A generic kernel function for interior point methods

Abstract

In this paper, a new class of kernel functions is introduced. We show that most existing kernel functions belong to this class. All functions in the new class are eligible in the sense of Bai et al. (SIAM J Optim 15(1):101–128, 2004), and hence the analysis of the resulting interior-point methods can follow the scheme proposed in Bai et al. (2004). We introduce five new kernel functions and by using this scheme we show that primal-dual IPMs based on these functions enjoy the best known iteration bound for large-update methods, i.e., $O(\sqrt{n}\log n\log \frac{n}{\epsilon }).$ Finally, to demonstrate the efficiency of IPMs based on the new kernel functions, some numerical results are provided.

Appendix

In this section, we prove that the five new kernel functions defined in Sect. 5 satisfy the conditions (9)–(11). To this end, it is clear that the first property, i.e., $f_i(1) = 0$ and $\lim _{t\mapsto 0^+}f_i(t)=+\infty$ ares true for all new functions $f_i(t)$. Thus we just demonstrate that these functions satisfy the second and third conditions.

Lemma 8.1

For the new proposed functions $f_i(t)$, $i=1,2,\ldots ,5$ the relations (10) and (11) are true:

Proof

To prove the first item, it is enough to prove the inequality $2t-(tf_i'(t)+1)\exp (f_i(t))>0,$ $i=1,2,\ldots ,5$ is true. Using $f_i(1)=0$ and the fact that $f_i$’s are strictly decreasing function (i.e., $f_i'(t)<0$ and $\exp (f_i(t))\le 1$), for all $t\ge 1$, we can conclude that:

$$\begin{aligned} 2t-(tf_i'(t)+1)\exp (f_i(t))\ge 2t-(tf_i'(t)+1) >0,\quad i=1,2,..5, \end{aligned}$$

in which the last inequality is obtained from the fact that $t\ge 1$.

On the other hand, when $t\in (0,1]$ it is sufficient to prove that $-(tf_i'(t)+1)>0$, $i=1,2,\ldots ,5.$ It is proved as follow:

• For the first function we have:

  $$\begin{aligned} -tf_1'(t)-1=\frac{te^tp(e-1)}{(e^t-1)^2}-1=\frac{te^tp(e-1)-(e^t-1)^2}{(e^t-1)^2}. \end{aligned}$$

  Since $(e^t-1)^2$is positive, we demonstrate that the function

  $$\begin{aligned} l_1:=te^tp(e-1)-(e^t-1)^2, \end{aligned}$$

  is positive. To this end, using the fact $t\in (0, 1]$ and we conclude that $(e-1)\ge (e^t-1)$, which implies that $l_1\ge (e^t-1)\left( te^tp-e^t+1\right)$. So, we deduce that the function $l_1$ is positive.

• For the function $f_2(t)$, we have:

  $$\begin{aligned} -tf_2'(t)-1=\frac{tpe^t}{e^t-1}+\frac{tpe^t(e-1)}{(e^t-1)^2}-1>0, \end{aligned}$$

  where the inequality is obtained by applying the first part of lemma and the fact that $\frac{pe^t}{e^t-1}$ is positive for all $t>0$.

• By considering the third function, we see that:

  $$\begin{aligned} -tf_3'(t)-1=2p+\frac{2p\pi t}{(2+2t)^2}\left( 1+\tan ^2\left( \frac{\pi }{2+2t}\right) \right) -1>0, \end{aligned}$$

  where the last inequality is derived from the fact that $p\ge 1$ and $\tan (\frac{\pi }{2+2t})$ is positive for all $t>0$.

• The forth function, i.e., $f_4(t)$ gives us:

  $$\begin{aligned} -tf_4'(t)-1= p+\frac{r p}{t}e^{p(\frac{1}{t}-1)}-1>0. \end{aligned}$$

• We can use the proof of $f_1$ for the $f_5$; so it is omitted here.

Using the fact the functions $f_i(t)$, $i=1,2,\ldots ,5$, are strictly decreasing and one can easily conclude that these functions satisfy the condition (11).

The next goal in this section is to prove that the functions $f_i(t)$, $i=1,2,\ldots ,5$ are convex. By regarding (Bai et al. 2004), it is enough to show that their second derivative are positive. Their second derivative are presented in Table 13. Obviously all of them are positive.

Table 13 The second derivative of the kernel functions in Table 2

$\square$