Z G Tegegne

C Viana

M Rosales

J L Polleux

C Algani

M Grzeskowiak

E Richalot

Substrate diode effect on the performance of Silicon Germanium phototransistors

Keywords: SiGe HPT, microwave-photonics, Silicon-based photodetectors, Radio-over-Fiber, phototransistor, opto-microwave performances, substrate effet I

This paper provides a study on the substrate effect on the opto-microwave behavior of Silicon-Germanium Heterojunction bipolar Photo-Transistors (HPT). An Opto-Microwave Scanning Near Field Optical Microscopy (OM-SNOM) is performed to observe the distribution of photocurrent and dynamic behavior over the structure of the phototransistor. The photocurrent generated in the photodiode created by a n++ sub-collector and p+ substrate is extracted and analyzed. A maximum substrate diode current of 700 A is observed at 850nm with a related cutoff frequency of 0.42GHz. We have extracted low frequency responsivity (at 50MHz) bandwidth product of 109.2 MHzA/W. Finally, this study will provide a design guide line for Si base phototransistors.

INTRODUCTION

Nowaday's wireless technologies have been developed to replace wire lines installed in the Home Area Network (HAN). The increase of new services and wireless devices leads us towards high data rate reaching Giga bits per second. For this purpose, new wireless network standards arise such as the IEEE802.11.ad [1] which is the extension of the Wi-Fi toward the millimeter wave ranges (60GHz). However, the propagation is limiting to a single room, due to both the high propagation attenuation at 60GHz and to the wall absorption and reflections [START_REF]NAVAIR Electronic Warfare/Combat Systems "Electronic warfare and radar systems engineering handbook[END_REF]. Therefore, an infrastructure is needed to cover the whole home area so as to distribute the signal from one room to another one. As a result an interest has been recently put to Radio-over-Fiber (RoF) home area network application [START_REF] Guillory | 60GHz Intermediate Frequency over Fiber using a passive Multipoint-to-Multipoint architecture[END_REF], for which low cost silicon based optoelectronics are highly desirable. SiGe hetrojunction bipolar phototransistor (SiGe HPT) are potential candidates for light detection that were proposed first in 2003 [START_REF] Polleux | A Strained SiGe layer Heterojunction BipolarPhototransistor for Short-Range Opto-Microwave Applications[END_REF] [START_REF] Polleux | An SiGe/Si Heterojunction Phototransistor for Opto-Microwave Applications: Modeling and first Experimental Results[END_REF] to be integrated in standard SiGe HBT technology. Since then, several laboratories are working on SiGe HPTs using different industrial process technologies like TSMC [START_REF] Pei | Bandwidth Enhancement in an Integratable SiGe Phototransistor by Removal of Excess Carriers[END_REF], AMS SiGe BiCMOS [START_REF] Egels | Design of an optically frequency or phase-controlled oscillator for hybrid fiber-radio LAN at 5.2 GHz[END_REF] and IBM SiGe BiCMOS [START_REF] Yin | Low-cost, high efficiency and high-speed SiGe phototransistors in commercial BiCMOS[END_REF]. Hybrid photoreceiver based on SiGe HPT for 60 GHz intermediatefrequency RoF applications was implemented in [START_REF] Viana | Hybrid photo-receiver based on SiGe heterojunction photo-transistor for low-cost 60 GHz intermediatefrequency radio-over-fibre applications[END_REF].

There is a continuous need to verify the ability of integrating phototransistors in newer commercial SiGe process technologies offering faster operating frequencies but also to improve the performances of the HPT without modification of the technology vertical stacks of layers. To optimize the speed of the phototransistor, [START_REF] Moutier | Frequency response enhancement of a single strained layer SiGe phototransistor based on physical simulations[END_REF] identified the fastest and slowest region of the structure based on physical simulations. References [START_REF] Helme | Analytical modeling of speed response of heterojunction bipolar phototransistors[END_REF] and [START_REF] Yuan | MEXTRAM modeling of Si-SiGe HPTs[END_REF] are investigating the perfomances of phototransitor through opto electric compact circuit modeling. M. D. Rosales et al [START_REF] Rosales | Opto-microwave experimental mapping of SiGe/Si phototransistors at 850nm[END_REF] verifies that the proximity of the base, emitter and collector contacts to the optical window has an influence on the dynamic response characteristic of the phototransistor. This paper investigates the effect of the substrate photodiode created by the p type substrate and n++ sub collector on the dynamic response the HPT. It also provides relavat information interms of bandwidth responsivity product to be used this HPT into microwave photonics applications. Finally a conclusion will be made on the design aspects of SiGe/Si HPT structure.

II. SIGE/SI HPT STRUCTURE UNDER TEST

The SiGe phototransistor (HPT) was fabricated using the existing SiGe2RF Telefunken GmbH SiGe Bipolar process technology. Indeed, the phototransistor fabrication does not modify the vertical stacks of layers that are used to define a standard SiGe2RF HBT technology. This ensures the compatibility with the process technology and potential integration of complete OE-RF circuits.

The basic HPT structure is designed by extending the emitter, base and collector layers of the reference HBT [1]. The optical opening is made through the emitter. To improve the optical illumination, the superficial Silicon oxide and nitride layers on top of the defined optical window are removed by using Reactive-Ion Etching (RIE) process. A cross-section representation of the phototransistor structure is given in Figure 1. The light paths goes through the polysilicon of the emitter before entering the Si emitter, SiGe base and Si collector region. This HPT is essentially one large HBT whose emitter metallization was removed on the side.

The optical opening of the phototransistor is of 10x10 m 2 .

The total emitter size is 11.3x9.2 m 2 and the total collector dimension is 16.5x10.6 m 2 . The base profile is thin 40-80nm abrupt SiGe layer with Ge content in the range of 20-25% and high p doping in the range of 10 19 cm -3 . The collector is typically 300-400nm thick with low doping.

III. EXPERIMENTAL RESULTS AND DISCUSSION

By using the bench setup described in [START_REF] Rosales | Opto-microwave experimental mapping of SiGe/Si phototransistors at 850nm[END_REF], we perform the experimental mapping of the HPT at two different bias conditions (photodiode (PD) and phototransistor mode) and, also, at different optical probe position. For this study, we use a multimode optical source at 850nm and optical probe in order to make sure that our study is more realistic for HAN application where multimode source and fiber is used. The phototransistor mode is studied at a fixed collector-emitter voltage of 3V and fixed base emitter voltage of 0.857V. The photodiode mode is studied by setting collector emitter voltage of 3V and base emitter voltage of 0V. These biasing conditions are the optimum biasing conditions in terms of opto-microwave responsivity.

Figure 2-a) shows the microscopic picture of the phototransistor where the ground (left and right) and signal (up and down) lines are clearly visible. The base contact is taken from the top side, collector contact is taken from the down side and the emitter contact is connected at its left and right side to the ground. The layout is accordingly sketched in Figure 2-b) which defines the optical probe coordinate with its origin given at the center of the optical window. respectively. The plot shows the current as a function of the optical probe position. The design of the phototransistor is superimposed to better observe the correspondence between the measurement and the different regions of phototransistor. The shape of the optical beam that is scanned over the HPT is Gaussian profile in the x and y axes. The resulting photocurrent is the correlation between the optical window and the Gaussian profile of the optical beam.

The base current mapping is symmetric along both x and y axis as shown in Figure 3 in both HPT (a) and PD (b) mode. In HPT mode, the sign of the base current is changed when the optical probe is moving in to the center of the phototransistor. The negative sign, when the active area is illuminated, indicates that parts of holes generated in the active area are flowing out through the base contact. However, parts of the holes are moving to the emitter for transistor amplification. In the PD mode, Ib has negative sign at all position of the optical probe as there is no any transistor action, all the holes generated are flowing out through the base contact. The base current (Ib) measured in PD mode operation is the sum of the dark current and the photogenerated current. Thus, the photocurrent generated in the structure, called the primary photocurrent, can be computed from the difference between the base current measured with and without light illumination.

