Yohan Wanderoild
email: yohan.wanderoild@cea.fr

Adrien Morel

Romain Grezaud

Gael Pillonnet

Dominique Bergogne

Hubert Razik

Optimal compensation capacitors maximizing coreless inductive power transfer

The use of coreless transformers is still limited because of the power they can transmit and the reactive power they absorb. This paper introduces a comprehensive theoretical approach of the transferable power and the related compensation topologies allowing to optimize the system used. Analytical equations are given in order to estimate the maximum power transmission of a coreless transformer. Based on the theoretical results the optimal topology can then be selected accordingly to the power source, the transformer and the sensibility over the deviations of the elements. This methodology has been validated experimentally and allowed to transmit 2W through a 2.25cm² coreless transformer under 1 MHz. Moreover this results can be broaden to any inductive power transfer system.

Introduction

Coreless transformers also known as coupled coils, allow to transmit power without any physical link, making them the perfect candidates for inductive power transfer (IPT). Indeed, this technology is currently being standardized for electrical vehicle chargers or biomedical implants. Furthermore, the transmission of energy through a magnetic field without a magnetic core allow the system to work at high temperature, high frequency while simplifying the fabrication process. As it also ensure a galvanic isolation, it appears to be an excellent solution for high temperature, high frequency isolated power converters [START_REF] Wanderoild | High Frequency, High Temperature designed DC/DC Coreless Converter for GaN Gate Drivers[END_REF], [START_REF] Bergogne | Integrated coreless transformer for high temperatures design and evaluation[END_REF].

Fundamentals of inductive power transfer

Firstly, we are going to introduce the fundamental differences between a transformer with or without a magnetic core. In magnetic core transformers, as explained by Witulski [START_REF] Witulski | Introduction to modeling of transformers and coupled inductors[END_REF], the use of a ferromagnetic core allows to drastically reduce the magnetic energy stored, since the magnetic permeability of the air is drastically lower than the magnetic material one. As a matter of fact, the magnetizing inductor 𝐿 𝑚 in Fig. 1 is several decades lower for coupled coils than in a transformer with a magnetic core. Moreover, without the use of a magnetic core, the magnetic field is no longer channeled between the two coils. It significantly decreases the coupling between the two coils and thus the magnetizing inductor, at the cost of an increase of the leakage inductors identified as 𝐿 𝑒1 and 𝐿 𝑒2 in Fig. 1.

Consequently, without proper compensation, the low magnetizing inductor 𝐿 𝑚 sinks a lot of current and adversely affects the power transfer from the primary to the secondary side. This inductor stores a significant amount of reactive power which needs to be compensated through harmonics oscillators. This compensation is made thanks to capacitors adequately tuned to the system.

In order to correctly adjust the compensation, the parasitic couplings need to be taken into account. If the working frequency is too close to the selfresonating frequency of the transformer, the interwinding coupling due to the electrostatic coupling between tracks can be substantial enough to affect the compensation. These capacitors which depend on the design of the transformer can be approximated by 𝐶 p1 , 𝐶 p2 and 𝐶 12 . For simplicity purposes the analytic solutions given are not taking into account these capacitances but the comprehensives solutions can be calculated using the same methodology.

A coreless transformer not only needs a significant amount of reactive energy, it also consumes active energy. The losses due to the copper, the current crowding and the skin effect must also be considered. These losses can be modeled by the frequency dependent resistances 𝑅 1 and 𝑅 2 .

The reactive power sunk by the transformer through the magnetizing inductor 𝐿 𝑚 is conventionally compensated with two impedance converters (Fig. 2.).

The impedance converters are set to maximize the available output power, they can be made with a serial or a parallel capacitor forming the four possible topologies shown in Fig. 3. The selection of the topology and associated components have been several time emphasized in literature. Wang's [START_REF] Wang | Power transfer capability and bifurcation phenomena of loosely coupled inductive power transfer systems[END_REF] and Auvigne [START_REF] Auvigne | About tuning capacitors in inductive coupled power transfer systems[END_REF] have presented methodologies which are available only when the losses modeled by 𝑅 1 and 𝑅 2 are negligible. in Fig. 1 are negligible. Wu [START_REF] Wu | Determining optimal tuning capacitor values of TET system for achieving maximum power transfer[END_REF] and Helpern [START_REF] Halpern | Optimal Tuning of Inductive Wireless Power Links: Limits of Performance[END_REF] have shown solutions for highly resistive transformers but only for SS and SP topologies.

This paper provides a new approach compatible with highly resistive transformers and generators for the four conventional topologies with their respective optimal load value.

Theoretical maximum power achievable

Based on an accurate model, the compensation can be optimized and the output power correctly estimated. First of all it is important to note that we are working with resonating oscillator, hence the compensation is valid at a given working frequency. Shifting the latter would detune the converters.

Considering that the optimized transformer acts as a bandpass filter, the first harmonic analysis allows to have a good estimation of the output power. This presumption has been validated thanks to a transient analysis showing a maximum 10% error with the first order approximation as shown in table 3. The generator is modeled by a resistive equivalent source. The maximum power that may be transferred can be calculated with a proper impedance matching and a Thevenin equivalent circuit composed by the generator, the primary impedance converter and the transformer as modeled in Fig. 4. For a given source and a given transformer, the values of 𝑍 𝑒𝑞 and 𝑉 𝑒𝑞 are not affected by the load or the secondary impedance converter 𝐶 2 , which mean that the equivalent model of the source is only dependent of value of 𝐶 1 . This equivalent source can provide a limited power level to the secondary converter and the load 𝐶 2 and 𝑅 𝑙 which must be chosen to absorb as much power as possible from the upstream conversion chain. Thanks to the maximum power theorem we can estimate the maximum power deliverable by the equivalent source constituted by 𝑍 𝑒𝑞 and 𝑉 𝑒𝑞 to the secondary. The maximum power transmission is obtained when the load modeled by 𝑍 𝑙𝑜𝑎𝑑 , is perfectly matched to the source [START_REF] Wanderoild | High Frequency, High Temperature designed DC/DC Coreless Converter for GaN Gate Drivers[END_REF].

𝑍 𝑒𝑞 * = 𝑍 𝑙𝑜𝑎𝑑 . (1)
When ( 1) is respected, the maximum power transmitted to the secondary 𝑃 𝑚𝑎𝑥 can expresses with 𝑍 𝑒𝑞 and 𝑉 𝑒𝑞 regardless of the load and the secondary impedance converter.

𝑃 𝑚𝑎𝑥 (𝐶 1 ) = 𝑉 𝑒𝑞 2 (𝐶 1 ) 4 • 𝑅𝑒 (𝑍 𝑒𝑞 (𝐶 1 )) (2)
Therefore 𝐶 1 can then be chosen by solving (3) in order to maximize the power transmitted.

𝜕𝑃 𝑚𝑎𝑥 𝜕𝐶 1 = 0 (3)
The load 𝑅 𝑙 associated with the secondary impedance converter 𝐶 2 are then chosen in order to respect [START_REF] Wanderoild | High Frequency, High Temperature designed DC/DC Coreless Converter for GaN Gate Drivers[END_REF].

For ease of reading, these shorted variables will be used.

𝑍 𝐿 1 = 𝐿 𝑚 𝜔 + 𝐿 𝑒1 𝜔 𝑍 𝐿 𝑚 = 𝐿 𝑚 𝜔 𝑅 𝑔1 = 𝑅 𝑔𝑒𝑛 + 𝑅 1 For a serial primary compensation, SS and SP topologies, the optimal value of C1 is given by

𝐶 1 = 1 𝑍 𝐿 1 𝜔 (4)
For a parallel primary compensation, PP & PS topologies the optimal value of C1 is given by

𝐶 1 = 𝑍 𝐿 1 (𝑍 𝐿 1 2 + 𝑅 1 𝑅 2 𝑍 𝐿 𝑚 2 + 𝑅 1 2 )𝜔 (5)
If we consider that the power supply provides an rms tension 𝑉 𝑖𝑛 , for SS and SP topologies the maximum power given by the equivalent source is:

𝑃 𝑚 = 𝑉𝑖𝑛 2 4𝑅 𝑔1 (1 + 𝑅 2 𝑅 𝑔1 𝑍 𝐿 𝑚 2 ) (6)
For PP and PS topologies, the maximum power becomes:

𝑃 𝑚 = 𝑉𝑖𝑛 2 ((𝑍 𝐿 1 2 + 𝑅 1 2 ) + 𝑅 1 𝑅 2 𝑍 𝐿 𝑚 2 ) 4 [ A 4 R 2 𝑍 𝐿 𝑚 2 + A²(𝑅 1 + 𝑅 𝑔1 ) + 𝑍 𝐿 𝑚 2 𝑅 1 𝑅 2 𝑅 𝑔1 ]
Where 𝐴² = 𝑍 𝐿 1 2 + 𝑅 1 𝑅 𝑔1

The secondary compensation is then calculated in order to match the equivalent source. When a rectifier is used, it can be modeled with the first harmonic approximation, as shown in Fig. 5. In order to match the equivalent source, for a serial secondary compensation, SS and PS topologies the load and the impedance converter are given by

𝐶 2 = 1 -𝐼𝑚(𝑍 𝑒𝑞 ) • 𝜔 𝑅 𝑙 = 𝑅𝑒(𝑍 𝑒𝑞 ) (10)
For a parallel secondary compensation, SP and PP topologies the optimal values become

𝑅 𝑙 = 𝑅𝑒(𝑍 𝑒𝑞 ) • (1 + ( 𝐼𝑚(𝑍 𝑒𝑞 ) 𝑅𝑒(𝑍 𝑒𝑞 ) ) 2 ) (11) 𝐶 2 = 𝐼𝑚(𝑍 𝑒𝑞 ) 𝜔 * (𝑅𝑒(𝑍 𝑒𝑞 ) 2 + 𝐼𝑚(𝑍 𝑒𝑞 )²) (12)
A combination between a parallel and a series compensation allows to have more flexibility regarding the optimal load.

Topology choice

The choice of the secondary compensation has no effect on the output power available as expressed by [START_REF] Bergogne | Integrated coreless transformer for high temperatures design and evaluation[END_REF]. It can be proven mathematically by comparing ( 6) and ( 7) that serial compensation for the primary side will be able to provide more power if the condition given by ( 13) is respected.

𝑅 𝑔𝑒𝑛 2 < 𝑍 𝐿1 2 + 𝑅 1 𝑅 2 𝑍 𝐿𝑚 2 + 𝑅 1 2 (13)
In most cases the power source is less resistive than the transformer, making the serial compensation for the primary the ideal choice for maximizing the power transferred.

In addition, the primary compensation has a clear effect on the current drown by the converter. On the one hand, the serial compensation avoids DC current passing through the transformer. On the other hand, when the converter is supplied by a square source, the voltage transient can create a substantial current peak in the parasitic and compensation capacitances. As a result, this may stress the source and increase drastically the losses in the converter, and, as a consequence lower the efficiency of the whole conversion chain. This current peak is due to the low impedance seen by the source at high frequency.

