Dr Alexandre Alanio
email: alexandre.alanio@pasteur.fr

Maud Gits-Muselli

Fanny Lanternier

Aude Sturny-Leclère

Marion Benazra

Samia Hamane

Anderson Messias Rodrigues

Dea Garcia-Hermoso

Olivier Lortholary

Françoise Dromer

Stéphane Bretagne

Françoise Foulet

Françoise Botterel

Nawel Ait Ammar

Marjorie Cornu

Severine Loridant

Guillaume Foulon

Sebastien Lefrançois

Maxime Dougados

Emilie Cardot-Martin

Chrisian Hiesse

Hélène Salvator

Claire Aguilar

Anne Gigandon

Marie-Elisabeth Bougnoux

Emilie Sitterlé

Arnaud Fekkar

Sébastien Imbert

Alexandre Bleibtreu

Blandine Denis

Jean-Michel Molina

Geoffroy Liegeon

Anne-Lise Munnier

Marion Malphettes

Evaluation of a new Histoplasma spp. reverse transcriptase quantitative PCR assay

Keywords: Histoplasmosis, qPCR, reverse transcriptase, whole nucleic acids, Whole blood, fungal load, quantification cycle

Laboratory diagnosis of histoplasmosis is based on various methods including microscopy, culture, antigen and DNA detection of Histoplasma capsulatum var. capsulatum (Hcc) or H. capsulatum var. duboisii (Hcd). To improve sensitivity of existing quantitative PCR assays, we developed a new reverse transcriptase qPCR (RTqPCR) assay allowing amplification of whole nucleic acids of Histoplasma spp.. and validated on suspected cases.

The limit of detection was 20 copies and the specificity against 114 fungal isolates/species was restricted to Histoplasma spp.. Whole nucleic acids of 1,319 prospectively collected consecutive samples from 907 patients suspected of histoplasmosis were tested routinely between May 2015 and May 2019 in parallel with standard diagnostic procedures performed in parallel. 44 had proven histoplasmosis due to Hcc (n=40) or Hcd (n=4) infections. RTqPCR was positive in 43/44 patients (97.7% sensitivity), in at least one specimen. Nine out of 863 cases (99% specificity) were RTqPCR positive and therefore classified as possible cases.

RTqPCR was positive in 13/30 (43.3%) blood tested in proven cases. A positive RTqPCR in blood was significantly associated with Hcc progressive disseminated histoplasmosis with a positive RTqPCR in 92.3% of the immunocompromised patients with disseminated disease. This new Histoplasma RTqPCR assay enabling amplification of hcc and hcd is highly sensitive and allows the diagnosis of histoplasmosis advantageously from blood and BAL.

