cos \left( {24\pi x_{1} + \frac{4j\pi }{n}} \right) + 0.6x_{1} } \right]\sin \left( {6\pi x_{1} + \frac{j\pi }{n}} \right) \, j \in J_{2} \hfill \\ \end{aligned} \right. \hfill \\ \end{aligned}$	Convex, the Pareto set (PS) is a nonlinear curve
UF3	30	[0,1]n	$\begin{aligned} f_{1} = x_{1} + \frac{2}{{\left| {J_{1} } \right|}}\left( {4\sum\limits_{{j \in J_{1} }} {y_{j}^{2} - 2\prod\limits_{{j \in J_{1} }} {\cos \left( {\frac{{20y_{j} \pi }}{\sqrt j }} \right) + 2} } } \right) \hfill \\ f_{2} = 1 - \sqrt {x_{1} } + \frac{2}{{\left| {J_{2} } \right|}}\left( {4\sum\limits_{{j \in J_{2} }} {y_{j}^{2} - 2\prod\limits_{{j \in J_{2} }} {\cos \left( {\frac{{20y_{j} \pi }}{\sqrt j }} \right) + 2} } } \right) \hfill \\ y_{j} = x_{j} - x_{1}^{{0.5\left( {1.0 + \frac{3(j - 2)}{n - 2}} \right)}} ,\quad j = 2, \ldots ,n \hfill \\ (J_{1} \;{\text{and}}\;J_{2} \;{\text{are}}\;{\text{the}}\;{\text{same}}\;{\text{as}}\;{\text{those}}\;{\text{of}}\;{\text{UF1)}} \hfill \\ \end{aligned}$	Convex, the Pareto set (PS) is a nonlinear curve
UF4	30	[0,1] × [-2,2]n−1	$\begin{aligned} f_{1} = x_{1} + \frac{2}{{\left| {J_{1} } \right|}}\sum\limits_{{j \in J_{1} }} {h(y{}_{j})} \hfill \\ f_{2} = 1 - x_{1}^{2} + \frac{2}{{\left| {J_{2} } \right|}}\sum\limits_{{j \in J_{2} }} {h(y{}_{j})} \, \hfill \\ y_{j} = x_{j} - \sin \left( {6\pi x_{1} + \frac{j\pi }{n}} \right)\quad j = 2, \ldots ,n \hfill \\ h(t) = \frac{\left| t \right|}{{1 + e^{2\left| t \right|} }}\quad (J_{1} \;{\text{and}}\;J_{2} \;{\text{are}}\;{\text{the}}\;{\text{same}}\;{\text{as}}\;{\text{those}}\;{\text{of}}\;{\text{UF1)}} \hfill \\ \end{aligned}$	Non-convex, the Pareto set (PS) is a nonlinear curve