# The topological Petersson product.

L. Candelori  and  A. Salch

###### Abstract.

The nondegeneracy of the Petersson inner product on cusp forms, and the fact
that Hecke operators are self-adjoint with respect to the Petersson product,
together imply that the cusp forms have a basis consisting of Hecke
eigenforms. In the literature on topological modular forms, no topological
analogue of the Petersson product is to be found, and it is not known which
topological spaces have the property that their topological cusp forms admit a
basis consisting of eigenforms for the action of Baker’s topological Hecke
operators. In this note we define and study a natural topological Petersson
product on complexified topological cusp forms, whose value on a one-point
space recovers the classical Petersson product. We find that the topological
Petersson product is usually degenerate: in particular, if $X$ is a space with
nontrivial rational homology in any positive degree, then the Petersson
product on complexified $tcf^{*}(X)$ is degenerate. Despite $tcf$-cohomology
being a stable invariant, the topological Petersson product is an unstable
invariant, vanishing on all suspensions. Nevertheless, we demonstrate
nontriviality of the topological Petersson product by giving an explicit
calculation of the topological Petersson product on complexified
$tcf$-cohomology of the complex projective plane. We show that, for a compact
Kahler manifold $X$, the Petersson product is nondegenerate on complexified
$\operatorname{{\it tcf}}$-cohomology of $X$ in a range of Atiyah-Hirzebruch
filtrations (essentially one-third of the possibly-nonzero Atiyah-Hirzebruch
filtrations in the $tcf$-cohomology of $X$).

###### Contents

  1. 1 Introduction
  2. 2 Topological cusp forms.
  3. 3 Hecke operators on rational topological cusp forms.
  4. 4 The Petersson product on complexified topological cusp forms.

##  1\. Introduction

The classical Petersson product of level $1$ cusp forms is the positive-
definite Hermitian form given by the formula

$\langle
f,g\rangle:=\int_{\mathbb{H}/\mathrm{SL}_{2}(\mathbb{Z})}f\bar{g}v^{k-2}du\,dv,\quad\tau=u+iv\in\mathbb{H}.$

Each Hecke operator $T_{m}$ is self-adjoint with respect to the Petersson
product on $S_{k}\otimes_{\mathbb{Z}}\mathbb{C}$. Consequently the action of
each Hecke operator on $S_{k}$ is diagonalizable, and $S_{k}$ has a basis
consisting of Hecke eigenforms; this is a fundamental, important result, since
(among other reasons) the modular forms with $L$-functions expected to occur
from irreducible degree $2$ Galois representations are the cusp forms, and for
a normalized Hecke eigenform $f$ the $p$-th $L$-series coefficient is simply
the eigenvalue of the action of $T_{p}$ on $f$. Having a basis for weight $k$
cusp forms consisting of Hecke eigenforms gives a powerful handle on their
$L$-functions: we write a cusp form as a linear combination of Hecke
eigenforms, and then on each summand we can recover the $L$-series
coefficients from the Hecke eigenvalues.

It is quite satisfying to be able to get this fundamental result as
consequences of the existence of a piece of structure, namely, an inner
product on cusp forms with respect to which the Hecke operators are self-
adjoint. One would like to see this same structure occur in the context of
topological modular forms, with some similar consequences, like
diagonalizability of the action of the Hecke operations on topological cusp
forms.

We define, in the most direct and naïve possible way, a “weight $j$
topological Petersson product”

$\displaystyle\operatorname{{\it
tcf}}^{-2k}(X)\otimes_{\mathbb{Z}}\operatorname{{\it tcf}}^{-2k}(X)$
$\displaystyle\rightarrow H^{2(j-k)}(X;\mathbb{C})$

for each unpointed finite CW-complex $X$. When $X$ is a single point, this
recovers the classical Petersson product in the case $j=k$, and is zero when
$j\neq k$. When $X$ is the complex projective plane, the weight $j$
topological Petersson product on $\operatorname{{\it tcf}}^{-2k}(X)$ vanishes
unless $j=k$ or $j=k+1$, and recovers the classical Petersson product when
$j=k$ or $j=k+1$. These examples are given in Examples 4.2.

Base-changing to $\mathbb{C}$, the topological Petersson product yields an
$\mathbb{R}$-bilinear form

$\displaystyle\left(\operatorname{{\it
tcf}}^{-2k}(X)\otimes_{\mathbb{Z}}\mathbb{C}\right)\otimes_{\mathbb{R}}\left(\operatorname{{\it
tcf}}^{-2k}(X)\otimes_{\mathbb{Z}}\mathbb{C}\right)$ $\displaystyle\rightarrow
H^{2(j-k)}(X;\mathbb{C})$

which is explicitly calculated in Proposition 4.3 up to the Atiyah-Hirzebruch
filtration, i.e., we calculate the topological Petersson product on the
associated graded of the Atiyah-Hirzebruch filtration on $\operatorname{{\it
tcf}}^{-2k}(X)\otimes_{\mathbb{Z}}\mathbb{C}$, where it depends only on the
cup product on $H^{*}(X;\mathbb{Q})$ and the classical Petersson product.

We compute examples of topological Petersson products in Examples 4.2:

  * •

On a single point, the topological Petersson product coincides with the
classical Petersson product of cusp forms.

  * •

On a sphere of positive dimension, the topological Petersson product again
coincides with the classical Petersson product of cusp forms, supported
entirely on the basepoint. That is, the topological Petersson product is
completely insensitive to the topology of the sphere.

  * •

However, the topological Petersson product over a space $X$ does detect some
of the topology on $X$: we show that the topological Petersson product of
topological cusp forms on $\mathbb{C}P^{2}$ agrees with the classical
Petersson product on the basepoint, and it agrees with a weight-shifted copy
of the classical Petersson product on the $2$-cell
$S^{2}\subseteq\mathbb{C}P^{2}$. It is trivial on the topological cusp forms
supported on the $4$-cell in $\mathbb{C}P^{2}$, however.

We give formula (10), which provides an explicit and highly usable way to
calculate the topological Petersson product of topological cusp forms over a
space $X$, at least up to terms of higher Atiyah-Hirzebruch filtration. That
is, formula (10) is an explicit formula for the topological Petersson product
on the associated graded of the Atiyah-Hirzebruch (i.e., skeletal) filtration
of $tcf^{*}(X)\otimes_{\mathbb{Z}}\mathbb{C}$. One consequence is that the
topological Petersson product is highly sensitive to the cup product structure
of the cohomology of a space, and is consequently not a stable invariant of
the space.

As another consequence, we get Corollary 4.4, which shows that, unfortunately,
the topological Petersson product is degenerate on $\operatorname{{\it
tcf}}^{*}(X)\otimes_{\mathbb{Z}}\mathbb{C}$ for finite CW-complexes $X$ with
any nontrivial rational cohomology in positive degrees at all.

Here is the moral: in this note, we have carried out the most naïve kind of
investigation into what a Petersson product on topological modular forms might
be, by simply considering the rational case. Rationalizing dramatically
simplifies stable homotopy types, so one might expect this to trivialize all
questions about topological Petersson products. Yet we see that the most basic
and important question about the Petersson product—is it degenerate, so that
we can use it to show that the topological Hecke operators are diagonalizable,
like how the argument goes for classical cusp forms?—remains quite nontrivial
even after our simplifications, and indeed, over any finite CW-complex with
any nontrivial rational cohomology in positive degrees, the topological
Petersson product is degenerate. Evidently some more involved techniques would
be required to establish diagonalizability of topological Hecke operators on
elliptic (co)homology.

Nevertheless, nondegeneracy of the topological Petersson product is not an
entirely lost cause. We end on a small positive note: in Theorem 4.7, we show
that when $X$ is a compact Kähler manifold of complex dimension $d$, the
topological Petersson product of $X$ is nondegenerate when restricted to the
elements of $\operatorname{{\it tcf}}^{*}(X)\otimes_{\mathbb{Z}}\mathbb{C}$ of
even Atiyah-Hirzebruch filtration $\leq 2d/3$. (When the rational homology of
$X$ is concentrated in even degrees, then all elements of $\operatorname{{\it
tcf}}^{*}(X)\otimes_{\mathbb{Z}}\mathbb{C}$ have even Atiyah-Hirzebruch
filtration.)

Throughout this note, on various occasions, we have an $H\mathbb{C}$-module
spectrum $X$ and a map of $H\mathbb{C}$-module spectra of the form

(1) $X\wedge_{H\mathbb{R}}X\rightarrow H\mathbb{C}.$

Naturally, this represents an $\mathbb{R}$-bilinear form on the spectrum $X$.
If one is willing to keep track of a $\operatorname{{\rm
Gal}}(\mathbb{C}/\mathbb{R})$-action on the relevant spectra, then one could
consider a map of the form

(2) $X\wedge_{H\mathbb{C}}\overline{X}\rightarrow H\mathbb{C},$

i.e., a sesquilinear form on $X$. We suspect that, using equivariant spectra
to keep track of the Galois action, the methods we pursue in this note in
order to get an $\mathbb{R}$-bilinear Petersson product on $\operatorname{{\it
tcf}}$-cohomology ought to also yield a sesquilinear Petersson product on
$\operatorname{{\it tcf}}$-cohomology. However, bringing in ideas from
equivariant stable homotopy would significantly increase the length and/or
expected background knowledge of this note, while making only a modest
improvement in the strength of the results, so we elected to consider only an
$\mathbb{R}$-bilinear (rather than sesquilinear) Petersson product here.

We are grateful to Theo Johnson-Freyd for a helpful conversation early on
during this project.

##  2\. Topological cusp forms.

In this section we recall the construction of topological cusp forms. We
learned about this construction from Theo Johnson-Freyd. Our understanding is
that it is originally due to Lennart Meier (unpublished). Let $F$ denote the
homotopy fiber of the usual $E_{\infty}$-ring spectrum map $tmf\rightarrow
ko$. The rational stable homotopy groups of $F$ are then given by

$\pi_{*}(F)\otimes_{\mathbb{Z}}\mathbb{Q}\cong
S_{2*}\otimes_{\mathbb{Z}}\mathbb{Q}\oplus\Sigma^{3}\mathbb{Q},$

the graded $\mathbb{Q}$-vector space of level $1$ cusp forms111The cusp forms
do not form a ring, since, for example, there is no weight $0$ cusp form,
hence no multiplicative unit., with grading given by weight
multiplied222Throughout, we will consistly write $S_{2*}$ to mean the graded
$\mathbb{Z}[\frac{1}{6}]$-module of level $1$ cusp forms over
$\mathbb{Z}\left[\frac{1}{6}\right]$ with grading given by weight multiplied
by $2$. We write $S_{-2*}$ for the same graded
$\mathbb{Z}[\frac{1}{6}]$-module but with the grading degrees also multiplied
by $-1$. We write $S_{k}$ for the $\mathbb{Z}\left[\frac{1}{6}\right]$-module
of weight $k$ level $1$ cusp forms over $\mathbb{Z}\left[\frac{1}{6}\right]$.
by $2$, along with an additional copy of $\mathbb{Q}$ in degree $3$ coming
from the Bott class in $\pi_{4}(ko)$ failing to be in the image of the map
$\pi_{*}(tmf)\rightarrow\pi_{*}(ko)$, essentially because the weight $2$
Eisenstein series fails to be a modular form.

In fact the above is true after inverting the primes $2$ and $3$; it is not
necessary to pass all the way to the rationalization. In particular, we have
$\pi_{n}(F\left[\frac{1}{6}\right])\cong 0$ if $n<3$, while
$\pi_{3}(F\left[\frac{1}{6}\right])\cong\mathbb{Z}\left[\frac{1}{6}\right]$.
We then have $\pi_{n}(F\left[\frac{1}{6}\right])\cong 0$ for
$n=4,5,6,7,8,\dots,23$, and then we get
$\pi_{24}(F\left[\frac{1}{6}\right])\cong\mathbb{Z}\left[\frac{1}{6}\right]$
generated by $\Delta$, and thereafter (i.e., for $n\geq 24$) we have that
$\pi_{n}(F\left[\frac{1}{6}\right])$ is a free
$\mathbb{Z}\left[\frac{1}{6}\right]$-module of rank $j$ if $n=24j+r$ with
$r\equiv 0,8,12,16,20$ modulo $24$, rank $j-1$ if $n=24j+r$ with $r\equiv
2,4,6,10,14,18,22$ modulo $24$, and rank $0$ otherwise. To narrow our focus to
the rank of degrees where the homotopy groups of $F$ describe (away from $6$)
exactly the classical level $1$ cusp forms, we define $\operatorname{{\it
tcf}}$ as the $4$-connective cover of $F$. This yields the isomorphism of
graded $\mathbb{Z}[\frac{1}{6}]$-modules $\pi_{*}(\operatorname{{\it
tcf}}[\frac{1}{6}])\cong S_{2*}$. So, after inverting $6$, the homotopy groups
of $\operatorname{{\it tcf}}$ are precisely the classical level $1$ cusp forms
over $\mathbb{Z}[\frac{1}{6}]$.

