\section{Introduction}
The Large Hadron Collider (LHC) has been in operation since the end of 2009
and has collided  protons at center of mass energies
$\sqrt{s}=0.9$~TeV, 2.76~TeV and 7~TeV and lead nuclei at
$\sqrt{s_{NN}}=2.76$~TeV. One of the experiments at the LHC is ALICE
which focuses on measuring heavy-ion collisions and minimum-bias \pp
collisions. The main goal of the analysis is to study the Quark-Gluon
Plasma (QGP) which is postulated to be created in Pb-Pb interactions. One of
the reasons to study \pp collisions is to provide a ``baseline'' for
the heavy-ion measurements, so direct comparisons between the two
collision types are especially interesting.

The QGP has been observed at lower energies at Relativistic Heavy-Ion
Collider
(RHIC)~\cite{Adams:2005dq,Adcox:2004mh,Back:2004je,Arsene:2004fa} and
possibly also at the Super Proton Synchrotron (SPS) at CERN. One of
the defining properties of the system created in such collisions is
its rapid evolution, which was found to be well described by the
collective models employing a hydrodynamics
framework~\cite{Kisiel:2008ws}. Specifically, strong radial expansion
(radial flow) as well as its elliptic asymmetry (elliptic flow) for
non-central collisions have been observed and explained in the frame
of such models.

In this work we focus on a particular observable, the femtoscopic
correlation function~\cite{Kopylov:1972qw,Kopylov:1974uc}. It uses the width of the
Bose-Einstein enhancement for identical pions at low relative momentum
to measure the size of the region emitting particles. Such
measurements at lower energies in heavy-ion collisions have shown
several features~\cite{Lisa:2005dd}. Firstly, the extracted radii, in three
dimensions in the Longitudinally Co-Moving System (LCMS) scale linearly with
the cube root of the pseudorapidity density $\left <dN_{ch}/d\eta\right
>^{1/3}$. Secondly the radii decrease with increasing pair transverse
momentum, which is interpreted via the ``homogeneity lengths''
mechanism~\cite{Akkelin:1995gh} as a signature of strong collective radial
flow. Both of these effects are expected at the LHC. At the same time
femtoscopic radii have also been extracted in \pp collisions at RHIC.
The scaling between the minimum-bias \pp and heavy-ion data was
observed. We aim to test such scalings in this work in the new energy
regime.

We note that the results for the central heavy-ion collisions have
been published by the ALICE Collaboration~\cite{Aamodt:2011mr}, while some
results for the \pp collisions have been also made
public~\cite{Aamodt:2011kd}. However the preliminary results for the \pp
collisions at $\sqrt{s}=2.76$~TeV are shown for the first time in this
work and the comparison of the \pp dataset for all $\sqrt{s}$ to the
heavy-ion collision data, including the ALICE central collision point,
is also discussed for the first time.

\section{Data analysis}

The analysis presented here was performed at the LHC with the ALICE
detector. In particular the Time Projection Chamber (TPC) was used for
tracking and particle identification, the Inner Tracking System (ITS)
was used for tracking, vertexing and trigger, and the VZERO detectors
were used for triggering and centrality determination in Pb-Pb
collisions~\cite{Aamodt:2008zz}. For the charged kaon analysis the
ALICE Time Of Flight (TOF) detector~\cite{Aamodt:2008zz} was used for
particle identification.  The \pp dataset at $\sqrt{s}=0.9$~TeV has
been collected in May 2010 and consisted of roughly 4 million events
passing the selection criteria. The dataset at 7~TeV has been
collected in June and consisted of roughly 60 million events. The
dataset at 2.76~TeV has been collected in March 2011 and consisted of
roughly 20 million events. The Pb-Pb dataset at $\sqrt{s_{NN}}=2.76$~TeV has
been collected at the end of 2010, the subsample of approximately 16 thousand
events with centrality 0-5\% of the total inelastic cross-section was
selected for the analysis.

\begin{figure}
\centerline{
\includegraphics[width=0.60\textwidth]{alifemto2.fig3.pres.m2k1.eps}
}
\caption{Correlation functions for \pp collisions at three collision
  energies for the same multiplicity and pair momentum range. The
  three panels show the first three non-vanishing components of the
  spherical harmonics decomposition.}
\label{Figtwo}
\end{figure}

A minimum-bias trigger using the combination of the response from the
inner layer of the ITS as well as the VZERO detector was used for \pp
collisions. In this analysis, the events were also required to have an
event vertex reconstructed within 12~cm of the center of the TPC and
to have at least one pion identified. The events were categorized
according to the raw total event multiplicity in $|\eta|<1.2$, the
value for the normalized pseudorapidity density $\left <dN_{ch}/d\eta
\right >$ was also determined for each of the ranges. For Pb-Pb
collisions the events were required to have a vertex reconstructed
within 12~cm of the center of the TPC  and to be in the top 5\%
centrality bin according to the VZERO detector.

The tracks were reconstructed with the TPC (and ITS for the \pp
analysis), and were required to pass standard criteria for a good
tracking fit. Primary particles were selected by requiring that they
point back to the primary vertex within 0.25~cm in the transverse
plane. Using the specific
energy loss in the TPC the particles were identified as pions, with
some electrons and kaons remaining. Particles
within $|\eta|<1.2$ were accepted for \pp analysis and those within
$|\eta|<0.8$ for
Pb-Pb collisions. The $p_{T}$ range was limited by the TPC acceptance and was
from 0.15(0.2)~GeV/c to 2.0~GeV/c for \pp(Pb-Pb) collisions.

We have also performed preliminary analysis with identical
kaons. Charged tracks with $p>0.6$~GeV/$c$ were required to be consistent
with the kaon mass hypothesis in the TOF detector to be accepted. A
small contamination from electrons remained in the sample below this
momentum. The $\mathrm{K}^0_{s}$ were identified using the ``V0'' decay topology
in the TPC from two oppositely charged pions. The resulting purity was
estimated to be 96\%.

For Pb-Pb collisions a specific two-track separation cut was used to
correct for the inefficiency of the reconstruction of two close tracks
in the detector~\cite{Aamodt:2011mr}.

\section{Correlation functions}

\begin{figure}
\centerline{
\includegraphics[width=0.60\textwidth]{alifemto2.fig8v2.qm1.eps}
}
\caption{Comparison of the femtoscopic radii for selected multiplicity
and pair transverse momentum ranges for different collision energies. The
points for different energies and multiplicities have been slightly
shifted in the horizontal direction for visibility.}
\label{Figthree}
\end{figure}

The identified primary pions in each event were combined into pairs to
form the ``signal'' sample $A$. The same was done for pair of pions
where each of them came from a different event to form the
uncorrelated ``background'' sample $B$. The events used for the mixing
had to have similar acceptance, so they were required to have event
vertexes within 1~cm of each other, and be in the same
multiplicity(centrality) range. The correlation functions were
constructed separately for each multiplicity/pair momentum range and
collision energy by dividing $A$ over $B$. Both $A$ and $B$ were created
as functions of the 3 components of the pair relative momentum $q$ in
LCMS, where the pair momentum along
the beam axis vanishes. Simple Cartesian representation was used for
Pb-Pb collisions while for \pp collisions the analysis was also done
in the Spherical Harmonics (SH) decomposition~\cite{Kisiel:2009iw}. An example
of the correlation function is shown in Fig.~\ref{Figtwo}. The first
three non-vanishing SH components are shown; they are enough to
perform the fitting procedure described later.

In Fig.~\ref{Figtwo} the correlation functions are shown for all three
collision energies for the same multiplicity/pair momentum range. They
are all similar, specifically the width of the low-$q$
enhancement is comparable. Some differences in the magnitude of the
enhancement is expected due to slightly different data taking
conditions for the three samples. The correlation functions
already suggest that the emission region size is independent of
collision energy if measured for collisions with same multiplicity.

\section{Emission region sizes}

The correlation functions can be analyzed quantitatively to extract
the sizes of the emitting region. We use the assumption of the
Gaussian shape of the source which gives the following fit function:
\begin{eqnarray}
C(\vec q) &=& N [ (1-\lambda) \\
&+& \lambda K(q_{inv}) (1 +   \exp(-R_{out}^{2} q_{out}^2 - R_{side}^2 q_{side}^2 - R_{long}^2   q_{long}^2 )  ] B(\vec q) \nonumber
\label{eq:cffit}
\end{eqnarray}
where $N$ is the normalization factor, $K$ is the Coulomb wave
function squared averaged over a Gaussian source with radius 1~fm, $q$
is the pair relative momentum, and $R$'s are the emission source
sizes. $B$ is accounting for non-femtoscopic correlations measured in
\pp collisions at the LHC which are described in detail
in~\cite{Aamodt:2011kd}. Both $q$ and $R$ have three components, the
``long'' along the beam axis, ``out'' along the pair
transverse momentum, and ``side'' perpendicular to the other two.