In the case of a parallel primary compensation the power source will be directly connected to the capacitance which will drain high frequency currents. Nevertheless, in the case of a serial primary the high frequency currents are stopped by the primary 𝐿 𝑒1 and 𝐿 𝑚 , although it may pass though the interwinding capacitance 𝐶𝑝1.

This phenomenon can be avoided by using a sinusoidal source, or by adding shock inductors between the generator and the impedance converter, this would increase the impedance seen by the power source at high frequency. However the primary compensation capacitance would have to be changed accordingly and the output power would be affected by the losses in the added inductors.

Finally it must be pointed out that the sensibility over the choice of the impedance converter's elements values changes according to the generator, transformer and the topology chosen. For example, for the primary impedance converter unlike the parallel compensation the optimal value of 𝐶 1 for a serial compensation is independent of the coupling and 𝑅 1 and 𝑅 2 . It can be also noted that the parallel primary capacitor has no effect as long as the generator used presents a negligible impedance in front of the one of the compensated transformer. The sensibility analysis allows to choose the topology in order to adapt the converter to the physical constraint and to focus the attention on the choice of the key components.

Transformer characterization

To validate the proposed methodology, the PCB coreless transformer shown in In order to perfectly tune the matching elements the transformer used has been measured with a Keysight E4990A impedance analyzer.

𝐶 12 . is determined by shorting the primary and the secondary and measuring the impedance between the two. An algorithm has then been applied in order to match the model with the measurement of the impedance of the transformer with open and short secondary. The open secondary measurement presents two peaks, which is a direct consequence of the bifurcation phenomena [START_REF] Wang | Power transfer capability and bifurcation phenomena of loosely coupled inductive power transfer systems[END_REF]. In our case, the first peak is due to the resonating RLC circuit composed of 𝐶 1 , 𝐿 𝑒1 , 𝐿 𝑚 then the second peak is mainly due to the RLC composed of 𝐶 1 , 𝐶 2 , 𝐶 12 , 𝐿 𝑒1 , 𝐿 𝑒2 .

To consider the skin effect, the resistors have been modeled by 𝑅 = 𝑅 0 + 𝑅 𝑠𝑘𝑖𝑛 • √𝑓. The estimated values can be found in Table 1. The real part of the impedance measured being several decades lower than the imaginary part at the working frequency, the resistive values are difficult to measure. The measurement of the resistive parts would be improved by tuning the resonant devices at the working frequency. Keeping in mind that the current crowding effect is linked to the current passing through the magnetizing inductor, meaning that the values of the model's elements are dependent on the source and the load. 8,2 pF

Table .1 Estimated values for the model (Fig. 1)

Experimental validation of the proposed method. The first harmonic analytic analysis has been validated with a transient simulation and experimental measurements (table 3). An error lower than 7% has been noticed between the simulations. The discrepancy is due to the harmonics neglected in the analytical analysis. The compensated transformer act as a bandpass filter. Unlike the SS topology, the PP topology allow a part of the current to pass through the capacitor without passing through the resistors 𝑅 1 and 𝑅 2 , increasing the quality factor of the filter formed. As a result, the SS topology allows more harmonics to pass through the transformer and the output waveform display stronger harmonics than the others topologies as seen on Fig 9 and10 The experimental measurements are showing an output power up to 33% lower. This discordance is due to a mismeasurement of the coils resistors, and the lack of current crowding simulation.

Conclusion

For a given transformer the maximum transferable power can be calculated. In order to reach this power an addition of two impedance converters is required. The choice of compensation used for the primary side impact the output power, but also the current draw by the system. Knowing the transformer and the source used, the optimal compensation can be calculated thought equation (13).

Using the proposed methodology and models, 2W has been transferred through a 2.25cm² coreless transformer under 1MHz, with a parasitic capacitance between the primary and secondary lower than 10pF. Knowing the maximum output power delivered by a given transformer, an optimization can easily be done during the earliest stages of the transformer design can be easily done. Since we are using a resonant compensation, the slightest variation of several parameters can drastically affects the power delivered. These variations can be created by production fluctuation, temperature changes, coupling with surrounding materials or by the current crowding phenomena. A sensibility analysis seems to be a key element in developing a closed loop approach to industrial production and will be investigated in our future works.

Fig. 1

 1 Fig.1 Conventional model of coreless transformers (T; ideal transformer).

Fig. 3

 3 Fig. 3 Conventional compensation topologies

Fig. 2

 2 Fig. 2 Conversion chain based on a coreless transformer.

Fig. 4

 4 Fig. 4 Thevenin equivalent circuit of the first part of the conversion chain

Fig. 5

 5 Fig. 5 Secondary impedance converter equivalent model.

  Fig 6 has been used. The primary as well as the secondary have been made with 18 square turns on two layers. The copper tracks are 17µm thick and 200 µm width and 150 µm isolation between each turn. The primary and secondary are separated with 710 µm of FR4.

Fig. 6

 6 Fig. 6 Transformer used for experimental validation of the proposed design methodology

Fig. 7

 7 Fig. 7 Results of the matching algorithm between the model and the measurements.

Fig. 8

 8 Fig. 8 Experimental set up

Fig. 9

 9 Fig. 9 Experiment and numerical simulation (dashed) of the SS topology

Fig. 11

 11 Fig. 11 frequency spectrum of the current measured

Table 2 .

 2

		SS	SP	PS	PP	unit
	𝐶 1	4.56	4.56	2.84	2.84	nF
	𝐶 2	3.99	0.156	7.16	6.72	nF
	𝑅 𝑙	197	205	5.66	92.8	Ω

Table 2

 2 Values calculated and used for the compensation with each topology

Table 3 Output

 3

power and error vs first harmonic simulation for first harmonic simulation (proposed methodology), circuit simulation and measured for each topology.

  Fig. 12 Impedance seen by the power source obtained by simulation with the converter tuned at 1MHz

		SS	SP	PS	PP
	First	3.02	3.02	1.17	1.17	W
	harmonic
	simulation
	Transient	3.30	3.00	1.12	1.16	W
	simulation	6.6	0.7	4	0.8	%
	Experiment 2.02	2.02	0.85	0.99	W
		33	33	27	15	%


Armande Capitaine

Gael Pillonnet

Thibaut Chailloux

Adrien Morel

Bruno Allard

Impact of switching of the electrical harvesting interface on microbial fuel cell losses

Keywords: microbial fuel cell, flyback converter, energy harvesting, dynamic electrical model, impedance spectroscopy I

Microbial fuel cells (MFCs) are sources harvesting energy from organic matters and showing great promise in powering environmental sensors. Because of the low power and voltage issues (100µW at 0.3V for 20 cm² electrodes), an electrical interface is required to extract the maximum power delivered by the MFC and boost the output voltage. However the switching operation of most converters induces a pulsed sinking current which may cause additional dynamic losses inside the MFC. Following a previous study on a flyback converter in discontinuous conduction mode, this paper analyzes the effect of switching of the converter on the MFC internal losses. A dynamic model of the MFC is deduced from an impedance spectroscopy characterization: it reveals a double RC behavior, one with a time constant of 10s of s, the other one of 100s of µs. Then in the frequency-domain, the MFC dynamic losses, induced by a previously optimized flyback, are calculated: they represent 50% of the maximum power that can be extracted from the MFC. Eventually in order to reduce these losses, we study the impact of three flyback parameters (primary inductance, duty cycle and decoupling capacitance). Adding a capacitance of 10 µF at the converter input, the MFC dynamic losses become negligible.

INTRODUCTION

Discovered in the last decades, microbial fuel cells (MFCs) are promising sources for scavenging energy from organic substrates (e.g. compost, sediment, wastewater…) by converting chemical energy into electricity thanks to catalysis properties of electro-active bacteria [START_REF] Logan | Microbial fuel cells: methodology and technology[END_REF]. They can be seen as an attractive alternative to polluting batteries to power sensors in hardly reachable areas (e.g. seafloors). Regarding the low voltage and low power the MFCs generate (100 µW at 0.3 V for 20 cm² electrodes [START_REF] Chailloux | Autonomous sensor node powered by cm-scale benthic microbial fuel cell and low-cost and off-the-shelf components[END_REF]), an electrical interface, i.e. a DC/DC converter, is required to extract the maximum power, PMPP, from the MFC and boost the voltage to the level required by the sensor. However these converters, because of their inherent switching operation, impose a pulsed sinking current which may cause additional losses inside the MFC, on top of the well-known losses due to the DC current (static losses). These additional losses due to AC currents are called MFC dynamic losses. The paper follows a previous work [START_REF] Capitaine | Loss analysis of flyback in discontinuous conduction mode for sub-mW harvesting systems[END_REF][START_REF] Wanderoild | 100 µW coreless flyback converter for microbial fuel cells energy harvesting[END_REF] where the MFC interface is a flyback converter in discontinuous conduction mode (DCM) extracting energy from a benthic MFC with 20 cm² electrodes and a maximum power of 90 µW. So far only the MFC static losses have been considered in literature. The paper studies the MFC additional dynamic losses induced by the switching operation of the flyback converter.

First a dynamic model of the MFC will be determined from experimental data obtained with electrochemical impedance spectroscopy (EIS) [START_REF] Yuan | Electrochemical impedance spectroscopy in PEM fuel cells[END_REF]. Then we will study the impact of the dynamic current imposed by the converter on the MFC dynamic losses. After developing a calculation method of the MFC dynamic losses, we will analyze the influence of three converter parameters (primary inductance, duty cycle, input capacitance) on these losses for minimization, i.e. avoid inefficient consumption of the organic substrate.

II. ELECTRICAL MODELING OF MFC

A. Static behavior

A typical way to model the MFC static behavior is shown in Fig. 1. The polarization curve is comparable to a straight line corresponding to a Thevenin circuit. Our lab-scale MFC generates PMPP = 90 µW at the maximum power point (MPP) and corresponds to an open-circuit voltage source VOC of 0.6 V and a series resistance RDC of 1 kΩ [START_REF] Chailloux | Autonomous sensor node powered by cm-scale benthic microbial fuel cell and low-cost and off-the-shelf components[END_REF]. However this model does not reflect the dynamic behavior of the MFC when an AC stimuli is applied.