It will not be important for the rest of this note, but it is perhaps
satisfying to know that the spectrum $\operatorname{{\it tcf}}$ is also a
$\operatorname{{\it tmf}}$-module, and the resulting action of
$\pi_{*}(\operatorname{{\it tmf}}[\frac{1}{6}])$ on $\operatorname{{\it
tcf}}[\frac{1}{6}]$ coincides with the usual action, by multiplication, of the
ring $M_{*}$ of holomorphic level $1$ modular forms on its submodule of cusp
forms. To see that this is so, first observe that the map $\operatorname{{\it
tmf}}\rightarrow ko$ is a ring spectrum map, hence in particular a
$\operatorname{{\it tmf}}$-module map, so its homotopy fiber $F$ is also a
$\operatorname{{\it tmf}}$-module. Since $\operatorname{{\it tmf}}$ is a
connective $E_{\infty}$-ring spectrum, we can take any $\operatorname{{\it
tmf}}$-module $X$ and attach $\operatorname{{\it tmf}}$-cells (rather than
$S^{0}$-cells, as in the stable version of the familiar “attaching cells to
kill higher homotopy” construction from classical homotopy theory) to kill all
the homotopy above some degree $n$, yielding a map of $\operatorname{{\it
tmf}}$-modules $X\rightarrow X(-\infty,n]$. Our spectrum $\operatorname{{\it
tcf}}$ is the fiber of the map $F\rightarrow F(-\infty,3]$, so
$\operatorname{{\it tcf}}$ is the fiber of a map of $\operatorname{{\it
tmf}}$-modules, so $\operatorname{{\it tcf}}$ is a $\operatorname{{\it
tmf}}$-module. It is routine to check that the resulting
$\pi_{*}(\operatorname{{\it tmf}}[\frac{1}{6}])$-action on
$\pi_{*}(\operatorname{{\it tcf}}[\frac{1}{6}])$ is as expected.

##  3\. Hecke operators on rational topological cusp forms.

When we speak of an “action of Hecke operators” on a spectrum $E$, we will
always mean that the relevant Hecke operator $\tilde{T}_{n}$ ought to be a
homotopy class of maps of spectra $E\rightarrow E$; we do not require Hecke
operators to be defined “on the nose,” and we only require the usual Hecke
relations $\tilde{T}_{m}\tilde{T}_{n}=\tilde{T}_{n}\tilde{T}_{m}$ and

(3)
$\tilde{T}_{p^{r+2}}(x)=\tilde{T}_{p}\left(\tilde{T}_{p^{r+1}}(x)\right)-\frac{1}{p}\Psi^{p}\tilde{T}_{p^{r}}(x)$

to hold up to homotopy333The classical Hecke relation is
$T_{p^{r+2}}=T_{p}T_{p^{r+1}}-p^{k-1}T_{p^{r}}$, which holds for the action of
the Hecke operators on the ring of holomorphic modular forms. However, Baker
points out in [Bak90] that his topological Hecke operators on elliptic
homology instead satisfy the more general relation (3), where $\Psi^{p}$ here
is the $p$th Adams operation. In the case of the holomorphic elliptic homology
of the zero-sphere $S^{0}$, the action of $\Psi^{p}$ on $\ell_{2k}(S^{0})\cong
M_{k}$ is simply by multiplication by $p^{k}$, so (3) reduces to the classical
relation $T_{p^{r+2}}=T_{p}T_{p^{r+1}}-p^{k-1}T_{p^{r}}$ when the topological
Hecke operators are evaluated on the zero-sphere. See [CS22] for further
discussion.. Consequently, in the rest of this section, we will always be
working in the stable homotopy category.

Getting an action of Hecke operators on $tcf$ is quite nontrivial: in [Bak90],
Baker constructs an action of Hecke operators on
$TMF\left[\frac{1}{6}\right]$, and essentially the same method yields an
action of Hecke operators on $tmf\left[\frac{1}{6}\right]$. Baker’s method
uses the complex-orientability of $TMF\left[\frac{1}{6}\right]$ in an
essential way, and it does not yield Hecke operations on $TMF$ (or on $Tmf$ or
$tmf$) without inverting the primes $2$ and $3$, since it is only after
inverting $6$ that these spectra become complex-oriented.

Like $TMF$ and $Tmf$ and $tmf$, the spectrum $tcf$ is not complex-oriented;
but unlike $TMF$ and $Tmf$ and $tmf$, inverting $6$ still does not make $tcf$
complex-oriented444This is simply because a spectrum cannot be complex-
oriented without having a ring structure. Since $\pi_{0}(tcf)$ remains trivial
after inverting any set of primes, it is not possible to make $tcf$ into a
ring spectrum by inverting any set of primes.. Some other idea is needed to
build Hecke operators on $tcf$, even on $tcf\left[\frac{1}{6}\right]$.

In this note we take the path of least resistance: we simply rationalize, and
then try to show that Baker’s Hecke operators on rational $tmf$ restrict to
operators on rational $tcf$. This is, however, straightforward: for each
positive integer $n$, we have the solid arrows in the diagram of spectra whose
rows are homotopy fiber sequences

(4) $\textstyle{F\wedge
H\mathbb{Q}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{tmf\wedge
H\mathbb{Q}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{T}_{n}}$$\textstyle{ko\wedge
H\mathbb{Q}}$$\textstyle{F\wedge
H\mathbb{Q}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{tmf\wedge
H\mathbb{Q}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{ko\wedge
H\mathbb{Q}}$

and the composite $F\wedge H\mathbb{Q}\rightarrow ko\wedge H\mathbb{Q}$ from
the upper left corner to the lower right corner induces the zero map in
$\pi_{*}$, since Baker’s $\tilde{T}_{n}$ agrees with the classical Hecke
operator on $T_{n}$ on $\pi_{*}(tmf\left[\frac{1}{6}\right])$. In rational
spectra, a map that induces zero in $\pi_{*}$ is nulhomotopic, so there exists
a map filling in the dotted line in diagram (4) and making it commute. The
number of such maps is the order of the cokernel of the map

$[F\wedge H\mathbb{Q},\Sigma^{-1}tmf\wedge H\mathbb{Q}]\rightarrow[F\wedge
H\mathbb{Q},\Sigma^{-1}ko\wedge H\mathbb{Q}].$

Since the rational homotopy groups of $\Sigma^{-1}ko$ are concentrated in odd
degrees and since the only odd-degree rational homotopy group of $F$ is the
copy of $\mathbb{Q}$ in rational $\pi_{3}(F)$ coming from the failure of
$\pi_{4}(tmf)\rightarrow\pi_{4}(ko)$ to be rationally surjective, the dotted
arrow in (4) is well-defined up to the question of how $T_{n}$ ought to act on
$\pi_{3}(F)\otimes_{\mathbb{Z}}\mathbb{Q}$; a reasonable way to make this
choice might be to use the action of $T_{n}$ on the quasimodular form $E_{2}$,
as in [Mov15]. For the sake of this note, one can make whatever choice one
wants, since we make no use of the Hecke action on $\pi_{3}(F)$; since taking
the $4$-connective cover is a functor, each map $F\wedge
H\mathbb{Q}\rightarrow F\wedge H\mathbb{Q}$ yields a map $\operatorname{{\it
tcf}}\wedge H\mathbb{Q}\rightarrow\operatorname{{\it tcf}}\wedge H\mathbb{Q}$
with the same effect on homotopy in degrees $\geq 4$, yielding our Hecke
action on $\operatorname{{\it tcf}}\wedge H\mathbb{Q}$.

Consequently, for each positive integer $n$, we have a topological Hecke
operator $\tilde{T}_{n}$ on $tcf$. It is a map of spectra
$\tilde{T}_{n}:tcf\wedge H\mathbb{Q}\rightarrow tcf\wedge H\mathbb{Q}$, well-
defined up to homotopy. By applying the base-change functor
$-\wedge_{H\mathbb{Q}}H\mathbb{C}$ we get a map of spectra $tcf\wedge
H\mathbb{C}\rightarrow tcf\wedge H\mathbb{C}$, which by abuse of notation we
will also call $\tilde{T}_{n}$, and which plays an important role in the next
section.

##  4\. The Petersson product on complexified topological cusp forms.

Given a connective spectrum $X$ and an integer $n$, let $X[0,n]$ denote $X$
with cells attached to kill all homotopy groups in degrees $>n$. For each
integer $k$, let $\operatorname{{\it tcf}}^{\operatorname{{\rm
wt}}k}_{\mathbb{C}}$ be the fiber of the map from $\left(\operatorname{{\it
tcf}}\wedge H\mathbb{C}\right)[0,2k]$ to $\left(\operatorname{{\it tcf}}\wedge
H\mathbb{C}\right)[0,2k-1]$. Since $\operatorname{{\it tcf}}\wedge
H\mathbb{C}$ is a rational spectrum, the fiber inclusion map
$\operatorname{{\it tcf}}^{\operatorname{{\rm
wt}}k}_{\mathbb{C}}\rightarrow\left(\operatorname{{\it tcf}}\wedge
H\mathbb{C}\right)[0,2k]$ splits, and a degree argument shows that the
splitting is unique. Hence $\operatorname{{\it tcf}}^{\operatorname{{\rm
wt}}k}_{\mathbb{C}}$ is uniquely a wedge summand of $\operatorname{{\it
tcf}}\wedge H\mathbb{C}$, homotopy equivalent to the $2k$th suspension of the
Eilenberg-Mac Lane spectrum of the $\mathbb{C}$-vector space
$S_{k}\otimes_{\mathbb{Z}}\mathbb{C}$ of weight $k$ level $1$ classical cusp
forms.

For each integer $k$, let

$\operatorname{{\rm Petersson}}_{k}:\Sigma^{-2k}\operatorname{{\it
tcf}}^{\operatorname{{\rm
wt}}k}_{\mathbb{C}}\wedge_{H\mathbb{R}}\Sigma^{-2k}\operatorname{{\it
tcf}}^{\operatorname{{\rm wt}}k}_{\mathbb{C}}\rightarrow H\mathbb{C}$

be the Eilenberg-Mac Lane spectrum functor $H$ applied to the Petersson
product map
$\left(S_{k}\otimes_{\mathbb{Z}}\mathbb{C}\right)\otimes_{\mathbb{R}}\left(S_{k}\otimes_{\mathbb{Z}}\mathbb{C}\right)\rightarrow\mathbb{C}$.

