Foundational Studies on ML-Based Enhancements

Abstract

Many efficient evolutionary multi- and many-objective optimization algorithms, jointly referred to as EMâOAs, have been proposed in the last three decades. However, while solving complex real-world problems, EMâOAs that rely only on natural variation and selection operators may not produce an efficient search [14, 33, 45]. Therefore, it may be desirable or essential to enhance the capabilities of EMâOAs by introducing synergistic concepts from probability, statistics, machine learning (ML), etc. This chapter highlights some of the key studies that have laid the foundations for ML-based enhancements for EMâOAs and inspired further research that has been shared in subsequent chapters.

Appendices

Appendices for in this Chapter

The following sections discuss examples of the manual innovization task, the automated innovization task, and the innovized repair operator.

3.6 Examples of Manual Innovization Task

The basic working of the manual innovization procedure has been illustrated here through a three-variable, two-objective truss design problem, which was originally studied using the $\epsilon$-constraint approach [5, 41] and later using an evolutionary approach [11]. In that, the truss (Fig. 3.3) must carry a certain load without incurring an elastic failure. The two conflicting design objectives are to (i) minimize the total volume of the truss members and (ii) minimize the maximum stress developed in both members (AC and BC) due to the application of the load 100 kN.  Furthermore, the three decision variables are cross-sectional area of AC and BC ($x_1$ and $x_2$, respectively), measured in square meters, and the vertical distance between A (or B) and C (y), measured in meters. The optimization problem formulation is given as follows:

$$\begin{aligned} \begin{array}{rl} \text {Minimize} &{} f_1(\vec {x},y) = x_1\sqrt{16+y^2} + x_2\sqrt{1+y^2},\\ \text {Minimize} &{} f_2(\vec {x},y) = \max (\sigma _{AC},\sigma _{BC}),\\ \text {Subject to } &{} \max (\sigma _{AC},\sigma _{BC}) \le S_{\max }, \\ &{} 0\le x_1, x_2 \le A_{\max }, \\ &{} 1\le y \le 3.\\ \end{array} \end{aligned}$$

(3.8)

Fig. 3.3

(taken from [16])

The design of two-bar truss

Using the dimensions and loading specified in Fig. 3.3, it can be observed that member AC is subjected to $20\sqrt{16+y^2}/y$ kN load and member BC is subjected to $80\sqrt{1+y^2}/y$ kN load. The stress values are calculated as follows:

$$\begin{aligned} \sigma _{AC} = \frac{20\sqrt{16+y^2}}{yx_1}, \quad \sigma _{BC} = \frac{80\sqrt{1+y^2}}{yx_2}. \end{aligned}$$

(3.9)

Here, the stress values and the cross-sectional areas are limited to $S_{\max }=1(10^5)$ kPa and $A_{\max }=0.01$ m$^2$, respectively. All three variables are treated as real-valued. Simulated binary crossover (SBX) with $\eta _c=10$ and polynomial mutation operator with $\eta _m=50$ have been used. All constraints are handled using the constraint-tournament approach [11]. Figure 3.4 shows the final set of non-dominated solutions obtained by an algorithmic run of NSGA-II. Although the trade-off between the two objectives is evident in Fig. 3.4, these solutions are further analyzed, using two different studies, to gain more confidence in the Pareto-optimality of these solutions. First, a single-objective RGA is used to find the optimum of individual objective functions, subject to the same constraints and variable bounds. Figure 3.4 marks these two solutions (one per objective) as 1-obj solutions. It is evident that the front obtained using NSGA-II extends to these two extreme solutions. Next, the normal constraint method (NCM) [40] is used with different starting points from a line that joins the two extreme solutions. The solutions thus obtained, one at the end of each NCM procedure, are also shown in Fig. 3.4. Since these solutions lie on the front obtained using NSGA-II, it is confirmed that the non-dominated solutions obtained using NSGA-II are close to the true $P\!F$.

