A novel optimization hardness indicator based on the relationship between optimization hardness and frequency features of real-parameter problems

Abstract

For evolutionary algorithms with the ability to self-adapt, linking the algorithmic operators and the problem features is one of the most interesting topics. One of the best ways to begin a study of this topic is to explore the relationship between the optimization hardness and the problem features. This paper attempts to interpret the relationship between optimization hardness and frequency features of real-parameter problems through a qualitative analysis based on an idealized model. Based on the results of a theoretically qualitative analysis, the effective high-frequency ratio (EHFR) is subsequently proposed to measure the optimization hardness of real-parameter problems. Finally, three aspects to the performance of EHFR are evaluated: stability, precision and ability to distinguish. Test results show that the EHFR is relevant not only for the results of theoretical analysis, but also for the other features related to the optimization hardness.

Appendices

Appendix A: Discussion of general solutions to $f'_{i+j=0}$

The imaginary unit is denoted by $I$ in this appendix, and Eq. (21a) is the decomposed form of Eq. (5) using the Euler equation.

$$\begin{aligned}&\frac{1}{{2I}}\left\{ - iw_{1} c_{i} \mathrm{e}^{{I\left( {iw_{1} x + \varphi _{i} } \right) }} + iw_{1} c_{i} \mathrm{e}^{{ - I\left( {iw_{1} x + \varphi _{i} } \right) }}\right. \nonumber \\&\quad \left. - jw_{1} c_{i} \mathrm{e}^{{I\left( {jw_{1} x + \varphi _{j} } \right) }} + jw_{1} c_{j} \mathrm{e}^{{ - I\left( {jw_{1} x + \varphi _{j} } \right) }} \right\} = 0\end{aligned}$$

(21a)

$$\begin{aligned}&\frac{1}{2I}\left\{ iw_1 c_i \mathrm{e}^{-Iiw_1 x} \mathrm{e}^{-I\varphi _i }+jw_1 c_j \mathrm{e}^{-Ijw_1 x} \mathrm{e}^{-I\varphi _j }\right. \nonumber \\&\quad \left. -iw_1 c_i e^{Iiw_1 x} \mathrm{e}^{I\varphi _i }-jw_1 c_i \mathrm{e}^{Ijw_1 x} \mathrm{e}^{I\varphi _j } \right\} =0 \end{aligned}$$

(21b)

$$\begin{aligned}&Q= \mathrm{e}^{I({w_1 x})}\end{aligned}$$

(21c)

$$\begin{aligned}&x=\frac{\ln Q}{Iw_1 } \end{aligned}$$

(21d)

$$\begin{aligned}&\frac{1}{2I}\left\{ iw_1 c_i Q^{-i} \mathrm{e}^{-I\varphi _i }+jw_1 c_j Q^{-j} \mathrm{e}^{-I\varphi _j }\right. \nonumber \\&\quad \left. -iw_1 c_i Q^ie^{I\varphi _i }-jw_1 c_j Q^j \mathrm{e}^{I\varphi _j } \right\} =0 \end{aligned}$$

(21e)

$$\begin{aligned}&\frac{-1}{2IQ^j}\left\{ jw_1 c_j \mathrm{e}^{I\varphi _j }Q^{2j}+iw_1 c_i \mathrm{e}^{I\varphi _i }Q^{j+i}\right. \nonumber \\&\quad \left. -iw_1 c_i \mathrm{e}^{-I\varphi _i }Q^{j-i}-jw_1 c_j \mathrm{e}^{-I\varphi _j } \right\} =0 \end{aligned}$$

(21f)

According to Sect. 3.1, $j$ and $i$ are natural numbers and $j>i>0$, and $2j$ is the largest exponent of $Q$. Evidently, Eq. (21f) is a high-order equation of $Q$. According to the Abel–Ruffini theorem, the general solution for Eq. (21f) does not exist when $j>2$. In the idealized model, $j$ cannot be restricted below two, and arithmetical solutions to Eq. (21f) are useless for the analysis in Sect. 3. Thus, defining the intermediate variable $t$ is a necessary step in the theoretical analysis of Sect. 3.

Appendix B: Details of the OR estimating method

The method for estimating the OR begins from the global optimum of the test function. Each dimension of the global optimum will be searched in two directions to find the points of inflection, which are found within the boundary of the OSC. The OSC for the test functions can be estimated with the coordinates of these inflecting points. The ratio for the estimated size of OSC to the size of the entire search space is the estimated value of OR. This method, however, leads unavoidably to the OR that is inaccurate. Two reasons example the inaccuracy: (1) the real points of inflection may be in the interval of two sampling points, and (2) the real figure of the OSC may differ from the estimation. These two reasons are illustrated in Fig. 11.

Fig. 11

Reasons for inaccuracy with the OR estimating method

In Fig. 11a, the solid line denotes the real curve of the fitness function, and the dashed line denotes the false curve of the sampling process. In Fig. 11b, the solid line denotes the real region of the OSC, and the dashed line denotes the estimated OSC results. Given the first reason for the inaccuracy, increasing the sampling point is an effective way to solve the problem, but with high computational costs. Increasing the search directions can partially resolve the inaccuracy, given the second reason. This, however, also comes at the price of increased computational costs.