Now we are ready to define the Petersson product on topological cusp forms:

######  Definition 4.1.

For each integer $j$, the weight $j$ topological Petersson bicharacter is the
composite map of spectra

$\displaystyle\Sigma^{-2j}\operatorname{{\it
tcf}}\wedge\Sigma^{-2j}\operatorname{{\it tcf}}$
$\displaystyle\rightarrow\left(\Sigma^{-2j}\operatorname{{\it tcf}}\wedge
H\mathbb{C}\right)\wedge_{H\mathbb{R}}\left(\Sigma^{-2j}\operatorname{{\it
tcf}}\wedge H\mathbb{C}\right)$
$\displaystyle\rightarrow\Sigma^{-2j}\operatorname{{\it
tcf}}^{\operatorname{{\rm
wt}}j}_{\mathbb{C}}\wedge_{H\mathbb{R}}\Sigma^{-2j}\operatorname{{\it
tcf}}^{\operatorname{{\rm wt}}j}_{\mathbb{C}}$
$\displaystyle\stackrel{{\scriptstyle\operatorname{{\rm
Petersson}}_{j}}}{{\longrightarrow}}H\mathbb{C}.$

Consequently, given an unpointed topological space555We cannot make the same
construction for an arbitrary spectrum $X$, since we are required to have a
diagonal map on $X$. $X$, an integer or half-integer $k$, and a pair of
elements $f,g\in[\Sigma^{\infty}X_{+},\Sigma^{-2k}\operatorname{{\it
tcf}}]\cong\operatorname{{\it tcf}}^{-2k}(X)$, the composite map

$\displaystyle\Sigma^{\infty}X_{+}$
$\displaystyle\stackrel{{\scriptstyle\Sigma^{\infty}\Delta_{+}}}{{\longrightarrow}}\Sigma^{\infty}(X\times
X)_{+}$
$\displaystyle\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\Sigma^{\infty}X_{+}\wedge\Sigma^{\infty}X_{+}$
$\displaystyle\stackrel{{\scriptstyle f\wedge
g}}{{\longrightarrow}}\Sigma^{-2k}\operatorname{{\it
tcf}}\wedge\Sigma^{-2k}\operatorname{{\it tcf}}$
$\displaystyle\stackrel{{\scriptstyle}}{{\longrightarrow}}\Sigma^{-2k}\operatorname{{\it
tcf}}\wedge_{H\mathbb{R}}\Sigma^{-2k}\operatorname{{\it tcf}}$
$\displaystyle\stackrel{{\scriptstyle\Sigma^{4(j-k)}\operatorname{{\rm
Petersson}}_{j}}}{{\longrightarrow}}\Sigma^{2(j-k)}H\mathbb{C}$

is an element of $\tilde{H}^{4(j-k)}(X_{+};\mathbb{C})\cong
H^{4(j-k)}(X;\mathbb{C})$. We define $\langle f,g\rangle_{j}$, the weight $j$
topological Petersson product of $f$ and $g$, to be that element of
$H^{4(j-k)}(X;\mathbb{C})$.

Note that the topological Petersson products are not stable invariants. In
particular, suspending $X$ tends to cause topological Petersson products not
supported on the basepoint to become zero.

######  Examples 4.2.

  * •

When $X=\operatorname{{\rm pt.}}$, an element $f$ of
$\left[\Sigma^{\infty}X_{+},\Sigma^{-2k}\operatorname{{\it
tcf}}\left[\frac{1}{6}\right]\right]\cong\pi_{2k}\left(\operatorname{{\it
tcf}}\left[\frac{1}{6}\right]\right)$ is given by a weight $k$ level $1$
classical cusp form. Since $\pi_{0}(\Sigma^{-2k}\operatorname{{\it
tcf}}^{\operatorname{{\rm wt}}j}_{\mathbb{C}})$ vanishes unless $j=k$, the
composite

$\displaystyle\Sigma^{\infty}X_{+}$
$\displaystyle\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\Sigma^{\infty}S^{0}$
$\displaystyle\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\Sigma^{\infty}S^{0}\wedge\Sigma^{\infty}S^{0}$
$\displaystyle\stackrel{{\scriptstyle f\wedge
g}}{{\longrightarrow}}\Sigma^{-2k}\operatorname{{\it
tcf}}\wedge\Sigma^{-2k}\operatorname{{\it tcf}}$
$\displaystyle\rightarrow\Sigma^{-2k}\operatorname{{\it
tcf}}^{\operatorname{{\rm
wt}}j}_{\mathbb{C}}\wedge_{H\mathbb{R}}\Sigma^{-2k}\operatorname{{\it
tcf}}^{\operatorname{{\rm wt}}j}_{\mathbb{C}}$

is nulhomotopic unless $j=k$. Consequently $\langle f,g\rangle_{j}=0$ unless
$j=k$. When $j=k$, the topological Petersson product is nonzero: it is
immediate from Definition 4.1 that $\langle f,g\rangle_{k}=\langle
f,g\rangle\in\mathbb{C}\cong H^{0}(\operatorname{{\rm pt.}};\mathbb{C})$, the
classical Petersson product of the cusp forms $f$ and $g$.

  * •

More generally, when $X$ is a finite discrete space with $n$ points, then
$\operatorname{{\it tcf}}^{2k}(X)$ splits as a direct sum of $n$ copies of
$\operatorname{{\it tcf}}^{2k}(\operatorname{{\rm pt.}})$, and the weight $k$
topological Petersson product on each summand of $\operatorname{{\it
tcf}}^{2k}(X)\left[\frac{1}{6}\right]$ is simply the classical Petersson
product, while the weight $j$ topological Petersson product is zero for $j\neq
k$.

  * •

When $X=S^{n}$ for $n>0$, it is standard that the composite map of spectra

$\displaystyle\Sigma^{\infty}S^{n}\vee\Sigma^{\infty}S^{0}$
$\displaystyle\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\Sigma^{\infty}S^{n}_{+}$
$\displaystyle\stackrel{{\scriptstyle\Sigma^{\infty}\Delta_{+}}}{{\longrightarrow}}\Sigma^{\infty}(S^{n}\times
S^{n})_{+}$
$\displaystyle\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\Sigma^{\infty}S^{2n}\vee\Sigma^{\infty}S^{n}\vee\Sigma^{\infty}S^{n}\vee\Sigma^{\infty}S^{0}$

sends the summand $\Sigma^{\infty}S^{n}$ in its domain via the categorical
diagonal map to the summands $\Sigma^{\infty}S^{n}\vee\Sigma^{\infty}S^{n}$ in
the codomain, and sends the summand $\Sigma^{\infty}S^{0}$ in its domain via
the identity map to the summand $\Sigma^{\infty}S^{0}$ in its codomain.
Consequently, given elements
$f,g\in\left[\Sigma^{\infty}S^{n}_{+},\Sigma^{-2k}\operatorname{{\it
tcf}}\left[\frac{1}{6}\right]\right]$, we decompose $f$ into a weight $k$ cusp
form $f_{0}\in\left[\Sigma^{\infty}S^{0},\Sigma^{-2k}\operatorname{{\it
tcf}}\left[\frac{1}{6}\right]\right]$ and a weight $k+\frac{n}{2}$ cusp form
$f_{n/2}\in\left[\Sigma^{\infty}S^{n},\Sigma^{-2k}\operatorname{{\it
tcf}}\left[\frac{1}{6}\right]\right]$, and similarly we decompose $g$ into a
weight $k$ cusp form
$g_{0}\in\left[\Sigma^{\infty}S^{0},\Sigma^{-2k}\operatorname{{\it
tcf}}\left[\frac{1}{6}\right]\right]$ and a weight $k+\frac{n}{2}$ cusp form
$g_{n/2}\in\left[\Sigma^{\infty}S^{n},\Sigma^{-2k}\operatorname{{\it
tcf}}\left[\frac{1}{6}\right]\right]$, so that the composite map

$\displaystyle\Sigma^{\infty}S^{n}_{+}$
$\displaystyle\longrightarrow\Sigma^{\infty}S^{n}_{+}\wedge\Sigma^{\infty}S^{n}_{+}$
$\displaystyle\stackrel{{\scriptstyle f\wedge
g}}{{\longrightarrow}}\Sigma^{-2k}\operatorname{{\it
tcf}}\left[\frac{1}{6}\right]\wedge\Sigma^{-2k}\operatorname{{\it
tcf}}\left[\frac{1}{6}\right]$
$\displaystyle\rightarrow\Sigma^{-2k}\operatorname{{\it
tcf}}^{\operatorname{{\rm
wt}}j}_{\mathbb{C}}\wedge_{H\mathbb{R}}\Sigma^{-2k}\operatorname{{\it
tcf}}^{\operatorname{{\rm wt}}j}_{\mathbb{C}}$
$\displaystyle\rightarrow\Sigma^{4(j-k)}H\mathbb{C}$

is nulhomotopic unless $j=k$. To be clear: the composite map

(5)
$\displaystyle\Sigma^{\infty}S^{2n}\vee\Sigma^{\infty}S^{n}\vee\Sigma^{\infty}S^{n}\vee\Sigma^{\infty}S^{0}$
$\displaystyle\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\Sigma^{\infty}(S^{n}\times
S^{n})_{+}$ (6)
$\displaystyle\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\Sigma^{\infty}S^{n}_{+}\wedge\Sigma^{\infty}S^{n}_{+}$
(7) $\displaystyle\stackrel{{\scriptstyle f\wedge
g}}{{\longrightarrow}}\Sigma^{-2k}tcf^{\operatorname{{\rm
wt}}j}_{\mathbb{C}}\wedge_{H\mathbb{R}}\Sigma^{-2k}tcf^{\operatorname{{\rm
wt}}j}_{\mathbb{C}}$ (8) $\displaystyle\rightarrow\Sigma^{4(j-k)}H\mathbb{C}$

has the following effect on each wedge summands in its domain:

    * –

On $\Sigma^{\infty}S^{2n}$, the composite map

$\Sigma^{\infty}S^{2n}\rightarrow\Sigma^{-2k}tcf^{\operatorname{{\rm
wt}}j}_{\mathbb{C}}\wedge_{H\mathbb{R}}\Sigma^{-2k}tcf^{\operatorname{{\rm
wt}}j}_{\mathbb{C}}\rightarrow\Sigma^{4(j-k)}H\mathbb{C}$

is $\langle f_{n/2},g_{n/2}\rangle_{j}$, a classical Petersson product of
weight $k+n/2$ cusp forms, hence certainly capable of being nonzero if (and
only if) $j=k+n/2$.

    * –

On one summand $\Sigma^{\infty}S^{n}$, the composite map

$\Sigma^{\infty}S^{n}\rightarrow\Sigma^{-2k}tcf^{\operatorname{{\rm
wt}}j}_{\mathbb{C}}\wedge_{H\mathbb{R}}\Sigma^{-2k}tcf^{\operatorname{{\rm
wt}}j}_{\mathbb{C}}\rightarrow\Sigma^{4(j-k)}H\mathbb{C}$

is $\langle f_{0},g_{n/2}\rangle_{j}$, a classical Petersson product of cusp
forms of distinct weights, hence is zero.

    * –

On the other summand $\Sigma^{\infty}S^{n}$, the composite map

$\Sigma^{\infty}S^{n}\rightarrow\Sigma^{-2k}tcf^{\operatorname{{\rm
wt}}j}_{\mathbb{C}}\wedge_{H\mathbb{R}}\Sigma^{-2k}tcf^{\operatorname{{\rm
wt}}j}_{\mathbb{C}}\rightarrow\Sigma^{4(j-k)}H\mathbb{C}$

is $\langle f_{n/2},g_{0}\rangle_{j}$, again a classical Petersson product of
cusp forms of distinct weights, hence again zero.

    * –

On the summand $\Sigma^{\infty}S^{0}$, the composite map

$\Sigma^{\infty}S^{0}\rightarrow\Sigma^{-2k}tcf^{\operatorname{{\rm
wt}}j}_{\mathbb{C}}\wedge_{H\mathbb{R}}\Sigma^{-2k}tcf^{\operatorname{{\rm
wt}}j}_{\mathbb{C}}\rightarrow\Sigma^{4(j-k)}H\mathbb{C}$

is $\langle f_{0},g_{0}\rangle_{j}$, a classical Petersson product of weight
$k$ cusp forms, hence capable of being nonzero if (and only if) $j=k$.

However, the topological Petersson product of $f$ and $g$ is the composite of
(6) and (7), precomposed with the diagonal map
$\Delta:\Sigma^{\infty}S^{n}_{+}\rightarrow\Sigma^{\infty}(S^{n}\times
S^{n})_{+}$. While the composite of (5) through (7) is non-nulhomotopic on the
top-dimensional summand $\Sigma^{\infty}S^{2n}$, precomposing with $\Delta$
misses that top-dimensional summand entirely. (Of course this follows
immediately from the triviality of $\pi_{n}(S^{2n})$. Recall that $n>0$
throughout this example.)

Consequently, on the $n$-sphere for $n>2$, the weight $j$ topological
Petersson product of $f$ and $g$ is trivial unless $j=k$. If $j=k$, then the
topological Petersson product of $f$ and $g$ is supported entirely on the
basepoint of $S^{n}$, where it agrees with the classical Petersson product of
$f_{0}$ and $g_{0}$. On the $n$-sphere for $n>0$, the topological Petersson
product is completely insensitive to $f_{n/2}$ and $g_{n/2}$. In particular,
the topological Petersson product is degenerate on $\operatorname{{\it
tcf}}^{*}(S^{n})\otimes_{\mathbb{Z}}\mathbb{C}$.