The fungus Histoplasma was first described in Panama by Samuel Taylor Darling in a lung sample from a patient from Martinique (French Caribbean islands) [START_REF] Darling | A protozoön general infection producing pseudotubercles in the lungs and focal necroses in the liver, spleen and lymphnodes[END_REF] by observation of free or intracellular small yeasts in a smear of respiratory specimen. Since then, two distinct varieties have been described in humans initially identified by morphological aspects in tissues and epidemiological criteria (i) Histoplasma capsulatum var. capsulatum (Hcc, New World human pathogen) associated with small ovoid (2-3 µm) yeasts, and (ii) Histoplasma capsulatum var. duboisii (Hcd, Old World human pathogen) associated with large citrus shaped yeasts (5-10 µm). A third variety was described in Africa, Histoplasma capsulatum var. farciminosum (Old World animal pathogen) associated with epizootic lymphangitis in horses. However, this classification has been revisited based on recent multilocus sequencing revealing several cryptic species, which are structured according to their geographical origin. [START_REF] Rodrigues | The global epidemiology of emerging Histoplasma species in recent years[END_REF] Histoplasma capsulatum is now composed of at least four phylogenetically different groups, [START_REF] Kasuga | Phylogeography of the fungal pathogen Histoplasma capsulatum[END_REF] including a specific group for var. duboisii. Histoplasma capsulatum var. farciminosum appeared to be polyphyletic suggesting this variety does not correspond to a specific phylogenetic group. [START_REF] Kasuga | Phylogeography of the fungal pathogen Histoplasma capsulatum[END_REF] Recent analyses of multiple isolates by whole genome sequencing recognized the existence of four phylogenetic species corresponding to the geographical clades of North America (2 species), Latin America and Panama. The African variety (Hcd) is still not considered a different species mainly due to the small number of isolates studied so far. [START_REF] Sepúlveda | Genome Sequences Reveal Cryptic Speciation in the Human Pathogen Histoplasma capsulatum[END_REF] Nowadays, diagnosis of histoplasmosis still relies on microscopic direct examination and culture, together with antibody or antigen detection. [START_REF] Almeida-Silva | Current Aspects of Diagnosis and Therapeutics of Histoplasmosis and Future Trends: Moving onto a New Immune (Diagnosis and Therapeutic) Era?[END_REF] In addition, PCR methods based on the amplification of various DNA targets and using different methodologies have been reported 6- 14 with a sensitivity ranging from 67 to 100% depending on the studies. [START_REF] Azar | Laboratory Diagnostics for Histoplasmosis[END_REF] More recently, realtime quantitative PCR (qPCR) methods based on DNA detection [START_REF] Muraosa | Detection of Histoplasma capsulatum from clinical specimens by cycling probe-based real-time PCR and nested real-time PCR[END_REF][START_REF] Buitrago | Detection of imported histoplasmosis in serum of HIV-infected patients using a real-time PCR-based assay[END_REF][START_REF] Simon | Detection of Histoplasma capsulatum DNA in human samples by real-time polymerase chain reaction[END_REF][START_REF] Babady | Detection of Blastomyces dermatitidis and Histoplasma capsulatum from Culture Isolates and Clinical Specimens by Use of Real-Time PCR †[END_REF][START_REF] Vasconcellos Ic Da | The Role of Molecular Tests in the Diagnosis of Disseminated Histoplasmosis[END_REF] have been developed for human diagnosis to overcome the limits of the microbiological culture. Indeed, qPCR assays are expected to be more sensitive, [START_REF] Simon | Detection of Histoplasma capsulatum DNA in human samples by real-time polymerase chain reaction[END_REF] more rapid than culture and prevent laboratory transmission of the mold form of Histoplasma spp. in the laboratory environment. [START_REF] Weinstein | Laboratory-Acquired Infections[END_REF][START_REF] Sewell | Laboratory-associated infections and biosafety[END_REF] A Histoplasma spp. RTqPCR assay was developed and validated based on a repeated gene target and the detection of whole nucleic acids (RNA plus DNA) to improve the specificity and the sensitivity of microscopy and culture and accelerate results. After validation, this RTqPCR was performed prospectively during four years and then evaluated its performance in all patients suspected of histoplasmosis in this French network.

Material and methods

Nucleic acids extraction

Extraction procedures were initially optimized. Preliminary experiments on a patient with a positive qPCR in blood (Supplemental Table 1) showed a 100-fold increased analytical sensitivity upon bead beating compared to without mechanical lysis (∆Cq=6.54). Therefore, all specimens were processed with bead beating in this study. Another preliminary experiment showed that whole blood was 100-fold increased analytical sensitivity as compared to plasma (∆Cq=6.7) (Supplemental Figure 1A and 1B) to detect circulating Histoplasma spp. nucleic acids. Therefore, 1.3 mL of whole blood was used with bead beating instead of serum or plasma in this study. Serum and plasma specimens were not considered further as relevant sample type.

Whole nucleic acids (WNAs) were extracted from non-fixed specimens using initial bead beating in a Precellys bead beater (Bertin Technologies, Montigny-le-Bretonneux, France).

Then, the tubes were centrifuged 5 min at 10,000 g at 4°C. WNAs from the supernatant were extracted with addition of 10 µL/sample of 1:5 diluted internal control (DNA Virus Culture, DICD-CY-L100, Diagenode, Seraing, Belgium) using a Qiasymphony (Qiagen, Hilden, Germany) with the Virus Pathogen extraction Kit (Qiagen, Hilden, Germany) following the manufacturer's instructions. WNAs were eluted in 85 μL volume. For the paraffin-embedded tissue specimens, extraction of three 10 µm-sections was performed using the DNeasy Blood and Tissue kit (Qiagen, Hilden, Germany) following the manufacturer's recommendations with a volume of elution of 110 µL in DNA grade water. All extracts and aliquots were stored at +4°C until use (less than 5 days). In a specific preliminary experiment comparing WNA and DNA amplification, the RTqPCR kit was used without or with the reverse transcriptase step (15 min at 50°C) for DNA and WNA amplification, respectively.

Selection of target gene

The ribosomal small subunit RNA gene of H. capsulatum (GenBank accession number, GG663449.1) was used to design primers and probes using Primer3web v4.0.0 software. Outer primers (Hc_7F: GATGATGGCTCTGATTGAACG and Hc_361R: GATGATGGCTCTGATTGAACG) were first used to analyze the Histoplasma sequence in the region of interest of this gene for the presence of polymorphism. The amplicons obtained with two strains of Hcc (CNRMA16.205, GenBank MW317018 and CNRMA17.309, GenBank MW317017), one strain of Hcd (CNRMA16.638, GenBank MW317134) were sequenced resulting in a 100% (357/357bp) and 99.9% (356/357bp) similarity with the reference sequence (GenBank GG663449.1), respectively. The final assay was then designed in conserved regions excluding the single-nucleotide polymorphism observed in Hcd. Of note, no polymorphism at the primers and probe loci were observed upon sequencing of isolates from various lineages (GenBank MW317018, MW317017, MW317134)

