# Re-entrant phenomenon in diffuse ferroelectric, BaSn0.15Ti0.85O3 : Local
structural insights and FORC study

Akash Surampalli  UGC-DAE Consortium for Scientific Research, University
Campus, Khandwa Road, Indore 452001, India.  Ramon Egli  Central Institute for
Meteorology and Geodynamics (ZAMG), 1190, Vienna, Austria  Deepak Prajapat
UGC-DAE Consortium for Scientific Research, University Campus, Khandwa Road,
Indore 452001, India.  Carlo Meneghini  Dipartimento di Scienze, Universita di
Roma Tre, I-00146 Roma, Italy.  V.Raghavendra Reddy<EMAIL_ADDRESS><EMAIL_ADDRESS>UGC-DAE Consortium for Scientific Research, University
Campus, Khandwa Road, Indore 452001, India.

###### Abstract

From the phase diagram as proposed by Lei et.al., [1] a BaSn0.15Ti0.85O3 is
chosen showing a diffuse phase transition between cubic to rhombohedral (C-R)
near room temperature. Dielectric analysis confirms a phase transition near
room temperature ($T$C $\approx$ 290 K) and also frequency dispersion in
dielectric constant is observed towards low temperature. Polarization and
first-order reversal curves (FORC) suggest that the system is in re-entrant
phase at low temperatures. Put together, all these electrical characterization
results points toward the relaxor behavior in the re-entrant phase. Local
probe techniques such as x-ray absorption near edge spectroscopy, Raman and
M$\ddot{o}$ssbauer spectroscopy are employed to investigate the local
environment changes around this region of low temperature dielectric anomaly.
A simple ferroelectric exchange model explaining the low temperature re-
entrant behavior is presented from these results.

$BaTiO_{3}$ based relaxors and ferroelectrics, first-order reversal curves,
x-ray absorption spectroscopy, Raman spectroscopy, M$\ddot{o}$ssbauer
spectroscopy.

###### pacs:

77.80.Jk, 78.70.Dm, 76.80.+y, 74.25.nd

##  I Introduction

The ferroelectric (FE) transition with frequency independent broad dielectric
peak as a function of temperature is occasionally denoted as diffused phase
transition (DPT) in order to distinguish it from the relaxor one [2, 3, 4].
Relaxors are usually associated with frequency dependency in their dielectric
response and they are made of polar regions of microscopic size (usually in
nano-meter range i.e., polar nano-regions (PNR)) showing different electric
field response from usual domain like response as observed in FE materials.

Earliest models proposed for DPT systems are indistinguishable from relaxors,
in which the smearing of transition is explained in terms of statistical
compositional fluctuations that occur among equivalent crystallographic sites
due to doping [4]. Such fluctuations in composition leads to the distribution
of local Curie temperatures, $\langle T$C$\rangle$, in the system resulting in
the broad temperature range of transition. The authors argued that that
different local and global symmetries have stringent consequences on their
poling-depoling behavior and their low temperature dielectric dispersion [4].
For instance Setter et.al., demonstrated experimentally that introducing
disorder to lead-scandium-tantalate (PST, PbScxTa1-xO3) makes the
ferroelectric transition from sharp to diffuse [5].

On the other hand it has been suggested that the mechanism of DPT involved in
relaxors differ from the one’s exhibiting in FE materials [6]. Unlike FE,
relaxors do not exhibit any phase transition across dielectric maxima and
instead the DPT in relaxors is explained as an overlapping phenomenon of
volume increase of polar micro regions (which are manifested at high
temperatures above dielectric maximum temperature) & freezing of dipoles [6].
Comprehending on this idea, further models have been proposed taking analogies
with spin-glass models to explain the relaxor behavior [7]. Further,
experimental evidences of PNR and non-linear dielectric responses as predicted
from these models are found [8, 9, 10]. This led to characterizing relaxors as
a separate class from FE. However, with the observation of soft mode like
behavior & domain like responses in the relaxors, the earliest interpretations
of relaxors have been revived and relaxors are defined as FE with multiple
inhomogeneities [11].

Barium titanate (BaTiO3, BTO) and its modified systems have been well studied
for their intriguing physics and applicability [12, 13, 14, 15]. In
particular, the systems in which titanium replaced with an isovalent atom
exhibits dielectric and polarization responses that are different from FE
systems. In literature, the compositional phase diagrams of such systems are
studied in detail, particularly the phase diagram of BaTi1-xSnxO3 is
intensively studied for their excellent properties such as electro-caloric
effect [15, 16, 1]. With increasing Sn doping the system exhibits a
transformation from FE to relaxors showing diffuse ferroelectric phase
transitions for intermediate compositions [1].

The system under present investigation, the solid solution BaTi1-xSnxO3 for a
particular composition (x =0.15) exhibiting DPT, has been a subject to
research in the past decade [17, 16, 12, 18, 13, 19, 20]. These past studies
are usually focused on the phase segregation that exists in this compound and
existence of FE like regions above $\langle T$C$\rangle$. Xie et al., observed
both static and dynamic like polar regions above $\langle T$C$\rangle$,
transforming to a long range FE below $\langle T$C$\rangle$ [19]. Similar
observations are made by Shvartsman et al., in which they argued the phase
transition occurring in BaTi0.85Sn0.15O3 is different from conventional FE
transition [13]. Precursor polar like regions that are responsible for
transitions are seen above dielectric maxima ($T$m) and the bulk
transformation to FE is completed at a much lower temperature (T $<$ $T$m).
However, in the later work published by Sanjay et al., of the similar
composition, it is conjectured that towards low temperature relaxor phase sets
in co-existing with FE [12]. This is interpreted from the changes in
polarization hysteresis curves in the electric field cooled conditions and re-
entrant like behavior in remanent polarization variation with temperature.

There is no further consensus available on the above mentioned aspects and in
general the re-entrant phenomenon is very meagerly explored in the context of
FE materials as compared to their magnetic counterparts [21, 22, 23]. Re-
entrant phases are usually associated with disordered phase that is followed
by an ordered phase on lowering the temperature. Such disorder is likely to
occur by breaking the ordered state into a disordered state or by having
different phases co-exist at different scales [24]. Re-entrant phenomenon in
ferroelectrics are distinguished in materials which reveals atypical sequence
of phase transitions viz., paraelectric-ferroelectric-relaxor on lowering
temperature. These phase transitions are identified by decrease in
polarization, dispersion in dielectric constant and an associated loss peak,
aging at low temperatures [25, 26, 27, 28]. Investigating re-entrant
transitions in ferroelectrics is especially challenging since the properties
that are associated with them can be explained with external process i.e.,
domain wall & domain responses . Hence, the idea that re-entrant phases in
ferroelectrics are associated with a thermodynamic phase transition is
received with skepticism [26].

In view of this, in the present work we have taken the system BaSn0.15Ti0.85O3
in which the properties associated with re-entrant behavior are observed [12].
To understand the polarization evolution in the system, we have employed
first-order reversal curves (FORC) to reveal subtle changes that are
responsible for the re-entrant behavior or to find out if such behavior
exists. Along with it, x-ray absorption spectroscopy (XAS), M$\ddot{o}$ssbauer
and Raman spectroscopy techniques are further employed to understand the
structural changes occurring across this re-entrant temperature. It may be
noted that combined use of such microscopic and macroscopic experimental
techniques has revealed structural modifications associated to the relaxor
behavior in BTO based compounds in our recent work [29].