  * •

The example of the sphere $S^{n}$ for $n>0$ may give the reader the impression
that the topological Petersson product on $\operatorname{{\it
tcf}}^{*}(X)\otimes_{\mathbb{Z}}\mathbb{C}$ is simply given by the classical
Petersson product on the cusp forms supported on the basepoint of $X_{+}$.
This impression is not correct, as the following example demonstrates: let
$X=\mathbb{C}P^{2}$. The composite map

$\displaystyle\Sigma^{\infty}S^{0}\vee\Sigma^{\infty}\mathbb{C}P^{2}$
$\displaystyle\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\Sigma^{\infty}\mathbb{C}P^{2}_{+}$
$\displaystyle\rightarrow\Sigma^{\infty}(\mathbb{C}P^{2}\times\mathbb{C}P^{2})_{+}$
$\displaystyle\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\Sigma^{\infty}\mathbb{C}P^{2}_{+}\wedge\Sigma^{\infty}\mathbb{C}P^{2}_{+}$
$\displaystyle\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\left(\Sigma^{\infty}\mathbb{C}P^{2}\wedge\Sigma^{\infty}\mathbb{C}P^{2}\right)\vee\Sigma^{\infty}\mathbb{C}P^{2}\vee\Sigma^{\infty}\mathbb{C}P^{2}\vee\Sigma^{\infty}S^{0}$

is non-nulhomotopic: it is nulhomotopic on the $2$-cell
$S^{2}\simeq\mathbb{C}P^{1}\subseteq\mathbb{C}P^{2}$, but the top cell of
$\mathbb{C}P^{2}$ is mapped by a rational equivalence to the $2$-skeleton of
$\mathbb{C}P^{2}\wedge\mathbb{C}P^{2}$, since the cup square of a generator of
$H^{2}(\mathbb{C}P^{2};\mathbb{C})$ is a generator of
$H^{4}(\mathbb{C}P^{2};\mathbb{C})$.

Consequently, given elements
$f,g\in\left[\Sigma^{\infty}\mathbb{C}P^{2}_{+},\Sigma^{-2k}\operatorname{{\it
tcf}}\left[\frac{1}{6}\right]\right]]$, we decompose $f$ into a weight $k$
cusp form $f_{0}\in\left[\Sigma^{\infty}S^{0},\Sigma^{-2k}\operatorname{{\it
tcf}}\left[\frac{1}{6}\right]\right]$ and an element $f^{\prime}$ of
$\left[\Sigma^{\infty}\mathbb{C}P^{2},\Sigma^{-2k}\operatorname{{\it
tcf}}\left[\frac{1}{6}\right]\right]$, and similarly we decompose $g$ into a
weight $k$ cusp form
$g_{0}\in\left[\Sigma^{\infty}S^{0},\Sigma^{-2k}\operatorname{{\it
tcf}}\left[\frac{1}{6}\right]\right]$ and an element $g^{\prime}$ of
$\left[\Sigma^{\infty}\mathbb{C}P^{2},\Sigma^{-2k}\operatorname{{\it
tcf}}\left[\frac{1}{6}\right]\right]$. For the same reasons as in the previous
examples, $\langle f_{0},g_{0}\rangle_{k}=\langle f,g\rangle_{k}$, and
$\langle f_{0},g_{0}\rangle_{j}=0$ if $j\neq k$.

The Atiyah-Hirzebruch spectral sequence

$\displaystyle E_{2}^{s,t}\cong H^{s}\left(\mathbb{C}P^{2};\operatorname{{\it
tcf}}^{t}\otimes_{\mathbb{Z}}\mathbb{C}\right)$
$\displaystyle\Rightarrow\operatorname{{\it
tcf}}^{s+t}(\mathbb{C}P^{2})\otimes_{\mathbb{Z}}\mathbb{C}$ $\displaystyle
d_{r}:E_{r}^{s,t}$ $\displaystyle\rightarrow E_{r}^{s+r,t-r+1}$

collapses at $E_{2}$; one way to see this is by Dold’s theorem that the
shortest nonzero Atiyah-Hirzebruch differential is torsion-valued, from
[Dol66] (see [Arl92] for a more easily-obtained English-language reference).
This yields a splitting of $\mathbb{C}$-vector spaces

$\displaystyle\operatorname{{\it
tcf}}^{-2k}(\mathbb{C}P^{2})\otimes_{\mathbb{Q}}\mathbb{C}$
$\displaystyle\cong\left(H^{0}(\mathbb{C}P^{2};\mathbb{C})\otimes_{\mathbb{Q}}S_{k}\right)$
$\displaystyle\oplus\left(H^{2}(\mathbb{C}P^{2};\mathbb{C})\otimes_{\mathbb{Q}}S_{k+1}\right)$
$\displaystyle\oplus\left(H^{4}(\mathbb{C}P^{2};\mathbb{C})\otimes_{\mathbb{Q}}S_{k+2}\right).$

Consequently we can decompose the image of $f$ in $\operatorname{{\it
tcf}}^{-2k}(\mathbb{C}P^{2})\otimes_{\mathbb{Z}}\mathbb{C}$ into a triple
$(f_{0},f_{2},f_{4})$, where $f$ is a weight $k$ cusp form and $f_{2}$ is a
weight $k+1$ cusp form and $f_{4}$ is a weight $k+2$ cusp form. This refines
the above decomposition of $f$ into $f_{0}$ and $f^{\prime}=(f_{2},f_{4})$. We
decompose $g$ into $(g_{0},g_{2},g_{4})$ similarly.

The composite

$\displaystyle\Sigma^{\infty}S^{2}$
$\displaystyle\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\Sigma^{\infty}\mathbb{C}P^{1}$
$\displaystyle\stackrel{{\scriptstyle}}{{\longrightarrow}}\Sigma^{\infty}\mathbb{C}P^{2}_{+}$
$\displaystyle\stackrel{{\scriptstyle}}{{\longrightarrow}}\Sigma^{\infty}\mathbb{C}P^{2}_{+}\wedge\Sigma^{\infty}\mathbb{C}P^{2}_{+}$
$\displaystyle\stackrel{{\scriptstyle f_{2}\wedge
g_{2}}}{{\longrightarrow}}\Sigma^{-2k}\operatorname{{\it
tcf}}\wedge\Sigma^{-2k}\operatorname{{\it tcf}}$
$\displaystyle\rightarrow\Sigma^{-2k}\operatorname{{\it
tcf}}^{\operatorname{{\rm
wt}}j}_{\mathbb{C}}\wedge\Sigma^{-2k}\operatorname{{\it
tcf}}^{\operatorname{{\rm wt}}j}_{\mathbb{C}}$

is nulhomotopic for all $j$, so the component of $\langle f,g\rangle_{j}$ on
the $2$-cell in $\mathbb{C}P^{2}_{+}$ is trivial for all $j$.

The $4$-cell in $\mathbb{C}P^{2}_{+}$ is the interesting part: by the above
cell-by-cell analysis of the homotopy class of the diagonal map on
$\mathbb{C}P^{2}_{+}$, the component of $\langle f,g\rangle_{j}$ on the
$4$-cell in $\mathbb{C}P^{2}_{+}$ is $\langle f_{2},g_{2}\rangle_{j+1}$.
Finally, we have

$\displaystyle\langle f,g\rangle_{k}$ $\displaystyle=\langle
f_{0},g_{0}\rangle\in\mathbb{C}\cong H^{0}(\mathbb{C}P^{2};\mathbb{C})$
$\displaystyle\langle f,g\rangle_{k+1}$ $\displaystyle=\langle
f_{2},g_{2}\rangle\in\mathbb{C}\cong H^{4}(\mathbb{C}P^{2};\mathbb{C})$
$\displaystyle\langle f,g\rangle_{j}$ $\displaystyle=0\mbox{\ if\ }j\neq
k,k+1.$

The above examples show that, since $\operatorname{{\it
tcf}}^{\operatorname{{\rm wt}}j}_{\mathbb{C}}$ is a rational spectrum, the
topological Petersson product on $\operatorname{{\it
tcf}}^{*}(X)\left[\frac{1}{6}\right]$ is determined on the $n$-skeleton
$\operatorname{{\it tcf}}^{*}(X^{n})\left[\frac{1}{6}\right]$ by the images of
the rational cells in $X^{n}\wedge H\mathbb{Q}$ under the rationalized
diagonal map $X\rightarrow X\times X$, up to elements of higher Atiyah-
Hirzebruch filtration666Recall that the Atiyah-Hirzebruch filtration on
$E^{*}(X)$, for a generalized cohomology theory $E^{*}$ and a CW-complex $X$,
is the decreasing filtration of $E^{*}(X)$ whose $n$th layer $F^{n}$ is the
set of elements of $E^{*}(X)$ whose image under the map $E^{*}(X)\rightarrow
E^{*}(X/X^{n-1})$ is zero. Here $X/X^{n-1}$ is $X$ with its $(n-1)$-skeleton
$X^{n-1}$ collapsed (“pinched”) down to a point. An element of $E^{*}(X)$ has
Atiyah-Hirzebruch degree $n$ if it has Atiyah-Hirzebruch filtration $n$ but
not Atiyah-Hirzebruch filtration $n-1$.; that is, on the associated graded of
the Atiyah-Hirzebruch filtration on $\operatorname{{\it
tcf}}^{*}(X)\otimes_{\mathbb{Z}}\mathbb{C}$, the topological Petersson product
is determined by the cup product on $H^{*}(X;\mathbb{C})$ and the classical
Petersson product. Consequently, we have:

######  Proposition 4.3.

Let $X$ be an unpointed finite CW-complex. Let $k$ be an integer or half-
integer, let $f,g\in\operatorname{{\it
tcf}}^{-2k}(X)\left[\frac{1}{6}\right]$, and let $j$ be an integer. Then
$\langle f,g\rangle_{j}$ is the element of
$H^{4(j-k)}(X;\mathbb{C})\cong\hom_{\mathbb{Q}}(H_{4(j-k)}(X;\mathbb{Q}),\mathbb{C})$
given, up to higher Atiyah-Hirzebruch filtration, by composing the rational
homology coproduct map

$\displaystyle H_{4(j-k)}(X;\mathbb{Q})$
$\displaystyle\rightarrow\coprod_{n}H_{n}(X;\mathbb{Q})\otimes_{\mathbb{Q}}H_{4(j-k)-n}(X;\mathbb{Q})$

with the summand projection

$\displaystyle\coprod_{n}H_{n}(X;\mathbb{Q})\otimes_{\mathbb{Q}}H_{4(j-k)-n}(X;\mathbb{Q})$
$\displaystyle\rightarrow
H_{2(j-k)}(X;\mathbb{Q})\otimes_{\mathbb{Q}}H_{2(j-k)}(X;\mathbb{Q})$

followed by the map

(9) $\displaystyle
H_{2(j-k)}(X;\mathbb{Q})\otimes_{\mathbb{Q}}H_{2(j-k)}(X;\mathbb{Q})$
$\displaystyle\rightarrow\left(S_{j}\otimes_{\mathbb{Z}}\mathbb{Q}\right)\otimes_{\mathbb{Q}}\left(S_{j}\otimes_{\mathbb{Z}}\mathbb{Q}\right)$

given by the chain of isomorphisms

$\displaystyle\operatorname{{\it
tcf}}^{-2k}(X)\otimes_{\mathbb{Z}}\mathbb{C}\cong\coprod_{n}H^{n}(X;\operatorname{{\it
tcf}}^{-n-2k}\otimes_{\mathbb{Z}}\mathbb{C})$
$\displaystyle\cong\coprod_{n}\hom_{\mathbb{Q}}(H_{n}(X;\mathbb{Q}),\operatorname{{\it
tcf}}^{-n-2k}\otimes_{\mathbb{Z}}\mathbb{C})$
$\displaystyle\cong\coprod_{n}\hom_{\mathbb{Q}}(H_{n}(X;\mathbb{Q}),S_{\frac{n}{2}+k}\otimes_{\mathbb{Z}}\mathbb{C}),$

and finally postcomposing (9) with the classical Petersson product map

$\displaystyle\left(S_{j}\otimes_{\mathbb{Z}}\mathbb{Q}\right)\otimes_{\mathbb{Q}}\left(S_{j}\otimes_{\mathbb{Z}}\mathbb{Q}\right)$
$\displaystyle\rightarrow\mathbb{C}.$