Figure~\ref{Figthree} shows the fitted femtoscopic radii in \pp
collisions as a function of pair momentum for three collision
energies, for selected multiplicity ranges where data from at least two
collision energies is available. At same $k_{T}$ and multiplicity the radii
do not depend on $\sqrt{s}$ which confirms the finding with raw
correlation functions. The radii universally grow with multiplicity
except the highest $k_{T}$ points for $R_{out}$. $R_{long}$ shows
a similar $k_{T}$ dependence at all multiplicities, with the overall
scale growing with multiplicity. $R_{side}$ shows a flat $k_{T}$
dependence in low multiplicity collisions, and develops a negative
slope as  it increases. For $R_{out}$ a similar but more dramatic
behavior is seen, the slope of the $k_{T}$ dependence is especially
strong at larger multiplicities. The difference in behavior between
$R_{out}$ and $R_{side}$ is most prominent in their ratio, which for
high multiplicities goes significantly below 1.0, much lower than any
value observed in heavy-ion collisions. The qualitative behavior of
radii, i.e. their decrease with $k_{T}$, is similar to the one observed
in heavy ions, however the details, especially the strong difference
in the transverse radii behavior, show that the radial flow
interpretation valid in heavy ions is not necessarily applicable to
\pp collisions. More development on the theory side is needed to give a
definite answer.

\section{Multiplicity dependence}

\begin{figure}
\centerline{
\includegraphics[width=0.48\textwidth]{alifemto2.fig9.qm.eps}
\includegraphics[width=0.48\textwidth]{alifemto2.fig10.qm.pbpb.kt3.CuAuPb.AC.from0.eps}
}
\caption{Left plot: the femtoscopic radii as a function of event
  multiplicity in \pp collisions. Lines show linear fits to all data
  for two pair momentum ranges separately. Right plot: the femtoscopic
  radii as a function of event multiplicity, for \pp and the heavy-ion
  collisions at a selected $k_{T}$. Two sets of dashed lines
  represent linear fits done separately to \pp and heavy-ion data, the
  dotted lines show the uncertainty of the fit parameters.}
\label{Figfour}
\end{figure}

In Fig.~\ref{Figfour} the multiplicity dependence of the femtoscopic
radii is shown for the \pp collisions in the left panel. All collision
energies are used, for two selected $k_{T}$ ranges. The lines
represent linear fits to both $k_{T}$ sets, done separately in
each direction. For all fits $\chi^2/N_{dof}$ is below 1.0; for
$R_{out}$ at lowest $k_{T}$ the lowest multiplicity points are not
included in the fit. The monotonic increase of $R_{side}$ and
$R_{long}$ with multiplicity is observed, while for $R_{out}$ the
behavior changes from the increase at low $k_{T}$ to flattening or
decreasing at higher $k_{T}$, which can also be inferred from the radii
seen in Fig.~\ref{Figthree}. Nevertheless a linear scaling for all
femtoscopic radii with multiplicity at collision energies different by
almost an order of magnitude is seen.

In the right panel of Fig.~\ref{Figfour} the linear dependence of the
radii for \pp collisions is compared to the similar dependence from
heavy-ion collisions, including the most recent results from the ALICE
Collaboration~\cite{Aamodt:2011mr}, which extends the reach in
$\left <dN_{ch}/d\eta\right >^{1/3}$ by 50\%. The linear fit is done
separately to heavy ion data, without including the \pp points. The
quality of the fit is worse than for \pp collisions, in fact fits to central
data only gives significantly different dependence. Regardless of
which subset of data is fitted, the heavy-ion radii do not scale in
the same way than the \pp ones, in all directions. We have checked other
$k_{T}$ ranges and the same discrepancy is seen there. We conclude
that the ``universal'' scaling of femtoscopic radii with final state
multiplicity is violated by the \pp data. Such strong statement is
possible for the first time with ALICE data, because in collisions at
$\sqrt{s}=7$~TeV multiplicities comparable to the peripheral heavy-ion
collisions are produced, so direct comparison without any extrapolation
can be done. It shows that any scaling law must take into account the
initial configuration of the collision. It will be interesting to see
if a scaling law can be proposed, that would accommodate both
datasets.

The ALICE most central datapoint is significantly further in the
horizontal direction than the most central
RHIC Au-Au collision point. It can serve as a very good check of
whether the approximate linear scaling of the radii seen at lower
energies persists at the LHC. $R_{long}$ is following the linear
scaling trend from lower energies, $R_{side}$ is also in agreement,
but at the lower edge of the uncertainty in the scaling
trend. $R_{out}$ however is clearly below the trend set by lower
energies. Such a discrepancy is not foreseen in simple scaling laws, but
can be explained in hydrodynamic models~\cite{Kisiel:2008ws}. It is
therefore important to provide results for other centralities of the
Pb-Pb collisions at the LHC to determine whether the scaling is
recovered at lower multiplicities or if it is a feature of the system
evolution at higher collision energy.

\section{Kaon femtoscopy}

\begin{figure}
\centerline{
\includegraphics[width=0.6\textwidth]{2011-May-16-KK_K0K0_kt_dep5.eps}
}
\caption{Charged and neutral kaon femtoscopy invariant radius as a
  function of   event multiplicity and pair momentum. Horizontal
  error bars correspond to the $k_{T}$ bin size, vertical error bars are
  statistical only.  }
\label{Figfive}
\end{figure}

The preliminary analysis has been performed both for charged and
neutral kaons. The available statistics allowed for 1-dimensional
analysis only, multiplicity and pair momentum binning was
possible. The results are show in Fig.~\ref{Figfive}.
 This is the first ever measurement of $\mathrm{K}^0_{s}$ correlations
in \pp collisions, and the first differential one (previous
measurements for $e^+ e^-$ and Au-Au collisions only produced a single
multiplicity and pair momentum integrated value).

The study of systematic errors is still ongoing; they are presently
estimated to be 20\% for neutral kaons and 30\% for the charged ones. The
kaon analysis extends the transverse momentum range for the radii
by a factor of 3 as compared to pions. The neutral kaon
radii extend the trend seen for identical pions smoothly, while the
charged kaon radii are above the pion ones. The next step in the
analysis is to obtain the expected radii from models of \pp collisions,
especially to to compare them to predictions from models with and
without a collective stage in the evolution of the elementary
collision.

\section{Summary}

We have presented the ALICE results of charged pion, charged kaon and
neutral kaon femtoscopy of \pp collisions at three collision
energies. The pion radii in three dimensions in LCMS scale linearly
with event multiplicity and show a decreasing trend with pair
momentum. In narrow multiplicity and pair momentum ranges they are
independent of the collision energy. The kaon results extend smoothly the
pair momentum dependence to larger pair momenta. The results were
compared to the ones from heavy-ion data, including the most central
Pb-Pb datapoint at $\sqrt{s_{NN}}=2.76$~TeV from ALICE. The \pp data
show a significantly different linear dependence than the heavy-ion
data. In addition deviations from the linear scaling are also observed
for the ALICE Pb-Pb point, in agreement with some predictions from
hydrodynamic models.

\section*{References}


\section{Introduction}\label{sec_I}
\IEEEPARstart{T}{raditional} fixed grid optical networks are not adaptive and therefore, they can not efficiently use systems resources such as power and spectrum according to the conditions of diverse heterogeneous traffic demands. To meet the requirements of constantly increasing and time-variable demands of data traffic, Elastic Optical Networks (EON) are used to adaptively assign routes, bandwidths, modulation levels and transmitted optical powers. Flexible assignment of such a wide range of system resources which are inter-dependent is a complex optimization problem and is conventionally referred to as the Routing and Spectrum Assignment (RSA). Providing fast and efficient RSA algorithms is an important topic of research \cite{chatterjee2015routing, abkenar2017study, sambo2015routing}.