##  II Experimental details

Polycrystalline 15% Sn doped BaTiO3 (BaSn0.15Ti0.85O3) is prepared with
conventional solid-state sintering method starting from high purity
($\geq$99.9%) oxide and carbonate precursors. X-ray diffration measurements
are carried out with Brucker D8-Discover system equipped with Cu K$\alpha$
radiation and LynxEye detector. Dielectric measurements were performed using
precision LCR meter E4980 and in-house developed sample holder with 1 V ac
signal at different frequencies in the temperature range 50-300 K, while
cooling the sample. Temperature dependent P-E hysteresis loops and first order
reversal curves (FORC) were measured using Radiant Precision Premier II
ferroelectric loop tracer using liquid nitrogen as cryogen in the temperature
range of 80-350 K and placing the sample in Delta-9023 chamber. Temperature
dependent Raman spectra were recorded with LabRam-HR800 in back scattering
geometry equipped with the diode laser ($\lambda$=473 nm) as excitation source
in the temperature range 80-325 K using Linkam cryostat. Temperature-dependent
Ti K edge x-ray absorption spectra (XAS) data were collected in transmission
geometry at ELETTRA (Trieste, Italy) 11.1R beamline [30]. Absorption spectra
were measured in transmission geometry using Ti foil as a reference sample for
energy calibration. The raw data were treated using standard procedures for
normalization and extraction of XAFS signal [31]. Temperature dependent 119Sn
M$\ddot{o}$ssbauer measurements are carried out in transmission mode using a
standard PC-based M$\ddot{o}$ssbauer spectrometer equipped with a WisEl
velocity drive in constant acceleration mode and placing the sample in a Janis
CCR system. The spectrometer is calibrated with natural iron and the reported
isomer shift values are with respect to SnO2.

Figure 1: Room temperature x-ray diffraction pattern of BaSn0.15Ti0.85O3
measured with $\lambda=0.154$ nm radiation. The experimental data is profile
fitted with FullProf[32] program.

##  III Results and discussions

The X-ray powder diffraction data obtained at room temperature (shown in Fig.
1) reveals a Pm$\overline{3}$m cubic symmetry consistent with the literature
[1]. The $\chi$2 value is found to be well within the accepted range. The XRD
data indicates the phase purity of the studied sample and the results of all
the measurements are discussed in the following sections.

Figure 2: (a) Temperature dependent dielectric constant ($\epsilon$) and
tan$\delta$ of BaSn0.15Ti0.85O3 at different frequencies. The inset shows the
Vogel-Fulcher fit to the variation of $T$m as a function of frequency. (b)
Variation of 1-($\epsilon$(50 kHz)/$\epsilon$(500 Hz)) as a function of
temperature of BaSn0.15Ti0.85O3 and remanent polarization, $P$r vs temperature
as obtained from PUND measurements, taken at a electric field of around 8
kV/cm. The inset shows the P-E loops at the indicated temperatures. Figure 3:
(a) Maximum electric polarization ($P$max) plotted as a function of electric
field, taken from the temperature dependent P-E data. The inset shows
polynomial fitting for 80 K as adopted in literature [33]. (b) Its derivative
& the inset shows the electric field corresponding to derivative maxima
against the temperature.

###  III.1 Temperature dependent dielectric and electric polarization studies

The temperature dependent dielectric constant ($\epsilon$) and dielectric loss
(tan$\delta$) are shown in Fig. 2. On cooling it can be seen from the Fig.
2(a) that at around 290 K ($T$C), a broad peak in dielectric constant is seen,
a characteristic of DPT, as well as a peak in dielectric loss near the same
temperature. This is a feature reminiscent of paraelectric-ferroelectric
transition ($T$C). Upon further lowering the temperature, frequency dispersion
in dielectric constant becomes evident and a steady increase in loss is seen
with frequency dependent dispersive peak. The loss peaks roughly follows
Vogel-Fulcher law indicating a possible existence of dipolar freezing dipole
dynamics in the system. The frequency dispersion in $\epsilon$ represented as
1-$\frac{\epsilon\textsubscript{50 kHz}}{\epsilon\textsubscript{500 kHz}}$
shows a behavior similar to tan$\delta$ as shown in Fig. 2(b). The frequency
dispersion start to appear above $T$C, showing a local maxima around $T$C. The
relaxation around dielectric maxima had been already discussed by shvartsman
et.al, that relaxing polar entities manifested at high temperatures
percolating to ferroelectric state are responsible for this dispersion around
dielectric maxima [13]. This FE transition is accompanied with increasing
dispersion on further lowering the temperature, peaking at a temperature and
starts to vanish below that. In order to investigate such behavior at low
temperatures, electric polarization versus electric field (P-E) loop evolution
with temperature and remanent polarization obtained from positive up and
negative down (PUND) measurements are performed and are also shown in Fig.
2(b) comparing with the trend in dielectric dispersion in the system.

Apart from the dispersion in dielectric constant, remanent polarization
obtained from PUND data is seen to be decreasing towards lower temperatures
(Fig. 2(b)), peaking at temperature $T$rg deviating from the trend in normal
ferroelectrics. It is also observed that this value of $T$rg is electric field
dependent (not shown here) and at higher fields the trend follows as such of
normal ferroelectrics. These results suggest that the system transforms into
metastable FE state at low temperature and can be stabilized into a FE state
on the application of high electric fields. Field cooling protocols have been
adopted in earlier studies to probe the metastable nature of this system [12].
The coercive switching fields are also seen to be increasing below $T$rg as
seen from hysteresis curves in Fig. 2. Our results are strikingly similar to
the dielectric and electric polarization studies in
(Ba0.925Bi0.05)(Ti1−xSnx)O3 system, where a re-entrant transition from
ferroelectric to glass like state is observed [28] from transmission electron
microscopic measurements. Adopting similar nomenclature used in the
literature, this characteristic temperature Trg is taken as re-entrant
temperature. However, with further analysis we establish the true nature
associated with this temperature.

Further, the electric polarization trend with electric fields obtained from
hysteresis curves taken at different temperatures are studied to realize the
nature of this re-entrant state and the data is shown in Fig. 3. The trend
obtained follows the polynomial that is adopted for relaxor studies [33] as
shown in the inset of Fig. 3(a). And the derivative as also shown in Fig.
3(b), which corresponds to static dielectric constant, is plotted against
electric field. The derivative, which shows a maxima corresponds to the
transformation from the relaxor ferroelectric state to long-range
ferroelectric [33]. It is also conjectured by Sanjay et al., [12] that this
system co-exists in ferroelectric & relaxor state towards low temperatures.
The results of electric polarization with electric field & dielectric
dispersion of Vogel-Fulcher like points towards the relaxor like signatures
that exist in the exist. These results led us to believe that the metastable
nature of this system is given by the relaxor like properties that are
observed. To further understand the evolution of electric polarization in such
systems first order reversal curves are measured and subsequently analyzed as
discussed in the following section.