We can rephrase Proposition 4.3 as the following method for calculating
$\langle-,-\rangle_{j}$, not on $\operatorname{{\it
tcf}}^{*}(X)\otimes_{\mathbb{Z}}\mathbb{C}$, but only on the associated graded
of the Atiyah-Hirzebruch filtration on $\operatorname{{\it
tcf}}^{*}(X)\otimes_{\mathbb{Z}}\mathbb{C}$. Given elements
$f,g\in\operatorname{{\it tcf}}^{-2k}(X)\otimes_{\mathbb{Z}}\mathbb{C}$, we
use the isomorphism $\operatorname{{\it
tcf}}^{*}(X)\otimes_{\mathbb{Z}}\mathbb{C}\cong\operatorname{{\it
tcf}}^{*}(S^{0})\otimes_{\mathbb{Z}}H^{*}(X;\mathbb{C})$ to express $f$ as a
sum $\sum_{h}f_{h}\otimes x_{h}$ and to express $g$ as a sum
$\sum_{i}g_{i}\otimes y_{i}$, with each $f_{h}$ and each $g_{i}$ a classical
cusp form, and with each $x_{h}$ and $y_{i}$ an element of
$H^{*}(X;\mathbb{C})$. Since we are only calculating $\langle-,-\rangle_{j}$
on the associated graded of the Atiyah-Hirzebruch filtration, we can assume
without loss of generality that $f$ and $g$ are each homogeneous with respect
to Atiyah-Hirzebruch filtration, i.e., that the integer $\left|x_{h}\right|$
is independent of $h$, and the integer $\left|y_{i}\right|$ is independent of
$i$. Then Proposition 4.3 yields the formula

(10) $\displaystyle\langle f,g\rangle_{j}$
$\displaystyle\equiv\sum_{(h,i):\left|x_{h}\right|=\left|y_{i}\right|=2(j-k)}\langle
f_{h},g_{i}\rangle\cdot(x_{h}\cup y_{i})$

modulo terms of greater Atiyah-Hirzebruch filtration. The form
$\langle-,-\rangle$ on the right-hand side of (10) is the classical Petersson
product of cusp forms, while $x_{h}\cup y_{i}$ is the usual cup product in
$H^{*}(X;\mathbb{C})$.

######  Corollary 4.4.

Let $X$ be a finite CW-complex such that $H^{m}(X;\mathbb{Q})$ is nonzero for
some positive integer $m$. Then there exists some integer $n$ such that, for
every integer $j$, the topological Petersson product of weight $j$ is
degenerate on $\operatorname{{\it tcf}}^{n}(X)\otimes_{\mathbb{Z}}\mathbb{C}$.

###### Proof.

Let $d$ be the greatest integer $m$ such that $H^{m}(X;\mathbb{Q})$ is
nonzero. Then $d>0$, so for degree reasons, there exists no element $y\in
H^{d}(X;\mathbb{Q})$ such that $x\cup y\neq 0$. Let $f$ be any nonzero
classical cusp form, of any weight $k$. Then

$0\neq f\otimes x\in\operatorname{{\it
tcf}}^{-2k}(S^{0})\otimes_{\mathbb{Z}}H^{d}(X;\mathbb{Q})\otimes_{\mathbb{Z}}\mathbb{C}\cong\operatorname{{\it
tcf}}^{d-2k}(X)\otimes_{\mathbb{Z}}\mathbb{C},$

and by formula (10) and the triviality of $x\cup y$ for all $y\in
H^{d}(X;\mathbb{Q})$, we have that $\langle f,g\rangle_{j}$ is zero modulo
terms of greater Atiyah-Hirzebruch filtration, for any $g$. But since $d$ is
the highest degree in which $H^{*}(X;\mathbb{C})$ vanishes, $f$ is already in
the highest possible Atiyah-Hirzebruch filtration. So $\langle f,g\rangle_{j}$
is zero for all $g$. So $\langle-,-\rangle_{j}$ is degenerate on
$\operatorname{{\it tcf}}^{d-2k}(X)\otimes_{\mathbb{Z}}\mathbb{C}$. ∎

A consequence of formula (10) is that the weight $j$ topological Petersson
product $\langle-,-\rangle_{j}$ vanishes on the weight $k$ cusp forms
supported on cells of dimension $>2(j-k)$. In light of that consequence, we
make the following definition of the topological Petersson product
simpliciter, rather than the topological Petersson product of a particular
weight:

######  Definition 4.5.

Let $X$ be a finite CW-complex. By the topological Petersson product on
$\operatorname{{\it tcf}}^{*}(X)\otimes_{\mathbb{Z}}\mathbb{C}$ we mean the
$\mathbb{R}$-bilinear form

$\displaystyle\left(\operatorname{{\it
tcf}}^{*}(X)\otimes_{\mathbb{Z}}\mathbb{C}\right)\otimes_{\mathbb{R}}\left(\operatorname{{\it
tcf}}^{*}(X)\otimes_{\mathbb{Z}}\mathbb{C}\right)$
$\displaystyle\rightarrow\mathbb{C}$

given on the elements of $\operatorname{{\it
tcf}}^{i}(X)\otimes_{\mathbb{Z}}\mathbb{C}$ of Atiyah-Hirzebruch degree $j$ by
the weight $(j-i)/2$ topological Petersson product
$\langle-,-\rangle_{(j-i)/2}$. (If $j-i$ is odd, then this product is defined
to be zero.) We write $\langle-,-\rangle_{X}$ for the topological Petersson
product on the space $X$.

######  Example 4.6.

In the case of the complex projective plane, collapse of the Atiyah-Hirzebruch
spectral sequence gives us that the complexified $tcf$-cohomology
$\operatorname{{\it tcf}}^{-*}(\mathbb{C}P^{2})\otimes_{\mathbb{Z}}\mathbb{C}$
is isomorphic to the direct sum of:

  * •

a copy of $S_{*}\otimes_{\mathbb{Z}}\mathbb{C}$ supported on the $0$-cell of
$\mathbb{C}P^{2}$,

  * •

a copy of $S_{2+*}\otimes_{\mathbb{Z}}\mathbb{C}$ supported on the $2$-cell of
$\mathbb{C}P^{2}$,

  * •

and a copy of $S_{4+*}\otimes_{\mathbb{Z}}\mathbb{C}$ supported on the
$4$-cell of $\mathbb{C}P^{2}$.

As in Examples 4.2, an element $f$ of $\operatorname{{\it
tcf}}^{-2k}(\mathbb{C}P^{2})\otimes_{\mathbb{Z}}\mathbb{C}$ is given by a
triple $(f_{k},f_{k+1},f_{k+2})$, where $f_{k},f_{k+1},f_{k+2}$ are weight
$k,k+1$, and $k+2$ classical cusp forms, respectively. These have Atiyah-
Hirzebruch filtrations $0,2,$ and $4$ respectively: this is simply a matter of
reading off the dimensions of the cells on which $f_{k},f_{k+1},$ and
$f_{k+2}$ are supported. Consequently it follows from Examples 4.2 and
Definition 4.5 that the topological Petersson product $\langle
f,g\rangle_{\mathbb{C}P^{2}}$ is the nonhomogeneous element $\langle
f_{k},g_{k}\rangle+\langle f_{k+1},g_{k+1}\rangle\in
H^{*}(\mathbb{C}P^{2};\mathbb{C})$. Here $\langle f_{k},g_{k}\rangle$ is the
classical Petersson product of $f_{k}$ and $g_{k}$, regarded as an element of
$\mathbb{C}\cong H^{0}(\mathbb{C}P^{2};\mathbb{C})$, while $\langle
f_{k+1},g_{k+1}\rangle$ is the classical Petersson product of $f_{k+1}$ and
$g_{k+1}$, regarded as an element of $\mathbb{C}\cong
H^{4}(\mathbb{C}P^{2};\mathbb{C})$.

Note that the topological Petersson product over $\mathbb{C}P^{2}$ is
degenerate, since it is insensitive to the part of the topological cusp form
supported on the top cell in $\mathbb{C}P^{2}$, a phenomenon we noticed
already in Examples 4.2.

######  Theorem 4.7.

Let $X$ be a compact Kähler manifold of complex dimension $d$. Write
$\operatorname{{\it tcf}}^{*}_{\mathbb{C}}(X)^{\leq 2d/3}$ for the
sub-$\mathbb{C}$-vector space of $\operatorname{{\it
tcf}}^{*}(X)\otimes_{\mathbb{Z}}\mathbb{C}$ spanned by the elements of Atiyah-
Hirzebruch degree $n$ such that $n$ is even and $n\leq 2d/3$. Then the
topological Petersson product is a nondegenerate $\mathbb{R}$-bilinear form on
$\operatorname{{\it tcf}}^{*}_{\mathbb{C}}(X)^{\leq 2d/3}$.

###### Proof.

Let $\omega$ be a Kähler $2$-form on $X$. Given an integer $n$ such that
$n<d$, the hard Lefschetz theorem gives us that multiplication by
$[\omega]^{d-n}$ yields an isomorphism $H^{n}(X;\mathbb{C})\rightarrow
H^{2d-n}(X;\mathbb{C})$. Since we assume $n\leq 2d/3$, we have $2n\leq 2d-n$,
and consequently multiplication by $[\omega]^{d-n}$ factors through the
multiplication by $[\omega]^{n/2}$ map

(11) $\displaystyle H^{n}(X;\mathbb{C})$ $\displaystyle\rightarrow
H^{2n}(X;\mathbb{C}).$

. Since an isomorphism factors through (11), we have that (11) is injective.
Now given a nonzero element

$\sum_{h}f_{h}\otimes x_{h}\in\operatorname{{\it
tcf}}^{*}(S^{0})\otimes_{\mathbb{Z}}H^{*}(X;\mathbb{C})\cong\operatorname{{\it
tcf}}^{*}(X)\otimes_{\mathbb{Z}}\mathbb{C}$

with the elements $\\{x_{h}\\}\subseteq H^{*}(X;\mathbb{C})$
$\mathbb{C}$-linearly independent, choose, for each $h$, a classical cusp form
$g_{h}\in\operatorname{{\it tcf}}^{*}(S^{0})$ such that the classical
Petersson product $\langle f_{h},g_{h}\rangle$ is nonzero. This is possible
since the classical Petersson product is nondegenerate. Let
$g=\sum_{h}g_{h}\otimes[\omega]^{n/2}\in\operatorname{{\it
tcf}}^{*}(S^{0})\otimes_{\mathbb{Z}}H^{*}(X;\mathbb{C})\cong\operatorname{{\it
tcf}}^{*}(X)\otimes_{\mathbb{Z}}\mathbb{C}$. Formula (10) yields

$\displaystyle\langle f,g\rangle_{X}$ $\displaystyle\equiv\sum_{h}\langle
f_{h},g_{h}\rangle\cdot(x_{h}\cup[\omega]^{n/2})\neq 0,$

since each $\langle f_{h},g_{h}\rangle$ is nonzero and since the elements
$\\{x_{h}\cup[\omega]^{n/2}\\}\subseteq H^{2n}(X;\mathbb{C})$ are
$\mathbb{C}$-linearly independent due to the hard Lefschetz theorem. ∎

## References

  * [Arl92] Dominique Arlettaz.  The order of the differentials in the Atiyah-Hirzebruch spectral sequence.  $K$-Theory, 6(4):347–361, 1992.
  * [Bak90] Andrew Baker.  Hecke operators as operations in elliptic cohomology.  J. Pure Appl. Algebra, 63(1):1–11, 1990.
  * [CS22] L. Candelori and A. Salch.  Topological Hecke eigenforms, 2022.  available on arXiv.
  * [Dol66] Albrecht Dold.  Halbexakte Homotopiefunktoren, volume 12 of Lecture Notes in Mathematics.  Springer-Verlag, Berlin-New York, 1966.
  * [Mov15] Hossein Movasati.  Quasi-modular forms attached to elliptic curves: Hecke operators.  J. Number Theory, 157:424–441, 2015.