B. Dynamic characterization

The EIS method is adopted in order to to appreciate the MFC dynamic behavior at different DC operating point (UDC, IDC). This method consists in adding a sinusoidal perturbation uAC with variable frequency f to the polarization voltage UDC (as schematized in Fig. 1). The amplitude of uAC is fixed to 50 mV so that the MFC can accurately be apprehended as a linear system. The resulting current, expressed as the sum of the static current IDC and a sinusoidal perturbation iAC, is then measured. From these measurements, we can deduce an MFC equivalent dynamic impedance Z(f) at different frequency f for a specific DC point (UDC, IDC). Its amplitude |Z| and phase φ, when working at MPP (UDC = 0.3V), are represented in Fig. 2. Similar AC characteristic is obtained for other polarization voltages. Weassume the MFC dynamic behavior is independent of the polarization point. Moreover the frequency range is large since some physical aspects characterized at low frequency (e.g. mHz) may have a significant impact even when using a kHz switching frequency converter. The dynamic electrical circuit deduced from the previous characterization is represented in Fig. 3. To the static model is added two parallel RC branches, one (R1C1) being associated to the low cut-off frequency fBF, the other (R2C2) to the higher one fHF (Fig. 2). The circuit components are calculated using equations in Fig. 3 in which {RBF, RHF, fBF, fHF} are evaluated from AC measurements (Fig. 2). RDC is not readable in Fig. 2 as extended AC measurements at lower frequencies is very time consuming. RDC is thus deduced from a previous static characterization (Fig. 1). The results are listed in Table I..It shows an accurate fit with the experimental data (Fig. 2). The first branch R1C1 has a time constant of the order of 10s of seconds and can be associated to the transport mechanisms of the organic matter to the electrodes [START_REF] Dhirde | Equivalent electric circuit modeling and performance analysis of a PEM fuel cell stack using impedance spectroscopy[END_REF]. Whereas the second branch R2C2, having a time constant of the order of 100s of microseconds may be due to the double layer electrical behavior at the electrodes' surface [START_REF] Dhirde | Equivalent electric circuit modeling and performance analysis of a PEM fuel cell stack using impedance spectroscopy[END_REF]. Therefore the MFC presents two dynamic resistances R1 and R2 that may engender additional intrinsic losses.

C. Dynamic electrical model

III. INFLUENCE OF THE HARVESTING INTERFACE ON THE MFC DYNAMIC LOSSES

A. Harvesting interface

Using the previous work presented in [START_REF] Capitaine | Loss analysis of flyback in discontinuous conduction mode for sub-mW harvesting systems[END_REF], the flyback in DCM is chosen as an harvesting interface mainly for its isolation capability. As explained in [START_REF] Capitaine | Loss analysis of flyback in discontinuous conduction mode for sub-mW harvesting systems[END_REF], to work at the DC-MPP and extract the maximum DC energy from the MFC, the flyback input impedance RIN has to meet the following constraints:

𝑅 𝐼𝑁 = 2𝐿 1 𝑓 𝑠𝑤 𝐷 2 = 𝑅 𝐷𝐶 (1)
L1 is the primary inductance, D the duty cycle and fsw the switching frequency of the flyback. This means the three parameters {L1, D, fsw} are linked. In [START_REF] Wanderoild | 100 µW coreless flyback converter for microbial fuel cells energy harvesting[END_REF] it was chosen to work with a coreless transformer to avoid converter losses due to saturation and hysteresis and thus maximize the converter efficiency.

The same experimental set-up {L1, D, fsw} = {583 µH, 0.07, 4.23 kHz} will be used in the following study to evaluate the additionnal AC losses due to R1 and R2.

B. Calculation of the MFC dynamic losses

Transient simulations may be very time consuming regarding the 10s of second time constant at stake in the MFC dynamic model compared to kHz flyback switching operation and so the time needed to reach steady-state operation. An analytical frequency approach based on the Fourier decomposition is thus preferred. For the sake of simplification, the flyback input current, called Iflyback in Fig. 4, is considered of constant at the flyback input and independent of MFC energy production. In frequency domain, we calculate the Fourier series of currents I1 and I2 in Fig. 3 using the respective transfer functions regarding Iflyback and the Fourier series of Iflyback. Again in the time-domain, the losses P1 and P2 induced respectively by the branch R1C1 and R2C2 are deduced. The method is summarized for P1 in Fig. 4.

C. Results

Following this calculation process, in the chosen conditions {L1, D, fsw} = {583 µH, 0.07, 4.23 kHz}, P1 is equal to 4.4 µW and P2 to 42.9 µW, representing respectively 5% and 48% of the extracted DC power PMPP (equal to 90 µW). Therefore the dynamic part of the MFC undergoes non-negligible losses compared to the 90 µW corresponding to the DC losses due RDC.

This means a power equivalent to 53% of the one usefully delivered to the converter is lost inside the MFC, contributing to a useless depletion of the organic matter resources. In order to reduce these losses, we will vary different flyback parameters, e.g. L1 and D, to change the frequency spectrum of the currents I1 and I2 and thus the energy dissipation in respectively in R1 and R2.

Influence of L1: fixing D to 0.07, P1 and P2 are calculated for L1 varying from 2 µF to 200 mF. In order to respect (1) and work at the DC-MPP, fsw is constantly adapted (from 1.23 MHz to 12.3 Hz). The results are shown in Fig. 5. When increasing L1, i.e. decreasing fsw, I1 increases (|G1| increases in the lower frequencies, Fig. 4) while I2 decreases. Regarding the high value of R1 compared to R2, the increase in P1 is more significant than the decrease in P2, hence the total dynamic losses explode, especially for L1 > 100 mH. For lower values, L1 variations have no impact on the dynamic losses which remain equal to 52.6% of PMPP. Therefore considering our previous working conditions (L1 = 583 µH), no minimization of the dynamic losses can be made by varying L1. Influence of D: L1 is fixed to 583 µH and D varies from 0 to 1 while adapting fsw according to (1) as it was done in the previous case. The results are shown in Fig. 6. When decreasing D, i.e. decreasing fsw, the dynamic losses increase exponentially. These losses are particularly important when working with D lower than 0.2. Then working with a D higher than 0.5, i.e. fsw higher than 214 kHz, ensures negligible dynamic losses. However [START_REF] Capitaine | Loss analysis of flyback in discontinuous conduction mode for sub-mW harvesting systems[END_REF] has shown that such an increase in fsw causes critical switching losses inside the flyback and thus drastically reduces its conversion efficiency. Therefore another way to reduce P1 and P2 without impacting on the conversion efficiency is needed. Influence of CIN: using the initial parameters given in section III.A, a capacitance is added at the input of the flyback converter. The results are displayed in Fig. 7. We observe that the inclusion of this capacitance decreases the MFC dynamic losses. This decrease becomes significant for CIN larger than few µF for which the dynamic losses represent less than 1% of the extracted power PMPP. With CIN = 10 µF (which represents a negligible extra surface on a PCB), these losses are equal to 0.1% of PMPP and may be neglected. Indeed all the AC current is flows in CIN, causing no loss as ESR is neglected. IV. CONCLUSION

The paper analyzes the influence of the flyback converter in DCM on the intrinsic losses of an MFC. Because of the switching behavior, the converter imposes dynamic currents to the MFC which may induce dynamic losses inside the MFC. First a dynamic model composed of two parallel RC branches associated to time constants of the order of 100s of µs and 10s of s was deduced from measurements obtained with EIS. Second we analyzed the influence of the flyback design (always working at MPP) on the losses induced by these two dynamic branches. Varying L1 offers no possible optimization, while increasing D may significantly reduce the dynamic losses. However increasing D means increasing f which drastically reduces the flyback efficiency. Eventually we show that inserting a decoupling capacitance larger than 10 µF at the input of the converter reduces the MFC dynamic losses from 52.6% to 0.1% of the extracted power PMPP.

Fig. 1 .

 1 Fig. 1. Static characteristic of the MFC associated to the MFC electrical model

Fig. 2 .

 2 Fig. 2. Amplitude and phase of the MFC dynamic impedance obtained experimentally with the impedance spectroscopy method (blue solid line) and of the deduced dynamic model of the MFC (red dashed line).

Fig. 3 .

 3 Fig. 3. Dynamic model of the MFC

Fig. 4 .

 4 Fig. 4. Method of the dynamic losses calculation, example of P1 calculation

Fig. 5 .

 5 Fig. 5. Influence of L1 choice on the MFC dynamic losses (D=0.07)

Fig. 6 .

 6 Fig. 6. Influence of D choice on the MFC dynamic losses (L1=583µH)

Fig. 7 .

 7 Fig. 7. Influence of a capacitance CIN at the input of the flyback on the MFC dynamic losses ({L1, D, fsw} = {583 µH, 0.07, 4.23 kHz})

TABLE I .

 I VALUES OF THE DYNAMIC MODEL COMPONENTS

	RDC	R1	C1	R2	C2
	1 kΩ	347 Ω	30 mF	35 Ω	3 µF


M Plamondon

THE COMMISSIONING OF THE ATLAS CALORIMETERS WITH COSMIC MUONS

The commissioning of the ATLAS calorimeters is an ongoing process since early 2006. During this period, cosmic muons have been recorded in several runs combining both hadronic and electromagnetic calorimeters. Among the goals are the measuremement of the uniformity of the liquid argon electromagnetic calorimeter to the level of 1% and the intercalibration in time of its channels to 1 ns.

Commissioning with cosmics

The ATLAS EM calorimeters undergo in 2007 the last steps of their installation. During the previous two years of commissioning, several milestones have been successfully attained, most prominently the complete cabling of the whole electronics chain, the cryostats fill with liquid argon and the high voltage ramping. The calorimeters in this definitive setup could then detect the cosmic muons reaching them in spite of their location 150 m underground in the cavern.

ATLAS calorimetry during the commissioning

The commissioning involves all calorimeter subdetectors for which a detailed description can be found in the ATLAS Technical Design Report [START_REF]Atlas: Detector and physics performance technical design report[END_REF] . The endcap regions being operational only since a few months, the emphasis will be put on the barrel part from which more information can be extracted with this special data taking. The trigger signal is provided by the hadronic calorimeter detecting top-bottom coincidences at a rate of one event every 20 seconds. Montecarlo simulations had predicted a strong occurence of incoming muons making their way down the access shafts. The accumulated data corroborate this idea with a maximum of events originating from that direction at |η| ∼ 0.3. The coverage of the trigger ranges basically up to |η| < 0.8 as seen in fig. 1, the rates falling abruptly beyond this rapidity.

The liquid argon technology used as active material of the electromagnetic calorimeter was adopted for its high stability and radiation hardness. This sensitive part is located in the gaps between the absorbers, accordion shaped plates made of lead. The schema of fig. 1 shows the longitudinal segmentation of this detector in a front, middle and back compartments. Due to its larger depth, the middle layer collects most of the small signals left by muons and will hence be subject to a more detailed analysis.

Motivations

Since the ATLAS Collaboration unveiled the first data taking of cosmic muons in August 2006 (fig. 2), an effort has been initiated in order to improve the knowledge of the calorimeters. Though the preceding testbeam data have brought much to our understanding, it is an opportunity to assess the behaviour of still untested modules.

