Approximate Bilevel Optimization with Population-Based Evolutionary Algorithms

Abstract

Population-based optimization algorithms, such as evolutionary algorithms, have enjoyed a lot of attention in the past three decades in solving challenging search and optimization problems. In this chapter, we discuss recent population-based evolutionary algorithms for solving different types of bilevel optimization problems, as they pose numerous challenges to an optimization algorithm. Evolutionary bilevel optimization (EBO) algorithms are gaining attention due to their flexibility, implicit parallelism, and ability to customize for specific problem solving tasks. Starting with surrogate-based single-objective bilevel optimization problems, we discuss how EBO methods are designed for solving multi-objective bilevel problems. They show promise for handling various practicalities associated with bilevel problem solving. The chapter concludes with results on an agro-economic bilevel problem. The chapter also presents a number of challenging single and multi-objective bilevel optimization test problems, which should encourage further development of more efficient bilevel optimization algorithms.

Appendix: Bilevel Test Problems

1.1 Non-scalable Single-Objective Test Problems from Literature

In this section, we provide some of the standard bilevel test problems used in the evolutionary bilevel optimization studies. Most of these test problems involve only a few fixed number of variables at both upper and lower levels. The test problems (TPs) involve a single objective function at each level. Formulation of the problems are provided in Table 13.5.

Table 13.5 Standard test problems TP1–TP8

Table 6 Table 13.5 Continued.

1.2 Scalable Single-Objective Bilevel Test Problems

Sinha-Malo-Deb (SMD) test problems [46] are a set of 14 scalable single-objective bilevel test problems that offer a variety of controllable difficulties to an algorithm. The SMD test problem suite was originally proposed with eight unconstrained and four constrained problems [46], it was later extended with two additional unconstrained test problems (SMD13 and SMD14) in [54]. Both these problems contain a difficult φ-mapping, among other difficulties. The upper and lower level functions follow the following structure to induce difficulties due to convergence, interaction, and function dependence between the two levels. The vectors x _u = x and x _l = y are further divided into two sub-vectors. The φ-mapping is defined by the function f ₁. Formulation of SMD test problems are provided in Table 13.6.

$$\displaystyle \begin{aligned} \begin{array}{l} F(\mathbf{x}, \mathbf{y}) = F_1({\mathbf{x}}^{1}) + F_2({\mathbf{y}}^{1}) + F_3({\mathbf{x}}^{2}, {\mathbf{y}}^{2}), \\ f(\mathbf{x}, \mathbf{y}) = f_1({\mathbf{x}}^{1}, {\mathbf{x}}^{2}) + f_2({\mathbf{y}}^{1}) + f_3({\mathbf{x}}^{2}, {\mathbf{y}}^{2}), \end{array} \end{aligned} $$

(13.10.13)

where, x = (x ¹, x ²) and y = (y ¹, y ²).

Table 13.6 SMD test problems 1–14

Table 8 Table 13.6 Continued.

Table 9 Table 13.6 Continued.

1.3 Bi-objective Bilevel Test Problems

The Deb-Sinha (DS) test suite contains five test problems with two objectives at each level. All problems are scalable with respect to variable dimensions at both levels. Note that x _u = x and x _l = y. The location and shape of respective upper and lower level Pareto-optimal fronts can be found at the original paper [14].

1.3.1 DS 1

Minimize:

$$\displaystyle \begin{aligned} \begin{cases} F_1(\mathbf{x}, \mathbf{y}) = 1 + r - \cos{}(\alpha\pi x_1)) + \sum_{j=2}^{K}(x_j - \frac{j-1}{2})^2 + \tau\sum_{i=2}^{K}(y_i - x_i)^2 - r~\cos\Big(\gamma\frac{pi}{2}\frac{y_1}{x_1}\Big)\\ F_2(\mathbf{x}, \mathbf{y}) = 1 + r - \sin{}(\alpha\pi x_1)) + \sum_{j=2}^{K}(x_j - \frac{j-1}{2})^2 + \tau\sum_{i=2}^{K}(y_i - x_i)^2 - r~\sin\Big(\gamma\frac{pi}{2}\frac{y_1}{x_1}\Big)\\ \end{cases} \end{aligned}$$

subject to:

$$\displaystyle \begin{aligned} \mathbf{y}\in \mathop{\mathrm{argmin}}\limits_{\mathbf{y}} \begin{cases} f_1(\mathbf{x}, \mathbf{y}) = & y_1^2 + \sum_{i=2}^{K}(y_i - x_i)^2 + \sum_{i=2}^{K}10(1 - \cos{}(\frac{\pi}{K}(y_i - x_i))) \\ f_2(\mathbf{x}, \mathbf{y}) = & \sum_{i=1}^{K}(y_i - x_i)^2 + \sum_{i=2}^{K}10|\sin{}(\frac{\pi}{K}(y_i - x_i))| \end{cases} \end{aligned}$$

$$\displaystyle \begin{aligned}y_i \in [-K,K], ~i = 1,\ldots,K, ~x_1 \in [1,4], ~x_j \in [-K,K], ~j=2,\ldots,K.\end{aligned}$$

Recommended parameter setting for this problem, K = 10 (overall 20 variables), r = 0.1, α = 1, γ = 1, τ = 1. This problem results in a convex upper level Pareto-optimal front in which one specific solution from each lower level Pareto-optimal front gets associated with each upper level Pareto-optimal solution.