Several variants of optimization formulations and algorithms have been proposed for RSA in EON where system variables are optimally selected such that a cost function (such as spectrum usage or power consumption) is minimized,  physical constraints (such as spectrum continuity, spectrum contiguity and spectrum non-overlapping) are satisfied and pre-defined levels of QoS (which is usually translated to equivalent levels of Optical Signal to Noise Ratio (OSNR)) are guaranteed. Some of the proposed solutions consider the optical fiber as an ideal channel and do not consider the QoS requirements in their optimization analysis \cite{fallahpour2015energy, hadi2016improved, archambault2016routing}. This is an oversimplification of the real world issues and does not yield optimized practical results. Fiber is a non-ideal channel and optical fiber communication requires detailed attention to the channel effects such as Amplified Spontaneous Emission (ASE) noise and NonLinear Interferences (NLI). Optical transmit power is a very important parameters in ensuring proper QoS level. It affects the nonlinear behavior of the fiber, transmission distance, modulation level and spectrum usage. Although some researchers have considered QoS, they do not consider the transmit power as an optimization variable which results in inefficient power allocation because of ignoring one degree of freedom in the optimization \cite{beyranvand2013quality, khodakarami2016quality, khodakarami2014flexible, yan2015link, zhao2015nonlinear}. It has been shown that by optimizing the transmission power per connection, the spectrum usage can be reduced by around $20\%$ \cite{yan2017joint}. Power optimized Routing and Wavelength Assignment (RWA) in nonlinear Wavelength Division Multiplex (WDM) networks has been studied in \cite{roberts2016convex, ives2014physical, ives2014adapting, ives2015routing}. Although these algorithms improve network throughput, they are specifically designed for WDM networks and cannot be applied to EONs with variable spectrum bandwidths and carrier frequency locations. Few research works take into account the interaction between flexible resources, ASE noise and NLIs to optimally allocate available resources, especially transmission power, while satisfying QoS requirements \cite{yan2015resource, yan2017joint}. Yan \textit{et al}.  have studied resource allocation for EONs with nonlinear channel model \cite{yan2015resource}. Their work decomposes resource allocation problem into two sub-problems of 1) Routing/Traffic Ordering (RTO) and 2) Power/Spectrum Assignment (PSA) and provides a Mixed-Integer Nonlinear Program (MINLP) formulation for PSA. Unfortunately, MINLP is an NP-hard problem and such formulation is not a suitable choice for practical resource provisioning in EONs. To overcome the complexity of MINLP formulation, a Mixed Integer Linear Program (MILP) formulation has been proposed in \cite{yan2017joint} where a piecewise linear approximation for nonlinear terms of the conventional MINLP formulation is used. Although their MILP optimization problem can be more efficiently solved but its relative error may be high especially when the number of linear functions in piecewise linear approximation is low. Some researchers have also evaluated the impact of different RTO scenarios on spectrum usage \cite{yan2015resource, zhao2015nonlinear} but to the best of our knowledge, no one has provided a comprehensive investigation of the effect of RTO on NLIs and ASE noise.

A Geometric Program (GP) is a mathematical optimization problem characterized by simple and generalized posynomial objective and constraint functions that have a special form. A geometric program is simply converted to a convex optimization problem and therefore, large-scale GPs are efficiently and reliably solved using advanced convex optimizer software packages and algorithms \cite{boyd2007tutorial, boyd2004convex}. GP is successfully used in QoS-aware communication optimization problems \cite{chiang2005geometric}. Following the approach of \cite{yan2015resource} and many other papers, we have decomposed the resource allocation problem into two interconnected optimization problems, \textit{i}.\textit{e}.  RTO and PSA. In the first problem, we optimally assign routing and traffic orders and in the second problem, we assign spectrum and power parameters. We consider NLI and ASE noise in our formulation and provide posynomial expressions to describe their impact on QoS. We also use few generalized posynomial expressions to relate the value of modulation spectral efficiency to its corresponding minimum required OSNR. Then, we use the proposed posynomial expressions to formulate six GPs for PSA sub-problem which have different accuracy, efficiency and run time. We also consider three RTO procedures which are distinguished by their cost functions. The proposed heuristic provides accurate solutions with a considerable lower complexity in comparison with the conventional MINLP approach. As an example, we have used European Cost239 network topology and evaluated the complexity and error of the proposed algorithms. Results show that our proposed GP formulations for PSA can be more than one order of magnitude faster and have negligible error compared to the MINLP. We have also used simulations to compare the three proposed RTO procedures and show the best candidate.

The rest of the paper is organized as follows. System model is introduced in Section \ref{Sec_II}. We propose our two-stage heuristic for resource allocation in Section \ref{Sec_III}. RTO procedures are discussed in Section \ref{Sec_IV} while we develop our GP formulations for PSA sub-problem in \ref{Sec_V}. Simulation results are included in \ref{Sec_VI}. Finally, we conclude the paper in Section \ref{Sec_VII}.

\textbf{Notation}. Optimization variables are shown in lower case letters. Bold letters are used to denote vectors and sets. We use calligraphic font to show constants. Functions are also shown by capital letters and their arguments are given by lower indexes. Optical fiber characterizing parameters are shown by thier typical notations.

\section{System Model}\label{Sec_II}
Consider a coherent optical communication network characterized by topology graph $G(\mathbf{V}, \mathbf{L})$ where $\mathbf{V}$ and $\mathbf{L}$ are the sets of optical nodes and directional optical fiber links, respectively. The optical fiber bandwidth $\mathcal{B}$ is assumed to be gridless. $\mathbf{Q}$ is the set of connection requests and $\mathbf{Q}_l, l \in \mathbf{L}$ shows the set of connection requests that share optical fiber $l$ on their allocated routes. Each connection request $q \in \mathbf{Q}$ is given a contiguous spectrum bandwidth $\Delta_q$ around carrier frequency $\omega_q$. To facilitate optical switching and remove the high cost of spectrum conversion \cite{spectrum2017hadi}, we assume that the assigned spectrum bandwidth to connection request $q$ is continuous over its routed path. $q$-th connection request passes $\mathcal{N}_q$ fiber spans along its routed path and has $\mathcal{N}_{q,i}$ shared fiber spans with connection request $i\neq q$. There are pre-defined modulation formats where each format has spectral efficiency $c$ and requires minimum OSNR $\Theta(c)$ to get a pre-Forward-Error-Correction (FEC) Bit-Error-Rate (BER) value of $4\times 10^{-3}$, as shown in Tab. \ref{tab:snr_spc} \cite{yan2015resource}. The optical transponder of connection request $q$ is given a modulation format with spectral efficiency $c_q$ and fills its assigned optical bandwidth $\Delta_q$ with optical power $p_q$. Transponders have maximum information bit rate $\mathcal{C}$ and fill both of the polarizations with the same power. Assuming Nyquist spectrum shaping, we clearly have $\Delta_q = \frac{\mathcal{R}_q}{c_q}$ where $\mathcal{R}_q$ is required traffic volume. According to filtering and switching constraints in optical nodes, we consider a guard band $\mathcal{G}$ between any two adjacent connection requests on each link.

\begin{table}[t!]
\centering
\caption{Available modulation formats along with their corresponding spectral efficiency $c$ and minimum required OSNR $\Theta(c)$ to achieve a pre-FEC BER values of $4 \times 10^{-3}$.}\label{tab:snr_spc}
\begin{tabular}{ccc}
\hline
\hline
Modulation Format & Spectral Efficiency $c$ & Minimum Required OSNR $\Theta(c)$\\
\hline
PM-BPSK & 2 & 3.52\\
PM-QPSK & 4 & 7.03\\
PM-8QAM & 6 & 17.59\\
PM-16QAM & 8 & 32.60\\
PM-32QAM & 10 & 64.91\\
PM-64QAM & 12 & 127.51\\
\hline
\hline
\end{tabular}
\end{table}

\section{RTO/PSA Problem}\label{Sec_III}
Solving resource allocation problem, the values of system model variables are determined such that a mixed cost function of transmission optical power and spectrum usage is minimized, physical constrains are satisfied and desired levels of OSNR are guaranteed. In general, such a problem is very hard to solve in reasonable time \cite{yan2015resource}. Therefore, we propose the two-stage heuristic Alg. \ref{alg:rsa} in which the complex resource allocation problem is  decomposed into two sub-problems: 1) RTO, where the routing and the ordering of connection requests on each link are defined, and 2) PSA, where resources such as optical power, spectrum width and carrier frequencies are allocated \cite{yan2015resource} . Usually the search for a near optimal solution involves iterations between these two sub-problems. To save this iteration time, it is of great interest to hold the run time of each sub-problem at its minimum value. In this work, we mainly focus on the second sub-problem which is the most time-consuming one and formulate it as a GP problem to benefit from fast convex optimization algorithms.

In the first stage of Alg. \ref{alg:rsa}, connection request volumes are partitioned such that no traffic volume is greater than transponder information bit rate $\mathcal{C}$. Thus, a high volume connection request is replaced by multiple connection requests that have the same source and destination as the original request but their volumes are no greater than $\mathcal{C}$. At the next step, RTO procedures should be applied. We consider three RTO methods, named Shortest Path Routing (SPR), Shortest Common Path Routing (SCPR) and Shortest Common Path-Rate Routing (SCPRR), which have different cost function and performance. At the next stage of Alg. \ref{alg:rsa}, the problem of PSA is addressed. We propose six Mixed-Integer GP (MIGP) formulations for PSA which are referred to as Geometric PSA (GPSA).  One of the proposed GPSA formulations is selected for the second stage of Alg. \ref{alg:rsa} and its continuous relaxed version is solved in a local loop \cite{boyd2007tutorial}. At each epoch, the relaxed GPSA formulation is optimized and obtained values for relaxed integer variables (here, $c_q$'s) are rounded by precision $\mathcal{J}$. If none of the rounded variables is valid (based on column $c$ in Tab. \ref{tab:snr_spc}), the precision is increased by $\mathcal{J}$ and the rounding is again applied. This process continues to find at least one valid element for relaxed integer variables. Then, we fix the acceptable rounded values and solve the relaxed continuous GPSA again. The local loop continues until all the integer variables have valid and acceptable values. Note that the heuristic needs at most $\abs{\mathbf{Q}}$ iterations (number of integer variables) to evaluate the GPSA formulation before it converges to a solution and in most cases, the number of iterations is less than this maximum. With each iteration of the local loop since some of the variables are fixed, there are fewer variables to be optimized and therefore, the loop runs faster.