Muons propagating almost at the minimum of ionisation do not offer optimal conditions for measuring performances of the EM calorimeter. The small energy deposits they produce amount typically to 275 M eV in the middle, only a factor 8 above the electronic noise and two orders of magnitude below the energy scale relevant to the LHC physics. Unlike the LHC environment, cosmic muons produce asynchronous data and their trajectories are seldomly projective, i.e. passing close to the interaction point. On the other hand, muons have well-understood energy deposition mechanisms and, unaffected by non-uniformities in the absorbers, they give a direct sensitivity to the width of the liquid argon gaps and effects due to a wrong calibration or signal reconstruction. They also allow to scan a large fraction of the detector without concerns about the energy of the incoming particles. A simulation of projective muons has demonstrated that a precision of 1% on the value of the most probable energy deposit could be achieved for each η range of the middle with the statistics accumulated during 3 months.

Another motivation comes from the observation of large energy deposits in the calorimeter interpreted as muons emitting a photon which then produces an electromagnetic shower. In the figure 2, an example of a high amplitude signal is shown, representing an equivalent of 25 GeV . These events can be exploited in various ways, for instance to validate predictions of the physics pulse shapes or to be used as reference when intercalibrating in time the channels. From the current observed rate, we can expect at least one such event per cell after the planned data taking period (3 months). Reversely, the absence of signals identifies dead channels. An amount of 60k muon signals have been recorded during the 3 days long runs taken in fall 2006 with a limited coverage and a high voltage set to 1600V. In the first half of 2007, more regular runs were taken overnight and during weekends with a more complete setup.

Results

The challenges related to the analysis of cosmic muons had to be overcome. For instance, the optimal filtering method 2 used for the signal reconstruction in the EM calorimeter does require the knowledge of the pulse's time. Though it can be evaluated in situ with large signals, this information will be provided by the hadronic calorimeter. Another issue concerns the signal to noise ratio which should be improved as much as possible. The low event rate allows data taking with more samples than the nominal 5 at the LHC. A significant reduction of the noise is obtained when performing instead an optimal filtering with 29 samples (see fig. 3). Is also shown the Landau distribution of the deposited energy of the muons. Figure 3: The noise is reduced by a factor 1.8 if the number of samples for the optimal filtering is increased from 5 to 29 (left). The graph on the right is the Landau distribution of muon signals observed during the testbeam.

The electronic noise is taken into account by a convolution with a gaussian.

Uniformity in energy

Due to the accordion geometry, a signal is always shared between two neighbouring cells in φ.

Adding these two cells would contain the whole signal as long as the muon is projective enough to not leak in adjacent cells in η for instance. Criteria to assess the biases introduced by the non-projectivity are under study. At this point, signals in the front could be used to determine more precisely the paths and hence avoid this issue. Out of the accumulated data, around 5% are described as projective enough to be considered in a uniformity measuremement. Though each muon produces signals at the top and bottom of the calorimeter, the limited statistics of projective muons allows to seek for effects along η (fig. 4) only with the assumption of symmetries of the detector in φ, top versus bottom and between A or C sides.

Intercalibration in time

To perform an intercalibration in time of two channels, a single particle crossing both of them would be sufficient after a correction for the time-of-flight. Unfortunately, the time measurement remains very unprecise at low amplitudes (see fig. 4). Large signals allow to measure precisely the relative offsets between the tile and electromagnetic signals. With more data being analysed, these reference signals will help to disentangle the various sources of offsets and solve the system such that all the channels are intercalibrated to 1 ns.

Conclusion

With more components becoming operational in ATLAS, recent cosmic runs taken in Spring 2007 are already integrating parts of the inner detector (TRT) and the muon system. Uniformity analysis and time intercalibration are currently performed in order to understand as much as possible the calorimeters before the LHC start-up next year.

Figure 1 :

 1 Figure 1: The η coverage made available by the tile calorimeter (left) and a schematic view showing the accordion geometry of the LAr EM calorimeter (right).

Figure 2 :

 2 Figure 2: On the left, an event display from August 2006 cosmic data shows the energy deposits left by a projective muon in the hadronic (red) and electromagnetic (green) calorimeters. Large signals of several GeV like the one appearing on the right hand side are frequently encountered.

Figure 4 :

 4 Figure 4: On the left, this preliminary graph shows a comparison between data and MC of the peak of the Landau shape for various η ranges. The time resolution for muons as a function of the energy is presented on the right.


Annabella Astorino
email: astorino@icar.cnr.it

Manlio Gaudioso
email: gaudioso@deis.unical.it

Alberto Seeger
email: alberto.seeger@univ-avignon.fr

An illumination problem: optimal apex and optimal orientation for a cone of light

Keywords: Illumination problem, Revolution cone, Aperture angle, Sharpness coefficient, Max-min problem

An illumination problem: optimal apex and optimal orientation for a cone of light.

Introduction

There is a broad literature devoted to various sorts of illumination and/or visibility problems; see for instance [START_REF] Ahn | Aperture-angle and Hausdorff approximation of convex figures[END_REF][START_REF] Bose | Some aperture-angle optimization problems[END_REF][START_REF] Coban | Model-based synthesis of sensor location[END_REF][START_REF] Omana-Pulido | Aperture-angle optimization problems in three dimensions[END_REF]. Most illumination problems arising in practice are stated in dimensions two and three, but one could equally well consider higher dimensions. Let R n be equipped with its usual inner product •, • and the associated norm • . Mathematically speaking, a conic light beam is a generalized revolution cone

• The apex z of the cone is the geometric locus at which a source of light is placed.

Throughout this work, z is selected from a closed set Z ⊆ R n . One refers to (1) as a "generalized" revolution cone because the apex z is not necessarily the origin 0 of the space R n . • The axis y ∈ R n is a nonzero vector that determines the orientation of the light beam.

The axis can always be normalized, so one takes it from the unit sphere S n := {y ∈ R n : y = 1}. • The scalar s ∈ [0, 1] is the sharpness coefficient of the light beam. One refers to ϑ := 2 arccos s as the aperture angle of [START_REF] Ahn | Aperture-angle and Hausdorff approximation of convex figures[END_REF]. So, maximizing the sharpness coefficient amounts to minimizing the aperture angle.

One may think of (1) as the visible or illuminated region of the space R n . Thus, a sample A = {a i : i ∈ I } is illuminated by the conic light beam [START_REF] Ahn | Aperture-angle and Hausdorff approximation of convex figures[END_REF] if and only if A ⊆ C(z, y, s). The smallest aperture angle illumination problem reads then as follows:

(P) ⎧ ⎨ ⎩ maximize s (z, y, s) ∈ Z × S n × [0, 1] f i (z, y, s) ≤ 0 for all i ∈ I, ( 2
)
where each f i : R 2n+1 → R is defined by f i (z, y, s) := s a izy, a iz .

The above maximization problem will be studied in detail in Sects. 2 and 3, both theoretically and algorithmically. We shall be concerned also with a sort of dual version of [START_REF] Astorino | Nonsmooth optimization techniques for semisupervised classification[END_REF]. Up to a closure operation, the invisible or penumbra region D(z, y, s) := {x ∈ R n : s xzy, xz ≥ 0}

(3) corresponds to the set-complement of [START_REF] Ahn | Aperture-angle and Hausdorff approximation of convex figures[END_REF]. The formulation of the largest aperture angle penumbra problem is as follows:

(Q) ⎧ ⎨ ⎩ minimize s (z, y, s) ∈ Z × S n × [0, 1]
f i (z, y, s) ≥ 0 for all i ∈ I.

(4)

Despite their striking similarity, the problems (2) and ( 4) are structurally different. Note for instance that the illuminated region (1) is convex, while the penumbra region (3) is nonconvex. Section 4 will be devoted to the analysis of the penumbra problem. One should not view [START_REF] Astorino | Nonsmooth optimization techniques for semisupervised classification[END_REF] and (4) as a pair of dual problems in the sense of convex optimization theory. The word duality is used here in a loose sense.

Theoretical aspects of the illumination problem

Preliminary material on supporting cones

The symbols P c , int(P), and bd(P) denote respectively the set-complement, the interior, and the boundary of the polytope P := co{a i : i ∈ I }. We use also the notation (y, s) := C(0, y, s) and T (z) := R + (Pz).

(

) 5
Both sets in [START_REF] Astorino | Conic separation of finite sets. I. The homogeneous case[END_REF] are closed convex cones with apices at 0. The first one is a standard revolution cone, whereas the second one is a polyhedral (or finitely generated) convex cone:

T (z) = i∈I μ i (a i -z) : μ i ≥ 0 for all i ∈ I .
One refers to T (z) as the supporting cone to P at z. Such cone is of special interest when z belongs to P c . The continuity result stated in Proposition 2.1 is recalled here for the sake of convenience. One considers the set (R n ) of nontrivial closed convex cones in R n equipped with the spherically truncated Pompeiu-Hausdorff metric

(K 1 , K 2 ) := haus(K 1 ∩ S n , K 2 ∩ S n ),
where haus(•, •) refers to the classical Pompeiu-Hausdorff metric on the collection of nonempty compact sets, i.e., haus( 1 , 2 ) := max max

x∈ 1 dist(x, 2 ), max x∈ 2 dist(x, 1 ) .
That a closed convex cone in R n is nontrivial simply means that it is different from the singleton {0} and different from the whole space R n .

Proposition 2.1 The function T

: P c → (R n ) is continuous. In fact, for all z 1 , z 2 ∈ P c , one has (T (z 1 ), T (z 2 )) ≤ 2 dist(z 1 , P) + dist(z 2 , P) z 1 -z 2 . ( 6
)
Proof It is not difficult to check that

T (z) ⊆ lim inf v→z T (v) for all z ∈ R n , ( 7
)
T (z) ⊇ lim sup v→z T (v) for all z ∈ P c ,
where the lower and upper limits are understood in the Painlevé-Kuratowski sense. This proves already the continuity of T : P c → (R n ), because convergence in (R n ) with respect to the metric is equivalent to Painlevé-Kuratowski set-convergence. However, one can go beyond mere continuity and prove a Lipschitz type estimate like [START_REF] Aubin | Set-Valued Analysis[END_REF]. Let us examine the expression

e(K 1 , K 2 ) := max x∈K 1 ∩ S n dist(x, K 2 ∩ S n ) for K 1 = R + ( 1 ) and K 2 = R + ( 2 )
, where 1 , 2 ⊆ R n are nonempty convex compact sets not containing the origin 0. A direct computation shows that

e(K 1 , K 2 ) = max u 1 ∈ 1 min u 2 ∈ 2 u 1 u 1 - u 2 u 2 .
But the Dunkl-Williams inequality [START_REF] Dunkl | A simple norm inequality[END_REF] asserts that

u 1 u 1 - u 2 u 2 ≤ 2 u 1 -u 2 u 1 + u 2
for any pair u 1 , u 2 of nonzero vectors. One gets in this way

e(K 1 , K 2 ) ≤ max u 1 ∈ 1 min u 2 ∈ 2 u 1 -u 2 3 with := 2 [dist(0, 1 ) + dist(0, 2 )] -1 .
A similar estimate holds for the symmetric term e(K 2 , K 1 ), so one ends up with

(K 1 , K 2 ) ≤ haus( 1 , 2 ). ( 8
)
If one takes j = Pz j , then dist(0, j ) = dist(z j , P) and haus( 1 , 2 ) = z 1z 2 . Substituting this information in (8) one gets [START_REF] Aubin | Set-Valued Analysis[END_REF].