The primary photocurrent topological map as a function of optical probe position is shown in Figure 4. It reveals to be symmetrical to the x and y axes; and its peak appears at the center of the optical window.

Optical window

Substrate contact

Emitter contact

P+

The collector current topological mapping is presented in Figure 5 in HPT (a) and PD (b) mode. The collector current mapping is not symmetrical along both x and y axes. In HPT mode the Ic peak is located at x=0 m and y=0 m. In the PD mode, the peaks appear outside the active region near to the base and collector contact. Those peaks on the side of the contacts are meanly due to the parasitic photocurrent in the substrate.

In phototransistors such as InGaAs/InP; when operating in photodiode mode(base emitter junction is off), the number of electrons collected at the collector contact is equal to the number of holes collected at the base contact at every position of the optical probe as InP substrate is not light sensitive [START_REF] Schiellein | Analysis of opto-microwave paths into a InP/InGaAs UTC-HPT[END_REF]. However, in Si based phototransistors in PD mode operation, the current measured at the base contact is much less than measured at collector contact at x=0 m and y=0 m (Ib=30 m and Ic=1mA). This difference can be explained from the structure of the phototransistor. As shown in Figure 1, we intentionally design a substrate contact to collect the holes which are generated in the substrate diode. Thus, the current measured at the base contact (Ib) does not include the substrate photocurrent. The electrons which are generated in the substrate are still collected at the collector contact.

The difference in the value between collector and base current at Vbe=0V in is the photocurrent from the substrate diode called substrate photocurrent. The topological mapping of the substrate photocurrent as a function of the probe position is given in Figure 6. There are two main peaks outside the active area, one near to the base contact (top) and the other near to the collector contact (down).

The peaks of the substrate photocurrent can be explained from the vertical and lateral structure of the HPT shown in Figure 1. When the optical probe moves over the structure, the optical beam passes through different stacks depending on the position of the probe. The two peaks of the substrate photocurrent, one near the collector contact and the other near the base contact, are due to the illumination of the photodiode created by n++ sub-collector and p type Si substrate.

Figure 7 shows the frequency behavior of HPT at the center of the optical window. We recognize a slope close to -20dB / decade in phototransistor mode, characteristic of the behavior of HPT, while the photodiode mode has a slope of about -10 dB/decade. The latter characterizes the behavior of the substrate, due to the difference in transit time of the photocarriers based on the detection depth into the substrate.

Figure 8-a) and b) shows the Optomicrowave Scanning Near Field Optical Microscopy (OM-SNOM) view of the low frequency opto-microwave responsivity under 50 condition in HPT and PD mode, respectively. In HPT mode, the peak of the responsivity appears inside the optical window. Whereas, in PD mode, the peak appears when the optical probe position starts to move outside the optical window and eventually creates a ring shape as shown in Figure 8-b). The ring is due to the illumination of the substrate diode created by n++ subcollector and p substrate. Indeed, it is much stronger than the photodiode created by the base and collector in the active area region. As shown in Figure 8-a) the ring shape does not appear in the phototransistor mode topological map as the substrate diode effects are hidden by the transistor amplification. A peak responsivity of 0.26A/W is measured at 50MHz and under Vce=3V and Vbe=0.857V (HPT mode).

The dynamic behavior of the phototransistor over the surface of the structure is analyzed through the measurement of the cutoff frequency of the HPT on both PD and HPT mode. Figure 9 presents the slice plot of the -3dB cutoff frequency at x=0 m. The cutoff frequency is usually small in phototransistor mode due to the large capacitance found in the forward biased base-emitter junction. Thus, it is assumed that the cutoff frequency is higher in the photodiode mode. However, the experimental result shown in Figure 9 indicates that the HPT mode cutoff frequency is much higher than PD mode cutoff frequency. This is due to the substrate diode that dominates over the diode created by the base and collector in PD mode operation. In HPT mode, the substrate diode is dominated by the transistor effect. Thus the substrate diode effect is stronger in PD mode operation than HPT mode.

. The cutoff frequency for y>-6 m and y<+6 m is meanly the dynamic response of the active area region. We reach up to 420MHz in HPT mode and 260MHz in PD mode operation. For y<-6 m and y>+6 m the cutoff frequency getting smaller and becomes equal in both modes. It indicates that in this region the cutoff frequency is extracted from the substrate.

IV. CONCLUSION This paper considered a phototransistor developed by using the existing SiGe HBT technology without modifying the vertical stacks and layers. The OM SNOM analysis revealed to be crucial to understand the behavior of SiGe HPTs. This allowed the extraction of the DC and dynamic behavior over the surface of the device. Slow carriers from the substrate highly affect the dynamic response of the structure in PD mode operation and it affects less in HPT mode of operation. We extract a maximum -3dB cutoff frequency of 0.42GHz in HPT mode and 0.26GHz in PD mode. We also extract a bandwidth responsivity product of 109.2MHzA/W. An alternative to get rid of the substrate contribution could be through a proper design of the optical window with proper metal diaphragms around it. Indeed, the substrate photodiode is hidden either by metal contacts or by upper layers of the intrinsic HPT, or through the use of a lateral illumination of the HPT.

Figure 3

 3 Figure 3 shows the experimental topological map of the base current in phototransistor mode (a) and photodiode mode (b)

Figure 4 :Figure 3 :Figure 2 : 5 Figure 1 :Emitter

 43251 Figure 4: Primary photocurrent distribution over the HPT structure.

Figure 8 :Figure 7 :Figure 6 :X 4 Figure 5 :

 87645 Figure 8: Low frequency opto-microwave responsivity in a) HPT mode, b) PD mode

Figure 9 :

 9 Figure 9: Opto-microwave -3dB frequency of the photodiode and phototransitor mode at X=0m for dc bias Vbe=0.857V and Vce=2V


T Cavoué

A Caravaca
email: angel.caravaca@ircelyon.univ-lyon1.fr

I Kalaitzidou

F Gaillard

M Rieu

J P Viricelle

P Vernoux

Ethylene epoxidation on Ag/YSZ electrochemical catalysts: Understanding of oxygen electrode reactions

Keywords: Electrochemical Impedance Spectroscopy (EIS), oxygen electrode reaction, Ag/YSZ, Ethylene epoxidation

In this study, we have investigated, for the very first time, the oxygen electrode reactions on Ag/YSZ electrochemical catalysts both in oxygen and under reaction conditions compatible with the ethylene epoxidation reaction. Electrochemical Impedance Spectroscopy (EIS) in combination with in-situ Raman spectroscopy and catalytic activity measurements were used to identify and understand the main oxygen reaction pathways. The results obtained suggested that the rate limiting step under an O2 reaction atmosphere (at 300 o C) is the O2 adsorption/dissociation process on the Ag catalyst-electrode. In addition, the polarization resistance increased with time under the presence of O2. This was attributed to the formation of Ag2O on the catalyst surface or near surface, which limits the oxygen electrode reactions.

Finally, we observed that the addition of ethylene in the feed stream hinders the electrode reaction, due to its competitive chemisorption with oxygen on Ag. These results give new insights into the design of selective Ag/YSZ catalyst for ethylene epoxidation.

Introduction

Ag-based electrodes interfaced with O 2-conductor solid electrolytes (e.g. Yttria Stabilized Zirconia, YSZ) have been extensively studied in literature for electro-catalytic processes [START_REF] Van Herle | Oxygen diffusion through silver cathodes for solid oxide fuel cells[END_REF][START_REF] Sakito | Silver infiltrated La0.6Sr0.4Co0.2Fe0.8O3 cathodes for intermediate temperature solid oxide fuel cells[END_REF][START_REF] Li | Electrochemical performance of La1.6Sr0.4NiO4-Ag composite cathodes for intermediate-temperature solid oxide fuel cells[END_REF]. This is mostly due to the fact that Ag exhibits excellent properties as an effective cathode for solid oxide fuel cells (SOFC), including its excellent electrical conductivity and high performance for the oxygen reduction reaction.