In the coming section, we provide a detailed study of the RTO procedures, \textit{i}.\textit{e}.  SPR, SCPR and SCPRR. Then, we focus on PSA and  show how it can be formulated as a GP.

\begin{algorithm}[t!]
\small
\caption{GPSA Heuristic Algorithm}\label{alg:rsa}
\begin{algorithmic}[1]
\Input{network topology, traffic matrix, link parameters, noise and nonlinearity parameters, transponder parameters, rounding precision}
\Output{routes, frequency spectrums, modulation levels, transmission powers, OSNR margins}
\Hline{\hspace{-.6 cm}\hrulefill}
\Statex  \hspace{-4mm}\textbf{Stage 1:}RTO (use one of the SPR, SCPR or SCPRR formulations)
\State partition traffic matrix in module of transponder capacity $\mathcal{C}$;
\State route new traffic requests;
\State order routed traffic requests;
\Statex \hspace{-4mm}\textbf{Stage 2:} PSA (use one of the GPSA1, GPSA2, GPSA3, GPSA4, GPSA5, GPSA6 formulations)
\Do
\State solve continuous relaxed GP to obtain $\mathbf{m}, \mathbf{c}$, $\mathbf{p}$, $\bm{\omega}$;
\State $\mathcal{I}\leftarrow 0$;
\Do
\For{all connection requests $q$}
\For{all spectral efficiency values $c$ of Tab. \ref{tab:snr_spc}}
\If {if $c_q$ falls in $\mathcal{I}$-neighborhood of $c$}
\State round and fix $c_q$ to $c$;
\State \textbf{Break};
\EndIf
\EndFor
\EndFor
\State $\mathcal{I} \leftarrow \mathcal{I}+\mathcal{J}$;
\doWhile{there is no valid spectral efficiency value in $\mathbf{c}$}
\State remove rounded values of $\mathbf{c}$;
\doWhile{there is an inacceptable spectral efficiency value in $\mathbf{c}$}
\end{algorithmic}
\end{algorithm}
\section{RTO Sub-Problem}\label{Sec_IV}
SPR is the most familiar routing algorithm that can be formulated as a Binary Linear Program (BLP):
\begin{align}
&\min_{\mathbf{f}}  \sum\limits_{\substack{q \in \mathbf{Q},l \in \mathbf{L}}}f_{q,l} \mathcal{L}_l \label{eq:routing1_g} \\
&\text{s.t.} \sum\limits_{\substack{l \in \mathbf{L},B_l = S_q}}f_{q,l} = 1, \quad \forall q \in \mathbf{Q} \label{eq:routing1_c1}\\
&\text{\quad} \sum\limits_{\substack{l \in \mathbf{L},E_l = D_q}}f_{q,l} = 1, \quad \forall q \in \mathbf{Q} \label{eq:routing1_c2}\\
&\text{\quad} \sum\limits_{\substack{l \in \mathbf{L},E_l = v\\v \neq S_q, v \neq D_q}}f_{q,l}=\sum\limits_{\substack{l \in \mathbf{L},B_l = v\\v \neq S_q, v \neq D_q}}f_{q,l}, \quad \forall q \in \mathbf{Q}, \forall v \in \mathbf{V} \label{eq:routing1_c3}
\end{align}
where $f_{q,l}$ is a binary variable that is $1$ if connection request $q$ is routed over link $l$ and $0$ otherwise. $S_q$ and $D_q$ denote $q$-th connection request source and destination nodes while $B_l$ and $E_l$ show the begin and end nodes of link $l$. The cost function is the route length where $\mathcal{L}_l$ shows $l$-th link length. Constraints \eqref{eq:routing1_c1}-\eqref{eq:routing1_c2} state that traffic should be added and dropped at its source and destination nodes. Constraint  \eqref{eq:routing1_c3} guarantees route continuity.  SPR is suitable when the communication noise is related to route length such as ASE noise. When fiber nonlinearities are important, the cost function should be revised to address the nonlinear interferences. The number of common links, spectrum width and transmission optical power are the main parameters that affect the amount of NLIs \cite{johannisson2014modeling, poggiolini2014gn}. To simultaneously minimize NLIs and ASE noises, one idea is to add minimization of the number of common links to the SPR goal function:
\begin{align}
\quad \sum\limits_{\substack{q,i \in \mathbf{Q},l \in \mathbf{L}}}f_{q,l}  f_{i,l}\mathcal{L}_l \label{eq:routing2_g}
\end{align}
We refer to Binary Quadratic Programming (BQP) formulation of \eqref{eq:routing2_g} and \eqref{eq:routing1_c1}-\eqref{eq:routing1_c3} as SCPR. Traffic volume $\mathcal{R}_q$ can be considered as an indicator of the NLI power since higher traffic volumes need more spectrum width and higher modulation levels and consequently higher transmitted optical power due to required minimum OSNR. Therefore, in SCPRR the goal function is set to:
\begin{align}
\quad \sum\limits_{\substack{q,i \in \mathbf{Q},l \in \mathbf{L}}}f_{q,l}  f_{i,l}\mathcal{R}_i\mathcal{L}_l \label{eq:routing3_g}
\end{align}
Following each routing algorithm, the request traffics are descendingly ordered according to their corresponding cost function. As a result, for SPR, SCPR and SCPRR, traffics are respectively ordered according to cost functions \eqref{eq:routing1_g}, \eqref{eq:routing2_g} and \eqref{eq:routing3_g}.
\section{PSA Sub-Problem}\label{Sec_V}
As mentioned, we formulate PSA as a GP to save run time and reduce optimization complexity. To develop the GPSA formulation, a posynomial expression for OSNR and a posynomial curve fitting for $(c, \Theta(c))$ pairs are required.
\subsection{Posynomial Expressions}\label{Sec_V_A}
The Gaussian noise model considers the NLIs between channels caused by the Kerr effect as additive Gaussian noise and combines this incoherently with ASE noise due to optical fiber amplifiers. The model holds for polarization-multiplexed coherent systems in which equal-length fiber span losses are compensated by optical amplifiers and there is no inline compensation for chromatic dispersion \cite{johannisson2014modeling, poggiolini2014gn}. Nonlinear cross-channel interference for $q$-th connection request $X_q$ (nonlinear kerr effect of connection request $q$ from other connection requests) is expressed as:
\begin{align}
X_{q} = \varsigma p_q \sum\limits_{\substack{i \in \mathbf{Q}\\i \neq q}}\frac{p_{i}^{2}}{\Delta_i^2}\mathcal{N}_{q,i}\log\abs{\frac{\abs{\omega_q-\omega_i}+\frac{\Delta_i}{2}}{\abs{\omega_q-\omega_i}-\frac{\Delta_i}{2}}}, \forall q \in \mathbf{Q}
\end{align}
$\varsigma$ is a constant coefficient and equal to:
\begin{equation}
\varsigma = \frac{3\gamma^2}{2\alpha\pi\abs{\beta_2}}
\end{equation}
where $\alpha$, $\beta_2$ and $\gamma$ are optical fiber attenuation, dispersion and nonlinear constants. Nonlinear self-channel interference power for $q$-th connection request $Y_q$ (nonlinear kerr effect of connection request $q$ on itself) can also be expressed as:
\begin{align}
Y_{q} = \varsigma\mathcal{N}_q  \frac{p_q^3}{\Delta_q^2} \sinh^{-1}(\iota\Delta_q^2), \forall q \in \mathbf{Q}
\end{align}
where $\iota$ is:
\begin{equation}
\iota = \frac{\pi^2\abs{\beta_2}}{2\alpha}
\end{equation}
In addition to the NLIs, each optical amplifier adds white Gaussian noise with power $E_q$:
\begin{equation}
E_{q} = \zeta \mathcal{N}_{q}\Delta_q,\quad \forall q \in \mathbf{Q}
\end{equation}
$\zeta$ is ASE noise coefficient and defined as:
\begin{equation}
\zeta = (e^{\alpha L}-1)h\nu n_{sp}
\end{equation}
where $L$ is the fiber length per span, $n_{sp}$ is the spontaneous emission factor, $\nu$ is the light frequency, and $h$ is Planck’s constant. Combining the linear and nonlinear noises, $q$-th connection request OSNR is \cite{johannisson2014modeling, poggiolini2014gn}:
\begin{equation}
\Psi_{q} = \frac{p_q}{E_{q}+X_{q}+Y_{q}}, \forall q \in \mathbf{Q}
\end{equation}