The inequality [START_REF] Aubin | Set-Valued Analysis[END_REF] shows that T is Lipschitzian on any compact set contained in P c . Indeed, if Z is the compact set under consideration, then

(T (z 1 ), T (z 2 )) ≤ (1/β) z 1 -z 2 for all z 1 , z 2 ∈ Z with β := min z∈Z dist(z, P). Remark 2.2 The set P c is open in R n . If {z k } k∈N is a sequence in P c
converging to z ∈ bd(P), then it does not follow necessarily that lim k→∞ (T (z k ), T (z)) = 0. In other words, is not always possible to predict the behavior of T (z) as z approaches to a point on the boundary of P. This issue is discussed at length in [21, Section 5].

Existence of solutions for the illumination problem

The optimal value of the illumination problem ( 2) is denoted by s * . The feasible set 2) is closed, but nonconvex in general. For making sure that W is nonempty one asks Z to contain a point that is not in the interior of P.

W := {(z, y, s) ∈ Z × S n × [0, 1] : A ⊆ C(z, y, s)} = {(z, y, s) ∈ Z × S n × [0, 1] : f i (z, y, s) ≤ 0 for all i ∈ I } of (

Proposition 2.3 The set W is nonempty if and only if Z is not included in int(P).

Proof For (z, y, s)

∈ Z × S n × [0, 1] one has A ⊆ C(z, y, s) ⇔ A -z ⊆ (y, s) ⇔ P -z ⊆ (y, s) ⇔ T (z) ⊆ (y, s).
Hence, the illumination problem (2) is feasible if and only if T (z) = R n for some z ∈ Z . In other words, W is nonempty if and only if there exists a point z ∈ Z that is not in the interior of P.

The next proposition shows that the z-component of a solution to the illumination problem (2) is necessarily on the boundary of Z .

Proposition 2.4

Assume that A is not on ray emanating from a point of Z . Let (z, y, s) be a solution to [START_REF] Astorino | Nonsmooth optimization techniques for semisupervised classification[END_REF].

Then Z ∩ int[z -(y, s)] = ∅. ( 9
)
In particular, z ∈ bd(Z ).

Proof Since A is not on ray emanating from a point of Z , the sharpness coefficient s is smaller than 1. Consequently, (y, s) has nonempty interior. Suppose that (9) is not true. Then there exists w ∈ int[ (y, s)] such that z 0 := z -w belongs to Z . Since Az ⊆ (y, s) and

(y, s) + int[ (y, s)] = int[ (y, s)],
one obtains

A -z 0 = (A -z) + w ⊆ int[ (y, s)].
In other words, for all i ∈ I , one can write

a i -z 0 ∈ int[ (y, s)]. ( 10
)
The interiority condition [START_REF] Danskin | The Theory of Max-Min and Its Application to Weapons Allocation Problems[END_REF] implies that each a i is different from z 0 . Moreover,

a i -z 0 ∈ bd[ (y, s i )] ⊆ (y, s i ) ⊆ (y, s),
where s := min i∈I s i and

0 ≤ s < s i := a i -z 0 a i -z 0 , y ≤ 1.
Note that the sharpness coefficient s is larger than s and that

A ⊆ z 0 + (y, s) = C(z 0 , y, s).
This contradicts the maximality of s.

The next lemma provides an easily computable lower bound for s * . The symbols c A and r A indicate, respectively, the Chebishev center and the Chebishev radius of the sample A. By definition, (c A , r A ) is the unique solution to

⎧ ⎨ ⎩ minimize r x ∈ R n , r ≥ 0 A ⊆ x + r B n ,
where

B n := {x ∈ R n : x ≤ 1} is the n-dimensional closed unit ball.
Lemma 2.5 For any z ∈ Z that is not in the Chebishev ball c A + r A B n , one can write

1 - r A z -c A 2 1/2 ≤ s * . ( 11
)
Proof The Chebishev ball contains the sample A. In fact, it is the closed ball of smallest radius that contains A. It follows that

A -z ⊆ M(y, r ) := R + (y + r B n ), ( 12
)
where y := c Az -1 (c Az) and r := c Az -1 r A . A matter of computation shows that M(y, r ) = (y, s) [START_REF] Frangioni | Solving semidefinite quadratic problems within nonsmooth optimization algorithms[END_REF] with s = 1r 2 1/2 . The formula (13) can be found for instance in [START_REF] Henrion | Condition number and eccentricity of a closed convex cone[END_REF]Lemma 2.1]. The combination of ( 12) and ( 13) yields the inequality [START_REF] Dörrie | Hundred Great Problems of Elementary Mathematics: Their History and Solution[END_REF].

The issue concerning the existence of solutions to illumination problem (2) is settled in Theorem 2.6. The proof of the part (a) is based on Lemma 2.5, but one could use also a more direct argument. Theorem 2.6 Suppose that A is not on ray emanating from a point of Z . Proof Suppose that Z is unbounded. Then there exists a sequence {z k } k∈N in Z such that lim k→∞ z k = ∞. Lemma 2.5 implies that

1 - r A z k -c A 2 1/2 ≤ s * ( 14
)
for all k sufficiently large. By passing to the limit in [START_REF] Fuduli | Minimizing nonconvex nonsmooth functions via cutting planes and proximity control[END_REF] one gets s * = 1. Supposed now that Z is bounded and not included in the interior of P. In such a case the problem ( 2) is about maximizing a linear function on a nonempty compact set.

Link with the theory of incenters and circumcenters

As explained next, the illumination problem ( 2) is somewhat related with the problem of finding the circumcenter of a polyhedral convex cone. Our first observation is that

s * = sup z∈Z (z), (15)
where (z) denotes the optimal value of the lower dimensional problem

(P) z ⎧ ⎨ ⎩ maximize s (y, s) ∈ S n × [0, 1] f i (z, y, s) ≤ 0 for all i ∈ I. ( 16
)
The solution set to ( 15) is given by

Z * := {z ∈ Z : (z) = s * } = {z : (z, y, s) solves (2)}.
For a fixed z ∈ Z , the reduced problem ( 16) has a very interesting geometric interpretation. Note that ( 16) can be rewritten as

⎧ ⎨ ⎩ maximize s (y, s) ∈ S n × [0, 1] T (z) ⊆ (y, s). ( 17
)
In other words, one must find a standard revolution cone of smallest aperture angle that contains the polyhedral cone T (z). The optimization problem [START_REF] Henrion | Inradius and circumradius of various convex cones arising in applications[END_REF] has been studied in [START_REF] Henrion | On properties of different notions of centers for convex cones[END_REF][START_REF] Henrion | Inradius and circumradius of various convex cones arising in applications[END_REF][START_REF] Iusem | Normality and modulability indices. I. Convex cones in normed spaces[END_REF] in the more general context in which T (z) is changed by an arbitrary

K ∈ (R n ). The optimal value of ⎧ ⎨ ⎩ maximize s (y, s) ∈ S n × [0, 1] K ⊆ (y, s) (18)
is denoted by τ (K ) and called the sharpness coefficient of K (cf. [START_REF] Iusem | Normality and modulability indices. I. Convex cones in normed spaces[END_REF]Section 5]). If K is pointed, then the maximization problem [START_REF] Henrion | Condition number and eccentricity of a closed convex cone[END_REF] admits exactly one solution. The y-component of such solution is called the circumcenter of K .

Theorem 2.7 For all z ∈ P c , the term (z) is equal to the sharpness coefficient of T (z). In particular, the function : P c → R is not merely continuous but satisfies

| (z 1 ) -(z 1 )| ≤ 2 dist(z 1 , P) + dist(z 2 , P) z 1 -z 2 (19) for all z 1 , z 2 ∈ P c .
Proof For all z ∈ P c , the cone T (z) is pointed. Recall that (z) is equal to the optimal value of ( 17), so the first part of the theorem is nothing new. For proving the last part of the theorem we view z ∈ P c → (z) = τ (T (z)) as a composite function. Proposition 13 of [START_REF] Iusem | Normality and modulability indices. I. Convex cones in normed spaces[END_REF] asserts that

|τ (K 1 ) -τ (K 1 )| ≤ (K 1 , K 2 ) for all K 1 , K 2 ∈ (R n ).
One gets in this way

| (z 1 ) -(z 1 )| ≤ (T (z 1 ), T (z 2 ))
for all z 1 , z 2 ∈ P c . Proposition 2.1 completes the proof of ( 19).

The inequality [START_REF] Iusem | Normality and modulability indices. I. Convex cones in normed spaces[END_REF] implies that is Lipschitzian on any compact set contained in P c . In particular, : P c → R is locally Lipschitz and one obtains the upper estimate

∂ (z) ⊆ [dist(z, P)] -1 B n for the Clarke subdifferential ∂ (z) := co lim k→∞ ∇ (z k ) : {z k } k∈N → z, z k ∈ dom(∇ )
of at z ∈ P c . In general, : P c → R is a nondifferentiable function, but the set dom∇ of differentiability points of has full Lebesgue measure in P c .

Remark 2.8

is finite-valued also on the boundary of P. For all z ∈ [int(P)] c one has

(z) = sup{s : (y, s) ∈ M(z)},
where M(z) is the nonempty compact set given by

M(z) := {(y, s) ∈ S n × [0, 1] : T (z) ⊆ (y, s)}.
By using [START_REF] Bose | Some aperture-angle optimization problems[END_REF] and the continuity of the function :

S n × [0, 1] → (R n ), one can check that M is upper-semicontinuous as set-valued map between the metric spaces [int(P)] c and S n × [0, 1]
. By relying on Berge's maximum theorem (cf. [6, Theorem 1.4.16]), one deduces that : [int(P)] c → R is upper-semicontinuous. However, (z) may not converge to (z) as z approaches to a point z on the boundary of P.

Corollary 2.9 For all z ∈ P c one has

max 0, 1 - diam(A) dist(z, P) ≤ (z),
where diam(A) = max i, j∈I a ia j . In particular,

lim z →∞ (z) = 1. ( 20
)
Proof The proof is inspired from Theorems 3.2 and 3.4 in [START_REF] Moreno | External analysis of boundary points of convex sets: supporting cones and drops[END_REF]. Since the sharpness coefficient of a ray is equal to 1, the nonexpansiveness of τ :

(R n ) → R yields | (z) -1| ≤ (T (z), R + y) (21)
for all y ∈ S n . If y is a unit vector taken from T (z), then

(T (z), R + y) = max u∈T (z)∩ S n dist(u, R + y ∩ S n ) = max u∈T (z)∩ S n u -y .
This and ( 21) yield 1

-(z) ≤ diam(T (z) ∩ S n ). By the Dunkl-Williams inequality, one has diam(T (z) ∩ S n ) = max u,v∈P u -z u -z - v -z v -z ≤ 1 dist(z, P) max u,v∈P u -v .
So, we have proven that

1 -(z) ≤ diam(P) dist(z, P) .
It suffices now to observe that diam(P) = diam(A), because a polytope admits always a pair of vertices attaining its diameter.