The electrochemical properties of these electrochemical cells have been therefore mostly studied at temperatures compatible with SOFC operation (600-1000 o C). According to literature, the oxygen electrode reactions include the following pathways (see Fig. S1) [START_REF] Bauerle | Study of solid electrolyte polarization by a complex admittance method[END_REF][START_REF] Vladikova | Impedance studies of cathode/electrolyte behaviour in SOFC[END_REF]: A) Transport of molecular oxygen into the pores of the electrode (O2 gas diffusion), B) Dissociative chemisorption of oxygen on the metal active sites (O2 gas  2 Oads), C) Diffusion of atomic oxygen to the electrode/electrolyte interface, D) Electron transfer reaction (Oads + 2e - O 2-). It is well understood that steps C and D should take place at the so-called triple phase boundaries (tpb): gas phase/catalyst-electrode (Ag)/solid electrolyte (YSZ).

However, Ag/YSZ materials could be also used for other applications at lower reaction temperatures. In this sense, since the discovery of the phenomenon of Electrochemical Promotion of Catalysis (EPOC), this kind of electrochemical cells could be considered as "electrochemical catalysts" [START_REF] Vayenas | Dependence of catalytic rates on catalyst work function[END_REF][START_REF] Vayenas | Non-faradaic electrochemical modification of catalytic activity: A status report[END_REF][START_REF] Vayenas | Electrochemical activation of catalysis: promotion, electrochemical promotion, and metal-support interactions[END_REF]. The particularity of these cells is based on the fact that the electrode behaves as a catalyst (catalyst-electrode) for a given reaction. This way, upon the application of an electrical current (or potential), a controlled amount of O 2-ions migrates through the solid electrolyte to the catalyst-electrode. From a catalytic point of view, the O 2- ions might behave as "electronic promoters", modifying the electronic density of the catalystelectrode, hence allowing to promote its activity and/or selectivity [START_REF] Stoukides | The effect of electrochemical oxygen pumping on the rate and selectivity of ethylene oxidation on polycrystalline silver[END_REF][START_REF] Hajar | Atomic layer deposition of highly dispersed Pt nanoparticles on a high surface area electrode backbone for electrochemical promotion of catalysis[END_REF][START_REF] Kalaitzidou | Electrochemical promotion of propylene combustion on Ag catalytic coatings[END_REF].

The EPOC phenomenon, discovered more than 30 years ago, can boost the catalytic activity and selectivity for a wide variety of environmental (e.g. reduction of NOx from mobile sources [START_REF] De Lucas-Consuegra | Use of potassium conductors in the electrochemical promotion of environmental catalysis[END_REF][START_REF] De Lucas-Consuegra | An electrochemically assisted NOx storage/reduction catalyst operating under fixed lean burn conditions[END_REF]) and industrial processes (e.g. ammonia synthesis [START_REF] Yiokari | Highpressure electrochemical promotion of ammonia synthesis over an industrial iron catalyst[END_REF], hydrogen production [START_REF] Lucas-Consuegra | Development of a new electrochemical catalyst with an electrochemically assisted regeneration ability for H2 production at low temperatures[END_REF]).

The promotional effect of O 2-ions on a catalyst-electrode was reported for the first time for the ethylene epoxidation reaction on a Ag/YSZ electrochemical catalyst [START_REF] Stoukides | The effect of electrochemical oxygen pumping on the rate and selectivity of ethylene oxidation on polycrystalline silver[END_REF][START_REF] Bebelis | Non-faradaic electrochemical modification of catalytic activity 6. Ethylene epoxidation on Ag deposited on stabilized ZrO2[END_REF][START_REF] Karavasilis | Non-faradaic electrochemical modification of catalytic activity: X. ethylene epoxidation on Ag deposited on stabilized ZrO2 in the presence of chlorine moderators[END_REF] for the production of ethylene oxide (EO). EO is an essential building block for chemical industry as a precursor of added-value molecules such as plastics and polyester fibers. EO is commercially produced at low temperatures (around 300°C) on Ag/α-Al2O3 catalysts using chlorinated hydrocarbons (1,2-dichloroethene (1,2-DCE) or vinyl chloride) in ppm quantities in the gas phase as moderators and alkali promoters (Cs ∼300 ppm). However, the competition with the direct oxidation process (leading to the production of CO2 instead of EO) leads to low EO yield and/or selectivity. Therefore, the industrial process to produce EO is neither efficient, nor eco-friendly (due to the use of 1,2-DCE). EPOC could be an alternative for a direct and environmentally-friendly new route for epoxidation of ethylene in order to produce EO with high selectivity and alkene conversion. For instance, in ref. [START_REF] Stoukides | The effect of electrochemical oxygen pumping on the rate and selectivity of ethylene oxidation on polycrystalline silver[END_REF] it was found that pumping O 2-ions towards the Ag electrode via applied polarization allowed to dramatically enhance the selectivity and yield of EO production at 400 o C and atmospheric pressure. It was demonstrated that the rate of EO production can exceed the rate of O 2- electrochemically supplied by a factor of 400. In other words, upon polarization, the enhancement of the EO production due to the catalytic process on the Ag catalyst-electrode (C2H4 + 0.5 O2 C2H4O) was much more pronounced than that for the electro-catalytic oxidation of C2H4 with the O 2-ions (C2H4 + O 2- C2H4O + 2 e -). The magnitude of this electrochemical activation strongly depends on the kinetics of the oxygen electrode reaction at the Ag/YSZ interface. A slow electrochemical process will increase the lifetime of ionic species. In that case, intermediate products such as partially discharged O δ-ionic species can migrate over the entire gas-exposed Ag surface and substitute Cl -anions. Furthermore, the lifetime of these O δ-ionic species is long enough if the coverage of "normal" chemisorbed oxygen species coming from the gas phase is high. Therefore, to predict the potential impact of EPOC for boosting the EO selectivity, it is crucial to unravel the oxygen electrode mechanism at the Ag/YSZ interface. Unfortunately, to the best of our knowledge, this has only been scarcely investigated below 400 °C, and never under ethylene epoxidation reaction conditions. Therefore, the aim of this work is, for the very first time in literature, to study and thoroughly understand the oxygen electrode mechanism at the Ag/YSZ interface at 300°C and under reaction conditions characteristic of the ethylene epoxidation catalytic process.

Experimental

In order to study the overall impedance of the electrochemical catalyst, a symmetric electrochemical cell was used with the following configuration: Ag//Yttria-Stabilized Zirconia (YSZ)//Ag, where Ag catalyst-electrodes with a geometrical area of 2.7 cm², were placed on the opposite sides of a YSZ solid electrolyte disk. Electrochemical properties were investigated in the same atmosphere for both electrodes and without any polarization. Therefore, both Ag electrodes have the same electrochemical properties and the impedance spectra represents the overall contribution of the two electrodes. First, a YSZ disk of 18.7 mm diameter and 1 mm thick was prepared by sintering a commercial YSZ powder (Tosoh, ZrO2-8 mol % Y2O3) at 1350 °C for 2 h (densification higher than 98%). We decided to prepare Ag coatings with similar morphology and porosity than those implemented in early studies on EPOC for ethylene epoxidation [START_REF] Stoukides | The effect of electrochemical oxygen pumping on the rate and selectivity of ethylene oxidation on polycrystalline silver[END_REF][START_REF] Bebelis | Non-faradaic electrochemical modification of catalytic activity 6. Ethylene epoxidation on Ag deposited on stabilized ZrO2[END_REF][START_REF] Karavasilis | Non-faradaic electrochemical modification of catalytic activity: X. ethylene epoxidation on Ag deposited on stabilized ZrO2 in the presence of chlorine moderators[END_REF]. Commercial silver-organic ink (Metalor ® HPS-FG32) was deposited on both faces via the screen printing method, followed by calcination at 600°C for 2 h in air (heating ramp of 2 °C min -1 ). The final Ag loading was ~20 mg on each electrode. The sample exhibits low porosity with Ag agglomerates in the micrometer range, which could be attributed to the high calcination temperature, as previously observed [START_REF] Kalaitzidou | Electrochemical promotion of propylene combustion on Ag catalytic coatings[END_REF].