Whole nucleic acid reverse transcriptase quantitative PCR assay (RT-qPCR)

WNA amplification was performed using the following conditions: 1´ Invitrogen RT-qPCR buffer mix (Superscript III [START_REF] Bustin | The MIQE Guidelines: Minimum Information for Publication of Quantitative Real-Time PCR Experiments[END_REF] . In addition, 50, 20 and 10 copies dilutions were tested in ten replicates to obtain the limit of detection.

Comparison of three qPCR assays

The evaluated in-house RTqPCR targeting mtSSU was compared on the LC480-II with two previously published qPCR assays targeting Internal Transcribed Spacer (ITS) 1 regions of H. capsulatum following the published protocols. ) (P=0.006, Figure 1). The limit of detection (LOD90) of the RTqPCR assay was 10 copies (Supplemental Table 4). The efficiency of the RTqPCR was 1.9 (95%) (Supplemental Table 4).

Analytical specificity

A total of 114 strains from 99 different species were tested showing no cross-reactivity except for Histoplasma species (Hcc and Hcd) and 3 closely related species Nannizziopsis sp.,

Emmonsia pasteuriana, and Emmonsia crescens (Supplemental Table 2 and3). The ability to detect 0.1 ng of DNA of various genetical clades of Histoplasma capsulatum including Hcd, LAm B, LAmE and RJ were similar (mean Cq = 22.51±0.82) whereas the 3 close relatives were not amplified in these conditions.

Comparison with qPCR assays

The RTqPCR assay was compared to two qPCR assays from the literature using 6 Hcc WNA and 2B, Supplemental Table 5).

Clinical evaluation

Over 4 years, 1,319 samples from 907 patients suspected of histoplasmosis were tested (Figure 3). A total of 44 (4.9%) patients had proven histoplasmosis based on a positive direct examination and/or culture (Table 1, Table 2). In these 44 proven cases, RTqPCR was positive in 43/44 (97.7% sensitivity) patients in at least one specimen (median: 2 RTqPCR-positive specimens/patient) (Table 2). In addition to proven cases, 9 patients out of 863 (99% specificity) with clinical presentation and exposure evocative of histoplasmosis were RTqPCR positive. Positive and negative predictive value were 0.82 and 0.99, respectively, resulting in a likelihood ratio of 93.7.

In proven patients, immunosuppression was reported in 25/44 (56.8%) patients, while 19/44 (43.2%) had no known major cause of immunosuppression and thereafter considered as immunocompetent. The underlying disease, the clinical presentation, the country of origin or travel history, and the species identified are summarized in Table 2.

In proven progressive disseminated histoplasmosis due to Hcc (23 patients), RTqPCR in respiratory specimens was positive in 13/14 (92.3%) of the patients sampled (17/19, 88.9% of the tested specimens) and in 7/7 patients with skin/mucosa sampled (8/9, 88.9% of tested specimens). RTqPCR in bone marrow aspirates was positive in 8/8 (100%) patients tested (9/10, 90% tested specimens) (Table 3). RTqPCR in blood was positive in 13/17 (76.5%) patients tested (25/48, 52.1% of tested specimens). In the subgroup of disseminated Hcc disease in immunocompromised patients (n=18) and HIV patients (n=12), the RTqPCR in blood the symptoms. Blood was tested negative in 7 cases out of 7 tested. No treatment and follow up were available for these patients.

One proven case of histoplasmosis has been reported out of the 855 (0.12%) RTqPCR-negative patients. This case had a negative RTqPCR test in BAL and blood but the diagnosis has been done on the microscopic visualization of yeasts evocative of Hcc in a lung biopsy (not tested by RTqPCR) before lung transplantation.

Blood RTqPCR for dissemination and follow-up analysis

A median of 1 blood specimen [range 1-9] per patient was tested (Table 1). RTqPCR in blood obtained within the first 7 days after diagnosis was positive only in 13 proven cases out of 30

cases with blood tested before treatment (43.3%). RTqPCR positivity in blood and not in all specimens was significantly associated with the underlying disease and the clinical presentation (<0.01, Table 4). Indeed, a positive RTqPCR in blood was significantly associated with proven progressive disseminated histoplasmosis due to Hcc with an increased rate of positive RTqPCR in disseminated (13/17, 76.5%) vs. localized (0/13, 0%) cases (P<0.0001). In acute pulmonary, CNS and localized histoplasmosis, the RTqPCR in blood was never positive (Table 3). The same feature was observed considering specimens (chi-square, P=0.001), with 25/58 (43.1%) positive RTqPCR in blood specimens from disseminated vs. 0/24 (0%) in blood specimens from localized cases (P<0.0001) (Table 4).