A problem closely related to [START_REF] Henrion | Condition number and eccentricity of a closed convex cone[END_REF] is that of finding a largest ball centered at a unit vector and contained in a given

Q ∈ (R n ): ⎧ ⎨ ⎩ maximize r (x, r ) ∈ S n × [0, 1] x + r B n ⊆ Q. ( 22
)
The optimal value of ( 22) is denoted by ρ(Q) and called the inradius of Q. If the cone Q has nonempty interior, then the maximization problem [START_REF] Omana-Pulido | Aperture-angle optimization problems in three dimensions[END_REF] admits exactly one solution. The x-component of such solution is called the incenter of Q.

Lemma 2.10 Suppose that K ∈ (R n ) is pointed or, equivalently, that its positive dual cone

Q := {v ∈ R n : v, u ≥ 0 for all u ∈ K } has nonempty interior. In such a case, τ (K ) = ρ(Q) and the circumcenter of K is equal to the incenter of Q.
Lemma 2.10 is borrowed from [START_REF] Henrion | On properties of different notions of centers for convex cones[END_REF][START_REF] Henrion | Inradius and circumradius of various convex cones arising in applications[END_REF]. A matter of computation shows that the positive dual cone of T (z) is equal to the polyhedral convex cone

R(z) := {x ∈ R n : a i -z, x ≥ 0 for all i ∈ I }. (23)
Hence, for all z ∈ P c , the term (z) can be seen also as the inradius of R(z). For notational convenience, we introduce the vector function y : P c → R n given by

y(z) := circumcenter of T (z) = incenter of R(z).
The next corollary is a direct consequence of Lemma 2.10 and does not need further explanation.

Corollary 2.11

Let Z be a compact set not intersecting P. Then (z, y, s) solves (2) ⇔ z solves [START_REF] Gourion | Solidity indices for convex cones[END_REF] and (y, s) = (y(z), (z)).

For each fixed z ∈ P c , the pair (y(z), (z)) is the unique optimal solution to [START_REF] Henrion | Inradius and circumradius of various convex cones arising in applications[END_REF]. So, one can view the set C(z, y(z), (z)) as the optimal conic light beam with prescribed apex z. We next explain how to evaluate the functions y(•) and (•) in practice. In what follows, the notation m refers to the standard simplex in R m . Theorem 2.12 Let m be the cardinality of I . Then admits on P c the lower hull representation

(z) = min λ∈ m m i=1 λ i c i (z) , ( 24
)
where each c i : P c → R n is given by

c i (z) := a i -z a i -z . ( 25
)
In particular, if Z is a compact set not intersecting P, then the optimal value of (2) admits the max-min formulation

s * = max z∈Z min λ∈ m m i=1 λ i c i (z) . ( 26
)
Proof We take z ∈ P c and write [START_REF] Wolfe | Finding the nearest point in a polytope[END_REF] in the equivalent form

R(z) = {x ∈ R n : c i (z), x ≥ 0 for all i ∈ I }. ( 27
)
Note that each a i is different from z and therefore the unit vector c i (z) is well defined. The representation formula (24) is obtained by applying [START_REF] Gourion | Solidity indices for convex cones[END_REF]Proposition 3.3] to the polyhedral convex cone (27). The second part of the theorem follows by combining (24) and (15).

Remark 2.13 One of the oldest illumination problem in the history of mathematics was formulated by the scholar Regiomontanus in the 15th century. Regiomontanus asked to find a point on the ground so that a perpendicularly suspended rod appears largest, i.e., subtends the greatest visual angle (cf. [START_REF] Dörrie | Hundred Great Problems of Elementary Mathematics: Their History and Solution[END_REF]). Regiomontanus' problem is a particular instance of the minimization problem

s := min z∈Z (z).
If Z is a compact set not intersecting P, then one gets

s = min z∈Z λ∈ m m i=1 λ i c i (z) .
In principle, computing s is simpler than computing s * .

An interesting aspect of (24) is that (z) can be evaluated by solving an "easy" convex optimization problem. To be more precise, one has to find the projection π(z) := Proj [0, (z)] of 0 onto the polytope (z) := co{c 1 (z), . . . , c m (z)}. By relying on [15, Proposition 3.1], one can write

(z) = min i∈I c i (z) , y(z) (28)
with y(•) admitting the explicit formula y(z) = π(z)/ π(z) . Note that ( 28) is yet another characterization of the function : P c → R.

Differentiability properties of

The lower hull representation (24) and the continuity of

(λ, z) ∈ m × P c → L(λ, z) := m i=1 λ i c i (z) ( 29
)
confirms that is continuous on P c . The function ( 29) is differentiable with respect to z and its partial gradient is given by

∇ z L(λ, z) = m i=1 λ i a i -z c i (z)(c i (z)) T -I n m i=1 λ i c i (z) m i=1 λ i c i (z)
, where I n is the identity matrix of order n and the superscript "T " stands for transposition.

Beware that the differentiability of each L(λ, •) does not imply the differentiability of . In fact, it is not difficult to construct a sample A for which : P c → R is nondifferentiable. The next theorem shows that is directionally differentiable and Clarke lower-regular.

Theorem 2.14 Let m be the cardinality of I . Then (a) : P c → R is directionally differentiable and its directional derivative at z ∈ P c is given by

h ∈ R n → (z; h) = min λ∈ (z) ∇ z L(λ, z), h , ( 30
)
where L is the function introduced in (29) and (z) is the solution set to (24). (b)

: P c → R is Clarke lower-regular, i.e., for all z ∈ P c the directional derivative (z; •) coincides with the Clarke lower derivative

h ∈ R n → ↓ (z; h) := lim inf v→z t→0 + (v + th) -(v) t .
In particular, the Clarke subdifferential of admits the representation

∂ (z) = co{∇ z L(λ, z) : λ ∈ (z)}. ( 31
)
Proof For proving (a) it suffices to apply Danskin's directional differentiability theorem [START_REF] Danskin | The Theory of Max-Min and Its Application to Weapons Allocation Problems[END_REF] to The formula (30) shows that is differentiable at each z ∈ P c for which (31) is a singleton. For instance, if the vectors c 1 (z), . . . , c m (z) are linearly independent, then the optimization problem (24) admits a unique solution, say λ(z), and, in such a case, one gets ∇ (z) = ∇ z L(λ(z), z). Linear independence of the c i (z)'s is however a very stringent assumption, because usually the size of the sample A is much larger than the dimension of the underlying space.

z ∈ P c → (z) = min λ∈ m L(λ, z).

Corollary 2.15

For all z ∈ P c , the directional derivative (z; •) : R n → R is concave and

max h =1 | (z; h)| ≤ 1 dist(z, P) . ( 32
)
Proof The first part follows from Theorem 2.14. The estimate (32) is a consequence of (19).

Directions of ascent for the function

By Proposition 2.4 one knows that all the solutions to the maximization problem [START_REF] Gourion | Solidity indices for convex cones[END_REF] are on the boundary of Z . As shown in the proposition below, the interior of Z is also free of local solutions to [START_REF] Gourion | Solidity indices for convex cones[END_REF].

Proposition 2. [START_REF] Henrion | On properties of different notions of centers for convex cones[END_REF] Let Z be a compact set not intersecting P. Assume that A is not on ray emanating from a point of Z . Then any local maximum of : Z → R is on the boundary of Z .

Proof Let z ∈ P c . Recall that y(z) stands for the circumcenter of T (z). One has

(z -t y(z)) > (z) (33)
for all positive real t. The above inequality is obtained by working out the proof of Proposition 2.4 with the particular choice w = t y(z). Now, if z ∈ int(Z ), then zt y(z) remains in Z for all t sufficiently small. Hence, a point z in the interior of Z cannot be a local maximum of : Z → R.

The inequality (33) says that -y(z) is a direction of ascent of at the point z. This information could be useful for algorithmic purposes. It is tempting to conjecture that -y(z) is a steepest ascent direction, but we have no evidence supporting this conjecture. The next proposition shows that (z) does not decrease if the argument of moves in a direction opposite to T (z).

Proposition 2.17 Let z ∈ P c and h ∈ T (z). Then for all positive real t one has

(z -th) ≥ (z). ( 34
)
Proof We start by proving a result that is of interest by itself:

T (z -w) ⊆ T (z) for all w ∈ T (z). ( 35
)
The statement (35) is somewhat implicit in [21, Lemma 5.1], but we give here a direct proof. Take

w = α(x -z) and u = β(v -(z -w)) (36)
with α, β ≥ 0 and x, v ∈ P. One must prove that u ∈ T (z). Let β = 0, otherwise u = 0 and we are done. Note that

(1/β)u = v -(z -α(x -z)) = v + αx -(1 + α)z.
Hence,

1 β(1 + α) u = 1 (1 + α) v + α (1 + α)
xz (37) belongs to Pz, because the term between brackets is a convex combination of vectors in P. This shows that u ∈ T (z) and completes the proof of (35). Now, since T (z) is a cone, one can apply (35) to w = th. This yields T (zth) ⊆ T (z), from where one derives (34).

Proof We suppose that s K < 1, otherwise (y K , s K ) is a ray and we are done. If the c i 's are all in the interior of (y K , s K ), then there exists s ∈]s K , 1] such that {c 1 , . . . c m } ⊆ (y K , s), contradicting the maximality of s K . Hence, at least one of the c i 's is in E(y K , s K ). Suppose for instance that

c 1 ∈ E(y K , s K ), {c 2 , . . . , c m } ⊆ int[ (y K , s K )]. ( 39
)
We must arrive to a contradiction. For t ∈]0, 1], consider the new axis

y t = y K + t (c 1 -y K ) y K + t (c 1 -y K )
and the new sharpness coefficient s t = c 1 , y t . By construction, one has c 1 ∈ E(y t , s t ). Since lim

t→0 + y t = y K lim t→0 + s t = c 1 , y K = s K c 1 = s K ,
the interiority condition (39) ensures the existence of t ∈]0, 1] such that {c 2 , . . . c m } ⊆ (y t , s t ) for all t ∈]0, t]. We have proven in this way that K ⊆ (y t , s t ) whenever t is sufficiently small. On the other hand, if one computes the right-derivative at 0 of the function t → s t , then one gets

ds t dt t=0 + = 1 -c 1 , y K > 0.
Hence, s t > s K for all t near 0, contradicting again the maximality of s K .

As a by-product of Lemma 2.21 one gets:

Proposition 2. [START_REF] Omana-Pulido | Aperture-angle optimization problems in three dimensions[END_REF] Let Z be a compact set not intersecting P. Let (z, y, s) be a solution to the illumination problem [START_REF] Astorino | Nonsmooth optimization techniques for semisupervised classification[END_REF]. Then the boundary of C(z, y, s) contains at least two of the a i 's.