Fig. 3.4

(taken from [16])

NSGA-II solutions obtained for the two-bar truss problem

3.1.1 3.6.1 Theoretical Innovized Principles and Manual Innovization Results

Before applying the manual innovization procedure to the solutions obtained using NSGA-II, an exact analysis of this problem is presented to identify the true $P\!F$, and the underlying innovized principles (theoretical), if any. The problem, although simple mathematically, is a typical optimization problem that has two resource terms in each objective, involving variables $x_1$ and $x_2$, and interlinking them with the third variable y. For such problems, the optimum occurs when identical resource allocation is made between the two terms in both objective and constraint functions, as shown below.

$$\begin{aligned} x_1\sqrt{16+y^2} = x_2\sqrt{1+y^2} &\implies & \frac{20\sqrt{16+y^2}}{yx_1} = \frac{80\sqrt{1+y^2}}{yx_2}. \end{aligned}$$

(3.10)

Fig. 3.5

(taken from [16])

Variation of $x_1$ and $x_2$ for the truss design problem

Thus, every optimal solution is expected to satisfy both of the above equations, resulting in the following innovated rules:

$$\begin{aligned} \frac{x_1}{x_2} = 0.5, & \text{ and } & y = 2. \end{aligned}$$

(3.11)

Substituting $y=2$ into the expression for the first objective (volume) leads to $x_2=V/2\sqrt{5}$ m$^2$, where V is the volume of the structure (in m$^3$). Similarly, substituting these values into the objective functions $V=f_1$ and $S=f_2$ leads to $SV=400$—an inverse relationship between the objectives. Thus, the solutions in the true $P\!F$ are given in terms of volume V, as follows:

$$x_1^{*} = \frac{V^{*}}{4\sqrt{5}}\ \text{ m}^2,\quad x_2^{*} = \frac{V^{*}}{2\sqrt{5}}\ \text{ m}^2,\quad y^{*}=2\ \text{ m },\quad S^{*}=400/V^{*}\ \text{ kPa }.$$

When the variable $x_2$ reaches its upper bound, that is, at the transition point T shown in Fig. 3.5, $V_T=0.04472$ m$^3$ and $S_T=8944.26$ kPa, since $x_2$ cannot be increased any further. The inset plot (drawn with a logarithmic scale of both axes) in Fig. 3.4 shows this interesting aspect of the front obtained. There are two distinct behaviors around the transition point T marked in the figure: (i) one that stretches from the smallest volume solution to a volume of about 0.04478 m$^3$ (point T), and (ii) another that stretches from this transition point to the smallest stress solution.

The extreme solutions and this intermediate solution, obtained by NSGA-II, are tabulated in Table 3.1.

An investigation of the values of the decision variables reveals the following:

• 1. The inset plot in Fig. 3.4 reveals that for optimal structures, the maximum stress (S) developed is inversely proportional to the volume (V) of the structure, that is, $SV=\textrm{constant}$, as predicted above. When a straight line is fitted through the logarithm of the two objective values, $SV=402.2$, a relationship is found between these solutions obtained using NSGA-II. The obtained relationship is close to the theoretical relationship computed above (from the true $P\!F$).

• 2. The inset plot also reveals that the transition occurs at $V=0.044779$ m$^3$, which is also close to the exact theoretical value computed above.

• 3. To achieve an optimal solution with a lower maximum stress (and larger volume), both cross-sectional areas (AC and BC) should increase linearly with volume, as shown in Fig. 3.5. The figure also plots the mathematical relationships ($x_1$ and $x_2$ versus V) obtained earlier with solid lines, which can barely be seen, as the solutions obtained using NSGA-II fall on top of these lines.

• 4. A further investigation reveals that the ratio between these two cross-sectional areas is almost 1:2, and the vertical distance (y) takes a value close to 2 for all solutions.

• 5. Figure 3.6 reveals that the stress values on both members (AC and BC) are identical for any Pareto-optimal solution (Fig. 3.7).