For electrochemical measurements, the Ag//YSZ//Ag sample was placed between two Au current collectors in an alumina holder inside a continuous-flow quartz reactor. The current collectors were attached to Au wires and then connected to a potentiostat/galvanostat VersaSTAT 3 (Princeton Applied Research). The impedance measurements were recorded by the Frequency Response Analyzer (FRA) Module of the potentiostat/galvanostat over a frequency range from 1 MHz to 1 mHz with an amplitude of the AC signal of 20 mV. All the tests were performed at 300 o C under an overall gas flow of 10 L h -1 . This temperature is compatible with the ethylene epoxidation reaction. Two different reaction atmospheres were investigated: O2 in He (0.8 to 8 % volume) and the ethylene epoxidation reaction mixture (4 % C2H4 / 1 % O2) at atmospheric pressure. The impedance data analysis was performed by the Z-view Software.

For catalytic measurements, the sample was placed in a quartz reactor described elsewhere [START_REF] Kalaitzidou | Electrochemical promotion of propylene combustion on Ag catalytic coatings[END_REF]. The reactants and products were analyzed by an on-line micro gas chromatograph (μGC-R3000, SRA), composed of a BF PPU 8m/PPQ 1m column (CO2 analysis) and a VAR PLOT Q 8m column (Ethylene Oxide analysis). The detection limit for Ethylene oxide analysis was ~ 2 ppm. A light-off experiment was performed from 100 °C to 300 °C under ethylene epoxidation reaction conditions (4 % C2H4, 1 % O2, 3 L h -1 , He balance).

Raman spectroscopy was performed in-situ at 300 o C under an 8 % O2 (in N2, 1.5 L h -1 ) reaction atmosphere in the region of 200-1600 cm -1 using a LabRam HR (Horiba) spectrometer equipped with a CCD detector cooled at -75 °C. Measurements were carried out under microscope with a laser wavelength of 514.53 nm. A low dispersion grating of 1800 grooves/mm was chosen to get a maximum signal (to improve the resolution). The spectra were recorded and treated using the Labspec software (Horiba). Prior to the analysis, the instrument was calibrated using a silicon wafer.

Results and discussion

Electrochemical properties in oxygen

The oxygen electrode reaction on the Ag electrode was carried out under an oxygen-based reaction atmosphere (in the absence of ethylene). Figure 1 shows a typical impedance spectrum of the Ag/YSZ electrochemical catalyst under 8 % O2 at 300°C. The spectrum could be separated into two arcs located in the high frequency zone (1 MHz to 1 KHz) and two arcs in the low frequency zone (1 KHz to 1 mHz). To determine the contribution of each process of the oxygen electrode reaction, a typical equivalent circuit [START_REF] Nielsen | Impedance of SOFC electrodes: A review and a comprehensive case study on the impedance of LSM:YSZ cathodes[END_REF] composed of a resistance and a Constant Phase Element (CPE) in parallel for every depressed-arc was used to fit the data (see inset on Fig 1). Usually, the YSZ solid electrolyte resistance produces at least two different arcs in the high frequency region, where the first one (highest frequency range) could be attributed to the charge transport through the bulk of the electrolyte, and the second arc is generally due to the transport through the grain boundaries [START_REF] Bauerle | Study of solid electrolyte polarization by a complex admittance method[END_REF][START_REF] Vladikova | Differential impedance analysis of single crystal and polycrystalline yttria stabilized zirconia[END_REF]. The low frequency contributions observed in the spectra could be therefore attributed to the oxygen electrochemical reaction over the electrodes, which seems to be composed of at least two different electrochemical processes. In the proposed equivalent circuit, R1 and R2 correspond to the resistance of the solid electrolyte, while the sum of R3 and R4 represents the total polarization resistance for the overall electrode reaction [START_REF] Van Herle | Oxygen diffusion through silver cathodes for solid oxide fuel cells[END_REF].

½ O2 gas + 2e - O 2-(1)
CPE are constant phase elements, which in general could be related to the double-layer capacitance [START_REF] Bauerle | Study of solid electrolyte polarization by a complex admittance method[END_REF]. Further experiments were performed in order to associate the main steps for the oxygen electrode reactions with the R3 and R4 contributions observed on Fig. 1. To this purpose, the effect of the oxygen partial pressure on the polarization resistance was explored at the same reaction temperature (300 o C). The oxygen concentration was varied from 0.8 % to 8 %. Fig. 2a shows the impedance spectra of the low frequency region (1 KHz to 1 mHz, electrode reactions). The experiments were performed in the chronological order indicated by the arrow in the legend. Hence, the spectra under 4 % and 8 % O2 were taken twice. It could be observed (Fig. 2a) a very important influence of the O2 concentration on the impedance spectra, increasing both R3 and R4 resistances as the oxygen partial pressure decreases.

Previous studies [START_REF] Li | Electrochemical performance of La1.6Sr0.4NiO4-Ag composite cathodes for intermediate-temperature solid oxide fuel cells[END_REF][START_REF] Siebert | Impedance spectroscopy analysis of La1 -xSrxMnO3-yttria-stabilized zirconia electrode kinetics[END_REF] have reported that the variation of the polarization resistance with the O2 partial pressure obeys the equation [START_REF] Sakito | Silver infiltrated La0.6Sr0.4Co0.2Fe0.8O3 cathodes for intermediate temperature solid oxide fuel cells[END_REF].

R = k PO2 -n (2)
Fig. 2b shows the log-log variation of R3 and R4 polarization resistances (obtained by treatment of Fig. 2a spectra) with oxygen partial pressure. The value of the slope "n" has been found to provide important insights regarding the species involved in the electrode reactions. For instance, Siebert et al. [START_REF] Siebert | Impedance spectroscopy analysis of La1 -xSrxMnO3-yttria-stabilized zirconia electrode kinetics[END_REF] found that n = 0.5 could be attributed to a concentration variation of adsorbed dissociated oxygen. In other words, it could be related with the dissociative chemisorption of O2 and the further diffusion of atomic oxygen to the "tpb" (Paths B and C in Fig. S1). On the other hand, n values equal to 0.25 and 1 have been attributed to the charge transfer step (Path D in Fig. S1) and the O2 gas diffusion in the electrode (Path A in Fig. S1), respectively [START_REF] Li | Electrochemical performance of La1.6Sr0.4NiO4-Ag composite cathodes for intermediate-temperature solid oxide fuel cells[END_REF]. Based on the results shown on Fig. 2b, a PO2 - 0.22 dependency has been observed for R3 (higher frequency part of the polarization resistance), which could be attributed to the charge transfer process at the Ag/YSZ interface. However, a PO2 -0.72 dependency was observed for R4 (lower frequency). According to the previous discussion, this resistance could be attributed to either the oxygen dissociative chemisorption (n = 0.5) or to the gaseous diffusion of O2 in the porosity of the Ag electrode (n = 1).

A pure unlimited O2 gas diffusion process (usually represented by the so-called Warburg element) manifests itself in the impedance spectra as a straight line making a 45 o angle with the Zreal axis [START_REF] Van Herle | Oxygen diffusion through silver cathodes for solid oxide fuel cells[END_REF]. Based on the shape of these spectra, it seems that the O2 gas diffusion resistance is negligible, and therefore the low frequency resistance R4 could be mostly attributed to the oxygen adsorption process. Similar results were obtained by Li et al. [START_REF] Li | Electrochemical performance of La1.6Sr0.4NiO4-Ag composite cathodes for intermediate-temperature solid oxide fuel cells[END_REF] with a La1.6Sr0.4NiO4-Ag/YSZ electrochemical cell. At 700 o C, they observed two different contributions to the electrode polarization resistance, which were attributed to the adsorption/desorption process (low frequency) and the charge transfer (high frequency).