Proof Let the c i (z)'s be defined as in (25). Note that T (z) = cone{c i (z) : i ∈ I } is pointed and that (y, s) is a solution to the problem (17). Lemma 2.21 implies that E(y, s) contains at least two of the c i (z)'s. In other words,

bd[C(z, y, s)] = {x ∈ R n : s x -z -y, x -z = 0} = z + E(y, s)
contains at least two of the a i 's.

Some particular cases

(z) = min α∈[0,1] (1 -α)c 1 (z) + α c 2 (z) .
The above minimum is attained at α = 1/2 and one gets the explicit formula

(z) = 1 √ 2 [1 + c 1 (z), c 2 (z) ] 1/2 . ( 40
)
The behavior of ( 40) is problematic if z approaches either to a 1 or to a 2 . If Z is a compact set not intersecting P, then an optimal apex z * ∈ Z is found by solving the maximization problem

κ * := max z∈Z c 1 (z), c 2 (z) .
A corresponding optimal axis y * ∈ S n is obtained by normalizing the vector

ξ * := c 1 (z * ) + c 2 (z * ) 2
and the optimal sharpness coefficient is given by s

* = [(1 + κ * )/2] 1/2 .
It is a great advantage to count on an explicit formula for as in (40). What happens, however, if the points a i are not on a line? The planar case is discussed in the example below.

Example 2.24 Consider an arbitrary sample {a i : i ∈ I } on the plane. What makes the planar case so simple is that, for each z ∈ P c , the supporting cone T (z) is already a revolution cone. If ϑ(z) denotes the aperture angle of T (z), then clearly

ϑ(z) = max i, j∈I i = j arccos c i (z), c j (z) .
Since the smallest revolution cone that contains T (z) is the cone T (z) itself, one has

(z) = cos ϑ(z) 2 = 1 + cos(ϑ(z)) 2 1/2 .
After simplification one ends up with

(z) = 1 √ 2 ⎡ ⎣ 1 + min i, j∈I i = j c i (z), c j (z) ⎤ ⎦ 1/2 (41)
for all z ∈ P c . Beware that the explicit formula (41) is no longer true for illumination problems in dimension three or more. In the nonplanar case, the expression on the right-hand side of ( 41) is a rather coarse upper bound of (z).

The next example shows that the number of optimal apices in a planar illumination problem is not necessarily finite. In fact, it may form a continuum. This phenomenon is well known in the literature.

Example 2.25 Consider a sample formed with two points on the plane, say a 1 = (0, 0) and

a 2 = (1, 0). Let Z ε = {(r cos θ, r sin θ) : θ ∈ [ε, π/2] and r = cos θ + sin θ } ,
where ε is a small positive parameter. Clearly, Z ε is a compact set not intersecting the line segment [a 1 , a 2 ]. A direct computation shows that c 1 (z), c 2 (z) = 1/ √ 2 for all z ∈ Z ε . Hence, the function is constant over Z ε . Every element of Z ε is therefore an optimal apex. If one sets ε = 0, then the situation is of course entirely different. Such choice of ε leads to an illumination problem admitting z * = a 2 as unique optimal apex. The most bothersome aspect of the optimization problem [START_REF] Gourion | Solidity indices for convex cones[END_REF] is the lack of convexity of the objective function . The nondifferentiability of is not truly a serious drawback. The proposition below is part of the folklore of constrained optimization theory. Proposition 2.26 Let Z be a convex compact set not intersecting P. If z * is a local solution to [START_REF] Gourion | Solidity indices for convex cones[END_REF], then z * satisfies the criticality condition

(z * ; h) ≤ 0 for all h ∈ T Z (z * ), ( 42
)
where T Z (z * ) denotes the tangent cone to Z at z * .

A point z * ∈ Z as in ( 42) is called a critical point of [START_REF] Gourion | Solidity indices for convex cones[END_REF]. If is differentiable at z * , then the condition (42) takes the simpler form

∇ (z * ) ∈ N Z (z * ), ( 43
)
where N Z (z * ) denotes the normal cone to Z at z * . The reader should be aware that the optimization problem (15) may have critical points that are not a solution to [START_REF] Gourion | Solidity indices for convex cones[END_REF], not even a local solution. This issue is illustrated with the help of the next example.

Example 2.27

Let A be formed with a 1 = (1, 0) and

a 2 = (0, 1). Consider Z = [-1, 0] × [-1, 0], which is a convex compact set not intersecting P. Let us examine κ(z) = c 1 (z), c 2 (z) around z * = (-1, 0). If α and β are the Cartesian coordinates of z, then κ(z) = κ(α, β) = α 2 + β 2 -α -β (1 -α) 2 + β 2 (1 -β) 2 + α 2 ,
and a matter of computation shows that

∂κ ∂α (-1, 0) < 0 and ∂κ ∂β (-1, 0) = 0. Since √ 2 (•) = √ 1 + κ(•), one has ∂ ∂α (α, β) = 1 2 √ 2 1 √ 1 + κ(α, β) ∂κ ∂α (α, β) ∂ ∂β (α, β) = 1 2 √ 2 1 √ 1 + κ(α, β) ∂κ ∂β (α, β).
Hence,

∂ ∂α (-1, 0) < 0 and ∂ ∂β (-1, 0) = 0. ( 44
)
The gradient ∇ (z * ) belongs to the boundary of

N Z (z * ) = R -× R + .
In particular, z * satisfies the criticality condition (43). However, z * is neither a local maximum nor a local minimum of on Z . Since is a nonconvex function, criticality alone is not enough to guarantee optimality. In this example, the critical point z * is not even a local extremum.

Numerical resolution of the illumination problem

This section concentrates on the numerical resolution of the illumination problem (2). We have adopted a simple structure for set Z , by defining it either as a box or as the parametric representation of a curve. We have focused on the max-min formulation

s * = max z∈Z (z) = max z∈Z min λ∈ m m i=1 λ i c i (z) ,
which has been tackled by resorting to a bundle type method (BM) suited for local minimization of nonconvex, nonsmooth functions in presence of linear constraints. The code used is NCVX which is based on the algorithm described in [START_REF] Fuduli | Minimizing nonconvex nonsmooth functions via cutting planes and proximity control[END_REF] and has been already used for solving classification and separation problems in [START_REF] Astorino | Nonsmooth optimization techniques for semisupervised classification[END_REF][START_REF] Astorino | DC models for spherical separation[END_REF][START_REF] Astorino | Margin maximization in spherical separation[END_REF][START_REF] Astorino | Conic separation of finite sets. I. The homogeneous case[END_REF]. NCVX is written in Fortran 77.

In addition we have tested a direct evaluation (DE) method which is merely based on a sampling of the feasible set Z and on direct calculation of (z). We remark that latter calculation, as explained in Sect. 2.3, requires to solve, at each tested point of Z , a least-norm problem on a polytope, a structured convex quadratic program widely studied in the literature (see [START_REF] Frangioni | Solving semidefinite quadratic problems within nonsmooth optimization algorithms[END_REF][START_REF] Kiwiel | A method for solving certain quadratic programming problems arising in nonsmooth optimization[END_REF][START_REF] Wolfe | Finding the nearest point in a polytope[END_REF]).

We have applied DE when Z (or its boundary) can be parameterized with the help of a single parameter ranging on a compact interval. Suppose for instance that

Z = {z(t) : t ∈ [0, 1]}, where z : [0, 1] → R n is a continuous function.
In such a case, the maximization problem ( 15) is nothing but a one dimensional optimization problem:

s * = max t∈[0,1] (z(t)). ( 45
)
Note that (45) is about maximizing the continuous function z (•) := (z(•)) on the interval [0, 1]. DE is quite easy to implement and reads as follows.

• Step 0. Initialization. A tolerance parameter ε > 0 is given. Choose two integers N and p max . Set p = 1 and s * = 0.

• Step 1. Set s ( p) = 0. For k = 1, . . . , p N -set t (k) := 1 2 k-1 p N + k p N
and z (k) := z(t (k) ), -solve the least-norm problem ( p) . Increase p by one unit. If p > p max , then STOP.

z (t (k) ) = min λ∈ m m i=1 λ i c i (z (k) ) . ( 46
) If z (t (k) ) > s ( p) , then set s ( p) = z (t (k) ). EndFor If |s ( p) -s * | ≤ ε, then STOP. • Step 2. If s ( p) > s * , then set s * = s
Go to Step 1.

For the numerical resolution of the least norm problem (46) we have used the function "DQPROG" (from IMSL Fortran Numerical Libraries) written in Fortran 77 for solving a quadratic programming problem subject to linear equality/inequality constraints. We have set ε = 10 -4 , N = 1000 and p max = 3. Both the codes BM and DE have been run on a 1.66 GHz Intel Core Duo, under Microsoft Windows XP operating system. For our experiments we have considered the following test problems:

-Dataset Nr1. For each dimension n, we have defined A, the set of points to be illuminated, as the set of the canonical vectors {e 1 , . . . , e n } of R n . The apex z is constrained to belong to the feasible set Z = [-1, 0] n . In this case the global optimal solution (z * , s * ) is the point

z * = (-1, . . . , -1), s * = (1 + 3/n) -1/2 (1 + 1/n).
-Dataset Nr2. The set A is formed with random points a 1 , . . . , a m uniformly distributed on [0, 1] n . The set Z is the line segment

Z = [u, v] := {(1 -t)u + tv : t ∈ [0, 1]},
where u and v are in turn uniformly distributed on [-1, 0] n , so that Z does not intersect the polytope P. We have tested the program for several combinations of n and m.

-Dataset Nr3. The set A is formed with random points a 1 , . . . , a m uniformly distributed on

[-1, 1] × [-1, 1] × {0}. The set Z is the ellipse Z = {(α cos(2π t), β sin(2π t), 1) : t ∈ [0, 1]} in R 3 .
The constants α and β are uniformly distributed on [0, 1]. Note that the ellipse Z and the polytope P are in parallel planes, so they do not intersect.

-Dataset Nr4. The set A is formed with random points a 1 , . . . , a m uniformly distributed on [-1, 0] n . The set Z is the moment curve

Z = {(t, t 2 , . . . , t n ) : t ∈ [0, 1]} in R n .
In the plane, the moment curve is a parabola, in the three-dimensional space it is a twisted cubic, and so on. Dataset Nr1 has been tackled exclusively by means of BM, Dataset Nr2 by both BM and DE, while Datasets Nr3 and Nr4 just by DE. In the following we present the results we have obtained.

BM has been run 100 times on Dataset Nr1, for randomly chosen starting points z 0 ∈ Z and for n ∈ {2, 3, . . . , 10}. In Table 1 we report for each n, the known global optimal solution s * , the average optimal value s provided by the program and the maximum error max i=1,...,100 (s *s i ), where s i is the optimal value of s relative to the i-th experiment. We have considered a 20 digit representation for the variable s.