###  III.2 First Order Reversal Curves

Figure 4:  FORC measurements at 293, 273, 223, 173, and 123 K (every fifth
curve is shown for clarity, electric field in kV/cm and polarization in
$\mu$C/cm2) . (b) FORC diagrams corresponding to the measurements in (a).
Contours enclose polarizations corresponding to the 15, 40, 60, and 80 %
quantiles of the total integral of the FORC function. (c) FORC diagrams
corresponding to the measurements in (a), expressed as a function of the total
internal field $E$i = $E$+$\alpha P$/$\epsilon$0, instead of the applied field
$E$, with coefficients $\alpha$ given in Table 1. The dashed lines connect the
peaks of the thermal relaxation feature on the left margin with the central
maximum along the trace of a single curve. Figure 5:  (a) Saturation
polarization $P$s (squares), remanent saturation polarization $P$rs and total
FORC polarization $P$forc (dots) obtained from the desheared FORC measurements
shown in 4. Lines are guides to the eye. (b) Temperature dependencies of the
exchange field $E$ex (diamonds), the median switching field $\langle
E$c$\rangle$ (circles), defined as the median of the coercivity distribution
represented by horizontal profiles of the FORC functions in 4c at $E$b=0 , and
the standard deviation of the random bias field $E$b (dots), obtained from
vertical profiles of the FORC function in 4c across the central maximum. Lines
are guides to the eye.

First-order reversal curves $\textit{P}(\textit{E}_{r},\textit{E})$ [34] have
been measured at selected temperatures, starting from room temperature
(Fig.4a). Each curve is preceded by a preparation step, where the applied
field is decreased from positive saturation to the so-called reversal field
Er, and consists in measurements of P while the applied field E is increased
from Er to positive saturation. The FORC function
$\rho$=$\partial$2$P$/($\partial$Er$\partial$E) has been calculated with the
VARIFORC protocol using local polynomial fits [35], and plotted in modified
coordinates Ec$=$(E-Er)/2 and Eb$=$(E+Er)/2 (Fig.4b). In Preisach theory [34],
Ec and Eb are the coercive and bias fields of elemental rectangular hysteresis
loops, called hysterons, and the FORC function is the joined probability
density of these fields, which is required to generate the measured curves.
The room temperature ($\sim$293 K) FORC function is concentrated along the
left margin (Ec=0 ) and has a large vertical spread with a single maximum
close to the origin. This signature is typical of a system containing regions
whose remanent polarization P(E=0) decays in a time comparable with that of
measurements, due to thermal relaxation [36]. The hysteresis becomes
completely closed above $T$C, yielding $\rho$=0. The approach to $T$C is
clearly visible on the temperature dependence of the remanent saturation
polarization $P$rs=P(Er, E=0) and of the total irreversible polarization
$P$forc obtained by integrating the FORC function over $E$c and $E$b (Table
1). Both quantities disappear at $T$C $\approx$297 K (Fig.5a), in
correspondence with the peak of the dielectric constant (Fig. 2). For
comparison, the saturation polarization $P$s displays only a moderate decrease
in proximity of $T$C.

Isolated regions undergoing thermal relaxation near $T$C produce FORC
functions characterized by a vertical ridge at $E$c=0, which extends mainly
over $E$b$\lesssim$ 0, since a negative reversal field is required to switch
these regions [36]. In the formalism of the Preisach theory, the almost
symmetric ridge observed at room temperature requires a random bias field $E$b
with a broad probability distribution, which acts as a thermal fluctuation
field with spatial and time correlation [37]. The existence of a strong random
coupling between relaxing regions is also suggested by the amplitude-
dependent, frequency independent maximum of tan$\delta$ in measurements of the
dielectric constant around $\sim$290 K (Fig. 2), which contrasts with the
amplitude-independent and frequency-independent behavior expected from thermal
relaxations in non-interacting systems [38].

A second FORC feature emerges at lower temperatures. This feature consists of
a second peak, hereafter called the central peak, which forms near the origin
(Fig.4b, 273 K), and moves towards higher values of Ec as the temperature
decreases. A similar feature, consisting of a horizontal ridge along $E$b=0,
is produced by the switching of isolated regions that become progressively
blocked [36]; however, the central maximum seen here is characterized by oval
contour lines, which, like in the case of the vertical ridge caused by thermal
relaxation, denote the presence of random interactions [39]. In the context of
Preisach theory, such interactions are represented by a local internal field
whose probability distribution density is given by vertical profiles through
the FORC function [40, 34]. The central peak is offset towards negative $E$b
values comprised between $\sim$0.2 kV/cm (273 K) and $\sim$1.2 kV/cm (123 K),
which can be understood as a positive mean internal field that promote
switching to positive saturation in external fields that are smaller than
those required without this internal field. A pair of positive and negative
ridges parallel to the descending diagonal departs from the central peak,
while the vertical ridge near the left edge of the FORC diagram is shifted
downwards. Similar signatures have been observed in polycrystalline exchange-
spring magnets consisting in a magnetically hard/soft bilayer [41], and in
polycrystalline iron thin films [42]. The development of the diagonal ridge
pair has also been associated with the formation of long-range ferromagnetic
order between ferromagnetic clusters embedded in a non-ferromagnetic matrix
[43].

Ferroelectric exchange coupling can be represented by a so-called exchange
electric field $E$ex=$\alpha P$ that is proportional to the bulk polarization
through a positive coefficient $\alpha$. In this case, switching of
ferroelectric regions depends on the total field $E$tot = $E$+$\alpha P$
resulting from the overlap of $E$ex with the field E applied during FORC
measurements. Electric polarization curves expressed as a function of $E$tot,
instead of the applied field, represent the intrinsic response of the
switching regions without any coupling between them. This operation is called
deshearing [34]. If FE exchange is representable by the above model, a value
of $\alpha$ can be found for which the diagonal ridge pair is eliminated from
the desheared FORC diagram, while the central maximum moves to the $E$b=0
position expected for unbiased switching regions. This is indeed the case
(Fig.4c), with increasingly large values of $\alpha$ required at lower
temperatures (Table 1). Hysteresis deshearing with $\alpha>$0 causes a
decrease of $P$rs, whereby $P$rs/$P$s $\approx$ 0.5 is obtained from desheared
$P$rs values at low temperatures. This result is expected for isolated and
completely blocked regions with uniaxial ferroelectric anisotropy.

The amplitude of the central maximum in desheared FORC diagrams decreases with
decreasing temperature; however, this is just due to its broadening, since
$P$forc increases monotonically as the sample is cooled down (Fig.5). The
median $\langle E$c$\rangle$ of the distribution obtained from horizontal
profiles of the FORC function (Table 1) is a measure for the mean switching
field. It is characterized by a strong temperature dependence and vanishes
near $T$C because of thermal relaxations. The vertical extension of the
desheared central maximum is a measure for the strength of random interactions
between switching regions, expressed for instance by the standard deviation
$\sigma$(Eb ) of a probability density function identified by vertical
profiles across the central maximum. Estimates of $\sigma$($E$b) have been
obtained for T$\leq$223 K (Table 1); above this temperature, the central
maximum overlaps with the vertical ridge caused by thermal relaxation yielding
unreliable results. The similarity between magnitude and temperature
dependence of $E$ex and $\sigma$($E$b) (Fig.5b) points to a common origin of
these two quantities, which can be understood as the mean and the standard
deviation of a positive random variate describing the intensity of local
ferroelectric exchanges. As in the case of thermal relaxation near $\sim$220
K, the presence of an interaction field is confirmed by the frequency-
dependent amplitude maximum of tan$\delta$ in measurements of the dielectric
constant in the re-entrant spin glass range (Fig.2a).