RTqPCR detection in blood was significantly associated with the underlying disease (P=0.005),

with RTqPCR positivity rate significantly increased in 10/18 (55.5%) immunocompromised proven cases vs. 1/10 (10%) in immunocompetent proven cases (P=0.041). The association was also significant if blood samples were considered with RTqPCR positivity observed in 24/60 (40.0%) specimens from immunocompromised vs. 1/22 (4.5%) in specimens from immunocompetent cases (P=0.003 (Table 4). Of note, blood was obtained from 3/4 immunocompetent patients with Hcd disseminated infections and RTqPCR was negative in all 3 cases (100%). Thirty-seven blood samples from 9 patients with at least one RTqPCR positive and one follow-up blood sample were studied for a median follow-up duration of 25 days [13.5-82] and with a median number of blood sample per patient of 3 (Table 5). All patients were treated with liposomal amphotericin B at D0 (date of the first positive blood specimen).

The kinetics of the fungal load is plotted in Figure 4A. The rate of negativation was dependent on the initial fungal load (Figure 4B) with a R 2 at 0.58 and a slope at -1.04 ± 0.44. This gain of 1 Cq per day corresponded approximately to a 2-fold decrease of the fungal load per day of treatment. Of note, for 3 of the 9 patients (Patients # 14, 38, 45), RTqPCR remained positive as the patient was lost (Table 5). In Patient#14 (HIV positive and inobservant), RTqPCR was still positive 334 and 611 days after the first positive result in blood with no intermediate testing.

Discussion

The present evaluation of this reverse transcriptase qPCR (RTqPCR) assay demonstrates an increased sensitivity of the Histoplasma detection using WNA instead of DNA and mtSSU gene instead of ITS as target gene. Thus, we were able to detect WNA in 97.7%

(43/44) of microscopy and/or culture proven cases in contrast with recent publications comparing ITS Loop-mediated Isothermal Amplification (LAMP) PCR and HSP100 nested PCR for which a positive PCR was observed in only in 54% and 64% of the culture positive specimens, respectively. [START_REF] Zatti M Da | Loop-mediated Isothermal Amplification and nested PCR of the Internal Transcribed Spacer (ITS) for Histoplasma capsulatum detection[END_REF] The high sensitivity of the present assay also allowed detection of Hcc disseminated diseases in the blood of 92.3% of immunocompromised patients and the detection of 9 patients including seven patients for whom the conventional mycology investigations were tested negative. Moreover, the high sensitivity of the test allows a rapid follow-up of treatment efficacy with the observation of a 2-fold decrease of the fungal load per day during optimal treatment.

To obtain a high sensitivity in proven cases not only confirms the diagnosis before the culture and without manipulating it and the hazard of laboratory-associated infection, but also allows a better delineation of dissemination in the patients with proven histoplasmosis.

Indeed, the RTqPCR was able to detect 10 more specimens than with conventional diagnosis in proven cases. Unsurprisingly, a positive RTqPCR in blood was found significantly associated with immunocompromised patients and with disseminated infection, as already observed. [START_REF] Dantas | Comparison of diagnostic methods to detect Histoplasma capsulatum in serum and blood samples from AIDS patients[END_REF] Indeed, more than 50% of HIV-positive patients with histoplasmosis had positive blood culture. [START_REF] Myint | HIV-Associated Histoplasmosis: Current Perspectives[END_REF] In this study, from the 30 patients with proven histoplasmosis and blood tested, 15

were HIV-positive patients and 10/15 (66.6%) had a positive RTqPCR in blood. No blood culture was positive in these patients. Of note, a positive RTqPCR in blood was also detected in a solid organ transplant recipient (#27), a patient with hematological malignancy (#04) and in an apparently immunocompetent patient (#10) (Table 5). All of them had a progressive disseminated disease. No positive RTqPCR in blood was observed in patients with Hcd and non-disseminated Hcc histoplasmosis (CNS, lung, localized mucocutaneous, localized osteoarticular and digestive histoplasmosis). This suggests that either the fungal load of circulating yeasts in blood is low resulting in a negative RTqPCR, or that dissemination occurred previously and was not present at diagnosis. The later hypothesis is more likely, since the patients presented at the hospital with very chronic disease without acute symptoms and since the volume of the lesions observed in tissues were high.