To have a posynomial expression for OSNR, we need to provide posynomial approximations for $\log$ and $\sinh^{-1}$ terms in $X_q$ and $Y_q$, respectively. Now consider the following $\log$ function and its two proposed posynomial approximations:
\begin{align}
&\log{\Big(\frac{1+0.5x}{1-0.5x}\Big)}\approx \kappa_1 x, \quad 0 \leqslant x \leqslant 1.2 \label{eq:xci_app1} \\
&\log{\Big(\frac{1+0.5x}{1-0.5x}\Big)}\approx  \kappa_1 x +  \kappa_2x^3, \quad 0 \leqslant x \leqslant 1.2 \label{eq:xci_app2}
\end{align}
where $ \kappa_1=0.4343$ and $ \kappa_2=0.0411$. Fig. \ref{fig:xci} shows the absolute relative error between the offered values of $\log{\Big(\frac{1+0.5x}{1-0.5x}\Big)}$ and its approximations in \eqref{eq:xci_app1} and \eqref{eq:xci_app2}. Clearly, there is a very good match between the function and its approximations for $0 \leqslant x \leqslant 1.2$. Using the proposed posynomial approximations:
\begin{align}\label{eq:xci_app_expr}
X_{q} & = \varsigma p_q \sum\limits_{\substack{i \in \mathbf{Q}\\i \neq q}}\frac{p_{i}^{2}}{\Delta_i^2}\mathcal{N}_{q,i}\log\abs{\frac{\abs{\omega_q-\omega_i}+\frac{\Delta_i}{2}}{\abs{\omega_q-\omega_i}-\frac{\Delta_i}{2}}}\\
\nonumber &= \varsigma p_q \sum\limits_{\substack{i \in Q\\i \neq q}}\frac{p_{i}^{2}}{\Delta_i^2}\mathcal{N}_{q,i}\log\abs{\frac{1+\frac{\Delta_i}{2d_{q,i}}}{1-\frac{\Delta_i}{2d_{q,i}}}}\\
\nonumber &\approx \varsigma p_q \sum\limits_{\substack{i \in Q\\i \neq q}}\frac{p_{i}^{2}}{\Delta_i^2}\mathcal{N}_{q,i}\Big[\frac{\kappa_1\Delta_i}{d_{q,i}}+\frac{\kappa_2\Delta_i^3}{d_{q,i}^3}\Big]\\
\nonumber &\approx \varsigma p_q \sum\limits_{\substack{i \in Q\\i \neq q}}p_{i}^{2}\mathcal{N}_{q,i}\Big[\frac{\kappa_1}{\Delta_i d_{q,i}}+\frac{\kappa_2\Delta_i}{d_{q,i}^3}\Big]\\
\nonumber &\approx \varsigma p_q \sum\limits_{\substack{i \in Q\\i \neq q}}p_{i}^{2}\mathcal{N}_{q,i}\frac{\kappa_1}{\Delta_id_{q,i}}, \forall q \in \mathbf{Q}
\end{align}
\begin{figure}[t!]
\center{\includegraphics[scale=0.45]{Fig/xci.pdf}}
\center{\caption{\label{fig:xci} Relative error between function $\log{\Big(\frac{1+0.5x}{1-0.5x}\Big)}$ and its posynomial approximations in \eqref{eq:xci_app1} and \eqref{eq:xci_app2}.}}
\end{figure}
where $d_{q,i}$ is the distance between carrier frequencies $\omega_q$ and $\omega_i$ and equals to $d_{q,i} = \abs{\omega_q-\omega_i}$. Note that for almost all practical scenarios, we have  $0 \leqslant \frac{\Delta_i}{2d_{q,i}} \leqslant 1.2$. For instance, a typical transponder with $\mathcal{R}_i=\mathcal{C}=100$ Gbps \cite{khodakarami2014flexible} needs at most $\Delta_i=50$ GHz bandwidth when it is assigned the lowest modulation format $c_i=2$. Considering a typical value of $\mathcal{G}=20$ GHz \cite{khodakarami2014flexible}, the maximum value of $\frac{\Delta_i}{d_{q,i}}$ is achieved when channels $q$ and $i$ are adjacent and is upper limited by $\frac{50}{20+50/2}=1.11$. Furthermore for non-adjacent channels, the value of  $\frac{\Delta_i}{d_{q,i}}$ is much lower than $1.11$ which is a good news since the accuracy of the approximations is improved for lower values of $\frac{\Delta_i}{2d_{q,i}}$. Unlink the piecewise linear approximated cross-channel interference expression in \cite{yan2017joint} where its relative error is a function of $p_i$, the relative error of \eqref{eq:xci_app_expr} is independent of $p_i$. Furthermore, for fixed value of $p_i$, the relative error of posynomial cross-channel interference is more than one order of magnitude lower than its corresponding value for linear approximation in \cite{yan2017joint}. As a result, the posynomial approximation provides more accurate and reliable results.

For sufficiently small values of $x$, we have $\sinh^{-1}(x) \approx x$. Thus, for practical values of $\beta_2$, $\alpha$ and $\Delta_q$, we can write:
\begin{align}\label{eq:sci_app_expr}
Y_{q} = \varsigma\mathcal{N}_q  \frac{p_q^3}{\Delta_q^2} \sinh^{-1}(\iota\Delta_q^2)\approx\varsigma\iota\mathcal{N}_qp_q^3
, \forall q \in \mathbf{Q}
\end{align}

The available modulation formats along with their corresponding spectral efficiency $c$ and minimum required OSNR $\Theta(c)$ have been collected in Tab. \ref{tab:snr_spc}. We fit the given $(c,\Theta(c))$ samples with the following posynomial expressions:
\begin{align}
& \Theta(c) \approx \kappa_3c^{\kappa_4}, \quad 2 \leqslant c \leqslant 12 \label{eq:snr_osnr_app1}\\
& \Theta(c) \approx (1+\kappa_5 c)^{\kappa_6}, \quad 2 \leqslant c \leqslant 12 \label{eq:snr_osnr_app2}\\
& \Theta(c) \approx (1+\kappa_5c)^{\kappa_7}, \quad 2 \leqslant c \leqslant 12 \label{eq:snr_osnr_app3}
\end{align}
where $\kappa_3=0.0351$, $\kappa_4=3.292$, $\kappa_5=0.0557$, $\kappa_6=10$ and $\kappa_7=9.4691$. Fig. \ref{fig:snr} show data samples and three posynomial expressions for data curve fitting. The curve fitting \eqref{eq:snr_osnr_app3} has the lowest relative error and the error increases for \eqref{eq:snr_osnr_app2} and \eqref{eq:snr_osnr_app1}, respectively.
\begin{figure}[t!]
\center{\includegraphics[scale=0.45]{Fig/snr.pdf}}
\center{\caption{\label{fig:snr} Different curve fittings \eqref{eq:snr_osnr_app1}, \eqref{eq:snr_osnr_app2} and \eqref{eq:snr_osnr_app3} for spectral efficiency-minimum required OSNR samples given in Tab. \ref{tab:snr_spc}.}}
\end{figure}

\subsection{Problem Formulation}\label{Sec_V_B}
Here we provide a concise review of the MINLP formulation for PSA and then introduce our GPSA counterparts. A MINLP formulation for PSA is as follows \cite{yan2015resource}:
\begin{align}
&\min_{\mathbf{c}, \mathbf{m}, \bm{\omega}, \mathbf{p}, \tau} \quad \mathcal{K}_1\tau+ \mathcal{K}_2\sum\limits_{q\in \mathbf{Q}}p_q+ \mathcal{K}_3 \sum\limits_{q\in \mathbf{Q}}m_q^{-1}\label{eq:nonlinear_g}\\
&\text{s.t.} \quad \Psi_q \geqslant m_q \Theta_q, \quad \forall q \in \mathbf{Q} \label{eq:nonlinear_c1}\\
\nonumber & \text{\quad}\quad \omega_{\Upsilon_{l, j}}+\frac{\Delta_{\Upsilon_{l, j}}}{2} + \mathcal{G}\leqslant \omega_{\Upsilon_{l, j+1}}-\frac{\Delta_{\Upsilon_{l, j+1}}}{2}, \quad \forall l \in \mathbf{L} \\
&\text{\quad} \quad, j = 1, \cdots, \abs{\mathbf{Q}_l}-1 \label{eq:nonlinear_c2}\\
&\text{\quad} \quad \frac{\Delta_q}{2} + \omega_q \leqslant \tau, \quad \forall q \in \mathbf{Q} \label{eq:nonlinear_c3}\\
&\text{\quad} \quad m_q \geqslant \mathcal{M}, \quad \forall q \in \mathbf{Q} \label{eq:nonlinear_c4}\\
&\text{\quad} \quad \tau \leqslant \mathcal{B} \label{eq:nonlinear_c5}
\end{align}
where $\tau$ is the penalty term for occupied frequency upper bound and $\mathcal{M}$ is minimum OSNR margin. $\Upsilon_{l, j}$ is a function that shows which connection request occupies $j$-th channel on link $l$ and its values are determined during RTO sub-problem. Constraint \eqref{eq:nonlinear_c1} is the QoS constraint that forces $q$-th connection request OSNR to be at least  $m_q$ times greater than its required minimum OSNR where $m_q$ is an optimization variable that shows OSNR margin of $q$-th connection request. Constraint \eqref{eq:nonlinear_c2} is nonoverlapping-guard constraint that prevents two connections requests to share a same frequency spectrum. It also guarantees the required guard band between any two adjacent connection requests. Constraint \eqref{eq:nonlinear_c3} compresses the used spectrum to decrease the penalty term for occupied frequency upper bound. Constraint \eqref{eq:nonlinear_c4} holds all OSNR margins greater than its minimum value $\mathcal{M}$ and finally, constraint \eqref{eq:nonlinear_c5} blocks spectrum assignment outside the fiber bandwidth $\mathcal{B}$. The formulation is flexible enough to minimize the maximum bandwidth usage on all the links (to increase remaining resources for more traffic demands), the total transmitted optical power (to reduce nonlinearity in the fibers) and maximize the total SNR margins (to improve the robustness and capacity of the network) \cite{yan2015resource}.