Further, the peak of the thermal relaxation feature along the left edge of the
desheared FORC diagrams is connected to the central maximum by a $45^{\circ}$
line (dotted in Fig.4c), which means that curves with most switching are also
those affected by the largest relaxation effects associated with the hook-
shaped initial part (Fig. 4a). The polarization decrease at the beginning of
FORCs can be explained by the thermally activated switching of positively
saturated regions in proximity of the negative reversal field [36]. The same
regions are switched back to positive saturation in correspondence of the
central maximum of the FORC diagram.

Further, to demonstrate the structural changes that are associated with the
electric polarization evolution, spectroscopic techniques viz., Mossbauer,
Raman & XANES are employed and are discussed in the following sections.

Table 1:  Parameters derived from FORC measurements: $P$s — saturation polarization, $P$rs — remanent saturation polarization, $P$rsd — remanent saturation polarization obtained from desheared measurements, $P$forc — integral of the FORC function over the measured domain, $\langle E$c$\rangle$ — median switching field calculated from horizontal profiles of the desheared FORC function at $E$b=0 and over $E$c$\geq$ 0.6 kV/cm (to exclude contributions from the vertical ridge), $\alpha$ — deshearing coefficient, $\sigma$($E$b) — standard deviation of the distribution of bias fields obtained from vertical profiles of the desheared FORC function through the central maximum, $E$ex=$\alpha P$ — mean exchange field. T |  $P$s |  $P$rs |  $P$rsd |  $P$forc |  $P$rsd/$P$forc |  $\langle E$c$\rangle$ | $\alpha$ |  $\sigma$($E$b) |  $E$ex
---|---|---|---|---|---|---|---|---|---
(K) |  $\mu$C/cm2 |  $\mu$C/cm2 |  $\mu$C/cm2 |  $\mu$C/cm2 | - | kV/cm | kVm/C | kV/cm | kV/cm
293 | 14.5 | 1.26 | 1.05 | 1.69 | 0.073 | 0.43 |  $\sim$ 820 | a | 1.19
273 | 16.3 | 6.16 | 4.28 | 6.72 | 0.26 | 0.76 |  $\sim$ 820 | a | 1.34
223 | 19.1 | 10.5 | 8.78 | 12.9 | 0.46 | 1.73 | 820 | 1.62 | 1.56
173 | 21.0 | 12.2 | 10.3 | 14.6 | 0.49 | 3.07 | 1150 | 2.19 | 2.41
123 | 21.3 | 13 | 11.3 | 15.2 | 0.53 | 5.83 | 1640 | 2.59 | 3.49
111Not determined because of the overlap with the vertical ridge. |  |  |  |  |  |  |  |  |

###  III.3 Raman spectroscopy investigation

Room temperature Raman spectroscopy as shown in Fig. 6 shows modes resembling
of rhombohedral/orthorhombic/ tetragonal modes which are symmetry forbidden
for the Pm$\overline{3}$m space group, as evidenced from XRD of the same
sample (Fig. 1). This is a consequence of the local structure deviating from
the average structure which is common in materials exhibiting DPT [29].
However, due to broad overlapping modes it is nearly impossible to distinguish
the exact local structure near room temperature. It can be still seen from the
spectra taken at 323 K ($>$ $T$C) modes still persist indicating local
structure is preserved well above $T$C. This is generally observed in DPT
where it is suspected that nucleation of low temperature phase starts
appearing at very high temperatures ($>$ $T$C).

Albeit abrupt changes in wave number is seen around this temperature range
($T$C), more temperature points above Tc are needed to conclude the phase
transition as observed from Raman spectroscopy. It is expected in DPT the
changes in local structure associated with phase transition are reflected at
higher temperatures than the $T$C observed from dielectric measurements.
However, this paper focuses on explaining the low temperature peculiar
behavior as observed from dielectric and electric polarization data, hence,
the spectroscopy analysis is limited to the temperatures below $T$C.

Figure 6: (a) Representative temperature dependent Raman spectra of
BaSn0.15Ti0.85O3. Temperature variation of (b) Raman shift of peak-1 and (c)
normalized area of peak-2.

Temperature dependent evolution of A(TO) mode wave number corresponding to
octahedral vibrations against barium atom is show in Fig. 6 (b). This mode is
susceptible to phase transitions in pure BTO showing discontinuous jump across
different phase transitions viz., Cubic - Tetragonal - Orthorhombic -
Rhombohedral [44]. The wave number of this mode shows a sharp decrease on
lowering temperatures near phase transitions. Similarly, a decrease in wave
number is observed in the present work, albeit gradually as a consequence of
diffuse nature of phase transition. These results suggest that system
transforms to rhombohedral [12] in a wide temperature range as expected for
DPT systems. Further, area under the polar TO mode is calculated ($\approx$535
cm-1) and it is seen to be decreasing below $T$C and increasing towards low
temperatures. All the spectra are corrected with bose-einstein factor before
calculating the area of the TO mode, to eliminate the varying population
contribution with temperature[45]. It is well known about the structural
inhomegenity in the compounds showing DPT and thus, this decrease can be
attributed to the system transforming to the much more ordered state by the
interplay between the increasing ferroelectric and decreasing paraelectric
nature present in the system even below $T$C. These results along with wave-
number trend with temperature emphasizes the DPT nature present in the system.
However, an unusual increase in area is observed towards low temperatures ($<$
$T$rg). Such increase might likely be due to the increase in structural
disorder, the reason which will be discussed in the further sections.
M$\ddot{o}$ssbauer technique is further employed to complement these results
and are discussed in the next section.

###  III.4 M$\ddot{o}$ssbauer analysis

Figure 7: (Left) Representative temperature dependent 119Sn M$\ddot{o}$ssbauer
spectra of BaSn0.15Ti0.85O3. (Right) Temperature variation of width of 119Sn
M$\ddot{o}$ssbauer, considered to be proportional to the electric field
gradient.

M$\ddot{o}$ssbauer spectra taken at different temperatures between 100 - 300 K
are shown in Fig. 7. The width obtained from the spectral fitting against the
temperature is also shown in the same figure. The M$\ddot{o}$ssbauer spectral
width which is proportional to electric field gradient gives information about
the local environment around Sn nucleus. The usual trend in ferroelectrics
follow an increasing value of width below Curie temperature as a consequence
of polar ordering giving rise to a non-symmetrical environment around Sn
nucleus. However, below $T$C an anomalous trend is observed here where
decreasing width is observed on lowering temperature contrary to the observed
increase in ferroelectric transitions [15]. The reason for such observation is
similar to the one discussed in the earlier section and can be explained
within the context the interplay between paraelectric & ferroelectric relative
phase fractions, would result in such decrease in line width below $T$C. On
further lowering the temperature an increasing trend in width is observed,
below $T$rg, complementing again with Raman results viz., an onset of disorder
in the system. Such increase in width is also resembles to the increasing
width across the relaxor transition in relaxors [29], thereby correlating the
dispersion trend observed in dielectric.

