A modified multivariate spectral gradient projection method for nonlinear complementarity problems

Abstract

We present a sufficient condition for monotonicity of the nonlinear nonsmooth system generated by Fischer–Burmeister function associated with nonlinear complementarity problem. Based on the presented condition, the nonlinear complementarity problem considered in this paper is equivalently formulated to a nonsmooth monotone system. We then propose a modified multivariate spectral gradient projection method for the resulting system, and establish the global convergence without smoothness and Lipschitz condition. Preliminary numerical experiments show that, compared to some existing methods, the proposed method is effective.

Appendix

Example 1

(La Cruz 2017) The function f(x) is endowed with the component as follows:

$$\begin{aligned} f_i(x)=e^{x_i} -1, \quad i\in [n]. \end{aligned}$$

Example 2

The function f(x) is endowed with the component as follows:

$$\begin{aligned} f_i(x)=\arctan (x_i) - \frac{\pi }{6}, \quad i\in [n]. \end{aligned}$$

Example 3

This problem is a modification of Example 1, and the function f(x) is endowed with the component as follows:

$$\begin{aligned} f_i(x)=e^{\frac{i}{n}x_i} - 1, \quad i\in [n]. \end{aligned}$$

Example 4

(Abubakar and Kumam 2019) The function f(x) is endowed with the component as follows:

$$\begin{aligned} f_i(x)=\ln (x_i + 1) - \frac{x_i}{n}, \quad i\in [n]. \end{aligned}$$

Example 5

(Fan 2015) This is the Kojima–Shindo problem, and the function f(x) is given by

$$\begin{aligned} f(x)= \left( \begin{array}{c} 3x_1^2 + 2x_1x_2 + 2x_2^2 + x_3 + 3x_4 - 6 \\ 2x_1^2 + x_1 + x_2^2 + 3x_3 + 2x_4 - 2 \\ 3x_1^2 + x_1x_2 + 2x_2^2 + 2x_3 + 3x_4 - 1 \\ x_1^2 + 3x_2^2 + 2x_3 + 3x_4 - 3 \end{array}\right) . \end{aligned}$$

Example 6

(La Cruz 2017) The function f(x) is endowed with the component as follows:

$$\begin{aligned} f_i(x)=2x_i -\sin (|x_i|), \quad i\in [n]. \end{aligned}$$

Example 7

(Ou and Li 2018) The function f(x) is endowed with the component as follows:

$$\begin{aligned} f_i(x)=x_i - \sin \left( |x_i - 1|\right) , \quad i\in [n]. \end{aligned}$$

Example 8

(Ou and Li 2018) The function f(x) is endowed with the component as follows:

$$\begin{aligned} f_i(x)=\ln (|x_i| + 1) - \frac{x_i}{n}, \quad i\in [n]. \end{aligned}$$

Example 9

(Yang and Gao 1991) A tridiagonal exponential function $f_h(x)$, which is endowed with the component as follows:

$$\begin{aligned} f_i(x) = x_i - e^{\cos [h(x_{i-1} + x_i + x_{i+1})]}, \quad i\in [n], \end{aligned}$$

where $x_0 = x_{n+1} = 0$ and $h = \frac{1}{n+1}$.

Example 10

A nonsmooth modified of Example 9, which is endowed with the component as follows:

$$\begin{aligned} f_i(x) = x_i - e^{\cos [h|x_{i-1} + x_i + x_{i+1}|]}, \quad i\in [n], \end{aligned}$$

where $x_0 = x_{n+1} = 0$ and $h = \frac{1}{n+1}$.

Example 11

(Zhang and Peng 2020) The function f(x) is endowed with the component as follows:

$$\begin{aligned} f_i(x) = \min \left( \min \left( |x_i|, x_i^2 \right) , \max \left( |x_i|, x_i^3 \right) \right) , \quad i\in [n]. \end{aligned}$$

Example 12

The function f(x) is endowed with the component as follows:

$$\begin{aligned} \begin{array}{l} f_1(x) = \frac{1}{n+1}x_1^3 + \frac{1}{n}x_2^2,\\ f_i(x) = -\frac{1}{n}|x_{i-1}| + \frac{i}{n+1}x_i^3 + \frac{1}{n}x_{i+1}^2, \quad i=2,3,\dots ,n-1,\\ f_n(x) = -\frac{1}{n}|x_{n-1}| + \frac{n}{n+1}x_n^3. \end{array} \end{aligned}$$

Example 13

A nonsmooth modification of Example 1, where f(x) is endowed with the component as follows:

$$\begin{aligned} f_i(x)=e^{|x_i|} -1, \quad i\in [n]. \end{aligned}$$

Example 14

The function f(x) is endowed with the component as follows:

$$\begin{aligned} \begin{array}{l} f_1(x) = x_1 + \arctan (x_n) - \cos (x_2) - 1,\\ f_i(x) = x_i + \arctan (x_i) - \cos \left( \frac{x_{n+1} + x_{n-1}}{n}\right) - 1, \quad i=2,3,\ldots ,n-1\\ f_n(x) = x_n + \arctan (x_n) - \cos (\frac{x_n}{n}) - 1. \end{array} \end{aligned}$$

Example 15

The function f(x) is endowed with the component as follows:

$$\begin{aligned} \begin{array}{l} f_1(x) = \sinh (\frac{1}{n} x_1) + x_2 + \frac{1}{n},\\ f_i(x) = \sinh (\frac{i}{n} x_i) + x_{i-1} + x_{i+1} \frac{i}{n}, \quad i=2,3,\dots ,n-1\\ f_n(x) = \sinh (\frac{1}{n} x_1) + x_{n-1} + 1, \end{array} \end{aligned}$$

where $sinh(\cdot )$ is the hyperbolic sine function, i.e., $\sinh (t) = \frac{e^t - e^{-t}}{2}$.