Now, consider the simplest posynomial approximations, \textit{i}.\textit{e}.  \eqref{eq:xci_app1} and \eqref{eq:snr_osnr_app1} and note the following GP:
\begin{align}
&\min_{\mathbf{d}, \mathbf{c}, \bm{\omega}, \mathbf{p}, \tau} \mathcal{K}_1\tau+\mathcal{K}_2\sum\limits_{q\in \mathbf{Q}} p_q+\mathcal{K}_3 \sum\limits_{q\in \mathbf{Q}}m_q^{-1}+\mathcal{K}_4\sum\limits_{\substack{q,i\in \mathbf{Q}\\i \neq q\\ \mathcal{N}_{q,i} \neq 0}}d_{q,i}^{-1} \label{eq:gp_1_g} \\
\nonumber &\text{s.t.} \quad \kappa_3\zeta m_q\mathcal{N}_{q}\mathcal{R}_{q}c_{q}^{\kappa_4-1}p_{q}^{-1}+\kappa_3\varsigma\iota m_q \mathcal{N}_{q}p_{q}^2c_q^{\kappa_4}+\\
& \text{\quad}\quad  \kappa_3\kappa_1 \varsigma m_q c_q^{\kappa_4}\sum\limits_{\substack{i \in \mathbf{Q}\\i \neq q}}p_{i}^{2}\mathcal{N}_{q,i}\mathcal{R}_i^{-1} c_i d_{q,i}^{-1}\leqslant 1, \quad \forall q \in \mathbf{Q} \label{eq:gp_1_c1} \\
\nonumber & \text{\quad} \quad \omega_{\Upsilon_{l, j}}\omega_{\Upsilon_{l, j+1}}^{-1}+0.5\mathcal{R}_{\Upsilon_{l, j}} c_{\Upsilon_{l, j}}^{-1} \omega_{\Upsilon_{l, j+1}}^{-1}+\mathcal{G}\omega_{\Upsilon_{l, j+1}}^{-1}\\
\nonumber & \text{\quad}\quad +0.5\mathcal{R}_{\Upsilon_{l, j+1}} c_{\Upsilon_{l, j+1}}^{-1} \omega_{\Upsilon_{l, j+1}}^{-1}\leqslant 1\\
& \text{\quad}\quad , \quad \forall l \in \mathbf{L}, j = 1, \cdots, \abs{\mathbf{Q}_l}-1 \label{eq:gp_1_c2}  \\
&\text{\quad} \quad 0.5\mathcal{R}_q c_q^{-1}\tau^{-1} + \omega_q\tau^{-1} \leqslant 1, \quad \forall q \in \mathbf{Q} \label{eq:gp_1_c3} \\
& \text{\quad}\quad \mathcal{M} m_q^{-1} \leqslant 1, \quad \forall q \in \mathbf{Q} \label{eq:gp_1_c4}  \\
& \text{\quad}\quad \tau \mathcal{B}^{-1} \leqslant 1 \label{eq:gp_1_c5}  \\
\nonumber &\text{\quad} \quad d_{\Upsilon_{l, i},\Upsilon_{l, j}} \omega_{\Upsilon_{l, i}}^{-1}+ \omega_{\Upsilon_{l, j}}\omega_{\Upsilon_{l, i}}^{-1}\leqslant  1,\quad \forall l \in \mathbf{L} \label{eq:gp_1_c6} \\
&\text{\quad} \quad, j = 1, \cdots, \abs{\mathbf{Q}_l}-1, i = j+1, \cdots, \abs{\mathbf{Q}_l}\\
\nonumber &\text{\quad} \quad d_{\Upsilon_{l, i},\Upsilon_{l, j}}\omega_{\Upsilon_{l, j}}^{-1}+\omega_{\Upsilon_{l, i}}\omega_{\Upsilon_{l, j}}^{-1} \leqslant  1 ,\quad \forall l \in \mathbf{L} \\
&\text{\quad} \quad, j = 2, \cdots, \abs{\mathbf{Q}_l}, i = 1, \cdots, j-1 \label{eq:gp_1_c7}
\end{align}
Ignoring constraints of \eqref{eq:gp_1_c6} and \eqref{eq:gp_1_c7} and the fourth penalty term of the goal function \eqref{eq:gp_1_g}, the above formulation is equivalent GP of the previous MINLP in which posynomial expressions of \eqref{eq:snr_osnr_app1} and \eqref{eq:xci_app1} have been used for QoS constraint \eqref{eq:gp_1_c1}. The added constraints and penalty term are to guarantee the equality of $d_{q,i} = \abs{\omega_q-\omega_i}$. In fact, constraints \eqref{eq:gp_1_c6} and \eqref{eq:gp_1_c7} specifies that $d_{q,i} \leqslant \abs{\omega_q-\omega_i}$ but added posynomial penalty term $k_4\sum d_{q,i}^{-1}$ forces $d_{q,i}$ takes their maximum values, \textit{i}.\textit{e}.  $\abs{\omega_q-\omega_i}$. Also this GP formulation has more variables and constraints compared to its MINLP counterpart but its solution is rapidly achieved using convexification techniques and fast convex optimization algorithms.