Figure 8:  (Top) Representative temperature dependent x-ray absorption spectra
(XAS) of BaSn0.15Ti0.85O3. The data is vertically shifted. Inset shows the
representative fitting of different modes in pre-edge region for 300 K data
(Bottom) Variation of integral area of peak-b with temperature.

###  III.5 Ti K- edge x-ray absorption spectroscopy investigation

X-ray Near Edge Spectroscopy (XANES) performed near Ti- K edge at various
temperatures is presented in Fig. 8. The inset shows pre-edge region with the
peaks labeled as a, b, c, c’. The details corresponding to the origin of these
peaks is discussed in literature widely [29]. In particular the behavior of
peak b, its intensity with temperature reveals information about TiO6
distortions. In fact it corresponds to electronic excitation to empty Ti d
states below the continuum, having quadrupolar transition rule. The p-d
hybridization with oxygen levels enhances the dipole allowed transition
probability. Such hybridization is forbidden in regular TiO6 octahedron but
increases for distorted octahedra and/or off center Ti displacement[46]. And
it is demonstrated that across the phase transitions in BTO, the octahedra are
susceptible to distortions, reflecting in the changes in peak position and
intensity of peak b[47]. The pre-edge features were analysed as a combination
of gaussian peaks. The intensity of the peak b is plotted against the
temperature in the Fig. 8 and it is evident that two different regions exist
below $T$C, separated by dashed lines in the Fig. 8, showing different
behavior in intensity of peak b with temperature. In particular lowering below
$T$rg the peak b raises, pointing out the raising of TiO6 octahedra
distortions, such off center Ti displacement allowing establishing the FE
order. Below $T$rg is evident a different slope of b peak with respect to
temperature change suggesting a different behaviour that may originate from
establishing local inhomogeneities related to polar nanoregions in the relaxor
phase.

##  IV Summary and Conclusions

In summary, we studied the diffuse phase transition occurring in
BaSn0.15Ti0.85O3. The dielectric data confirms the system’s transformation of
paraelectric-ferroelectric over a broad range of temperature, a characteristic
of DPT. Within this region spatial inhomogeneities in structure viz., co-
existing in different phases, and is confirmed from the trends in
M$\ddot{o}$ssbauer and Raman spectral analysis. The remanent polarization
obtained from P-E hysteresis loops, is seen increasing on cooling till $T$rg,
correlating with the trend observed in ferroelectrics. Below $T$rg, the
remanent polarization starts to decrease concomitantly a significant
dielectric dispersion becomes evident. This low temperature phase ($<$ $T$rg)
is denoted as re-entrant phase with relaxor like properties matching with the
literature viz., Vogel-Fulcher like dispersion in dielectric loss, decrease in
remenant polarization and the trend in maximum polarization with electric
field. The increasing line widths & area of Mossbauer spectra and Raman
spectroscopy at low temperatures on cooling further signals a much disordered
state at low temperatures.

Further, the polarization evolution with temperature is studied from FORC
distribution. The FORC distribution reveals clusters with ferroelectric
ordering start to form within the range of Curie temperatures. A similar
phenomenon is described by Davies et.al., [43], where they call this an
“isolated clusters phase”. If really isolated, they produce a FORC signature
consisting of a vertical ridge along the left edge of the FORC space, entirely
confined in the lower quadrant [36]. In reality, we see a ridge that extends
symmetrically in the upper and lower quadrant, and this can only be explained
by the presence of a random bias field originating from interactions between
these regions. So, these are definitively not isolated regions. Also, unlike
in Ref[43], ordering starts in a blocking state, as shown by their FORC
diagram, while we start in an unblocked state, where the polarization of
ordered regions decay in time. The room-temperature FORC diagram thus shows
that ordering does not start at the level of single, totally isolated
crystals, as one might expect from compositional variations. Instead, a sort
of loose network with FE ordering is forming (maybe along grain boundaries).

The FORC studies do not reveal any thermodynamic phase transition near $T$rg.
Instead, it is seen that ferroelectric exchange field that represents the
interaction strength between different regions in the sample is increasing
with temperature. This behavior suggests no breaking of long-range order in
the system at low temperatures and ferroelectric nature is increasing at all
temperature. On the other, the regions that are susceptible to fluctuate with
electric field are observed at all measured temperatures viz., relaxor like
entities that exist at all temperatures.

In conclusion the FORC analysis provides a picture of polar regions those
switch like expected from an ideal single-domain behavior, but the switching
field depends on time : the more time is spent at a certain field, the more
likely some particles will be switched. Contrary to many magnetic nanoparticle
systems, the regions are affected by exchange coupling, probably across the
grain boundaries. Cooling below 100 K is expected to progressively block the
entire specimen volume and produces the decrease of tan$\delta$ seen in Fig.
2. A very noticeable result is that the ratio between saturation remanent
polarization and saturation polarization approaches 0.5 if calculated from
desheared data, that is, as if the regions would be uncoupled. This ratio is
expected for randomly oriented domains with uniaxial anisotropy. Such
randomness of anisotropies in the system might result either from the micro
structure of the polycrystalline or from having to contain different polar
orders in the system. The latter reason might explain the increasing line
widths of Raman and Mossbauer spectra, XANES anomalies at low temperatures, as
this can be explained by different polar orders are percolating throughout out
the measured temperature range. The work done by Fang et.al., [28] on barium
titanate based disordered systems comes close to our results in which they
demonstrated visually re-entrant glass as co-existing orthorombhic &
tetragonal domain structures. This might explain the existence of randomly
oriented domains of uni-axial anisotropy. Hence, re-entrant glass is unlike a
spin glass system, but a heterogeneous system that behaves as a glass (i.e.,
the disorder occurs at a much larger scale). Finally, we want to point that
since the low temperature relaxor like behavior is explained within
ferroelectric exchange model and thus investigating proper relaxor
compositions & other established re-entrant materials in future employing FORC
studies provides basis for understanding such complex materials.

##  V Acknowledgments

AS and VRR thank Department of Science and Technology (DST) and IISc,
Bangalore for travel support for performing experiments at XAFS beamline,
ELETTRA. Authors thank Dr. V. G. Sathe for the Raman spectroscopy
measurements. Thanks are due to Dr. Giuliana Aquilanti and Dr.Luca Olivi for
the help during XAFS measurements at Elettra. Mr. Anil Gome is acknowledged
for the help with M$\ddot{o}$ssbauer spectroscopy measurements.

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For our analysis of SBLO, we use the same simplified model that we used in our analysis of SLO (see <ref>). In particular, we still assume that a server can at most process a single job during a time slot. The dynamic and static offline optimum are similar to those in our analysis of SLO. In particular, the static offline optimum still only uses 8-GPU-servers. <Ref> shows the schedule obtained by lazy budgeting ($\epsilon = 1/4$) in comparison with the offline optimal. Lazy budgeting achieves normalized costs 1.284 and a cost reduction of 11%. The runtime of the algorithm is shown in <ref>.