The EIS technique could be also used to clarify the rate limiting step regarding the O2 electrode reaction. Based on the results shown on Figs 2 a and b, it is evident that R4 is larger than R3. It seems to indicate that the oxygen adsorption/dissociation process is the major rate determining step for the Ag electrode in the whole range of O2 partial pressure studied, at the reaction temperature of 300 o C, in good agreement with a previous study performed by Huang et al. [21] for low temperatures SOFC (300-500 o C) on Ag-based electrodes.

The second spectra recorded under O2 4 % and 8 % exhibit an increased overall polarization resistance (R3 + R4) with respect to the first analysis. It seems to indicate that the electrode suffers some modification with time in the presence of oxygen. In order to verify this effect, Fig. 3a shows the evolution of the impedance spectra with time under an 8 % O2 reaction atmosphere. An important increase of the electrode resistance is observed, with a more pronounced effect on the R4 resistance (previously attributed to the oxygen dissociative adsorption process). In addition, Fig. 3b shows the comparison of the impedance spectra recorded under 8 % O2 for the Ag/YSZ electrochemical catalyst before and after a reducing treatment under a 5 % H2 reaction atmosphere (He balance, 300 o C, 10 L h -1 ) at 300°C.

Comparing the two spectra, it could be clearly observed that the R4 resistance strongly decreased (it almost disappeared) after the H2 treatment. A small shift was observed in the spectrum after H2 treatment, which could be attributed to a slightly lower reaction temperature (~ 5 o C). These results suggest that the interaction of the Ag catalyst-electrode with O2 leads to a significant modification of the electrode performance, mainly affecting the O2 dissociative adsorption path. reaction atmosphere was switched to O2 (8 % in N2, flow 1.5 L h -1 ) and Raman spectra were obtained every 2 hours for 15.5 hours at a constant temperature of 300 o C. We observe similar bands in all the Raman spectra but the area and the intensity of the bands at 800 cm -1 and 542 cm -1 significantly change as a function of time. The band at ~ 800 cm -1 was assigned by G. Waterhouse et al. [START_REF] Waterhouse | Oxygen chemisorption on an electrolytic silver catalyst: a combined TPD and Raman spectroscopic study[END_REF] to an Ag-O stretching mode, due to surface chemisorbed oxygen formed by the segregation of atomic oxygen coming from the Ag subsurface layers. This atomic oxygen strongly adsorbed on Ag was demonstrated to be stable even at high temperatures (up to 923 K) in the presence of O2 by in-situ Raman studies at different reaction temperatures. On the other hand, B. Pettinger et al. [START_REF] Pettinger | Thermal decomposition of silver oxide monitored by raman spectroscopy: From AgO Units to oxygen atoms chemisorbed on the silver surface[END_REF] performed Raman experiments on Ag-based materials at different reaction temperatures. They concluded that a band at approximately 530 cm -1 could be representative of bulk Ag2O species. The formation of Ag2O species was also observed by M.Fee et al. [START_REF] Fee | Electrochemical behavior of silver thin films interfaced with yttria-stabilized zirconia[END_REF] on Ag-based electrodes interfaced with a YSZ solid electrolyte under similar reaction conditions (O2-based reaction atmosphere at 350 o C). Therefore, the bands observed at 800 cm -1 and 542 cm -1 in Fig. 4a could be attributed to strongly chemisorbed oxygen on the Ag catalyst-electrode and to the oxidation of silver to silver oxide, respectively. Fig. 4b shows the evolution of the ratio of the areas of the two bands (800 cm -1 /542 cm -1 ) with time. The trend observed in this figure seems to indicate that the proportion of Ag2O with respect to that of chemisorbed oxygen increased with time at 300 o C in an oxygen-based reaction atmosphere. Other bands observed at 425 cm -1 and 910 cm -1 on the Raman spectra could be attributed to different kind of adsorbed oxygen species on the Ag catalyst-electrode [START_REF] Deng | In situ surface Raman spectroscopy studies of oxygen adsorbed on electrolytic silver[END_REF][START_REF] Boghosian | In situ high temperature SERS study of Ag catalysts and electrodes during ethylene epoxidation[END_REF].

R1 CPE1 R2 CPE2 R3 CPE3 R4 CPE4 Element Freedom Value Error Error % R1 Fixed(X) 0 N/A N/A CPE1-T Fixed(X) 0 N/A N/A CPE1-P Fixed(X) 1 N/A N/A R2 Fixed(X) 0 N/A N/A CPE2-T Fixed(X) 0 N/A N/A CPE2-P Fixed(X) 1 N/A N/A R3 Fixed(X) 0 N/A N/A CPE3-T Fixed(X) 0 N/A N/A CPE3-P Fixed(X) 1 N/A N/A R4 Fixed(X) 0 N/A N/A CPE4-T Fixed(X) 0 N/A N/A CPE4-P Fixed(X) 1 N/A N/
The results shown on Figs. 3 and4 suggest that the formation of silver oxide species strongly modifies and limits the oxygen electrode reaction on Ag under the reaction conditions studied. It increased the polarization resistance, mainly concerning the oxygen dissociative adsorption process (R4), which has been identified as the rate determining step. A reduction step in H2 (Fig. 3b) can reduce the silver oxide species, regenerating the Ag surface and promoting therefore the O2 electrode reaction.

S3

). In addition, Fig. S4 shows a light-off experiment to assess the catalytic performance of the Ag/YSZ electrochemical catalyst.

Under ethylene epoxidation conditions, with a significant excess of ethylene in the feed (as typically used for this catalytic process), the electrode polarization resistance significantly increased (Fig. 5a, Fig S3). In parallel, the catalytic performances are poor, reaching a very low ethylene conversion of around 0.045 % at 300°C. We did not detect any ethylene oxide production, suggesting that the Ag electrode is only active for ethylene combustion. These results indicate that an important amount of ethylene is adsorbed on the Ag catalyst-electrode under these conditions, hindering in a significant manner the O2 chemisorption and then the electrode reaction and the catalytic reaction. In addition, a new impedance spectrum was obtained under 1 % O2 reaction atmosphere (Fig. 5a), right after stopping the flow of ethylene. In comparison with the spectrum under similar conditions before the ethylene epoxidation treatment, an important decrease in the polarization resistance (mostly concerning R4) is observed. This effect was similar but less pronounced to that observed after a H2 treatment (Fig. 3b). It could be attributed to the consumption of the silver oxide species with the ethylene present in the reaction atmosphere. In other words, ethylene seems to have an important reducing effect on the Ag catalyst-electrode. This effect was further verified by following the evolution of the impedance spectra under 1 % O2 with time, right after ethylene epoxidation reaction conditions (Fig. 5b). A significant increase of the polarization resistance is observed with time, similar to that observed on Fig. 3a, which could be most likely attributed to the slow and gradual formation of silver oxide species.