As for Dataset Nr2, we have considered all possible combinations of m ∈ {5, 10, 15, 20} and n ∈ {3, 4, 5}. For each combination (m, n) we have generated 100 times the interval endpoints u, v. In Table 2 we report comparison of the results provided by BM and DE. For each combination (m, n) we report the ordered list (t BM , tDE , W BM , W DE ), where tBM is the average CPU time for BM, tDE is the average CPU time for DE, W BM is the number of times BM has provided a better objective function value at stop, and W DE is the number of times DE has provided a better objective function value at stop. The results of the numerical experiments indicate that the max-min formulation (26) provides an effective computational scheme for numerically solving the illumination problem (2).

On Datasets Nr3 and Nr4 we have applied DE for m ∈ {5, 10, 15, 20}. Figures 1 and2 show typical shapes of z when Z is an ellipse and a moment curve in R 3 .

The planar case

The analysis of the illumination problem ( 15) is considerably easier in a two dimensional context. The particular case n = 2 is mentioned only briefly in Example 2.24, but it has been the object of a discussion in [START_REF] Bose | Some aperture-angle optimization problems[END_REF]. Instead of recalling the specially tailored techniques developed in [START_REF] Bose | Some aperture-angle optimization problems[END_REF], we mention below a few useful guidelines for the resolution of (15) in the planar case. There are two interesting subcases for consideration: I. Let A = {a 1 , a 2 } and Z be a polytope whose vertices {z k } r k=1 are known. One assumes that the line passing through a 1 and a 2 does not intersect Z . As seen in Example 2.23, solving the illumination problem [START_REF] Gourion | Solidity indices for convex cones[END_REF] amounts to maximize the function

z ∈ Z → κ(z) := c 1 (z), c 2 (z) . ( 47
) Fig. 2 A typical shape of z when Z is a moment curve in R 3 (m = 15)
Thanks to Proposition 2.16, any maximum of (47) belongs to the boundary of Z . On the other hand, it is not difficult show that (47) is a strictly quasiconvex function, i.e.,

κ((1 -t)u + tv) < max{κ(u), κ(v)} if t ∈]0,
for each k ∈ {1, . . . , r } separately. Note that (49) can be treated for instance with the DE method.

The penumbra problem

For the penumbra problem (4) it is natural to assume that Z is included in the interior of P, otherwise the problem admits a separating half-space as trivial solution. Of course, the hypothesis that Z is closed remains always in force. In particular, z ∈ bd(Z ).

Proof The first part of the proposition is evident. Let (z, y, s) be a solution to [START_REF] Astorino | Margin maximization in spherical separation[END_REF]. Then s = s and s a iz ≥ a iz, y for all i ∈ I . Suppose that (50) is not true. Then there exists w ∈ int[ (y, s)] such that z 0 := z + w belongs to Z . Since a iz 0 = a iz -w ≥ a iz -w , a iz 0 , y = a iz -w, y = a iz, y -w, y , one obtains s a iz 0 ≥ s a izs w ≥ a iz, ys w , ≥ a iz 0 , y + w, ys w But w, ys w > 0, because w belongs to the interior of (y, s). Hence, s a iz 0 > a iz 0 , y for all i ∈ I . By setting s := max i∈I s i with s i := max 0, a iz 0 a iz 0 , y , one gets 0 ≤ s < s and A ⊆ D(z 0 , y, s). This contradicts the minimality of s.

The next result is similar in spirit to Proposition 2.22. We must arrive to a contradiction. If one sets c 1 := a 1z (a 1z) , y t := yt (c 1y) yt (c 1y) , s t := c 1 , y t , then one can prove that {a 1 , . . . a m } ⊆ D(z, y t , s t ) for all t > 0 small enough. On the other hand, the right-derivative at 0 of the function t → s t is given by ds t dt t=0 + = c 1 , y -1 < 0.

Hence, s t < s for all t near 0, contradicting again the minimality of s.

( a )

 a If Z is unbounded, then s * = 1 and (2) has no solution. (b) If Z is bounded and not included in the interior of P, then (2) has at least one solution.

The set m

  is compact and the function (λ, z) ∈ m × P c → ∇ z L(λ, z) is continuous. Hence, the hypotheses of Danskin's theorem are in force. Part (b) is obtained by applying [8, Theorem 2.1].

Fig. 1 A

 1 Fig. 1 A typical shape of z when Z is an ellipse (m = 15)

1 [

 1 and u, v ∈ Z are distinct. Hence, any maximum of (47) is in fact a vertex of Z . Thus, everything boils down to evaluate the function κ(•) at each z k and keep the vertex that yields the highest function value. Such numerical task can be performed in linear time in the number of vertices. II. Let Z be a polytope with vertices {z k } r k=1 and A = {a i } m i=1 be a sample with more than two points. One assumes that Z does not intersect the convex hull of A. As seen in Example 2.24, solving (15) amounts to maximizez ∈ Z → κ(z) := min 1≤i< j≤m c i (z), c j (z) . (48)It may happen that no vertex of Z maximizes (48), so one needs to evaluate κ on the whole boundary of Z . Without loss of generality, we assume that Z has nonempty interior and label the vertices of Z in such a way that bd(Z ) = r k=1 [z k , z k+1 ]with z r +1 := z 1 . So, everything boils down to solve maximize z∈[z k ,z k+1 ] κ(z)

Proposition 4 . 1

 41 Assume that Z ⊆ int(P). Then the optimal value s of the problem (4) is positive and the solution set is nonempty. Furthermore, if (z, y, s) is a solution to (4), then Z ∩ int[z + (y, s)] = ∅.(50)

Proposition 4 . 2

 42 Let Z ⊆ int(P) and (z, y, s) be a solution to the penumbra problem[START_REF] Astorino | Margin maximization in spherical separation[END_REF].Then the boundary of D(z, y, s) contains at least two of the a i 's.Proof Let m be the cardinality of I . If the a i 's are all in the interior of D(z, y, s), then there exists s ∈]0, s] such that {a 1 , . . . a m } ⊆ D(z, y, s), contradicting the minimality of s. Hence, at least one of the a i 's belongs to the boundary of D(z, y, s). Suppose for instance that a 1 ∈ bd[D(z, y, s)], {a 2 , . . . , a m } ⊆ int[D(z, y, s)].

  This short section displays a few useful examples and counter-examples. Example 2.23 If the a i 's are arranged on a line, then one can drop the intermediate points and keep only the two extreme points. Renumbering the a

i 's if necessary, one may suppose that the extreme points are a 1 and a 2 . Hence, P is the closed line segment [a 1 , a 2 ] and the function : P c → R is given by

Table 1

 1 BM on Dataset Nr1

	n	s *	s	max i=1,...,100	(s * -s i )
	2	0.9486833	0.9486833	0.0
	3	0.9428090	0.9428090	1.0 × 10 -15
	4	0.9449112	0.9449112	1.6 × 10 -10
	5	0.9486833	0.9486833	1.1 × 10 -7
	6	0.9525793	0.9525793	1.0 × 10 -15
	7	0.9561829	0.9561829	1.0 × 10 -15
	8	0.9594032	0.9594032	1.0 × 10 -15
	9	0.9622504	0.9622504	1.0 × 10 -15
	10	0.9647638	0.9647638	1.0 × 10 -15

Table 2

 2 BM versus DE on Dataset Nr2

	m	n= 3	n = 4	n = 5
	5	(0.0022, 0.1075, 89, 11)	(0.0019, 0.1145, 86, 14)	(0.0030, 0.1228, 88, 12)
	10	(0.0028, 0.3131, 89, 11)	(0.0036, 0.3381, 87, 13)	(0.0033, 0.3589, 88, 12)
	15	(0.0042, 0.6719, 88, 12)	(0.0027, 0.6950, 89, 11)	(0.0025, 0.7116, 91, 9)
	20	(0.0041, 1.2427, 87, 13)	(0.0034, 1.2600, 86, 14)	(0.0044, 1.2850, 93, 7)

We state next a sort of "strict" version of Proposition 2.17. To do this, we need first to prove a technical lemma concerning the function τ : (R n ) → R. It is plain to see that τ (•) is nonincreasing with respect to set inclusion. The following property is a bit more subtle, but not altogether surprising.

Proof Let ( ȳ, s) be a solution to [START_REF] Henrion | Condition number and eccentricity of a closed convex cone[END_REF]

. This completes the proof.

Proposition 2.19 Suppose that P has nonempty interior. Let z ∈ P c and h ∈ int(T (z)).

Then for all positive real t one has

Proof For all z ∈ P c one has

In view of Lemma 2.18, for proving (38) it is enough to check that

The above condition corresponds to a strict version of (35). Let w ∈ int(T (z)) and u be a unit vector in T (z -w). One can represent w and u as in (36), with α > 0, β > 0, x ∈ int(P), and v ∈ P. The term between brackets in (37) belongs now to the interior of P. Hence, u ∈ int(T (z)).

Remark 2.20 Let z ∈ P c . Beware that y(z) belongs to T (z), but not necessarily to int(T (z)). So, the condition (33) cannot be inferred directly from Proposition 2.19.

2.6 How many a i 's are on the boundary of an optimal conic light beam?

We open a parenthesis and address the question raised above. We do this just out of curiosity. The next proposition shows that at least two points of the sample {a i : i ∈ I } must be on the boundary of the optimal conic light beam. The following lemma is intuitively clear. generated by a finite collection {c i } m i=1 of unit vectors. Suppose that K is pointed and let (y K , s K ) be the solution to [START_REF] Henrion | Condition number and eccentricity of a closed convex cone[END_REF]. Then the boundary

of the revolution cone (y K , s K ) contains at least two of the c i 's.

We end this paper by stating a min-max formulation of the penumbra problem. Following the same pattern as in Sect. 2.3, we start by writing

where (z) denotes the optimal value of the lower dimensional problem

Unfortunately, the function is not as simple to evaluate as . The following result corresponds to the penumbra counterpart of Theorem 2.12. The notation

Theorem 4.3 Let m be the cardinality of I . Then : A c → R is continuous and admits the representation

where σ (z, •) is the nonnegative sublinear function given by σ (z, y) := max {0, c 1 (z), y , . . . , c m (z), y } .

In particular, if Z is included in int(P) and does not intersect A, then the optimal value of (4) admits the min-max formulation

Proof Since the variable s is to be minimized in (52), one can drop the constraint s ≤ 1. On the other hand, the last constraint in (52) can we written in the form c i (z), y ≤ s for all i ∈ {1, . . . , m}. Keeping in mind that s must also satisfy the contraint s ≥ 0, one deduces that (z) is equal to the optimal value of the minimization problem ⎧ ⎨ ⎩ minimize s y = 1 max {0, c 1 (z), y , . . . , c m (z), y } ≤ s.

This leads to the characterization (53) and proves the continuity of on A c . The second part of the theorem is obtained by combining (51) and (53).