§.§ Descent Methods

We also evaluated the performance of P-OBD and D-OBD on the Microsoft Fiddle trace under our original models (see <ref>). In our analysis, we use the squared $\ell_2$ norm as the distance-generating function, i.e. $h(x) = \frac{1}{2} \norm{x}_2^2$, which is strongly convex and Lipschitz smooth in the $\ell_2$ norm. In our data center model, we use the $\ell_1$ norm to calculate switching costs, however, we observe that this approximation still achieves a good performance when compared against the dynamic offline optimum. The negative entropy $h(x) = \sum_{k=1}^d x_k \log_2 x_k$, which is commonly used as a distance-generating function for the $\ell_1$ norm cannot be used in the right-sizing data center setting as $\mathbf{0} \in \mathcal{X}$. <Ref> and <ref> show the achieved normalized cost and cost reduction. The resulting schedules under the first model are compared with the offline optimal in <ref>. Remarkably, P-OBD and D-OBD obtain the exact same schedule under our second model. <Ref> visualizes the runtime of P-OBD and D-OBD under our first model.

Normalized cost

Cost reduction

Performance of P-OBD  ($\beta = 1/2$) and D-OBD  ($\eta = 1$) for the Microsoft Fiddle trace when compared against the offline optimum. $h(x) = \frac{1}{2} \norm{x}_2^2$ is used as the distance-generating function.

Comparison of the schedules obtained by OBD for the Microsoft Fiddle trace under our first model and the offline optimum. P-OBD ($\beta = 1/2$) is shown in blue and D-OBD ($\eta = 1$) is shown in red. The two time slots during which P-OBD and D-OBD differ are marked in yellow ($t \in \{6, 20\}$). In both time slots, D-OBD is slightly less “sticky”, resulting in a slightly better normalized cost. Also observe that under our first model, the dynamic offline optimum strictly prefers 2-GPU-servers over 8-GPU-servers In contrast, under our second model, 8-GPU-servers are slightly preferred by the dynamic offline optimum (see <ref>). $h(x) = \frac{1}{2} \norm{x}_2^2$ is used as the distance-generating function.

Although, OBD incurs an increased cost compared to the static offline optimal (the optimal static choice in hindsight) albeit by a factor less than 1, our very conservative model indicates that in practice, significant cost savings are possible.

Runtime of OBD algorithms.

§ PREDICTIONS

In our analysis, we use the Alibaba trace under our second model to evaluate the effects of using predictions. We consider two different types of predictions: perfect predictions, and actual predictions that are obtained using Prophet as described in <ref>. The obtained predictions are based on the four preceding days of the Alibaba trace up until the second to last day. <Ref> visualizes the used prediction for the most common job type (i.e., short jobs). Note that in our analysis, we use the mean predicted load.

Prediction of the load of short jobs for the second to last day of the Alibaba trace. The mean prediction is shown as the black line. The interquartile range of the predicted distribution is shown as the shaded region. The marks represent actual loads.

<Ref> shows the effect of the prediction window $w$ when used with LCP for perfect and actual predictions. We observe that in practice, the prediction window can significantly improve the algorithm performance. Additionally, we find that this effect is also achieved with imperfect (i.e., actual) predictions. Previously,  <cit.> only showed this effect for perfect predictions with additive white Gaussian noise.

Interestingly, RHC and AFHC achieve equivalent results for perfect and imperfect predictions. <Ref> shows how the achieved normalized cost changes with the prediction window. Crucially, note that the MPC-style algorithms do not necessarily perform better for a growing prediction window.  <cit.> showed this effect previously for an adversarially chosen example, however, we observe this behavior with AFHC in a practical setting. In fact, for the Alibaba trace, RHC and AFHC achieve their best result when used without a prediction window, i.e. $w = 0$.

LCP and Int-LCP

RHC and AFHC

Performance of online algorithms with a prediction window for the Alibaba trace. The left figure shows the performance of LCP and Int-LCP. The right figure shows the performance of RHC and AFHC. Note that LCP does not continuously approach the normalized cost $1$ as $w \to 24$ because of numerical inaccuracies solving the convex optimizations and as the obtained is compared to the integral offline optimum rather than the fractional offline optimum. For RHC and AFHC, the achieved normalized cost is independent of whether perfect predictions or the predictions from <ref> are used.

CHAPTER: FUTURE WORK

Our implementation separates the model layer from the problem and algorithm layer. Therefore, testing the empirical performance of the discussed online algorithms in other application areas is a natural and exciting direction for future research. In <ref>, we have given an overview of some promising applications.

Algorithms for Convex Body Chasing We have mentioned in <ref> that smoothed convex optimization and convex body chasing are equivalent. Therefore, an interesting research project would be to extend our library of implemented algorithms by the known algorithms for convex body chasing to compare their empirical performance. In particular, the $\mathcal{O}(d)$-competitive algorithm obtained by  <cit.> is highly relevant for the application of right-sizing data centers as it does not impose a restriction on cost functions beyond their convexity.

Dynamic Bounds and Dimensions In practice, the number of available servers (and even server types) in a data center is likely to change over time. An unexplored area of research is how the discussed approaches for online algorithms can be extended to a setting where the bounds on the decision space $\mathcal{X}$ and the dimension of $\mathcal{X}$ are allowed to change over time.  <cit.> discuss how their offline algorithm solving the multi-dimensional integral case can be extended to a setting with time-dependent bounds.

Optimal Valley Filling In <ref>, we have discussed the advantages and disadvantages of valley filling, i.e., scheduling low-priority tasks during periods of low loads to reduce the peak-to-mean ratio. Therefore, an interesting problem is finding optimal server configurations and job scheduling times such that the operating costs and switching costs of the data center are balanced with the revenue loss of delaying specific jobs. This problem extends smoothed convex optimization in the data center setting by allowing for incoming loads to be postponed to a later time slot.

Optimal Assignments of Jobs to Servers Smoothed convex optimization determines the optimal assignment of jobs to a collection of servers of the same type. In <ref>, we have seen that an optimal dispatching rule is to distribute jobs across all servers evenly. However, this approach does not have to be optimal in practice as job arrival times may vary and jobs are discrete, i.e., it may not be possible to distribute them evenly. Therefore, an interesting problem is determining an optimal scheduling of jobs that accounts for the approximations of smoothed convex optimization.

Long-Running Jobs An vital research problem is to extend the data center model we discussed in <ref> to support jobs with a longer runtime than the length of a time slot. In our case studies from <ref>, we have seen that for a time slot length on the order of tens of minutes to an hour, a significant fraction of jobs (in practical scenarios more than 50%) fulfill this criterion. This can be achieved by “memorizing” which jobs have not been completed yet and ensuring that enough servers of each type are active in the following time slot so that jobs do not have to be rerouted to a server of a different type.

Gang Scheduling Requirement A similar problem is to extend the model to allow for mutual job requirements. For example, requirements ensuring jobs are processed simultaneously or on servers of the same type.

Without Lookahead The classical smoothed online convex optimization problem assumes that the convex cost function is known before a move has to be made. This is reasonable to separate the problem of anticipating movement costs with current hitting costs from estimating the current hitting costs. However, in practice, many applications require action before the hitting costs are known. For example, in the setting of right-sizing data centers, enough servers need to be available before jobs arrive as the powering up of servers requires some time. We have seen in <ref> that predictions can be used to get around this restriction. Nonetheless, it would be interesting to see which algorithms perform well in a smoothed setting without lookahead.