The results obtained in this study allowed, for the very first time, to relate the catalytic properties of an electrochemical catalyst (Ag/YSZ) with electrochemical measurements (Electrochemical Impedance Spectroscopy) concerning the oxygen electrode reactions. The experiments were performed under reaction conditions compatible with the ethylene epoxidation reaction (low reaction temperature: 300 o C, and high ethylene excess). The presence of ethylene seems to hinder the oxygen adsorption on the Ag catalyst-electrode, increasing the overall polarization resistance attributed to the oxygen electrode reactions. A slow electrochemical process is a requirement for achieving significant electrochemical promotion. However, our results also emphasize a low oxygen coverage on Ag which will be detrimental for the lifetime of O δ-ionic promoters. In addition, this low oxygen coverage also explains the poor catalytic performances of the catalyst. We decided to prepare this design of Ag/YSZ catalysts with a low porosity and large micrometric Ag agglomerates to mimic early studies on EPOC for ethylene epoxidation [START_REF] Stoukides | The effect of electrochemical oxygen pumping on the rate and selectivity of ethylene oxidation on polycrystalline silver[END_REF][START_REF] Bebelis | Non-faradaic electrochemical modification of catalytic activity 6. Ethylene epoxidation on Ag deposited on stabilized ZrO2[END_REF][START_REF] Karavasilis | Non-faradaic electrochemical modification of catalytic activity: X. ethylene epoxidation on Ag deposited on stabilized ZrO2 in the presence of chlorine moderators[END_REF]. However, according to the findings of this study, this design has to be improved to enhance the oxygen coverage on Ag by preparing a catalytic layer containing Ag nanoparticles. Our results also emphasize that micrometric Ag particles interfaced on YSZ can stabilize Ag2O on their surface at 300°C as shown by Raman spectroscopy. It has been found that the dissolution of oxygen in the Ag bulk or near surface plays a key role [START_REF] Böcklein | High-pressure scanning tunneling microscopy of a silver surface during catalytic formation of ethylene oxide[END_REF] in the stabilization of electrophilic oxygen weakly adsorbed on Ag which are considered as the active sites for the production of EO [START_REF] Bukhtiyarov | Combined in situ XPS and PTRMS study of ethylene epoxidation over silver[END_REF].

Therefore, the combination of micrometric and nanometric Ag particles supported on YSZ could lead to a selective catalyst for the ethylene epoxidation.

Conclusions

EIS measurements were performed to relate the O2 electrode reaction kinetics with the catalytic performances of Ag/YSZ electrochemical catalysts. Experiments were implemented in oxygen and in an oxygen/ethylene mixture representative of the ethylene epoxidation reaction (300°C, ethylene rich composition). Under O2, two different polarization resistances were observed, which were attributed to the electron transfer and the oxygen dissociative adsorption, the latter being the rate determining step. This polarization resistance was found to increase with time, most likely due to the formation of silver oxide species on the surface or sub-surface, as confirmed by in-situ Raman characterizations. The presence of high ethylene excess in the feed hinders the O2 electrode reaction. This was attributed to the preferential chemisorption of ethylene (in competition with O2) on the Ag catalyst-electrode, in good with poor measured catalytic performances. These results give new insights for the design of active and selective electro-promoted Ag/YSZ catalysts for ethylene epoxidation.

  Fig. S2 a) and b) show SEM images of the "fresh" electrochemical catalyst. The thickness of the Ag catalyst-layer is in the range 10-25 μm.

Figure 1 .

 1 Figure 1. Impedance spectrum of the Ag/YSZ/Ag symmetric electrochemical catalyst. Reaction conditions: 8 % O2 in He, overall flow = 10 L h -1 , Temperature = 300 o C

Figure 3 .

 3 Figure 3. Impedance spectra of the Ag/YSZ/Ag symmetric electrochemical catalyst under 8% O2 (10 L h -1 , Temperature = 300 o C): a) Influence of time on stream, and b) Influence of a pretreatment under a 5 % H2 reaction atmosphere (He balance, 10 L h -1 , 300 o C, 34 hours).

Figure 4 .

 4 Figure 4. a) in-situ Raman spectra of a Ag/YSZ electrochemical catalyst. Pretreatment: 6 % H2 in N2, total flow 1.5 L h -1 , 1 hour, 300 o C. Spectra taken at 300 o C under O2 8 % (total flow 1.5 L h -1 , N2 balance). b) Evolution of the ratio of the band areas at 800 cm -1 and 542 cm -1 with time.

Figure 5 .

 5 Figure 5. a) Impedance spectra of the Ag/YSZ/Ag symmetric electrochemical catalyst under different reaction atmospheres: ) 1 % O2 in He (10 L h -1 , 300 o C), ) 1 % O2 + 4 % C2H4 in He (10 L h -1 , 300 o C), ) 1 % O2 in He (10 L h -1 , 300 o C) after ethylene epoxidation). b) Impedance spectra of the Ag/YSZ/Ag symmetric electrochemical catalyst under 8% O2 (10 L h -1 , Temperature = 300 o C): Influence of time in stream after ethylene epoxidation conditions. The first spectrum started at t = 0, the second one at t= 1.5 hours and the third one at t = 3 hours.

  Fig. S2. SEM images of the Ag/YSZ electrochemical catalyst after calcination at 600 o C: a) cross-view, b) surface of the Ag catalyst-electrode.

Acknowledgments

This study was performed in the "EPOX" project, funded by the French National Research Agency (ANR), ANR-2015-CE07-0026.


Arthur Marmin

Marc Castella

Jean-Christophe Pesquet

HOW TO GLOBALLY SOLVE NON-CONVEX OPTIMIZATION PROBLEMS INVOLVING AN APPROXIMATE 0 PENALIZATION

Keywords: polynomial and rational optimization, global optimization, 0 penalization, sparse modeling

For dealing with sparse models, a large number of continuous approximations of the 0 penalization have been proposed. However, the most accurate ones lead to non-convex optimization problems. In this paper, by observing that many such approximations are piecewise rational functions, we show that the original optimization problem can be recast as a multivariate polynomial problem. The latter is then globally solved by using recent optimization methods which consist of building a hierarchy of convex problems. Finally, experimental results illustrate that our method always provides a global optimum of the initial problem for standard 0 approximations. This is in contrast with existing local algorithms whose results depend on the initialization.

INTRODUCTION

When rich statistical or variational models are employed, it is often necessary to select the most economical one (i.e. involving the fewest parameters) so as to avoid overfitting. In this context, compressive sensing has become a key solution to represent data in a sparse form in order to compress them efficiently and extract the salient information.

To promote sparse solutions and estimators, a common approach consists of adding an 0 penalization to a data-fit cost function [START_REF] Nikolova | Description of the minimizers of least squares regularized with 0 norm. Uniqueness of the global minimizer[END_REF]. Nevertheless, this is known to lead to NP hard optimization problems. Consequently, several surrogates to the 0 penalization have been suggested, the simplest one being the 1 norm. The latter has the enjoyable property of being convex, which simplifies the optimization task [START_REF] Combettes | Proximal thresholding algorithm for minimization over orthonormal bases[END_REF][START_REF] Combettes | Proximal Splitting Methods in Signal Processing[END_REF], but it also strongly penalizes high values of the variables and thus introduces a bias in the solutions. Therefore further relaxations of 0 function have been investigated (see [START_REF] Fan | Variable selection via nonconcave penalized likelihood and its oracle properties[END_REF]). A major drawback is that those relaxations are non-convex and result in optimization problems which are difficult to solve globally. Most of the current available algorithms are limited in the sense that they only converge to local solutions due to the non-convexity of the cost function and they are therefore highly dependent on their initialization. The iterative hard thresholding [START_REF] Blumensath | Iterative thresholding for sparse approximations[END_REF] is a well-known example of such an algorithm that provides a local minimum for an 0 penalized cost function. The work in [START_REF] Patrascu | Random coordinate descent methods for 0 regularized convex optimization[END_REF] suggests stronger optimality conditions to provide local minima close to the global one. Non-convex approximations to the 0 function that maintain the convexity of the overall cost function have also been proposed [START_REF] Selesnick | Sparse regularization via convex analysis[END_REF] to tackle this issue but they require specific assumptions to be met.