Table 3.1 Two extreme solutions and an interesting intermediate solution (T) for the two-bar truss design problem are presented (taken from [16])

Fig. 3.6

(taken from [16])

Variation of stresses in AC and BC of the two-bar truss problem

Fig. 3.7

(taken from [16])

Variation of y for the two-bar truss design problem

The innovized rules illustrated above are some interesting properties of the original optimization problem that may not be intuitive to the designer. However, these principles can be explained from the mathematical formulation described above. Thus, although these optimality conditions can be derived mathematically from the problem formulation given in Eq. 3.8 in this simple problem, such optimality conditions may often be tedious and difficult to achieve exactly for large and complex problems. The application of a numerical optimization procedure and then investigating the obtained optimal solutions have the potential to reveal such important innovative design principles.

3.7 Examples of Automated Innnovization Task

The  same two-bar truss problem, discussed above, is chosen to illustrate the working of the AutoInn procedure. For the solutions obtained using NSGA-II, the AutoInn procedure finds four rules common to $87\%$ to $92\%$ of the non-dominated dataset:

$$\begin{aligned} SV = 400.770, \quad \frac{x_1}{V} = 0.111, \quad \frac{x_2}{V} = 0.224, \quad \frac{x_2}{x_1} = 1.984. \end{aligned}$$

(3.12)

Figure 3.8 shows the relevant non-dominated solutions obtained using NSGA-II. Some unclustered solutions are marked as red points. Figure 3.9 shows the distribution of $c_k$ values for one of the rules obtained $V^{-0.997}x_1^{1.000}=c$. It is clear that the values V and $x_1$ of the majority ($87\%$) non-dominated solutions satisfy the rule. The clustering algorithm inbuilt in the AutoInn procedure found three clusters with slightly different $c_k$-values. But the non-dominated solutions that do not satisfy the rule have very different $c_k$ values. For ease of understanding, the $c_k$-values are sorted from low to high in the figure shown.

Fig. 3.8

Pareto front for the two-bar truss design problem. Red points are a few unclustered points for the $V^{-0.997}x_1^{1.000}=c$ rule

Fig. 3.9

(taken from [2])

$c_k$ distribution for the rule $V^{-0.997}x_1^{1.000}=c$ is shown, found to 87% of the non-dominated dataset. Unclustered non-dominated data are shown with a ‘X’

The respective distributions of the $c_k$ values for two other rules are shown in Figs. 3.10 and 3.11.

Fig. 3.10

c-value distribution for $S^{1.0000}V^{0.9999}=c$ found to 92% of the non-dominated dataset

Fig. 3.11

(taken from [2])

Cluster plot for $V^{-0.9999}x_2^{1.0000}=c$ found to 88% of the non-dominated dataset

Although the AutoInn procedure finds multiple clusters, the respective $c_k$ values are close to each other, and the difference in the c values from the unclustered points is significant.

3.8 Examples of Innovized Repair Operator

Fig. 3.12 shows the median generational distance (GD) and inverse generational distance (IGD) metrics [11] for the two-bar truss design problem. Notably, GD is an indicator of convergence, and IGD is a combined indicator of convergence and diversity. The plots in Fig. 3.12 reveal that NSGA-II-IR with repair preference given to short rules (SN repair strategy) performs much better than the no repair strategy (NI, i.e., base NSGA-II), in terms of GD (smaller the better). However, in terms of the IGD metric, the NI strategy performs marginally better. This is expected, as the NSGA-II with SN repair strategy is expected to focus more on improving the convergence than on maintaining the diversity.

Fig. 3.12

Median GD and IGD results for the two-bar truss design problem over 30 runs

The rules extracted at the end of the NSGA-II run with the SN strategy, provided below, closely match the theoretical property of variables stated in Eq. 3.11:

$$\begin{aligned} x_1^{-1.008} x_2 = 2.0750, \quad x_3 = 1.9449 \pm 0.0674. \end{aligned}$$

(3.13)