Bilevel optimization based on iterative approximation of multiple mappings

Abstract

A large number of application problems involve two levels of optimization, where one optimization task is nested inside the other. These problems are known as bilevel optimization problems and have been studied by both classical optimization community and evolutionary optimization community. Most of the solution procedures proposed until now are either computationally very expensive or applicable to only small classes of bilevel optimization problems adhering to mathematically simplifying assumptions. In this paper, we propose an evolutionary optimization method that tries to reduce the computational expense by iteratively approximating two important mappings in bilevel optimization; namely, the lower level rational reaction mapping and the lower level optimal value function mapping. The algorithm has been tested on a large number of test problems and comparisons have been performed with other algorithms. The results show the performance gain to be quite significant. To the best knowledge of the authors, a combined theory-based and population-based solution procedure utilizing mappings has not been suggested yet for bilevel problems.

Appendices

Appendix A: Standard test problems

In this section, we provide some of the standard bilevel test problems chosen from the literature. Most of these test problems are small with only small number of variables at both levels.

Appendix B: Additional SMD test problems

SMD test problems (Sinha et al. 2014) are a set of 12 scalable test problems that offer a variety of controllable difficulties to an algorithm. We add two more test problems to the previous test-suite in this paper (Table 12). Both these problems contain a difficult $\varphi$-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 $x_u$ and $x_l$ are further divided into two sub-vectors. The $\varphi$-mapping is defined by the function $f_1$.

$$\begin{aligned} \begin{array}{l} F(x_u,x_l) = F_1(x_{u1}) + F_2(x_{l1}) + F_3(x_{u2},x_{l2}) \\ f(x_u,x_l) = f_1(x_{u1}, x_{u2}) + f_2(x_{l1}) + f_3(x_{u2},x_{l2})\\ \text{ where }\\ \quad \quad x_u = (x_{u1}, x_{u2}) \quad \text{ and } \quad x_l = (x_{l1}, x_{l2}) \end{array} \end{aligned}$$

(5)

Table 12 SMD Test Problems. (Note that $(x_{u1}, x_{u2}) = (a,b)$ and $(x_{l1}, x_{l2})=(c,d)$)

Appendix C: Stackelberg duopoly formulation

In this section, we provide the complete formulation of the Stackelberg duopoly problem. Each player produces 5 products. The profit functions for the leader and the follower involve both x and y, which means that the price of the product is influenced by the produce from both the leader and the follower. Each of the players have their own resource constraints that is provided by their respective constraints.

The bilevel optimum for the above problem is not readily available. Therefore, we solved the above problem multiple times using nested approach and then performed a refined grid search to locate the bilevel optimum. The best solution obtained has been provided below. The decision vectors have been rounded to three decimal digits and the function values have been rounded to two decimal digits.

$$\begin{aligned} (x_1, x_2, x_3, x_4, x_5)^{*}= & {} (12.016, 9.333, 15.667, 10.625, 9.375),\\ (y_1, y_2, y_3, y_4, y_5)^{*}= & {} (8.868, 6.132, 9.868, 5.000, 2.656),\\ \Pi _u (x,y)^{*}= & {} 1684.10,\\ \Pi _l (x,y)^{*}= & {} 490.77. \end{aligned}$$

Table 13 Median function evaluations on TP test suite

Appendix D: Additional results

In this section, we provide results for all the test problems with the variance-based termination criterion through Tables 13, 14 and 15. The termination parameters used for the runs are $\alpha _{u}^{stop} = \alpha _{l}^{stop} = 10^{-5}$. Once an algorithm terminates based on the variance-based termination criterion, the best point reported by the algorithm is compared with the true bilevel optimum, and the run is considered successful only if the objective function accuracy of $10^{-2}$ is achieved at both levels.

Table 14 Median function evaluations on low dimension SMD test suite

Table 15 Median function evaluations on high dimension SMD test suite