# A new search direction for full-Newton step infeasible interior-point method
in linear optimization

B. Kheirfam
Department of Applied Mathematics
Azarbaijan Shahid Madani University
Tabriz, Iran
<EMAIL_ADDRESS>
###### Abstract

In this paper, we study an infeasible interior-point method for linear
optimization with full-Newton step. The introduced method uses an algebraic
equivalent transformation on the centering equation of the system which
defines the central path. We prove that the method finds an
$\varepsilon$-optimal solution of the underlying problem in polynomial time.

keywords: Linear optimization, infeasible interior-point methods, new search
directions, polynomial complexity.

AMS: 90C05, 90C51.

##  1 Introduction

Interior-point methods (IPMs) for linear optimization (LO) began when
Karmarkar [6] published his exceptional paper in 1984\. After that, several
variants of this algorithm were presented. In the meantime, we can talk about
feasible and infeasible IPMs. In feasible IPMs we presume that a strictly
feasible point is at hand which the algorithm can be immediately beginning.
Usually find such a starting point is not simple. In that case an infeasible
IPM (IIPM) should be used. These methods begin from an arbitrary positive
point and try to reach both feasibility and optimality. IIPMs were first
introduced by Lustig [18] and Tanabe [22]. The first feasible IPM with full-
Newton step for LO was presented by Roos et al. [21]. Determining the search
directions plays a very important role in IPMs. In 2003, Darvay [2] utilizes
the AET technique on the centering equation of the system defining the central
path for LO. He uses the square root function in the AET strategy and then
applies the Newton method for obtain the search directions. This method is
extended in [1, 23, 24, 25], respectively, to convex quadratic optimization
(CQO), second-order cone optimization (SOCO), symmetric optimization (SO) and
the Cartesian $P_{*}(\kappa)$ linear complementarity problem (LCP). Kheirfam
and Haghighi [16] have proposed an IPM for $P_{*}(\kappa)$-LCP which uses the
function $\psi(t)=\sqrt{t}/2(1+\sqrt{t})$ in the AET technique. An infeasible
version of the method proposed in [21] has presented by Roos in [19] which
needs a feasibility step and three centering steps in each main iteration.
Some generalizations and versions of the method can be seen in [15, 8, 9, 17,
27, 10]. The author is improved this algorithm so that the algorithm performs
only one feasibility step in each iteration and does not need centering steps
[20]. Kheirfam [11, 12, 13, 14] extended the algorithm proposed in [20] to
HLCP, the Cartesian $P_{*}(\kappa)$-LCP, the convex quadratic symmetric cone
optimization (CQSCO) and SO. By considering the AET technique based on the
function $\psi(t)=t-\sqrt{t}$, Darvay et al. [3] have introduced a full-Newton
step IPM for LO. Kheirfam [7] has presented an infeasible version of this
algorithm for SDLCP. Darvay et al. [5] published a corrector-predictor IPM
(CP-IPM) for LO using the function $\psi(t)=t-\sqrt{t}$ for AET. Darvay and
Takács [4] proposed an IPM for LO based on a new type of AET on the centering
equation of the central path.

Motivated by the aforementioned works, in this paper we aim to present a full-
Newton step IIPM for LO using the AET $\psi(t)=\psi(\sqrt{t})$ for the
centering equation of the central path. The method uses the function
$\psi(t)=t^{2}$ in order to determine the new search directions and performs
only one feasibility step in a main iteration. In fact, our method is an
infeasible version of the method proposed in [4]. We prove that the proposed
algorithm enjoys the best-known iteration complexity for IIPMs.

The paper is organized in the following way. In the next section, we remember
the problem pair (P) and (D). We state the perturbed problems corresponding to
(P) and (D) and then provided the central path. In Sect. 3, the new search
directions based on the new type of AET using the function $\psi(t)=t^{2}$ is
discussed, and finally the algorithm is presented. Section 4 consists of the
complexity analysis of the introduced IIPM with the new search directions. In
Section 5, some concluding remarks are followed.

##  2 Preliminaries

Let us consider the LO problem in the standard form

$\displaystyle(P)~{}~{}~{}~{}~{}~{}~{}~{}~{}\min~{}\\{c^{T}x:~{}Ax=b,~{}x\geq
0\\},~{}~{}~{}~{}~{}$

where $A\in{R}^{m\times n}$ with $rank(A)=m,b\in R^{m}$ and $c\in R^{n}$. The
dual of this problem can be written in the following standard form:

$\displaystyle(D)~{}~{}~{}~{}~{}~{}~{}~{}~{}\max~{}\\{b^{T}y:~{}A^{T}y+s=c,~{}s\geq
0\\}.$

In accordance with the routine of IIPMs, we consider the starting point
$(x^{0},y^{0},s^{0})=\xi(e,0,e)$ such that $\|(x^{*};s^{*})\|_{\infty}\leq\xi$
for some primal-dual optimal solution $(x^{*},y^{*},s^{*})$, where $e$ is the
all-one vector and $\xi$ is a positive scalar. It should be noted that for the
optimal solution $(x^{*},y^{*},s^{*})$ the inequality
$\|(x^{*};s^{*})\|_{\infty}\leq\xi$ is true if and only if

$\displaystyle 0\leq x^{*}\leq\xi e,~{}~{}0\leq s^{*}\leq\xi e.$ (1)

For an IIPM, a triple $(x,y,s)$ is called an $\varepsilon$-solution of (P) and
(D) if

$\max\big{\\{}x^{T}s,\|b-Ax\|,\|c-A^{T}y-s\|\big{\\}}\leq\epsilon,$

where $\varepsilon$ is a accuracy parameter. Following [19], for any
$0<\nu\leq 1$ we consider the perturbed problem pair ${\rm(P_{\nu})}$ and
${\rm(D_{\nu})}$ as follows:

$\displaystyle(P_{\nu})~{}~{}~{}~{}~{}~{}~{}~{}~{}\min~{}\\{(c-\nu
r^{0}_{c})^{T}x:~{}b-Ax=\nu r_{b}^{0},~{}x\geq 0\\},~{}~{}~{}~{}~{}$
$\displaystyle(D_{\nu})~{}~{}~{}~{}~{}~{}~{}~{}~{}\max~{}\\{(b-\nu
r^{0}_{b})^{T}y:~{}c-A^{T}y-s=\nu r^{0}_{c},~{}s\geq 0\\},$

where $r^{0}_{b}:=b-A\xi e$ and $r^{0}_{c}:=c-\xi e.$ It is simply seen that
$(x^{0},y^{0},s^{0})=\xi(e,0,e)$ is a feasible solution of the problem pair
${\rm(P_{\nu})}$ and ${\rm(D_{\nu})}$ if $\nu=1$. We conclude that if $\nu=1$,
then ${\rm(P_{\nu})}$ and ${\rm(D_{\nu})}$ satisfy the interior point
condition (IPC). We recall the following lemma.

######  Lemma 1.

(Theorem 5.13 in [26]) The original problems, (P) and (D) are feasible if and
only if for each $\nu$ satisfying $0<\nu\leq 1$ the perturbed problems
${\rm(P_{\nu})}$ and ${\rm(D_{\nu})}$ satisfy the IPC.

In the view of Lemma 1, we assume that the original problem pair (P) and (D)
is feasible and $\nu\in(0,1]$, the central path of the perturbed pair
${\rm(P_{\nu})}$ and ${\rm(D_{\nu})}$ exists;

$\displaystyle\begin{array}[]{ccccccc}b-Ax=\nu r^{0}_{b},~{}~{}x\geq 0,\\\
c-A^{T}y-s=\nu r^{0}_{c},~{}~{}s\geq 0,{}{}{}{}{}{}{}\\\ xs=\mu
e,{}{}{}{}{}{}\end{array}$ (5)

has a unique solution $(x(\mu,\nu),y(\mu,\nu),s(\mu,\nu))$ for every $\mu>0$.
This solution consists of the $\mu$-centers of the perturbed problems
${\rm(P_{\nu})}$ and ${\rm(D_{\nu})}$. Note that for $x,s>0$ and $\mu>0$ from
the third equation of system (5) we deduce that

$\displaystyle xs=\mu
e\Leftrightarrow\frac{xs}{\mu}=e\Leftrightarrow\sqrt{\frac{xs}{\mu}}=e\Leftrightarrow\frac{xs}{\mu}=\sqrt{\frac{xs}{\mu}}.$
(6)

Now the perturbed central path can be equivalently stated as follows:

$\displaystyle\begin{array}[]{ccccccc}b-Ax=\nu r^{0}_{b},~{}~{}x\geq 0,\\\
c-A^{T}y-s=\nu r^{0}_{c},~{}~{}s\geq 0,{}{}{}{}{}{}{}\\\
\dfrac{xs}{\mu}=\sqrt{\dfrac{xs}{\mu}}.{}{}{}\end{array}$ (10)

In the sequel, the parameters $\mu$ and $\nu$ always satisfy the relation
$\mu=\nu\mu^{0}=\nu\xi^{2}$.

##  3 New search directions

In accordance with the Darvay’s idea, we consider the function $\psi$ defined
and continuously differentiable on the interval $(k^{2},\infty)$, where $0\leq
k<1$, such that $2t\psi^{{}^{\prime}}(t^{2})-\psi^{{}^{\prime}}(t)>0,\forall
t>k^{2}$. Now, if we apply the AET method to (10), then we get

$\displaystyle b-Ax=\nu r^{0}_{b},~{}~{}x\geq 0,$ (11) $\displaystyle
c-A^{T}y-s=\nu r^{0}_{c},~{}~{}s\geq 0,$ (12)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\psi\Big{(}\dfrac{xs}{\mu}\Big{)}=\psi\Big{(}\sqrt{\dfrac{xs}{\mu}}\Big{)}.~{}~{}~{}$
(13)

Let $(x,y,s)$ be a feasible solution of the perturbed pair ${\rm(P_{\nu})}$
and ${\rm(D_{\nu})}$. We consider the notation

$f(x,y,s)=\begin{bmatrix}\nu^{+}r^{0}_{b}-b-Ax\\\
\nu^{+}r^{0}_{c}-c+A^{T}y+s\\\
\psi\Big{(}\dfrac{xs}{\mu}\Big{)}-\psi\Big{(}\sqrt{\dfrac{xs}{\mu}}\Big{)}\\\
\end{bmatrix}=0,$

where $\nu^{+}=(1-\theta)\nu$ and $\theta\in(0,1)$. Applying Newton’s method
to this system, we get

$J_{f}(x,y,s)\begin{bmatrix}\Delta x\\\ \Delta y\\\ \Delta s\\\
\end{bmatrix}=-f(x,y,s),$

where $J_{f}(x,y,s)$ denotes the Jacobian matrix of $f$ at $(x,y,s)$. After
some computations, we obtain the following system:

$\displaystyle\begin{array}[]{ccccccc}A\Delta x=\theta\nu
r^{0}_{b},{}{}{}{}{}{}{}{}{}{}\\\ A^{T}\Delta y+\Delta s=\theta\nu
r^{0}_{c},{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}\\\
\frac{1}{\mu}\big{(}s\Delta x+x\Delta
s\big{)}=\dfrac{-\psi(\frac{xs}{\mu})+\psi(\sqrt{\frac{xs}{\mu}})}{\psi^{{}^{\prime}}(\frac{xs}{\mu})-\frac{1}{2\sqrt{\frac{xs}{\mu}}}\psi^{{}^{\prime}}(\sqrt{\frac{xs}{\mu}})}.\end{array}$
(17)

Defining the scaled search directions

$\displaystyle d_{x}:=\frac{v\Delta x}{x},~{}~{}d_{s}:=\frac{v\Delta
s}{s},~{}~{}{\rm where}~{}v=\sqrt{\frac{xs}{\mu}},$ (18)

we can give the scaled form of system (17):

$\displaystyle\begin{array}[]{ccccccc}{\bar{A}}d_{x}=\theta\nu
r^{0}_{b},{}{}{}{}{}{}\\\ {\bar{A}}^{T}\frac{\Delta y}{\mu}+d_{s}=\theta\nu
vs^{-1}r^{0}_{c},{}{}{}{}{}{}{}{}{}\\\
d_{x}+d_{s}=p_{v},{}{}{}{}{}{}{}{}{}{}{}{}{}\end{array}$ (22)

where

$p_{v}:=\frac{2\psi(v)-2\psi(v^{2})}{2v\psi^{{}^{\prime}}(v^{2})-\psi^{{}^{\prime}}(v)},~{}~{}{\rm
and}~{}~{}\bar{A}:=A{\rm diag}\big{(}\frac{x}{v}\big{)}.$

If we use the function
$\psi:(\frac{1}{\sqrt{2}},\infty)\rightarrow\mathbb{R},\psi(t)=t^{2}$
introduced in [4], then we obtain

$\displaystyle p_{v}=\frac{v-v^{3}}{2v^{2}-e}.$ (23)

After a full-Newton step, the new iterate is given by

$\displaystyle x_{+}:=x+\Delta x,~{}~{}~{}y_{+}:=y+\Delta
y,~{}~{}~{}s_{+}:=s+\Delta s.$ (24)

Furthermore, in each iteration of the algorithm, a quantity is needed to
measure how far an iterate is from the central path. We consider the proximity
measure defined by

$\displaystyle\delta(v):=\delta(x,s;\mu)=\frac{\|p_{v}\|}{2}=\frac{1}{2}\Big{\|}\frac{v-v^{3}}{2v^{2}-e}\Big{\|},$
(25)

which was first suggested for a feasible IPM in [4].