For the diagnosis of histoplasmosis due to Hcd, the RTqPCR gained in the detection of Hcd over the other classical tests, [START_REF] Buitrago | Detection of imported histoplasmosis in serum of HIV-infected patients using a real-time PCR-based assay[END_REF][START_REF] Simon | Detection of Histoplasma capsulatum DNA in human samples by real-time polymerase chain reaction[END_REF] which is of utmost importance because this infection is frequently underdiagnosed in Africa. [START_REF] Loulergue | Literature review and case histories of Histoplasma capsulatum var. duboisii infections in HIV-infected patients[END_REF] The clinical picture of histoplasmosis due to Hcd is clearly different than that of Hcc patients in patients without immunosuppression, but the clinical picture could present as a disseminated infection in HIV-positive patients. [START_REF] Theodoro | PRP8 intein in cryptic species of Histoplasma capsulatum: evolution and phylogeny[END_REF] In this cohort, Hcd infections was diagnosed in four patients considered immunocompetent but with multiple organ infections and in 3/3 patients tested, RTqPCR in blood was negative.

The overall positivity rate in the 169 specimens of the proven cases seems not so high (57.4%) but this overall analysis contains specimens taken from patients with no evidence of Histoplasma by culture of direct examination. Indeed, when focusing on culture/microcopy positive specimens, the rate of positivity was 92.2%. In addition, the RTqPCR was also positive in 10/30 (33.3%) culture/microscopy negative which were not useful for the diagnosis of the patients using culture/microscopy. RTqPCR positivity rate in blood was positive in 92.3% of the immunocompromised patients with dissemination.

The higher sensitivity of a PCR test over conventional methods raises the issue of the meaning of an isolated positive PCR result. Indeed, here, the RTqPCR detected seven patients for whom conventional diagnosis failed to detect Histoplasma microscopically or by culture.

These patients can be classified as possible cases. Six of these cases were immunocompetent and were not treated with antifungal. Indeed, five of them had mild acute pulmonary histoplasmosis and did not required treatment. [START_REF] Wheat | Clinical Practice Guidelines for the Management of Patients with Histoplasmosis: 2007 Update by the Infectious Diseases Society of America[END_REF] These cases are indeed debatable since no other diagnostic mean was positive for Histoplasma. Anti-Histoplasma antibody or Histoplasma antigen were not tested in these patients since no reproductible test are implemented in our lab or in our reference center due to lack or reproducibility of the immunodiffusion test and poor performances in immunocompromised patients. [START_REF] Wheat | Clinical Practice Guidelines for the Management of Patients with Histoplasmosis: 2007 Update by the Infectious Diseases Society of America[END_REF] However, antigen detection seems more adapted to diagnose histoplasmosis in immunocompromised patients as parallelized from what is known in patients with invasive aspergillosis. [START_REF] Pauw | Revised Definitions of Invasive Fungal Disease from the European Organization for Research and Treatment of Cancer/Invasive Fungal Infections Cooperative Group and the National Institute of Allergy and Infectious Diseases Mycoses Study Group (EORTC/MSG) Consensus Group[END_REF] Nevertheless, systematic review of the detection of antigen in patients with histoplasmosis failed to find studies dealing with large cohort of patients and homogenous group of patients. [START_REF] Nacher | Histoplasma capsulatum antigen detection tests as an essential diagnostic tool for patients with advanced HIV disease in low and middle income countries: A systematic review of diagnostic accuracy studies[END_REF] .

A highly sensitive assay on blood can help following the patients after treatment. These results demonstrated a decrease of the fungal load overtime under liposomal amphotericin B therapy with a rate of 2-fold decreased every day of treatment. Thus, a follow up using the present RTqPCR allows the determination of the fungal load in blood to help managing the patient. This follow-up has been already advocated several years ago based on blood culture in patients treated with liposomal amphotericin B or itraconazole. [START_REF] Wheat | Clearance of Fungal Burden during Treatment of Disseminated Histoplasmosis with Liposomal Amphotericin B versus Itraconazole[END_REF] The present RTqPCR provides a more suitable means to achieve this goal. Of note, in one patient, RTqPCR was persistently positive over a 2 years period in a context of very poor HIV replication control highlighting that antifungal treatment without improvement of the immune status is potentially insufficient, as described with leishmaniasis. [START_REF] Lindoso | Visceral leishmaniasis and HIV coinfection: current perspectives[END_REF] Because of the number of patients (n=907), It was not possible to get clinical data (country of origin, travel, antifungal treatment, outcome along with other microbiological investigations) for every patient but for the 54 patients with proven histoplasmosis (53 RTqPCR positive and 1 RTqPCR negative). Thus, clinical data from 854 patients (1135 samples of which 510 blood) with a negative RTqPCR result are lacking. Data from these patients has been collected only on case that had been diagnosed as a localized pulmonary histoplasmosis.