Using Predictions More research is needed to find online algorithms that use predictions robustly and consistently. Here, robustness refers to a bounded competitive ratio when predictions are adversarial, and consistency refers to an improved competitive ratio when predictions are accurate <cit.>. Note that clearly model predictive control-style algorithms can perform arbitrarily poorly when predictions are adversarial.

CHAPTER: CONCLUSION

This work surveyed numerous offline and online algorithms for fractional and integral smoothed convex optimization in single and high dimensions. We evaluated the performance of online algorithms for the application of dynamically right-sizing data centers using traces from various different sources. We found that the discussed online algorithms perform nearly optimally with respect to the observed normalized cost when compared to the dynamic offline optimum. Moreover, we observed that the normalized cost increases as potential cost savings increase. However, the achieved cost reduction depends primarily on the difference between the dynamic offline optimum and the strategy compared to (in our experiments the static offline optimum). Previous results show that changes in the data center model, such as increasing the energy cost or decreasing the normalized switching cost lead to significant increases in cost savings. We also observed that the difference between fractional and integral solutions is negligible in the application of right-sizing data centers, or generally when total costs are large. Further, we have seen that online algorithms using predictions of future loads can achieve a better normalized cost, even when predictions are non-perfect but obtained using standard methods. However, we have also seen that the achieved normalized cost does not necessarily improve for larger prediction windows.

Finally, we made our implementation of the discussed algorithms available. This implementation can be used to test the performance of algorithms for smoothed online convex optimization in other applications and compare their performance. Our data center model balances energy cost and revenue loss. Using past data, this model can be tuned to be safe (i.e., provide enough computing resources for incoming loads), fulfill service level agreements, and provide significant cost and energy savings in real-world settings.

[ignoreall,show=problem,title=List of Problems]

tocchapterList of Algorithms

§ PROBLEMS

$T \in \mathbb{N}$     number of time slots

$\delta \in \mathbb{R}_{>0}$     length of a time slot

$d \in \mathbb{N}$     number of dimensions

$\mathcal{X} \subset \mathbb{R}^d$     decision space

$m_k \in \mathbb{N}$     upper bound of dimension $k$

$M_k \subset \mathbb{N}_0$     allowed values of dimension $k$ (in the discrete case)

$\mathcal{M} = \mathcal{X}$     set of all configurations (in the discrete case)

$f_t(x) \in \mathbb{R}_{\geq 0}$     hitting cost of action $x \in \mathcal{X}$ during time slot $t$

$\beta_k \in \mathbb{R}_{>0}$     switching cost of dimension $k$

$\lambda_t \in \mathbb{N}_0^e$     load profile during time slot $t$

$c_k \in \mathbb{R}_{\geq 0}$     time-independent hitting cost of dimension $k$

§ DATA CENTER MODEL

§.§ Dispatching

$m_k \in \mathbb{N}$     maximum number of servers of type $k$

$l_k^{max} \in \mathbb{N}$     maximum number of jobs a server of type $k$ can process in a single time slot

$e \in \mathbb{N}$     number of load types

$\lambda_{t,i} \in \mathbb{N}_0$     number of jobs of type $i$ during time slot $t$

$\lambda_t \in \mathbb{N}_0$     total load during time slot $t$

$\lambda^{max} \in \mathbb{N}$     maximum total load of feasible load profiles

$\mathcal{Z}$     set of job assignments of jobs of individual load types to server types

$z_{t,k,i} \in [0,1]$     fraction of jobs of type $i$ assigned to servers of type $k$ during time slot $t$

$l_{t,k,i} \in [0,\lambda^{max}]$     (fractional) number of jobs of type $i$ assigned to servers of type $k$ during time slot $t$

$l_{t,k} \in [0,\lambda^{max}]$     (fractional) number of jobs assigned to servers of type $k$ during time slot $t$

$s_k(l) \in [0,1]$     utilization (or speed) of a server of type $k$ with load $l$

$\theta_k \in [0,1]$     maximum allowed utilization of a server of type $k$

$g_{t,k}(l) \in \mathbb{R}_{\geq 0}$     operating cost of a server of type $k$ during time slot $t$ when $l$ jobs are processed on the server

$q_{t,k,i}(l) \in \mathbb{R}_{\geq 0}$     cost of processing a job of type $i$ on a server of type $k$ during time slot $t$ when $l$ jobs are processed on the server

§.§ Energy Cost

$e_{t,k}(s) \in \mathbb{R}_{\geq 0}$     energy cost of a server of type $k$ with utilization $s$ during time slot $t$

$\nu_{t,k}(p) \in \mathbb{R}_{\geq 0}$     energy cost of a server of type $k$ with energy consumption $p$ during time slot $t$

$\phi_k(s) \in \mathbb{R}_{\geq 0}$     energy consumption of a server of type $k$ with utilization $s$

$\Phi_k(s) \in \mathbb{R}_{\geq 0}$     power consumption of a server of type $k$ with utilization $s$

$\Phi_k^{max} \in \mathbb{R}_{\geq 0}$     power consumption of a server of type $k$ on full load

$\Phi_k^{min} \in \mathbb{R}_{\geq 0}$     power consumption of an idling server of type $k$

$c_{t,i} \in \mathbb{R}_{\geq 0}$     energy cost of energy source $i$ per unit of energy during time slot $t$

§.§ Revenue Loss

$\gamma \in \mathbb{R}_{\geq 0}$     revenue loss factor, i.e., lost revenue per unit of delay

$r_{t,i}(d) \in \mathbb{R}_{\geq 0}$     revenue loss of jobs of type $i$ with an average delay $d$ during time slot $t$

$d_{t,k,i}(l) \in \mathbb{R}_{\geq 0}$     average delay of a job of type $i$ processed on a server of type $k$ during time slot $t$ where the total load on the server is $l$

$\delta_{t,k,i} \in \mathbb{R}_{\geq 0}$     constant delay when processing a job of type $i$ on a server of type $k$ during time slot $t$

$\mu_k \in \mathbb{R}_{\geq 0}$     service rate of a server of type $k$

$\delta_i \in \mathbb{R}_{\geq 0}$     minimal detectable delay of jobs of type $i$

$\eta_{k,i} \in \mathbb{R}_{\geq 0}$     processing time of a job of type $i$ on a server of type $k$

§.§ Switching Cost

$\epsilon_k \in \mathbb{R}_{\geq 0}$     additional energy consumed by toggling a server of type $k$ on and off

$\delta_k \in \mathbb{R}_{\geq 0}$     delay in migrating connections or data of a server of type $k$ before it can be powered down

$\tau_k \in \mathbb{R}_{\geq 0}$     wear-and-tear costs of toggling a server

$\rho_k \in \mathbb{R}_{\geq 0}$     perceived risk associated with toggling a server of type $k$

$\xi_k \in \mathbb{R}_{>0}$     normalized switching cost measuring the minimum duration a server of type $k$ must be asleep to outweigh the switching cost

§.§ Networks

$\iota \in \mathbb{N}$     number of data centers

$\zeta \in \mathbb{N}$     number of geographically centered request sources

$\delta_{t,j,s} \in \mathbb{R}_{\geq 0}$     network delay incurred by routing a request from source $s$ to data center $j$ during time slot $t$