Let $q_{v}=d_{x}-d_{s}$. Then

$\displaystyle
d_{x}=\frac{p_{v}+q_{v}}{2},~{}~{}~{}d_{s}=\frac{p_{v}-q_{v}}{2},~{}~{}~{}d_{x}d_{s}=\frac{p_{v}^{2}-q_{v}^{2}}{4},$
(26)

and

$\displaystyle\frac{\|q_{v}\|^{2}}{4}=\frac{\|d_{x}-d_{s}\|^{2}}{4}=\frac{\|d_{x}+d_{s}\|^{2}}{4}-d_{x}^{T}d_{s}=\frac{\|p_{v}\|^{2}}{4}-d_{x}^{T}d_{s}.$
(27)

Suppose that for some $\mu\in(0,\mu^{0}]$, our algorithm begins from a
feasible solution $(x,y,s)$ of the problem pair ${\rm(P_{\nu})}$ and
${\rm(D_{\nu})}$ with $\nu=\frac{\mu}{\mu^{0}}$, and such that
$\delta(x,s;\mu)\leq\tau,\tau\in(0,1)$. Then, the algorithm finds a feasible
solution $(x_{+},y_{+},s_{+})$ of ${\rm(P_{\nu}^{+})}$ and
${\rm(D_{\nu}^{+})}$, where $\nu^{+}=(1-\theta)\nu,\theta\in(0,1)$. In this
case, $\mu$ is decreased to $\mu^{+}=(1-\theta)\mu$ and such that
$\delta(x_{+},s_{+};\mu^{+})\leq\tau$. This procedure is repeated until an
$\varepsilon$-solution is found. We are now in a position to state the
theoretical framework of the infeasible interior-point algorithm as follows:

$\begin{array}[]{cccccc}\hline\cr{\rm{\bf
Algorithm1}:an~{}infeasible~{}interior-point~{}algorithm}\\\
\hline\cr~{}~{}~{}~{}~{}~{}{\bf
Input:{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}\\\
~{}{\rm~{}Accuracy~{}parameter~{}\varepsilon>0};{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}\\\
{\rm~{}barrier~{}update~{}parameter~{}\theta,~{}0<\theta<1;{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}\\\
{\rm~{}threshold~{}parameter~{}}\tau>0.{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}\\\
{\bf
begin{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}\\\
x:=\xi e;~{}y:=0;~{}s:=\xi
e;~{}\mu:=\nu\xi^{2};~{}\nu=1;{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}\\\
{\bf while}~{}\max(x^{T}s,\|r_{b}\|,\|r_{c}\|)>\varepsilon~{}{\bf
do{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}\\\
{\bf
begin{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}\\\
{\rm~{}solve~{}the~{}system~{}(\ref{eq8})~{}and~{}use~{}(\ref{eq7})~{}to~{}obtain}~{}(\Delta
x,\Delta y,\Delta s);\\\ (x,y,s):=(x,y,s)+(\Delta x,\Delta y,\Delta
s);{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}\\\
{\rm~{}update~{}of~{}}\mu~{}{\rm
and}~{}\nu:{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}\\\
\mu:=(1-\theta)\mu;{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}\\\
\nu:=(1-\theta)\nu;{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}\\\
{\bf
end{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}\\\
{\bf
end}.{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}\\\
\hline\cr\end{array}$

##  4 Analysis of the algorithm

Here, we will prove that Algorithm 1 is well-defined. The main goal of our
analysis is to find some values for the parameters $\tau$ and $\theta$ such
that $x_{+}>0$ and $s_{+}>0$, and we have
$\delta(x_{+},s_{+};\mu^{+})\leq\tau.$ In the following section, we obtain an
upper bound for the proximity measure after an iteration of the algorithm.

###  4.1 Upper bound for $\delta(v^{+})$

In the next lemma, we give a condition on the proximity measure which ensures
the feasibility of a full-Newton step. In what follows, we use the notation
$\omega=\frac{1}{2}\big{(}\|d_{x}\|^{2}+\|d_{s}\|^{2}\big{)}.$

######  Lemma 2.

The iterate $(x_{+},y_{+},s_{+})$ with $v>\frac{1}{\sqrt{2}}e$ is strictly
feasible if $\delta(v)^{2}+\omega<1$.

###### Proof.

Let $0\leq\alpha\leq 1$. We define $x(\alpha):=x+\alpha\Delta x$ and
$s(\alpha):=s+\alpha\Delta s.$ Using (18), the third equation of (22) and (26)
one can find

$\displaystyle\frac{x(\alpha)s(\alpha)}{\mu}=\frac{xs}{v^{2}}(v+\alpha
d_{x})(v+\alpha d_{s})=v^{2}+\alpha v(d_{x}+d_{s})+\alpha^{2}d_{x}d_{s}$
$\displaystyle=(1-\alpha)v^{2}+\alpha(v^{2}+vp_{v})+\alpha^{2}\Big{(}\frac{p_{v}^{2}-q_{v}^{2}}{4}\Big{)}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(28)
$\displaystyle\geq(1-\alpha)v^{2}+\alpha^{2}e+\alpha^{2}\frac{p_{v}^{2}}{4}-\alpha^{2}\frac{q_{v}^{2}}{4},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$

where the inequality is due to $\alpha\geq\alpha^{2}$ and the following
inequality:

$\displaystyle
v^{2}+vp_{v}-e=v^{2}+\frac{v^{2}-v^{4}}{2v^{2}-e}-e=\frac{v^{4}}{2v^{2}-e}-e=\frac{(v^{2}-e)^{2}}{2v^{2}-e}\geq
0.$ (29)

The inequality $x(\alpha)s(\alpha)>0$ holds if

$\displaystyle\Big{\|}-\frac{p_{v}^{2}}{4}+\frac{q_{v}^{2}}{4}\Big{\|}_{\infty}\leq\Big{\|}\frac{p_{v}^{2}}{4}\Big{\|}_{\infty}+\Big{\|}\frac{q_{v}^{2}}{4}\Big{\|}_{\infty}\leq\frac{\|p_{v}\|^{2}}{4}+\frac{\|q_{v}\|^{2}}{4}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle={\delta(v)^{2}}-{d_{x}^{T}d_{s}}\leq{\delta(v)^{2}}+\|d_{x}\|\|d_{s}\|\leq\delta(v)^{2}+\omega<1,$

where the equality is due to (27), the third inequality uses from the Cauchy-
Schwarz inequality and the last inequality holds due to the assumption of the
lemma. Thus, $x(\alpha)s(\alpha)>0$, for $0\leq\alpha\leq 1;$ $x(\alpha)$ and
$s(\alpha)$ do not change sign on the interval $[0,1]$. Consequently,
$x(0)=x>0$ and $s(0)=s>0$ yields $x(1)=x_{+}>0$ and $s(1)=s_{+}>0$. Thus, the
proof is completed. ∎

In correspondence to the definition (25), we have

$\delta(v_{+})=\delta(x_{+},s_{+};\mu^{+})=\frac{1}{2}\Big{\|}\frac{v_{+}-v_{+}^{3}}{2v^{2}_{+}-e}\Big{\|},~{}{\rm
where}~{}v_{+}=\sqrt{\frac{x_{+}s_{+}}{\mu^{+}}}.$

######  Lemma 3.

Let $\delta(v)^{2}+\omega<\frac{1}{2}(1-\theta)$ and $v>\frac{1}{\sqrt{2}}e$.
Then, $v_{+}>\frac{1}{\sqrt{2}}e$ and

$\delta(v_{+})\leq\frac{\sqrt{1-\delta(v)^{2}-\omega}\big{(}\theta\sqrt{n}+10\delta(v)^{2}+\omega\big{)}}{2\sqrt{1-\theta}(2(1-\delta(v)^{2}-\omega)-(1-\theta))}.$

###### Proof.

Let $\alpha=1$. Then from (4.1) it follows that

$\displaystyle
v_{+}^{2}=\frac{x_{+}s_{+}}{\mu^{+}}=\frac{v^{2}+vp_{v}+\frac{p_{v}^{2}}{4}-\frac{q_{v}^{2}}{4}}{1-\theta}=\frac{e+\frac{(v^{2}-e)^{2}}{2v^{2}-e}+\frac{p_{v}^{2}}{4}-\frac{q_{v}^{2}}{4}}{1-\theta}$
$\displaystyle=\frac{e+\big{(}\frac{9v^{2}-4e}{v^{2}}\big{)}\frac{p_{v}^{2}}{4}-\frac{q_{v}^{2}}{4}}{1-\theta}\geq\frac{e-\frac{q_{v}^{2}}{4}}{1-\theta},$

where the second equality is due to (29) and the inequality follows from the
fact that $9v^{2}-4e\geq 0.5e>0$. Consequently, we have

$\displaystyle\min(v_{+})\geq\sqrt{\frac{1-\frac{1}{4}\|q_{v}\|^{2}_{\infty}}{1-\theta}}\geq\sqrt{\frac{1-\frac{1}{4}\|q_{v}\|^{2}}{1-\theta}}\geq\sqrt{\frac{1-\delta(v)^{2}-\omega}{1-\theta}},$
(30)

where the last inequality follows from (27) and the Cauchy-Schwarz inequality.

From $\delta(v)^{2}+\omega<\frac{1}{2}(1-\theta)$ it follows that
$\min(v_{+})>\frac{1}{\sqrt{2}}$, hence $v_{+}>\frac{1}{\sqrt{2}}e$. Now, we
have

$\displaystyle\delta(v_{+})=\frac{1}{2}\Big{\|}\frac{v_{+}-v_{+}^{3}}{2v^{2}_{+}-e}\Big{\|}=\frac{1}{2}\Big{\|}\frac{v_{+}}{2v^{2}_{+}-e}\big{(}e-v_{+}^{2}\big{)}\Big{\|}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle\leq\frac{\min(v_{+})}{2(2\min(v_{+})^{2}-1)}\big{\|}e-v_{+}^{2}\big{\|}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle\leq\frac{\sqrt{(1-\theta)(1-\delta(v)^{2}-\omega)}}{2(2(1-\delta(v)^{2}-\omega)-(1-\theta))}\big{\|}e-v_{+}^{2}\big{\|}.~{}~{}~{}~{}~{}$
(31)

On the other hand, one has

$\displaystyle\big{\|}e-v_{+}^{2}\big{\|}=\Big{\|}\frac{e+\big{(}\frac{9v^{2}-4e}{v^{2}}\big{)}\frac{p_{v}^{2}}{4}-\frac{q_{v}^{2}}{4}}{1-\theta}-e\Big{\|}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle\leq\frac{1}{1-\theta}\Big{(}\|\theta
e\|+\Big{\|}\Big{(}\frac{9v^{2}-4e}{v^{2}}\Big{)}\frac{p_{v}^{2}}{4}-\frac{q_{v}^{2}}{4}\Big{\|}\Big{)}$
$\displaystyle\leq\frac{1}{1-\theta}\Big{(}\theta\sqrt{n}+9\frac{\|p_{v}\|^{2}}{4}+\frac{\|q_{v}\|^{2}}{4}\Big{)}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle=\frac{1}{1-\theta}\big{(}\theta\sqrt{n}+10\delta(v)^{2}+\omega\big{)}.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$

