Numerical examples

The system of nonsmooth equaitons usually appears in solving mathematical program-ming problems, such as the bilevel programprogram-ming and complementarity problems. During the process of solving the bilevel programming problem, we need to transform the lower level problem into the system of nonsmooth equations. Clearly, numerical results of solving bilevel programming problems are heavily dependent on the solution of system of nons-mooth equations. The complementarity problem can also be transformed into the system of nonsmooth equations.

In order to investigate the accuracy of solutions, the following criterion is adopted. Sup-pose that x^∗is a solution, the value of each function in the system of nonsmooth equation is expressed as

H(x^∗) = (H₁(x^∗), H₂(x^∗),· · · , Hm(x^∗))^T. Then, the average error of all the functions is defined as

e =

∑m i=1

|Hi(x^∗)|

m ,

where | · | stands for absolute value. H(x^∗) reflects the accuracy of a solution for each function in the system of nonsmooth equation, while the average error e reveals the whole accuracy of a solution.

The proposed algorithm is tested using some test problems with smooth and nonsmooth

system of equations. All test problems are run in an environment of MATLAB(2010a) in-stalled on an ACER ASPIRE4730Z laptop with a 2G RAM and a 2.16GB CPU. To present numerical results we use the following notation:

• k — index for different starting points;

• ¯x — parameter from the upper level problem;

• y0 and λ0 — starting point;

• y^∗and λ^∗— solution for the test problems;

• H(¯x, y^∗, λ^∗) — value of equation systems at the solutions;

• e — the average error of function values for solution x^∗.

The first three problems are bilevel programming problems which are from [85]. We first illustrate the process of transforming the lower level problems into systems of nonsmooth equations and then use the proposed method to solve the transformed systems of nonsmooth equations. The results are presented in Table 5.2 and compared with the known solution which is presented in Table 5.1.

Example 5.3.1 Consider the bilevel programming problem with the upper-level objective function

f (x, y) = 1

2(y1− 3)²+1

2(y2− 4)²

and the lower-level optimization problem (in variable y)

which is dependent on the parameter x∈ R.

Assume that the admissible set U_ad = [1, 10]. For x ∈ Uad, the lower-level problem can be transformed into the form of Generalized equation

0∈

Table 5.1.: Known solution for equation system (5.8)

k x¯ y₁ y₂ λ₁ λ₂ λ₃ λ₄

1 4.0604 2.6822 1.4871 0 0.6621 0 0

Table 5.2.: Numerical results for equation system (5.8)

k (¯x, y, λ) (¯x, y^∗, λ^∗) H(¯x, y^∗, λ^∗) e

It is clear that the equation system (5.8) is a system of nonsmooth equations since there are two minimum functions. When parameter from the upper level problem ¯x = 4.0604, the solution of the system (5.8) is presented in Table 5.1. For this solution, H(x^∗) = (−0.0017, −0.0003, 0.0009, 0.0004, 0, 0) and the average error e = 5.4999 × 10⁻⁴. Ta-ble 5.2 presents the numerical solutions obtained by our proposed method. From TaTa-ble 5.2, the proposed method can achieve the same solution from different starting points generated randomly. By direct comparison, we can see that our solutions are much better than the solutions given in [85].

Example 5.3.2 Consider the bilevel programming problem with the upper-level objective function

f (x, y) = x²₁− 2x1 − x²2 + y²₁+ y²₂

and the lower-level optimization problem (in terms of variable y)

which is dependent on the parameter x∈ R².

Assume that the admissible set U_ad = [0, 2]× [0, 2]. For x ∈ Uad, the lower-level problem can be transformed to the form of Generalized equation

0∈

We solve this system of nonsmooth equations with different starting points ¯x (which are generated randomly). The numerical results are presented in Table 5.3. From this table, we can see that for different ¯x and randomly generated starting points, our proposed method can solve this system of nonsmooth equations efficiently.

Example 5.3.3 Consider the bilevel programming problem with the upper-level objective

Table 5.3.: Numerical results for equation system (5.9)

where r = 100 is the penalty parameter. The lower-level optimization problem (in terms of variable y)  which is dependent on the parameter x∈ R².

Assume that the admissible set U_ad = [0, 50]× [0, 50]. For x ∈ Uad, the lower-level

problem can be transformed into the form of Generalized equation

which can be rewritten as



Table 5.4 presents the corresponding numerical results by our method. Among them, the first two results are obtained with ¯x = (0, 0). The others are obtained with ¯x generated randomly. For all of them, the starting points are generated randomly.

The last problem is a nonlinear complementarity problem from [128]. The strategy for solving the nonlinear complementarity problem is to transform it first into the system of nonsmooth equations (5.1) and then to the nonsmooth optimization problem (5.4).

Example 5.3.4 Consider the following nonlinear complementarity problem: Find x ∈ R⁴ such that

x≥ 0, f(x) ≥ 0, x^Tf (x) = 0

Table 5.4.: Numerical results for equation system (5.11)

where f :R⁴ → R⁴ is given by

f₁(x) = 3x²₁+ 2x₁x₂+ 2x²₂+ x₃+ 3x₄− 6, f₂(x) = 2x²₁+ x₁+ x²₂+ 10x₃+ 2x₄− 2, f3(x) = 3x²₁+ x1x2+ 2x²₂+ 2x3+ 9x4− 9, f₄(x) = x²₁+ 3x²₂+ 2x₃+ 3x₄− 3.

This problem is equivalent to solving the nonsmooth equation system

F (x) = 0, where F (x) = min[f (x), x], (5.13)

and min denotes the componentwise minimum. This problem has two solutions

x^∗₁ = (1, 0, 3, 0)^T, x^∗₂ = (√

6/2, 0, 0, 0.5)^T.

Clearly, F (x) is differentiable at x^∗₁but nondifferentiable at x^∗₂.

In [128], the Newton’s method was used to solve the system of nonsmooth equations (5.13). We solve the same problem by the proposed method and the numerical results are given in Table 5.5. Compared with the numerical results presented in Table 2 and Table 3 in [128], our method obtained more exact results than those by the Newton’s method.

5.4. Conclusion

This chapter presented a new method for solving the system of nonsmooth equations based upon the quasisecant method. Given the equivalence between the system of nonsmooth equa-tions and the nonsmooth optimization problem, the original system of nonsmooth equaequa-tions is handled by solving a nonsmooth optimization problem using the quasisecant method. Sev-eral numerical experiments are presented to test the proposed method. The numerical results

Table 5.5.: Numerical results for equation system (5.13)

show that the proposed method is efficient and robust for solving the system nonsmooth equations.

Chapter 6.

Applications

In this chapter, we investigate some applications of the hybrid method proposed in Chapter 4.