As a reference center with an implemented management stewardship of invasive fungal infection in France, suspicion or histoplasmosis cases are referred by clinician or mycologist to us (two referents and three referent mycologists) at the reference center. There is little chance that such unusual case in France would have been not referred for the purpose of diagnosis or clinical management, but it cannot be excluded that it could be the cases for some patients included in this study.

Consequently, to fully evaluate the RTqPCR and evaluate fully this diagnostic tool, a prospective study including the evaluation of conventional diagnosis, but also Histoplasma antigen testing in serum and/or urine should be implemented. Additionally, this tool will be used to investigate the rate of decrease of the fungal load under different therapeutic regimens aiming to show that rapid decrease will be associated with better outcome in patients.

In

5

 5 biopsy and one brain abscess (2 patients infected with Hcd). The three assays were performed the same day on the same LC480-II and the dilutions aliquots realized extemporaneously. A standard curve allowing PCR efficiency calculation was obtained for each assay based on the result of PCR duplicates in six serial dilutions WNAs.Patient classificationPatients included are consecutive patients admitted to hospitals from a French network from May 2015 to May 2019 as part of a routine procedure of diagnosis. All patients were at risk of histoplasmosis, as they all originated or travelled from endemic areas, and had compatible clinical symptoms. Routine diagnosis of histoplasmosis in metropolitan France does not include either detection of Histoplasma antigens because of the rarity of imported cases, nor detection of anti-Histoplasma antibodies for lack of accuracy to detect acute histoplasmosis in immunocompromised patients and even in immunocompetent patients from endemic areas. Clinical data were recorded retrospectively for all PCR positive patients. Each histoplasmosis case was classified according to the EORTC/MSG criteria[START_REF] Pauw | Revised Definitions of Invasive Fungal Disease from the European Organization for Research and Treatment of Cancer/Invasive Fungal Infections Cooperative Group and the National Institute of Allergy and Infectious Diseases Mycoses Study Group (EORTC/MSG) Consensus Group[END_REF] and described according to the 2007 Infectious Diseases Society of America guidelines[START_REF] Wheat | Clinical Practice Guidelines for the Management of Patients with Histoplasmosis: 2007 Update by the Infectious Diseases Society of America[END_REF] and clinical textbook on histoplasmosis.[START_REF] Kauffman | Histoplasmosis: a Clinical and Laboratory Update[END_REF] For included each patient, the normal clinical work up for suspicions of histoplasmosis (reference tests: histopathology and culture) have been performed according to the clinical presentation and the technical method available locally. The results of these tests have been collected retrospectively in all PCR-positive patients. In most cases, the results of the reference tests were not available at the time of the PCR testing. PCR results were delivered as a routine test to the prescriber. The probable case category defined as an isolated positive antigen test was not possible as antigen testing is not routinely performed in France. Disseminated cases were defined as those with ≥2 non-contiguous clinical locations associated with mycological diagnosis on at least one lesion. Localized cases were those with lesions observed only in one organ. Nevertheless, histoplasmosis cases where only qPCR was positive without other mycological criteria (histopathology, direct microscopy, culture) in a compatible epidemiological (living or having traveled in an endemic area for histoplasmosis) and clinical context (including sepsis, respiratory, digestive, neurological hematological, dermatological symptoms and/or lesions of deep organs) with a good outcome under adapted therapeutic management were classified as possible cases. Galactomanan test (Biorad, Platelia) have been performed as part of the routine procedure in histoplasmosis suspicion in centers were this test was available.SamplesThe samples were taken from the patients prospectively at the time of the clinical suspicion and sent to the Mycology-Parasitology laboratory, Saint-Louis Hospital, Paris, France for routine testing. They consisted in various specimens including whole blood, tissue biopsies (lungs, digestive tract, adrenal glands), liquids (bronchoalveolar lavage fluid, cerebrospinal fluid, bone marrow aspirate, pleural, peritoneal, pericardial exudates) or skin/mucous scrapings (skin or mucous ulcerations). Microscopic examination and culture and/or histology were performed in parallel to the RTqPCR. Paraffin-embedded tissue sections were also used in two cases requiring specific extraction procedures.Polyphasic identification of clinical strainsHistoplasma isolates were sent to the National Reference Center for Invasive Mycoses & Antifungals, Institut Pasteur, Paris, France for identification which was handled in a Biosafety Level-3 (BSL-3) laboratory. Strains were sub-cultured on Sabouraud chloramphenicol agar slants and incubated at 27-30°C until sufficient sporulation was attained. The observed macroscopic features included white colonies with fine aerial mycelia or flat tan colonies diffusing a dark brown pigment with age. Microscopic preparations on lactic blue revealed the presence of thick-walled tuberculate macroconidia and smooth-walled microconidia.Molecular analyses were performed by sequencing the ITS1-5.8S-ITS2 (ITS) region of the ribosomal deoxyribonucleic acid (rDNA)[START_REF] Garcia-Hermoso | Molecular and phenotypic evaluation of Lichtheimia corymbifera (formerly Absidia corymbifera) complex isolates associated with human mucormycosis: rehabilitation of L. ramosa[END_REF] to confirm the identification of the genus. The small region (772 bp) of the gene prp8 27 allowed the differentiation of Hcd from the other geographical clades of Histoplasma capsulatum with a 97% degree of similarity between them. Of note all hcd cases had a typical positive direct examination showing the characteristic 5-10µM ovoid yeasts in specific specimens.Statistical analysisIntended sample size was not determined as the accuracy was unknown as a new assay and as the clinical characteristics of the patients was not targeted in this study. No indeterminate reference and PCR tests and no missing data on evaluated items were observed in this study.Median and interquartile range are given for specific descriptions in parameters with non-normal distribution. Non-parametric paired t-test were performed to compare WNA and DNA Cq and for Cq comparison of the qPCR assays. Contingency tables and Fischer's exact test were performed to analyze the statistical link between clinical presentation, underlying disease, visited countries, positive galactomannan test and a positive qPCR test using Prism v9.0 (Graphpad, San Diego, CA, USA).EthicsAll specimens have been tested as part of the routine diagnostic procedure in Saint-Louis Hospital, Paris, France (current care) and were not part of a clinical trial. No specific ethic validation was required. Patients' data have been recorded retrospectively and collected in an anonymized file. The data collection has been declared at the French Commission Nationale de l'Informatique et des Libertés (CNIL) under the number: YNb2747338P.ResultsEvaluation and validation of the Histoplasma RTqPCRInterest of WNA over DNA -Limit of Detection -Efficiency.The analytical performance of the RT-qPCR amplifying WNA was first compared to qPCRs amplifying only DNA. A comparison of the results on 11 WNA extracts from 3 patients showed that the sensitivity on WNA (11/11, 100%) was higher than on DNA (9/11, 81% specimens detected). In addition, WNA gave significantly lower Cq values (median 40.3 [29.5-41.2]) compared to DNA (median 40.2[31.6-45]