To improve the accuracy of the proposed formulation, one can use more accurate posynomial approximations. For \eqref{eq:snr_osnr_app1} and \eqref{eq:xci_app2}, the previous formulation is valid if we replace \eqref{eq:gp_1_c1} with the following:
\begin{align}
\nonumber & \kappa_3\zeta m_q \mathcal{N}_{q}\mathcal{R}_{q}c_{q}^{\kappa_4-1}p_{q}^{-1}+\kappa_3\varsigma\iota m_q \mathcal{N}_{q}p_{q}^2c_q^{\kappa_4}\\
 &  + \kappa_3\kappa_1 \varsigma m_q c_q^{\kappa_4}\sum\limits_{\substack{i \in \mathbf{Q}\\i \neq q}}p_{i}^{2}\mathcal{N}_{q,i}\mathcal{R}_i^{-1} c_i d_{q,i}^{-1} \label{eq:gp_2_c1}\\
\nonumber &  + \kappa_3\kappa_2 \varsigma m_q c_q^{\kappa_4}\sum\limits_{\substack{i \in \mathbf{Q}\\i \neq q}}p_{i}^{2}\mathcal{N}_{qi}\mathcal{R}_ic_i ^{-1} d_{q,i}^{-3}\leqslant 1 , \quad \forall q \in \mathbf{Q}
\end{align}
For posynomial expressions \eqref{eq:snr_osnr_app2} and \eqref{eq:xci_app1}, the constraint \eqref{eq:gp_1_c1}  should be replaced with:
\begin{align}
\nonumber & m_q (1+\kappa_5c_q)^{\kappa_6}\Big[\zeta\mathcal{N}_{q}\mathcal{R}_{q}c_{q}^{-1}p_{q}^{-1}+\varsigma\iota \mathcal{N}_{q}p_{q}^2\\
 &+ \kappa_1\varsigma\sum\limits_{\substack{i \in \mathbf{Q}\\i \neq q}}p_{i}^{2}\mathcal{N}_{q,i}\mathcal{R}_i^{-1} c_i d_{q,i}^{-1}\Big]\leqslant 1, \quad \forall q \in \mathbf{Q} \label{eq:gp_3_c1}
\end{align}
while for approximations \eqref{eq:snr_osnr_app2} and \eqref{eq:xci_app2}:
\begin{align}
\nonumber &m_q(1+\kappa_5c_q)^{\kappa_6}\Big[\zeta\mathcal{N}_{q}\mathcal{R}_{q}c_{q}^{-1}p_{q}^{-1}+\varsigma\iota \mathcal{N}_{q}p_{q}^2\\
\nonumber &+ \kappa_1 \varsigma\sum\limits_{\substack{i \in \mathbf{Q}\\i \neq q}}p_{i}^{2}\mathcal{N}_{q,i}\mathcal{R}_i^{-1} c_i d_{q,i}^{-1}\\
&  + \kappa_2 \varsigma \sum\limits_{\substack{i \in \mathbf{Q}\\i \neq q}}p_{i}^{2}\mathcal{N}_{q,i}\mathcal{R}_ic_i ^{-1} d_{q,i}^{-3}\Big]\leqslant 1, \quad \forall q \in \mathbf{Q} \label{eq:gp_4_c1}
\end{align}
Note that in \eqref{eq:gp_3_c1} and \eqref{eq:gp_4_c1}, Newton binomial expansion of the term $(1+\kappa_5c_q)^{\kappa_6}$ should be multiplied by the remaining posynomial terms and the result is clearly posynomial. This fact is not true for \eqref{eq:snr_osnr_app3} in which $\kappa_7$ is not a natural number. Therefore, auxiliary variable $t_q$'s should be used to change the non-posynomial multiplier of $(1+\kappa_5c_q)^{\kappa_7}$ to a posynomial expression \cite{boyd2007tutorial}. Consequently for \eqref{eq:snr_osnr_app3} and \eqref{eq:xci_app1}, we should change constraint \eqref{eq:gp_1_c1} to:
\begin{align}
\nonumber &m_q t_q^{\kappa_7}\Big[\zeta\mathcal{N}_{q}\mathcal{R}_{q}c_{q}^{-1}p_{q}^{-1}+\varsigma\iota \mathcal{N}_{q}p_{q}^2\\
 &+ \kappa_1 \varsigma\sum\limits_{\substack{i \in \mathbf{Q}\\i \neq q}}p_{i}^{2}\mathcal{N}_{q,i}\mathcal{R}_i^{-1} c_i d_{q,i}^{-1}\Big]\leqslant 1, \quad \forall q \in \mathbf{Q} \label{eq:gp_5_c1}
\end{align}
and add a new constraint:
\begin{equation} \label{eq:gp_5_c8}
 t_q^{-1}+\kappa_5c_qt_q^{-1} \leqslant 1, \quad \forall q \in \mathbf{Q}
\end{equation}
Finally for approximations \eqref{eq:snr_osnr_app3} and \eqref{eq:xci_app2}, we similarly add constraint \eqref{eq:gp_5_c8} and change the QoS constraint to:
\begin{align}
\nonumber &m_q t_q^{\kappa_7}\Big[\zeta\mathcal{N}_{q}\mathcal{R}_{q}c_{q}^{-1}p_{q}^{-1}+\varsigma\iota \mathcal{N}_{q}p_{q}^2+ \kappa_1 \varsigma\sum\limits_{\substack{i \in \mathbf{Q}\\i \neq q}}p_{i}^{2}\mathcal{N}_{q,i}\mathcal{R}_i^{-1} c_i d_{q,i}^{-1}\\
&  + \kappa_2 \varsigma \sum\limits_{\substack{i \in \mathbf{Q}\\i \neq q}}p_{i}^{2}\mathcal{N}_{q,i}\mathcal{R}_ic_i ^{-1} d_{q,i}^{-3}\Big]\leqslant 1, \quad \forall q \in \mathbf{Q} \label{eq:gp_6_c1}
\end{align}

\section{Numerical Results}\label{Sec_VI}
In this section, we use simulation results to demonstrate the performance of the proposed formulations and algorithms. The European Cost239 optical network is considered with the topology and the normalized traffic matrix given by Fig. \ref{fig:euronet} and Tab. \ref{tab:traffic_matrix}, respectively \cite{muhammad2014introducing, khodakarami2014flexible}. The optical link parameters and other constant values are reported in Tab. \ref{tab:sim_param}. We consider the MINLP formulation of \eqref{eq:nonlinear_g}-\eqref{eq:nonlinear_c5} as a benchmark for evaluating the heuristic and formulations \cite{yan2015resource}. Various formulations for RTO and PSA are summarized in Tab. \ref{tab:forms} and their corresponding number of variables and constraints are specified.
\begin{figure}[t!]
\center{\includegraphics[scale=0.45]{Fig/EuroNet.pdf}}
\center{\caption{\label{fig:euronet} European Cost239 optical network with $11$ optical nodes and $26$ bi-directional links. Digits on each link shows its length in km.}}
\end{figure}
\begin{table}[t!]
\centering
\caption{European Cost239 optical network normalized traffic matrix.}\label{tab:traffic_matrix}
\begin{tabular}{cccccccccccc}
\hline
\hline
\textbf{Node} & $\mathbf{01}$ & $\mathbf{02}$ & $\mathbf{03}$ & $\mathbf{04}$ & $\mathbf{05}$ & $\mathbf{06}$ & $\mathbf{07}$ & $\mathbf{08}$ & $\mathbf{09}$ & $\mathbf{10}$ & $\mathbf{11}$\\
\hline
$\mathbf{01}$ & $00$ & $01$ & $01$ & $03$ & $01$ & $01$ & $01$ & $35$ & $01$ & $01$ & $01$\\
$\mathbf{02}$ & $01$ & $00$ & $05$ & $14$ & $40$ & $01$ & $01$ & $10$ & $03$ & $02$ & $03$\\
$\mathbf{03}$ & $01$ & $05$ & $00$ & $16$ & $24$ & $01$ & $01$ & $05$ & $03$ & $01$ & $02$\\
$\mathbf{04}$ & $03$ & $14$ & $16$ & $00$ & $06$ & $02$ & $02$ & $21$ & $81$ & $09$ & $09$\\
$\mathbf{05}$ & $01$ & $40$ & $24$ & $06$ & $00$ & $01$ & $11$ & $06$ & $11$ & $01$ & $02$\\
$\mathbf{06}$ & $01$ & $01$ & $01$ & $02$ & $01$ & $00$ & $01$ & $01$ & $01$ & $01$ & $01$\\
$\mathbf{07}$ & $01$ & $01$ & $01$ & $02$ & $11$ & $01$ & $00$ & $01$ & $01$ & $01$ & $01$\\
$\mathbf{08}$ & $35$ & $10$ & $05$ & $21$ & $06$ & $01$ & $01$ & $00$ & $06$ & $02$ & $05$\\
$\mathbf{09}$ & $01$ & $03$ & $03$ & $81$ & $11$ & $01$ & $01$ & $06$ & $00$ & $51$ & $06$\\
$\mathbf{10}$ & $01$ & $02$ & $01$ & $09$ & $01$ & $01$ & $01$ & $02$ & $51$ & $00$ & $81$\\
$\mathbf{11}$ & $01$ & $03$ & $02$ & $09$ & $02$ & $01$ & $01$ & $05$ & $06$ & $81$ & $00$\\
\hline
\hline
\end{tabular}
\end{table}
\begin{table*}[t!]
\centering
\caption{Simulation constant parameters.}\label{tab:sim_param}
\begin{tabular}{cccccccccccc}
\hline
\hline
$\abs{\substack{\beta_2}}$ & $\alpha$ & $L$ & $\nu$ & $n_{sp}$ & $\gamma$ & $\mathcal{G}$ & $\mathcal{B}$ & $\mathcal{C}$  & $\mathcal{J}$\\
$\text{fs}^2/\text{m}$ & dB/km & km & THz &  & 1/W/km & GHz & THz & Gbps & \\
\hline
$20393$ & $0.22$ & $80$ & $193.55$ & $1.58$ & $1.3$ & $20$ & $2$ & $100$ & 0.1\\
\hline
\hline
\end{tabular}
\end{table*}
\begin{table*}[t!]
\centering
\caption{ Various formulations for RTO and PSA considered in the paper and their corresponding number of variables and constraints.}\label{tab:forms}
\begin{tabular}{cccccc}
\hline
\hline
Formulation & Type & Goal & Constraint & \#Variables & \#Constraints\\
\hline
SPR & BLP  & \eqref{eq:routing1_g}  & \eqref{eq:routing1_c1}-\eqref{eq:routing1_c3} & $\abs{\mathbf{Q}}\abs{\mathbf{L}}$ & $2\abs{\mathbf{Q}}+\abs{\mathbf{Q}}\abs{\mathbf{V}}$ \\
SCPR & BQP  & \eqref{eq:routing2_g} & \eqref{eq:routing1_c1}-\eqref{eq:routing1_c3} & $\abs{\mathbf{Q}}\abs{\mathbf{L}}$ & $2\abs{\mathbf{Q}}+\abs{\mathbf{Q}}\abs{\mathbf{V}}$\\
SCPRR & BQP  & \eqref{eq:routing3_g}  & \eqref{eq:routing1_c1}-\eqref{eq:routing1_c3} & $\abs{\mathbf{Q}}\abs{\mathbf{L}}$ & $2\abs{\mathbf{Q}}+\abs{\mathbf{Q}}\abs{\mathbf{V}}$ \\
MINLPPSA & MINLP  & \eqref{eq:nonlinear_g} & \eqref{eq:nonlinear_c1}-\eqref{eq:nonlinear_c5} & $4\abs{\mathbf{Q}}+1$ & $3\abs{\mathbf{Q}}+\abs{\mathbf{Q}}\abs{\mathbf{L}}+1$ \\
GPSA1 & MIGP  & \eqref{eq:gp_1_g}  & \eqref{eq:gp_1_c1}-\eqref{eq:gp_1_c7} & $\abs{\mathbf{Q}}^2+4\abs{\mathbf{Q}}+1$ & $3\abs{\mathbf{Q}}+3\abs{\mathbf{Q}}\abs{\mathbf{L}}+1$ \\
GPSA2 & MIGP & \eqref{eq:gp_1_g}  &  \eqref{eq:gp_2_c1}, \eqref{eq:gp_1_c2}-\eqref{eq:gp_1_c7} & $\abs{\mathbf{Q}}^2+4\abs{\mathbf{Q}}+1$ & $3\abs{\mathbf{Q}}+3\abs{\mathbf{Q}}\abs{\mathbf{L}}+1$  \\
GPSA3 & MIGP  & \eqref{eq:gp_1_g}  &  \eqref{eq:gp_3_c1}, \eqref{eq:gp_1_c2}-\eqref{eq:gp_1_c7} & $\abs{\mathbf{Q}}^2+4\abs{\mathbf{Q}}+1$ & $3\abs{\mathbf{Q}}+3\abs{\mathbf{Q}}\abs{\mathbf{L}}+1$ \\
GPSA4 & MIGP  & \eqref{eq:gp_1_g}  &  \eqref{eq:gp_4_c1}, \eqref{eq:gp_1_c2}-\eqref{eq:gp_1_c7} & $\abs{\mathbf{Q}}^2+4\abs{\mathbf{Q}}+1$ & $3\abs{\mathbf{Q}}+3\abs{\mathbf{Q}}\abs{\mathbf{L}}+1$\\
GPSA5 & MIGP & \eqref{eq:gp_1_g} & \eqref{eq:gp_5_c1}, \eqref{eq:gp_1_c2}-\eqref{eq:gp_1_c7}, \eqref{eq:gp_5_c8} & $\abs{\mathbf{Q}}^2+5\abs{\mathbf{Q}}+1$ & $4\abs{\mathbf{Q}}+3\abs{\mathbf{Q}}\abs{\mathbf{L}}+1$\\
GPSA6 & MIGP  & \eqref{eq:gp_1_g} & \eqref{eq:gp_6_c1}, \eqref{eq:gp_1_c2}-\eqref{eq:gp_1_c7}, \eqref{eq:gp_5_c8} & $\abs{\mathbf{Q}}^2+5\abs{\mathbf{Q}}+1$ & $4\abs{\mathbf{Q}}+3\abs{\mathbf{Q}}\abs{\mathbf{L}}+1$ \\
\hline
\hline
\end{tabular}
\end{table*}

