A generalized direction in interior point method for monotone linear complementarity problems

Abstract

In this paper, we present a new interior point method with full Newton step for monotone linear complementarity problems. The specificity of our method is to compute the Newton step using a modified system similar to that introduced by Darvay in Stud Univ Babe-Bolyai Ser Inform 47:15–26, 2017. We prove that this new method possesses the best known upper bound complexity for these methods. Moreover, we extend results known in the literature since we consider a general family of smooth concave functions in the Newton system instead of the square root.

A Numerics

To validate the theoretical results, we first implemented the $\varphi$-FN-IPM described in Algorithm 2 with the sequence of update parameters given by Theorem 3.4 and with the functions $\varphi$ given in Table 1.

The datasets and the residuals after convergence are detailed in Table 2. The stopping criterion is $z^Ts \le n \epsilon$, where $\epsilon =10^{-6}$, and we compute the residuals as $\Vert zs\Vert _\infty \times 10^6$. For every $\varphi$ function that appears in Table 1, the method converged in the same number of iterations, so we only display the number of iterations once. This phenomenon is not surprising, since the $\varphi$-FN-IPM used here stays very close to the central path, as shown by Theorem 3.3. Therefore, this implementation does not exploit fully the new directions. Nonetheless, some small differences remain in the residuals as illustrated in Table 2.

Table 1 $\varphi$-functions used in the computational tests

Table 2 Value of Res. = $\Vert zs\Vert _\infty \times 10^6$, after the algorithm reaches $z^Ts\le n\epsilon =10^{-6}n$

Table 3 Value of Res. = $\Vert zs\Vert _\infty \times 10^6$ after the algorithm reaches $z^Ts\le n\epsilon =10^{-6}n$ and number of iterations (Iter.)

To illustrate the possible differences between the functions $\varphi$, we run another experiment that considers a fixed value for $\theta ^k=\frac{1}{\sqrt{n}}$ for all $k \in \mathbb {N}$.

Table 3 illustrates the different behaviours observed for different choices of $\varphi$. We notice that a smaller number of iterations seems to be required when the derivative in zero is larger for the methods with $\alpha =0.25,0.5$ and 0.75. Thus, further research exploring interior-point methods in large neighbourhood of the central path may get the best out of these differences.