Figure 1 :

 1 Figure 1: Comparison of the detection of Histoplasma spp. DNA by qPCR and WNA by

Figure 2 :

 2 Figure 2: Comparison of 3 quantitative PCR assay including the evaluated RTqPCR assay using

Figure 3 :Figure 4 :Supplemental Figure 1 :

 341 Figure 3: Flow-chart of the design of the study

  [START_REF] Buitrago | Detection of imported histoplasmosis in serum of HIV-infected patients using a real-time PCR-based assay[END_REF][START_REF] Simon | Detection of Histoplasma capsulatum DNA in human samples by real-time polymerase chain reaction[END_REF] Quantitative PCR reactions assay from

	of 25 μL containing 1x LC480-II Probes Master, 0.9 μM of each primer, 0.25 μM of the probe,
	7 μL of template DNA 16 . The qPCR assay from Buitrago et al. (qPCR 2) was carried out as
	advised by the authors in a final volume of 25 μL containing 2x LC480-II Probes Master, 0.5
	μM of each primer, 0.2 μM of the probe, 7 μL of template DNA 11 . A total of 10 WNAs were
	tested including four WNA extracted from the two CNRMA strains of each Hcc (CNRMA17.309
	and CNRMA16.205) and Hcd (CNRMA16.638 and CNRMA17.108) adjusted at 1 ng/µL, and
	positive WNA samples using the RTqPCR (n=6) from whole blood, one bone marrow
	Simon et al. publication (qPCR 1) was carried out as advised by the authors in a final volume

aspiration, one lymph node, one digestive biopsy (4 patients infected with Hcc), and one bone

  The RTqPCR gave significantly lower mean Cq values (27.26 and 22.1) than qPCR 1 (31.0 and 25.3) and qPCR 2 (31.6 and 28.8) assays targeting ITS for Hcc (P=0.008) or Hcd (P=0.042), respectively (Figure

4 Hcd WNA and gave significantly lower Cq values compared to qPCR 1 and qPCR 2 assays (P=0.004) (Figure

2A

). Of 6 clinical specimens, one was not detected with qPCR 2.