1.3.2 DS 2

Minimize:

subject to:

$$\displaystyle \begin{aligned} \mathbf{y}\in \mathop{\mathrm{argmin}}\limits_{\mathbf{y}} \begin{cases} f_1(\mathbf{x}, \mathbf{y}) = & y_1^2 + \sum_{i=2}^{K}(y_i - x_i)^2 \\ f_2(\mathbf{x}, \mathbf{y}) = & \sum_{i=1}^{K}i(y_i - x_i)^2 \end{cases} \end{aligned}$$

$$\displaystyle \begin{aligned}y_i \in [-K,K], ~i = 1,\ldots,K, ~x_1 \in [0.001,K], ~x_j \in [-K,K], ~j=2,\ldots,K.\end{aligned}$$

Recommended parameter setting for this problem, K = 10 (overall 20 variables), r = 0.25. Due to the use of periodic terms in v ₁ and v ₂ functions, the upper level Pareto front corresponds to only six discrete values of y ₁ = [0.001, 0.2, 0.4, 0.6, 0.8, 1]. Setting τ = −1 will introduces a conflict between upper and lower level problems. For this problem, a number of contiguous lower level Pareto-optimal solutions are Pareto-optimal at the upper level for each upper level Pareto-optimal variable vector.

1.3.3 DS 3

Minimize:

$$\displaystyle \begin{aligned} \begin{array}{l} \begin{cases} F_1(\mathbf{x}, \mathbf{y}) = x_1 + \sum_{j=3}^{K}(x_j - j/2)^2 + \tau\sum_{i=3}^{K}(y_i-x_i)^2 - R(x_1)\cos\big(4\tan^{-1}(\frac{x_2-y_2}{x_1-y_1})\big),\\ F_2(\mathbf{x}, \mathbf{y}) = x_2 + \sum_{j=3}^{K}(x_j - j/2)^2 + \tau\sum_{i=3}^{K}(y_i-x_i)^2 - R(x_1)\sin\big(4\tan^{-1}(\frac{x_2-y_2}{x_1-y_1})\big),\\ \end{cases} \end{array} \end{aligned}$$

subject to:

$$\displaystyle \begin{aligned} \begin{array}{l} \mathbf{y}\in \mathop{\mathrm{argmin}}\limits_{\mathbf{y}} \begin{cases} f_1(\mathbf{x}, \mathbf{y}) = y_1 + \sum_{i=3}^{K}(y_i - x_i)^2, \\ f_2(\mathbf{x}, \mathbf{y}) = y_2 + \sum_{i=3}^{K}(y_i - x_i)^2,\\ \mathop{\mbox{subject}\ \mbox{to}}: \quad g_1(y) = (y_1 - x_1)^2 + (y_2 - x_2)^2 \leq r^2, \end{cases}\\ G(\mathbf{x}) = x_2 - (1 - x_1^2) \geq 0, \\ y_i \in [-K,K], ~i = 1,\ldots,K, ~x_j \in [0,K], ~j = 1,\ldots,K, ~x_1 \mbox{ is a multiple of } 0.1. \end{array} \end{aligned}$$

In this test problem, the variable x ₁ is considered to be discrete, thereby causing only a few x ₁ values to represent the upper level Pareto front. Recommended parameter setting for this problem: $R(x_1)=0.1+0.15|\sin {}(2\pi (x_1 - 0.1)|$ and use r = 0.2, τ = 1, and K = 10. Like in DS2, in this problem, parts of lower level Pareto-optimal front become upper level Pareto-optimal.

1.3.4 DS 4

Minimize:

$$\displaystyle \begin{aligned} \begin{array}{l} \begin{cases} F_1(\mathbf{x}, \mathbf{y}) = (1 - y_1)(1 + \sum_{j=2}^{K} y_j^2)x_1, \\ F_2(\mathbf{x}, \mathbf{y}) = y_1(1 + \sum_{j=2}^K y_j^2)x_1, \\ \end{cases} \mathop{\mbox{subject}\ \mbox{to}}: \\ \mathbf{y}\in \mathop{\mathrm{argmin}}\limits_{\mathbf{y}} \begin{cases} f_1(\mathbf{x}, \mathbf{y}) = (1 - y_1)(1 + \sum_{j=K+1}^{K+L} y_j^2)x_1, \\ f_2(\mathbf{x}, \mathbf{y}) = y_1(1 + \sum_{j=K+1}^{K+L} y_j^2)x_1, \\ \end{cases}\\ G(\mathbf{x}) = (1-y_1)x_1 + \frac{1}{2} x_1 y_1 - 1 \geq 0, \\ 1 \leq x_1 \leq 2, \quad -1 \leq y_1 \leq 1, \quad -(K + L) \leq y_i \leq (K + L), \ i = 2, \ldots, (K + L). \end{array}\end{aligned}$$

For this problem, there are a total of K + L + 1 variables. The original study recommended K = 5 and L = 4. This problem has a linear upper level Pareto-optimal front in which a single lower level solution from a linear Pareto-optimal front gets associated with the respective upper level variable vector.

1.3.5 DS 5

This problem exactly the same as DS4, except

$$\displaystyle \begin{aligned}G(\mathbf{x}) = (1-y_1)x_1 + \frac{1}{2} x_1 y_1 - 2 + \frac{1}{5}\left[5(1-y_1)x_1+ 0.2\right] \geq 0.\end{aligned}$$

This makes a number of lower level Pareto-optimal solutions to be Pareto-optimal at the upper level for each upper level variable vector.