$\xi$     number of energy sources

$p_{t,i,j} \in \mathbb{R}_{\geq 0} \cup \{\infty\}$     energy from source $i$ available at data center $j$ during time slot $t$

$u_{t,i} \in \mathbb{R}_{\geq 0}$     average profit per unit of energy from source $i$ during time slot $t$

$q_{t,i} \in [0,1]$     minimum fraction for energy from source $i$ during time slot $t$

$\delta_{t,i,j} \in \mathbb{R}_{\geq 0}$     remaining energy requirement of data center $j$ during time slot $t$ after all energy sources up to source $i$ were used

§ COMPLEXITY

$\mathcal{O}(C)$     complexity of computing the hitting costs $f_t$

$\mathcal{O}(O_{\epsilon}^d)$     convergence rate of a convex optimization finding an $\epsilon$-optimal solution in $d$ dimensions

$\mathcal{O}(R_{\epsilon})$     convergence rate of a root finding method finding an $\epsilon$-optimal root

$\mathcal{O}(I_{\epsilon})$     convergence rate of a quadrature method finding an $\epsilon$-optimal integral

§ ALGORITHMS

$\tau$     current time slot

$\hat{x}$     optimal value of $x$ with respect to some optimization

$X \in \mathcal{X}^t$     schedule, i.e., sequence of configurations over the time horizon $t$

$x, y \in \mathcal{X}$     configuration

$i, j \in [m_k]_0$     value of dimension $k$

$x_{k \gets j}$     configuration $x$ after updating dimension $k$ to $j$

§ IMPLEMENTATION

$\chi_{\tau}(t, x) \in \bigcup_{n=1}^{\infty} \mathbb{R}_{\geq 0}^n$     hitting cost of action $x \in \mathcal{S}$ at time slot $t \in [T]$ given the information from time slot $\tau$

$\mathcal{P}_t \in \left(\bigcup_{n=1}^{\infty} \mathbb{N}_0^n\right)^e$     predicted load profile for time slot $t$, i.e., vector of predicted loads for each job type

§ MISCELLANEOUS

$[n] := \{1, \dots, n\}$     range of natural numbers from $1$ to $n$

$[n]_0 := \{0\} \cup [n]$     range of integers from $0$ to $n$

$[a : b] :=\newline \{a, a+1, \dots, b\}$     range of integers from $a$ to $b$

$(x)_a^b :=\newline \max\{a, \min\{b, x\}\}$     uni-dimensional projection of $x \in \mathbb{R}$ onto $[a,b]$

$(x)^+ := \max\{0, x\}$     uni-dimensional projection of $x \in \mathbb{R}$ onto $[0, \infty)$

$x_{\text{min}}$     smallest entry of the vector $x$

$x_{\text{max}}$     largest entry of the vector $x$

CHAPTER: TAXONOMY OF ONLINE ALGORITHMS

Here, we give an overview of the algorithms that were discussed in this chapter. The first table includes online algorithms for the one dimensional setting. The latter table includes online algorithms for the general setting.

name     problem     performance     time complexity
[alg:ud:lcp]LCP     [problem:simplified_smoothed_convex_optimization]SSCO     3-competitive     $\mathcal{O}((\tau + w) C O_{\epsilon}^{\tau + w})$

[alg:ud:lcp]Int-LCP     [problem:simplified_smoothed_convex_optimization]Int-SSCO     3-competitive     $\mathcal{O}((\tau + w)^2 C \log_2 m)$

[alg:ud:memoryless]Memoryless     [problem:simplified_smoothed_convex_optimization]SSCO     3-competitive     $\mathcal{O}(C O_{\epsilon}^1)$

[alg:ud:probabilistic]Probabilistic     [problem:simplified_smoothed_convex_optimization]SSCO     2-competitive     $\mathcal{O}(\tau^2 C I_{\epsilon} |B_{f_0}| R_{\epsilon} O_{\epsilon}^1)$

[alg:ud:rbg]RBG     [problem:smoothed_convex_optimization]SCO     $\alpha$-unfair competitive ratio $(1+\theta) / \min \{\theta, \alpha\}$ and regret $\mathcal{O}(\max \{T / \theta, \theta\})$; 2-competitive for $\theta = 1$     $\mathcal{O}(C (O_{\epsilon}^1)^{\tau+1})$

[alg:ud:randomized]Rand-Probabilistic     [problem:simplified_smoothed_convex_optimization]Int-SSCO     2-competitive     $\mathcal{O}(\tau^2 m C I_{\epsilon} R_{\epsilon} O_{\epsilon}^1)$

[alg:ud:randomized]Rand-RBG     [problem:smoothed_convex_optimization]Int-SCO     2-competitive     $\mathcal{O}(C (O_{\epsilon}^1)^{\tau+1})$

Taxonomy of uni-dimensional online algorithms.

name     problem     performance     time complexity
[alg:md:lazy_budgeting:det_slo]LB     [problem:slo]SLO     $2d$-competitive     $\mathcal{O}(m d^2 + C d \prod_{k=1}^d m_k)$

[alg:md:lazy_budgeting:det_slo]Randomized LB     [problem:slo]SLO     $\frac{e}{e-1}d$-competitive     $\mathcal{O}(m d^2 + C d \prod_{k=1}^d m_k)$

[alg:md:lazy_budgeting:sblo_c]LB     [problem:sblo]SBLO     $(2d + 1 + \epsilon)$-competitive     $\mathcal{O}(\widetilde{n}_{\tau} \widetilde{\tau}^2 |\mathcal{M}| C d)$

[alg:md:ogd]OGD     [problem:smoothed_convex_optimization]SCO     $\mathcal{O}(\sqrt{T})$-regret     $\mathcal{O}(d C O_{\epsilon}^d)$

[alg:md:obd]OBD     [problem:smoothed_convex_optimization]SCO     -     $\mathcal{O}(O_{\epsilon}^d)$

[alg:md:pobd]P-OBD     [problem:smoothed_convex_optimization]SCO     $3 + \mathcal{O}(1 / \alpha)$-competitive     $\mathcal{O}(C O_{\epsilon}^d + O_{\epsilon}^d R_{\epsilon})$

[alg:md:dobd]D-OBD     [problem:smoothed_convex_optimization]SCO     $\mathcal{O}(\sqrt{T})$-regret     $\mathcal{O}(C O_{\epsilon}^d + (O_{\epsilon}^d)^2 R_{\epsilon})$

[alg:predictions:rhc]RHC     [problem:smoothed_convex_optimization]SCO     $(1 + \mathcal{O}(1/w))$-competitive in one dimension; otherwise $(1 + \max_{k \in [d]} \beta_k / e_k(0))$-competitive     $\mathcal{O}(C O_{\epsilon}^{dw})$

[alg:predictions:afhc]AFHC     [problem:smoothed_convex_optimization]SCO     $(1 + \max_{k \in [d]} \frac{\beta_k}{(w+1) e_k(0)})$-competitive, i.e., $(1 + \mathcal{O}(1/w))$-competitive     $\mathcal{O}(w C O_{\epsilon}^{dw})$

Taxonomy of multi-dimensional online algorithms.

Missing 'biblatex' package
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