Substituting this bound into (4.1) gives us exactly the desired result. Thus,
the proof is completed. ∎

###  4.2 Upper bound for $\omega$

Following [20], let $\mathcal{N}:=\\{\zeta:{\bar{A}}\zeta=0\\}$ denote the
null space of the matrix $\bar{A}$. Then, the
$\\{\zeta:{\bar{A}}\zeta=\theta\nu r^{0}_{b}\\}$ affine space equals
$\mathcal{N}+d_{x}$. Since the row space of $\bar{A}$ is the orthogonal
complement ${\mathcal{N}}^{\perp}$ of $\mathcal{N}$, thus $d_{s}\in\theta\nu
vs^{-1}r^{0}_{c}+{\mathcal{N}}^{\perp}.$ Also note that
$\mathcal{N}\cap{\mathcal{N}}^{\perp}=\\{0\\}$, and the affine spaces
$\mathcal{N}+d_{x}$ and ${\mathcal{N}}^{\perp}+d_{s}$ meet in a unique point
$q$. Applying a similar argument to Lemma 3.4 in [20], we can find

$\displaystyle
2\omega\leq\|q\|^{2}+\bigg{(}\|q\|+\Big{\|}\frac{v-v^{3}}{2v^{2}-e}\Big{\|}\bigg{)}^{2}=\|q\|^{2}+\big{(}\|q\|+2\delta(v)\big{)}^{2}.$
(32)

Again from [20], we have

$\displaystyle\|q\|\leq\frac{\theta\big{(}n+\|v\|^{2}\big{)}}{\min(v)}.$ (33)

By definition (25) , we have

$\displaystyle
2\delta(v)=\Big{\|}\frac{v-v^{3}}{2v^{2}-e}\Big{\|}=\Big{\|}\frac{v^{2}+v}{2v^{2}-e}(e-v)\Big{\|}\geq\frac{1}{2}\big{\|}e-v\big{\|}\geq\frac{1}{2}\big{(}\|v\|-\|e\|\big{)},$

which implies

$\|v\|\leq\|e\|+4\delta(v)=\sqrt{n}+4\delta(v).$

Furthermore, we have

$4\delta(v)\geq\|e-v\|\geq|1-v_{i}|,i=1,\ldots,n.$

This gives $\min(v)\geq 1-4\delta(v)$. Combining these two inequalities with
(33), we will get

$\displaystyle\|q\|\leq\frac{\theta\Big{(}n+\big{(}\sqrt{n}+4\delta(v)\big{)}^{2}\Big{)}}{1-4\delta(v)}.$
(34)

###  4.3 Values for $\theta$ and $\tau$

In this section, we require finding values $\theta$ and $\tau$ such that if
$\delta(v)\leq\tau$ holds, then $\delta(v_{+})\leq\tau$. From Lemma 3, it
suffices to have

$\displaystyle\frac{\sqrt{1-\delta(v)^{2}-\omega}\big{(}\theta\sqrt{n}+10\delta(v)^{2}+\omega\big{)}}{2\sqrt{1-\theta}(2(1-\delta(v)^{2}-\omega)-(1-\theta))}\leq\tau,$
(35)

provided that $\delta(v)^{2}+\omega<\frac{1}{2}(1-\theta)$. One can easily see
the right-hand-side of (34) is monotonically increasing with respect to
$\delta(v)<1$. Hence, invoking $\delta(v)\leq\tau$, we have

$\displaystyle\|q\|\leq\frac{\theta\Big{(}n+\big{(}\sqrt{n}+4\tau\big{)}^{2}\Big{)}}{1-4\tau}.$

By substituting the above result into (32) and using again
$\delta(v)\leq\tau$, we obtain

$\omega\leq\frac{1}{2}\bigg{[}\bigg{(}\frac{\theta\big{(}n+\big{(}\sqrt{n}+4\tau\big{)}^{2}\big{)}}{1-4\tau}\bigg{)}^{2}+\bigg{(}\frac{\theta\big{(}n+\big{(}\sqrt{n}+4\tau\big{)}^{2}\big{)}}{1-4\tau}+2\tau\bigg{)}^{2}\bigg{]}=:f(\tau).$

We claim that

$\displaystyle\chi(t):=\frac{\sqrt{1-t}}{2(1-t)-(1-\theta)},~{}~{}0\leq
t\leq\frac{1}{2}(1-\theta),$ (36)

is increasing. Hence, $0\leq\delta(v)^{2}+\omega\leq\tau^{2}+f(\tau)$ implies
$\chi(\delta(v)^{2}+\omega)\leq\chi(\tau^{2}+f(\tau))$. Therefore,
$\delta(v)^{2}+\omega\leq\frac{1}{2}(1-\theta)$ and (35) will certainly hold
if

$\tau^{2}+f(\tau)\leq\frac{1}{2}(1-\theta),~{}~{}~{}y(\tau):=\frac{\chi(\tau^{2}+f(\tau))(\theta\sqrt{n}+10\tau^{2}+f(\tau))}{2\sqrt{1-\theta}}\leq\tau.$

If we take $\tau=\frac{1}{12}$ and $\theta=\frac{1}{22n},n\geq 4$, then
$\tau^{2}+f(\tau)=0.0645<0.4773\leq\frac{1}{2}(1-\theta)$ and $y(\tau)\leq
0.0827<\frac{1}{12}$. Hence, we may state the following result.

######  Lemma 4.

If $\tau=\frac{1}{12}$ and $\theta=\frac{1}{22n},n\geq 4$, then
$\delta(v)\leq\tau$ implies $\delta(v_{+})\leq\tau.$

###  4.4 Complexity analysis

Lemma 4 establishes the proposed algorithm is well-defined, in the sense that
the property $\delta(x,s;\mu):=\delta(v)\leq\tau$ is maintained in all
iterations.

In each main iteration, both the barrier parameter $\mu$ and the norms of the
residual vectors are reduced by the factor $1-\theta$. Hence, the total number
of main iterations is bounded above by

$\frac{1}{\theta}\log\frac{\max\\{n\xi^{2},\|r_{b}^{0}\|,\|r_{c}^{0}\|\\}}{\varepsilon}.$

Now, we state our main result.

######  Theorem 5.

If (P) and (D) are feasible and $\xi>0$ such that
$\|(x^{*};s^{*})\|_{\infty}\leq\xi$ for some optimal solutions $x^{*}$ of (P)
and $(y^{*},s^{*})$ of (D), then after at most

$22n\log\frac{\max\\{n\xi^{2},\|r_{b}^{0}\|,\|r_{c}^{0}\|\\}}{\epsilon}$

iterations, the algorithm finds an $\epsilon$-optimal solution of (P) and (D).

##  5 Conclusions

The method presented in this paper is a full-Newton step IIPM for LO based on
the AET proposed in [4]. The method is used in each iteration only one
feasibility step. Our method analysis is different from the existing IIPMs
based on the AET because it uses a different AET. The obtained complexity
bound coincides with the current best-known theoretical iteration bound for
IIPMs.

## References

  * [1] Achache, M., A new primal-dual path-following method for convex quadratic programming, Comp. Appl. Math., 25(1), 97-110 (2006).
  * [2] Darvay, Zs., New interior-point algorithms in linear programming, Adv. Model. Optim., 5(1), 51-92 (2003).
  * [3] Darvay, Zs., Papp, I.M., Takács, P.R., Complexity analysis of a full-Newton step interior-point method for linear optimization, Period. Math. Hung., 73, 27-42 (2016).
  * [4] Darvay, Zs., Takács, P.R., New method determining search directions for interior-point algorithms in linear optimization, Optim. Lett., 12, 1099-1116 (2018).
  * [5] Zs. Darvay, Zs., Illés, T., Kheirfam, B., Rigó, P.R., A corrector-predictor interior-point method with new search direction for linear optimization, Cent. Eur. J. Oper. Res., 28(3), 1123-1140 (2020).
  * [6] Karmarkar, N.K, A new polynomial time algorithm for linear programming, Combinatorica, 4, 375-395 (1984).
  * [7] Kheirfam, B., An infeasible interior point method for the monotone SDLCP based on a transformation of the central path, J. Appl. Math. Comput., 57(1), 685-702 (2018).
  * [8] Kheirfam, B., A new complexity analysis for full-Newton step infeasible interior-point algorithm for $P_{*}(\kappa)$-horizontal linear complementarity problems, J. Optim. Theory Appl., 161(3), 853-869 (2014).
  * [9] Kheirfam, B., A full Nesterov-Todd step infeasible interior-point algorithm for symmetric optimization based on a specific kernel function, NACO, 3(4), 601-614 (2013).
  * [10] Kheirfam, B., A new infeasible interior-point method based on Darvay’s technique for symmetric optimization, Ann. Oper. Res., 211(1), 209-224 (2013).
  * [11] Kheirfam, B., An improved full-Newton step $O(n)$ infeasible interior-point method for horizontal linear complementarity problem, Numer. Algorithms, 71(3), 491-503 (2016).
  * [12] Kheirfam, B., A full step infeasible interior-point method for Cartesian $P_{*}(\kappa)$-SCLCP, Optim. Lett., 10(3), 591-603 (2016).
  * [13] Kheirfam, B., An improved and modified infeasible interior-point method for symmetric optimization, Asian-Eur. J. Math., 9(2), 1650059 (13 pages) (2016).
  * [14] Kheirfam, B., An infeasible full-NT step interior point algorithm for ${\rm CQSCO}$, Numer. Algorithms, 74(1), 93-109 (2017).
  * [15] Kheirfam, B., Mahdavi-Amiri, N., A full Nesterov-Todd step infeasible interior-point algorithm for symmetric cone linear complementarity problem, Bull. Iranian Math. Soc., 40(3), 541-564 (2014).
  * [16] Kheirfam, B., Haghighi, M., A full-Newton step feasible interior-point algorithm for $P_{*}(\kappa)$-LCP based on a new search direction, Croat. Oper. Res. Rev., 7(2), 277-290 (2016).
  * [17] Liu, Z., Sun, W., Tian, F., A full-Newton step $O(n)$ infeasible interior-point algorithm for linear programming based on kernel function, Appl. Math. Optim., 60, 237-251 (2009).
  * [18] Lustig, I.J., Feasibility issues in a primal-dual interior-point method for linear programming, Math. Program., 49(1-3), 145-162 (1991).
  * [19] Roos, C., A full-newton step $O(n)$ infeasible interior- point algorithm for linear optimization, SIAM J. Optim., 16(4), 1110-1136 (2006).
  * [20] Roos, C., An improved and simplified full-Newton step $O(n)$ infeasible interior-point method for linear optimization, SIAM J. Optim., 25(1), 102-114 (2015).
  * [21] Roos, C., Terlaky, T., Vial, J-Ph., Theory and Algorithms for Linear Optimization. An Interior-Point Approach, John Wiley and Sons, Chichester, UK, (1997).
  * [22] Tanabe, K., Centered Newton method for linear programming: Interior and ’exterior’ point method (in Janpanese). In: New Methods for Linear Programming. K. Tone (Ed.) 3, pages 98-100, 1990.
  * [23] Wang, G.Q., Bai, Y.Q., A primal-dual path-following interior-point algorithm for second-order cone optimization with full Nesterov-Todd step, Appl. Math. Comput., 215(3), 1047-1061 (2009).
  * [24] Wang, G.Q., Bai, Y.Q., A new full Nesterov-Todd step primal-dual path-following interior-point algorithm for symmetric optimization, J. Optim. Theory Appl., 154(3), 966-985 (2012).
  * [25] Wang, G.Q., Fan, X.J., Zhu, D.T., Wang, D.Z., New complexity analysis of a full-Newton step feasible interior-point algorithm for $P_{*}(\kappa)$-LCP, Optim. Letters, 9(6), 1105-1119 (2015).
  * [26] Ye, Y., Interior point algorithms. Wiley-Interscience Series in Discrete Mathematics and Optimization. John Wiley $\&$ Sons Inc., New York, 1997. Theory and analysis, A Wiley-Interscience Publication.
  * [27] Zhang, L., Sun, L., Xu, Y., Simplified analysis for full-Newton step infeasible interior-point algorithm for semidefinite programming, Optimization, 62(2), 169-191 (2013).