Ruggedness Quantifying for Constrained Continuous Fitness Landscapes

Abstract

Constrained optimization problems appear frequently in important real-world applications. In this chapter, we study algorithms for constrained optimiation problems from a theoretical perspective. Our goal is to understand how the fitness landscape influences the success of certain types of algorithms. One important feature for analyzing and classifying fitness landscape is its ruggedness. It is generally assumed that rugged landscapes make the optimization process by bio-inspired computing methods much harder than smoothed landscapes, which give clear hints toward an optimal solution. We introduce different methods for quantifying the ruggedness of a given constrained optimization problem. They, in particular, take into account how to deal with infeasible regions in the underlying search space.

Appendix

The experimented benchmark functions described in Mallipeddi and Suganthan (2010) are summarised here. In this experiment $\varepsilon$ is considered as 0.0001.

2.1.1 C01

Minimize

$$\begin{aligned} f(x)= -\left| \frac{\sum _{i=1}^{D}\cos ^{4}(z_{i}) - 2\prod _{i=1}^{D}\cos ^{2}(z_{i})}{\sqrt{\sum _{i=1}^{D}iz_{i}^{2}}}\right| \quad z=x-o \end{aligned}$$

subject to

$$\begin{aligned}&g_{1}(x)=0.75-\prod \limits _{i=1}^{D} z_{i} \le 0 \\&g_{2}(x)=\sum \limits _{i=1}^{D}-0.75D\le 0 \\&x \in [0,10]^{D} \end{aligned}$$

2.1.2 C02

Minimize

$$\begin{aligned} f(x)=\mathrm{{max}}(z)\quad z=x-o, y=z-0.5 \end{aligned}$$

subject to

$$\begin{aligned}&g_{1}(x)=10-\frac{1}{D}\sum \limits _{i=1}^{D}[z_{i}^{2}-10\cos (2\pi z_{i})+10]\le 0\\&g_{2}(x)=\frac{1}{D}\sum \limits _{i=1}^{D}[z_{i}^{2}-10\cos (2\pi z_{i})+10]-15\le 0\\&h(x)\,\,=\frac{1}{D}\sum \limits _{i=1}^{D}[y_{i}^{2}-10\cos (2\pi y_{i})+10]-20\le 0\\&x \in [-5.12,5.12]^{D} \end{aligned}$$

2.1.3 C03

Minimize

$$\begin{aligned} f(x)=\sum \limits _{i=1}^{D-1}(100(z_{i}^{2}-z_{i+1})^2+(z_{i}-1)^2) \quad z=x-o \end{aligned}$$

subject to

$$\begin{aligned}&h(x)=\sum \limits _{i=1}^{D-1}(z_{i}-z_{i+1})^2=0\\&x \in [-1{,}000{,}1{,}000]^{D} \end{aligned}$$

2.1.4 C06

Minimize

$$\begin{aligned} f(x)&=\mathrm{{max}}(z)\quad z=x-o, \\ y&=(x+483.6106156535-o)M- 483.6106156535 \end{aligned}$$

subject to

$$\begin{aligned}&h_{1}(x)=\frac{1}{D}\sum \limits _{i=1}^{D}\left( -y_{i}\sin \left( \sqrt{|y_{i}|}\right) \right) =0\\&h_{2}(x)=\frac{1}{D}\sum \limits _{i=1}^{D}\left( -y_{i}\cos \left( 0.5\sqrt{|y_{i}|}\right) \right) =0\\&x \in [-600,600]^{D} \end{aligned}$$

2.1.5 C07

Minimize

$$\begin{aligned} f(x)&=\sum \limits _{i=1}^{D-1}(100(z_{i}^{2}-z_{i+1})^2+(z_{i}-1)^2) \\ z&=x+1-o, y= x-o \end{aligned}$$

subject to

$$\begin{aligned} g(x)&=0.5-\exp \left( -0.1\sqrt{\frac{1}{D}\sum \limits _{i=1}^{D}y_{i}^{2}}\right) -3\exp \left( \frac{1}{D}\sum \limits _{i=1}^{D}\cos (0.1y)\right) \\&\quad \, + \exp (1)\le 0 \\ x \in [&-140,140]^{D} \end{aligned}$$

2.1.6 C09

Minimize

$$\begin{aligned} f(x)&=\sum \limits _{i=1}^{D-1}(100(z_{i}^{2}-z_{i+1})^2+(z_{i}-1)^2) \\ z&=x+1-o, y= x-o \end{aligned}$$

subject to

$$\begin{aligned}&h_{1}(x)=\sum \limits _{i=1}^{D}\left( y_{i}\sin \left( \sqrt{|y_{i}|}\right) \right) =0 \\&x \in [-500,500]^{D} \end{aligned}$$

2.1.7 C10

Minimize

$$\begin{aligned} f(x)&=\sum \limits _{i=1}^{D-1}(100(z_{i}^{2}-z_{i+1})^2+(z_{i}-1)^2) \\ z&=x+1-o, y=(x-o)M \end{aligned}$$

subject to

$$\begin{aligned}&h_{1}(x)=\sum \limits _{i=1}^{D}\left( y_{i}\sin \left( \sqrt{|y_{i}|}\right) \right) =0\\&x \in [-500,500]^{D} \end{aligned}$$

2.1.8 C17

Minimize

$$\begin{aligned} f(x)=\sum \limits _{i=1}^{D}(z_{i}-z_{i+1})^2 \quad z=x-o \end{aligned}$$

subject to

$$\begin{aligned}&g_{1}(x)=\prod \limits _{i=1}^{D}z_{i}\le 0 \\&g_{2}(x)=\sum \limits _{i=1}^{D}z_{i}\le 0 \\&h(x)\,\,=\sum \limits _{i=1}^{D}\left( z_{i}\sin \left( 4\sqrt{|z_{i}|}\right) \right) =0 \\&x \in [-10,10]^{D} \end{aligned}$$

2.1.9 C18

Minimize

$$\begin{aligned} f(x)=\sum \limits _{i=1}^{D}(z_{i}-z_{i+1})^2 \quad z=x-o \end{aligned}$$

subject to

$$\begin{aligned}&g(x)=\sum \limits _{i=1}^{D}\left( -z_{i}\sin \left( \sqrt{|z_{i}|}\right) \right) \le 0 \\&h(x)=\sum \limits _{i=1}^{D}\left( z_{i}\sin \left( \sqrt{|z_{i}|}\right) \right) =0\\&x \in [-50,50]^{D} \end{aligned}$$