In this work, we propose a method to find the global minimum of a wide class of criteria involving non-convex approximations to 0 function [START_REF] Fan | Variable selection via nonconcave penalized likelihood and its oracle properties[END_REF][START_REF] Zhang | Analysis of multi-stage convex relaxation for sparse regularization[END_REF][START_REF] Zhang | Nearly unbiased variable selection under minimax concave penalty[END_REF][START_REF] Artina | Linearly constrained nonsmooth and nonconvex minimization[END_REF][START_REF] Jezierska | A fast solver for truncated-convex priors: Quantizedconvex split moves[END_REF][START_REF] Soubies | A continuous exact 0 penalty (CEL0) for least squares regularized problem[END_REF]. We show that such criteria are leading to optimization problems which are actually piecewise rational. We use the framework of Lasserre's hierarchy [START_REF] Lasserre | Global optimization with polynomials and the problem of moments[END_REF] which allows polynomial optimization problems to be globally solved. This framework has been extended to rational optimization problems in [START_REF] Bugarin | Minimizing the sum of many rational functions[END_REF][START_REF] Castella | Rational optimization for nonlinear reconstruction with approximate 0 penalization[END_REF]. Here, we further extend the method to find the global extrema of a piecewise rational optimization problem. We compare the obtained solutions with the ones provided by classical algorithms in the literature [START_REF] Fan | Variable selection via nonconcave penalized likelihood and its oracle properties[END_REF][START_REF] Ochs | On iteratively reweighted algorithms for nonsmooth nonconvex optimization in computer vision[END_REF][START_REF] Candès | Enhancing sparsity by reweighted 1 minimization[END_REF][START_REF] Breheny | Coordinate descent algorithms for nonconvex penalized regression, with applications to biological feature selection[END_REF]. In contrast with existing methods, we are able to find and certify the global optimum.

Our paper is organized as follows: Section 2 introduces the problem and gives examples of widely used approximations to 0 function. Section 3 explains how to recast a piecewise rational problem into a rational problem and how to apply it on our optimization problem. Simulation results are presented in Section 4. Finally, some concluding remarks are drawn in Section 5. In the following, the characteristic function of a given set X is denoted by 1 {•∈X } with 1 {x∈X } = 1 if x ∈ X and 0 otherwise.

PROBLEM STATEMENT

We tackle the minimization problem of the following composite criterion J :

(∀x ∈ R T ) J (x) = f y (x) + R λ (x) .
The optimization variable is thus a real-valued vector of dimension T , f y is a fitting function depending on a vector of observations y and assumed to be rational. R λ is a regularization that promotes sparsity and depends on a parameter λ ∈]0, +∞[. Ideally, we would like R λ to be the sparsity measure 0 but, in order to derive computationally efficient optimization techniques, a suitable separable approximation is substituted for it:

R λ (x) = T t=1 Ψ λ (x t ) .
The function Ψ λ : R → R then requires the following three properties [START_REF] Fan | Variable selection via nonconcave penalized likelihood and its oracle properties[END_REF]: unbiasedness for large values, sparsity to reduce the complexity of the model, and continuity to ensure the stability of the model. Those conditions lead to non-convex functions. Indeed, the unbiasedness condition implies that the penalization is constant for large values of the variable and the sparsity and continuity conditions lead to a null value at zero. Below and displayed in Figure 1, we give examples of some of the most famous penalizations that satisfy the above three properties.

• Capped p [START_REF] Zhang | Analysis of multi-stage convex relaxation for sparse regularization[END_REF][START_REF] Artina | Linearly constrained nonsmooth and nonconvex minimization[END_REF][START_REF] Jezierska | A fast solver for truncated-convex priors: Quantizedconvex split moves[END_REF]:

(∀p ∈ N\{0}) Ψ λ (x) = |x| p 1 {|x|≤λ} +λ p 1 {|x|>λ} .
• Smoothly clipped absolute deviation (SCAD) [START_REF] Fan | Variable selection via nonconcave penalized likelihood and its oracle properties[END_REF]:

Ψ λ (x) = λ |x| 1 {|x|≤λ} - λ 2 -2γλ |x| + x 2 2(γ -1) 1 {λ<|x|≤γλ} + (γ + 1)λ 2 2 1 {|x|>γλ} ,
where γ is a parameter taking values in ]2, +∞[.

• Minimax concave penalty (MCP) [START_REF] Zhang | Nearly unbiased variable selection under minimax concave penalty[END_REF]:

Ψ λ (x) = λ |x| - x 2 2γ 1 {|x|≤γλ} + γλ 2 2 1 {|x|>γλ} ,
where γ ∈]0, +∞[.

• Continuous exact 0 (CEL0) [START_REF] Soubies | A continuous exact 0 penalty (CEL0) for least squares regularized problem[END_REF]:

Ψ λ (x) = λ - γ 2 2 |x| - √ 2λ γ 2 1 |x|≤ √ 2λ γ , where γ ∈]0, +∞[.
Note that the lower the parameter γ, the closer the approximation to 0 function but the stronger also the non-convexity. Since those functions are non-convex, the existing first or second-order optimization algorithms are only guaranteed to deliver a local minimizer of the criterion J [START_REF] Fan | Variable selection via nonconcave penalized likelihood and its oracle properties[END_REF][START_REF] Zhang | Nearly unbiased variable selection under minimax concave penalty[END_REF]. However, these methods do not exploit the fact that the previous penalizations are all piecewise polynomial functions. We show in Section 3 that the minimization of a piecewise rational function like J can reformulated as the minimization of a polynomial function under polynomial constraints. Then, the latter is relaxed into a hierarchy of semi-definite programming (SDP) problems that yields the global minimizers of the original function.

RATIONAL FORMULATION

Reformulation of the considered class of penalties

Let us first show how to recast into a constrained rational optimization problem, the minimization of any piecewise rational function. Assume that this function reads

(∀x ∈ R) Ψ(x) = I i=1 g i (x)1 {σi-1≤x<σi} , (1)
where (g i ) 1≤i≤I are rational functions, I is a nonzero integer and (σ i ) 0≤i≤I is an increasing sequence of real values. The function Ψ is our basic block for building the class of piecewise rational approximation to 0 function. All the functions discussed in the previous section can be rewritten under this form.

We introduce the binary variables z (i) such that

(∀i ∈ 0, I ) z (i) = 1 {σi≤x} .
Note that we can possibly set σ 0 = -∞, z (0) = 1 and σ I = +∞, z (I) = 0 to define Ψ on the whole real line R. From the definition of z (i) , we deduce that

1 {σi-1≤x<σi} = z (i-1) (1 -z (i) ) . (2)
Finally, the constraint z (i) = 1 {σi≤x} is equivalent to two polynomial constraints

z (i) = 1 {σi≤x} ⇔ (z (i) ) 2 -z (i) = 0 z (i) -0.5 (x -σ i ) ≥ 0 . (3)
Indeed, the polynomial equality constraint enforces z (i) to be a binary variable while the polynomial inequality constraint ensures that it takes the same values as 1 {σi≤x} . Therefore, by combining (1)-(3), we obtain the minimization of a rational function depending on x and z (i) 0≤i≤I under polynomial constraints. In turn, it has been shown in [START_REF] Bugarin | Minimizing the sum of many rational functions[END_REF][START_REF] Castella | Rational optimization for nonlinear reconstruction with approximate 0 penalization[END_REF] how rational problems can be reduced to polynomial ones.

Reformulation of our optimization problem

By applying the previous reasoning to penalization R λ , we reformulate the original minimization of J . By introducing the binary variables z = (z

(i) t ) 0≤i≤I,0≤t≤T , the optimization problem becomes minimize x,z f y (x) + T t=1 I i=1 g i (x t )z (i-1) t (1 -z (i) t ) s.t. (∀(i, t) ∈ 0, I × 1, T ) (z (i) t ) 2 -z (i) t = 0 z (i) t -0.5 (x t -σ i+1 ) ≥ 0 . (4)
Problem ( 4) is a rational optimization problem since f y and (g i ) 1≤i≤I are rational by assumption and the constraints are polynomial. To find the global minimizers, we can thus exploit the problem structure and use the technique detailed in [START_REF] Castella | Rational optimization for nonlinear reconstruction with approximate 0 penalization[END_REF] which is based on Lasserre's method. It consists in relaxing a polynomial problem into a hierarchy of convex SDP problems indexed by an integer k corresponding to the relaxation order. Solving each SDP problem yields a lower bound J * k on the optimal value J * of the criterion J . Furthermore, the higher the order k, the tighter the bound J * k but the higher also the dimensions of the SDP problem. It has been proved [START_REF] Lasserre | Global optimization with polynomials and the problem of moments[END_REF] that (J * k ) k∈N is an increasing convergent sequence whose limit is J * . Finally, the solution x of the polynomial problem is extracted from the solution of the SDP problem [START_REF] Henrion | Detecting global optimality and extracting solutions in GloptiPoly[END_REF] and we can theoretically certify that x is a global minimizer by comparing J (x) to J * k .