Fig. \ref{fig:time} compares the run time of the different PSA formulations for various number of transmit transponders. For all of the formulations, the same RTO (here SPR) are applied. Traffics are randomly selected from traffic matrix in Tab. \ref{tab:traffic_matrix}. GPSAs are modeled and solved using YALMIP \cite{lofberg2005yalmip} and MOSEK \cite{mosek2010mosek} software packages while we use BONMIN \cite{bonami2007bonmin} for solving MINLP PSA (MINLPPSA). Simulations run over a Corei7 8 GB RAM computer. Clearly, the run time is considerably improved for GPSA formulations. Among GPSAs, the run time increases for higher precision posynomial expressions and the best time is obtained for GPSA1, the simplest GPSA. Based on Fig. \ref{fig:time}, the run time of GPSA1 is more than $59$ times shorter than MINLPPSA when there are $46$ transmit transponders. Although GPSA formulations have more number of constraints and variables, their special properties result in very short convergence time.
\begin{figure}[t!]
\center{\includegraphics[scale=0.45]{Fig/time.pdf}}
\center{\caption{\label{fig:time}Run time in terms of the number of transmit transponders for MINLP and GP formulations of PSA stage of Alg. \ref{alg:rsa}.}}
\end{figure}

For the European Cost239 optical network in Fig. \ref{fig:euronet} with $46$ transmit transponders, the distribution of OSNR relative errors (with respect to MINLPPSA) given by GPSA5 and GPSA6 solutions are shown in Fig. \ref{fig:err1} and Fig. \ref{fig:err2}, respectively.  The average relative error is $2.13\%$ for GPSA5 and $1.09\%$ for GPSA6. This shows that GPSA6 and GPSA5 provide accurate results with a much shorter run time compared to MINLPPSA.
\begin{figure}[t!]
\center{\includegraphics[scale=0.45]{Fig/error1.pdf}}
\center{\caption{\label{fig:err1} Distribution of OSNR relative errors of GPSA5 with respect to MINLPPSA in European Cost239 optical network with $46$ transmit transponders.}}
\end{figure}
\begin{figure}[t!]
\center{\includegraphics[scale=0.45]{Fig/error2.pdf}}
\center{\caption{\label{fig:err2} Distribution of OSNR relative errors of GPSA6 with respect to MINLPPSA in European Cost239 optical network with $46$ transmit transponders.}}
\end{figure}

Considering the product of $p_q\Delta_q$ as available communication resources for $q$-th connection request, Fig.  \ref{fig:spectral} shows the average transmission bit rate per available resources $\frac{\mathcal{R}_q}{p_q\Delta_q}$  in terms of total NLIs and ASE noise $X_q+Y_q+E_q$ for different values of minimum OSNR margin $\mathcal{M} = 1, 2, 4$ in European Cost239 optical network with $46$ transmit transponders. As it is seen, when the system noise is increased, the resource utilization of the network is decreased.
\begin{figure}[t!]
\center{\includegraphics[scale=0.45]{Fig/spectral.pdf}}
\center{\caption{\label{fig:spectral} Average transmission bit rate per available resources $\frac{\mathcal{R}_q}{p_q\Delta_q}$  in terms of total NLIs and ASE noise $X_q+Y_q+E_q$ for different values of OSNR margin $\mathcal{M} = 1, 2, 4$ in European Cost239 optical network with $46$ transmit transponders.}}
\end{figure}

Finally, to compare proposed RTO procedures, we run Alg. \ref{alg:rsa} for different RTO algorithms and fixed GPSA1 formulation and compare the total transmitted optical power and accumulated NLI and ASE noise power in terms of the total number of used fiber spans. We use CPLEX \cite{ilog2012cplex} software package for solving BLP and BQP formulations of SPR, SCPR and SCPRR. As shown in Fig. \ref{fig:power} and Fig. \ref{fig:noise}, the amount of transmitted optical power and accumulated NLIs and ASE noise powers are the lowest for SCPRR compared to two other RTO procedures.  The improvement gap is higher for low traffic networks. This shows that SCPRR is more suitable for RTO in EONs with considerable amount of fiber NLIs and ASE noise.
\begin{figure}[t!]
\center{\includegraphics[scale=0.45]{Fig/power.pdf}}
\center{\caption{\label{fig:power}  Total transmission optical powers in terms of the number of used fiber spans.}}
\end{figure}
\begin{figure}[t!]
\center{\includegraphics[scale=0.45]{Fig/noise.pdf}}
\center{\caption{\label{fig:noise}  Total accumulated NLI and ASE noise power in terms of the number of used fiber spans.}}
\end{figure}
\section{Conclusion}\label{Sec_VII}
In this paper, we provide a heuristic algorithm for routing, traffic ordering, power allocation and spectrum assignment in elastic optical networks in which mixed minimization of transmitted optical power and spectrum usage constrained to physical and quality of service requirements is addressed. Our proposed algorithm consists of two main stages for routing/traffic ordering, and power/spectrum assignment. We propose three routing/traffic ordering procedures and compare them in terms of the total transmission optical power, spontaneous emission noise and nonlinear interferences. We provide posynomial expressions for fiber nonlinear cross- and self-channel interferences and relate modulation spectral efficiency values to their corresponding minimum required optical signal to noise ratio through posynomial curve fitting. Then, we use the results to formulate different geometric optimization problems for power/spectrum assignment stage of the heuristic algorithm.  Although there is a trade-off between run time and accuracy of the proposed geometric formulations, for all of them a considerable reduction in convergence time compared to the well-known mixed integer nonlinear power/spectrum assignment is obtained. Simulation results also confirm that minimization of the number of common fiber spans to transmission bit rate product is the preferred cost function for routing/traffic ordering in elastic optical networks with considerable nonlinear interferences and additive amplification noise.

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