Examples

To clarify the previous reformulation, we demonstrate it on the regularizations given in Section 2. Note that since the considered penalizations are even, we can use the resulting symmetry to decrease the number of variables and hence reduce the computations. Absolute values can be handled with an additional variable constrained to be nonnegative and whose square is equal to the square of the absolute value of the original variable [START_REF] Castella | Rational optimization for nonlinear reconstruction with approximate 0 penalization[END_REF]. Since SCAD has three pieces, it requires to introduce variables z

(1) t and z

(2) t so leading to minimize

x,z

f y (x) + T t=1 (1 -z (1) t )λ |x t | -z (1) t (1 -z (2) t ) λ 2 -2γλ |x t | + x 2 t 2(γ -1) + z (2) t (γ + 1)λ 2 2 s.t. (∀(i, t) ∈ {1, 2} × 1, T ) (z (i) t ) 2 -z (i) t = 0 (∀t ∈ 1, T ) z (1) t -0.5 (|x t | -λ) ≥ 0 (∀t ∈ 1, T ) z (2) t -0.5 (|x t | -γλ) ≥ 0 .
A similar approach applies to Capped p , MCP, and CEL0; the details are omitted for conciseness.

NUMERICAL SIMULATIONS

Simulation scenario

We consider the following degradation model:

y = Hx + w ,
where w is a zero-mean white Gaussian noise and H is a Toeplitz band matrix corresponding to a Gaussian convolution filter. For each test, coefficients of the vector x were randomly drawn according to a uniform distribution on [0.6, 1]. We set the size T of the vectors to 200. We defined the ideal criterion J 0 as

J 0 (x) = 1 2 y -Hx 2 + λ 0 (x) ,
where λ is set to 0.1.

Our goal is to approximate this criterion using Capped 1 , SCAD, MCP, and CEL0 instead of 0 penalization. For each regularization, we compared out method to Forward-Backward (FB), Iteratively Reweighted 1 (IRL1) [START_REF] Ochs | On iteratively reweighted algorithms for nonsmooth nonconvex optimization in computer vision[END_REF][START_REF] Candès | Enhancing sparsity by reweighted 1 minimization[END_REF] and Coordinate Descent (CD) [START_REF] Breheny | Coordinate descent algorithms for nonconvex penalized regression, with applications to biological feature selection[END_REF] algorithms. IRL1 minimizes iteratively the fitting function penalized with a weighted 1 norm whose weights are computed by linearizing the nonconvex penalization function. CD solves a sequence of singledimensional problems by fixing all the variables except one in the original problem. We used GloptiPoly [START_REF] Henrion | Glop-tiPoly 3: moments, optimization and semidefinite programming[END_REF] to relax rational problems into SDP problems which are then solved with the solver SDPT3 [START_REF] Toh | SDPT3 -a Matlab software package for semidefinite programming, version 1.3[END_REF]. The relaxation was performed at order k equal to 2 or 3. The parameter γ was set to 0.5 for MCP and 2.1 for SCAD. Following [START_REF] Soubies | A continuous exact 0 penalty (CEL0) for least squares regularized problem[END_REF], we used the norm of the t-th column of H as parameter γ for the input variable x t in CEL0.

Local methods and initial points

We first show that FB, IRL1 and CD can get trapped in local minima whereas our method is guaranteed to provide a global minimizer. We ran the algorithms using different initial points x init : a random point, zero, and the true value x (which is not available in practice). Notice that our method does not need any initial point. Table 1 compares the criterion value at the solutions returned by the different algorithms.

We observe that all the local algorithms have different estimated optimal criterion values and thus return different minimizers depending on the starting point x init . Furthermore, it can be observed that local optimization algorithms often are not reliable. Hence for Capped 1 , CD yields a better minimizer than IRL1 when using a random starting point, whilst the converse holds when 0 is the starting point. Without a global method, we cannot assess the quality of the obtained solution. Interestingly, our method provides the lowest criterion value; the next section confirms the validity of the method.

Global vs local algorithms

We now compare the different values of the criterion at the solutions returned by each algorithm as well as the lower bound J * k . Figure 2 shows the values of the criterion for the SCAD regularization. We ran 200 tests on randomly generated data but, for the sake of clarity, only 100 are plotted. The results are ordered according to the value of the criterion at the minimizer x obtained with our approach. We observe that our method always yields a better minimizer of the criterion while the other methods are sensitive to local minimizers.

Moreover, we compare the estimated optimal value of the criterion J (x) given by our method with the lower bounds J * 2 and J * 3 . Table 2 shows statistics over the 200 tests about the difference between these values. Up to a numerical precision of 10 -4 , Ĵ * reaches the lower bound with a relaxation order of 2 for Capped 1 and 3 for the three other regularizations. The global optimality is therefore theoretically certified in these cases. It is remarkable that Lasserre's hierarchy converges for a low value of k. Test ID Number

Case of real-valued signals

We provide additional simulation results concerning realvalued signals x. In the case of positive signals, we could naturally drop the absolute values since |x| = x. In contrast, the case of real-valued signals is more intricate due to the introduction of additional variables to handle this absolute value. As a consequence, the convergence of Lasserre's hierarchy does not occur as fast as in the positive case. However, we can use the solution obtained with an order of relaxation k = 3 as an initial point of a local method and improve both solutions as pointed out by Table 3. The latter shows the statistics on J (x) for FB and FB initialized with Lasserre's solution (FBwL) using the SCAD and MCP regularizations.

CONCLUSION

We study the global optimization of cost functions penalized by classic non-convex continuous approximations to 0 regularization. By showing that the original optimization problem can be reformulated as a polynomial problem, we obtain sparse solutions through convex relaxation techniques. Finally our simulation results illustrate that the main benefit of our approach is to secure a global optimum of the criterion in contrast with state-of-the-art local methods.

Fig. 1 .

 1 Fig. 1. Examples of continuous relaxation of 0 penalization (λ = 1, γ SCAD = 2.5, γ MCP = 2, γ CEL0 = 1).

Fig. 2 .

 2 Fig. 2. Optimal criterion values with SCAD regularization.

Table 1 .

 1 Optimal criterion value depending on initial point.

	Alg.	Ψ λ	Capped 1 SCAD MCP CEL0
	Proposed (k = 3)	3.725	3.506 2.749 4.425
	Random x init
	FB		4.372	3.759 3.194 4.638
	IRL1	7.052	4.632 3.381 6.926
	CD		4.099	3.927 3.520 4.638
	x init = 0
	FB		4.208	3.763 3.149 4.638
	IRL1	4.839	4.566 3.013 6.879
	CD		5.455	4.371 3.717 5.316
	x init = x
	FB		3.867	3.639 2.871 4.637
	IRL1	4.766	4.567 2.914 6.874
	CD		4.093	3.639 3.015 5.365

Table 2 .

 2 Statistics on J (x) -J * k for positive signals.

	(200 runs)	Capped 1 SCAD	MCP	CEL0
	J (x) -J * 2	Avg 2.0e-4 3.4e-2 5.0e-2 2.3e-3 Med 2.0e-4 3.3e-2 5.0e-2 9.0e-4
	J (x) -J * 3	Avg 2.0e-4 4.0e-5 1.0e-4 2.0e-4 Med 2.0e-4 <1e-5 1.0e-4 <1e-5

Table 3 .

 3 Statistics on J (x) for real-valued signals.

	(200 runs)	FB FBwL
	SCAD	Average 4.317 3.977 Median 4.301 3.969
	MCP	Average 3.936 3.426 Median 3.920 3.419