Table 3 :

 3 non-endemic regions receiving patients originated for endemic regions, or travelers to endemic region, the rarity of histoplasmosis justify the use of a very sensitive test to assess the diagnosis more widely, more rapidly, and more safely than pathology and mycological culture. The present RTqPCR can confirm the diagnosis in respiratory specimens in 92.8 % of all disseminated cases and 100% of acute pulmonary cases and in blood in 92.3% of the immunocompromised patients with disseminated proven cases without waiting for a confirmation on culture saving several days or weeks. Moreover, Histoplasma spp. are Biosafety level 3 (BSL3) organisms with the recommendation not to handle the positive cultures when histoplasmosis is suspected unless a BSL3 facility is available.[START_REF] Sewell | Laboratory-associated infections and biosafety[END_REF] Based on these results, this RTqPCR should be considered as a very useful diagnostic test both for the diagnosis and the management of histoplasmosis. Proportion of RTqPCR positive specimens in patients belonging to the four most frequent clinical presentations

		Asia	4	7.5	4	9.1		0.0
		North America	3	5.7	3	6.8		0.0
		French Polynesia	1	1.9	0	0.0		11.1
		Species identified
		Hcc	/	/	40	90.9	/	/
		Hcd	/	/	4	9.1	/	/
		Galactomannan index
		≥0.5	12	22.6	11	25.0		11.1
		<0.5	22	41.5	22	50.0		0.0
		nd	19	35.8	11	25.0		88.9
		RTqPCR positive in blood
		Positive	13	24.5	13	29.5		0.0
		Negative	21	39.6	17	38.6		44.4
		nd	19	35.8	14	31.8		55.6
	604	nd: not done
	605
	606

Acknowledgement

The authors report no funding for this study. The authors have declared that no competing interests exist. The authors are grateful to the technical staff of the mycology-parasitology laboratory at Hospital Saint Louis, Paris, France for handling specimens and performing this assay routinely, in particular, Elodie Da Silva, Thierry Pautet, Dieyenaba Siby-Diakite, Sarah seng and Julie Bui, who performed optimization experiments.

Table 4 :

 4 Description of the RTqPCR results in the 184 specimens tested in the 53 patients

	612
	613
			Number of specimens tested		Number of Whole blood tested
			Total	Negative	Positive		Total	Negative	Positive
			(n=184)	PCR (n=77)	PCR (n=107)		(n=82)	PCR (n=57)	PCR (n=25)
			Total	n (%)	n (%)	p	Total	n (%)	n (%)	p
	Underlying disease
	Immunocompetent	72	32 (44.4)	40 (55.6)	0.80 23	22 (95.7)	(4.3)	0.003
	HIV		78	32 (41)	46 (59)		46	26 (56.5)	(43.5)
	Hematological
	malignancies	11	4 (36.4)	7 (63.6)		5	2 (40)	(60)
	Immunosuppressive
	therapy	14	7 (50)	7 (50)		6	6 (100)	(0)
	SOT		7	1 (14.3)	6 (85.7)		1	0 (0)	(100)
	Solid cancer	2	1 (50)	1 (50)		1	1 (100)	(0)
	Clinical presentation
	Progressive Disseminated	127	48 (37.8)	79 (62.2)	0.36 58	33 (56.9)	(43.1)	0.005
	Histoplasmosis
	Acute pulmonary	38	18 (47.4)	20 (52.6)		16	16 (100)	(0)
	histoplasmosis ±
	mediastinal lymphadenitis
	CNS Histoplasmosis	12	8 (66.7)	4 (33.3)		5	5 (100)	(0)
	Localized muco-cutaneous 2	1 (50)	1 (50.0)		1	1 (100)	(0)
	Localized adrenal Gland	1	0 (0)	1 (100)		0	0 (0)	(0)
	Localized osteoarticular	3	2 (66.7)	1 (33.3)		2	2 (100)	(0)
	Localized digestive tract	1	0 (0)	1 (100)		0	0 (0)	(0)
	Country of origin or travel history
	Sub-Saharan African	70	29 (41.4)	41 (58.6)	0.94 27	20 (74.1)	(25.9)	0.74
	Central and south America 48	19 (39.6)	29 (60.4)		21	15 (71.4)	(28.6)
	Carribean islands	31	12 (38.7)	19 (61.3)		14	9 (64.3)	(35.7)
	Asia		26	12 (46.2)	14 (53.8)		17	10 (58.8)	(41.2)
	North America	7	4 (57.1)	3 (42.9)		2	2 (100)	(0)
	French polynesia	2	1 (50)	1 (50)		1	1 (100)	(0)
	Species identified
	Hcc		157	64 (40.8)	93 (59.2)	0.53 74	49 (66.2)	(33.8)	0.10
	Hcd		27	13 (48.1)	14 (51.9)		8	8 (100)	(0)
	Galactomannan index
	≥0.5		53	18 (34)	35 (66)	0.37 27	13 (48.1)	(51.9)	0.11
	<0.5		83	35 (42.2)	48 (57.8)		32	23 (71.9)	(28.1)
	614	CNS, Central nervous system
	615

Table 1 -

 1 5: see specific file.

The authors declare no specific funding for this study performed based on standard care samples. AMR is supported by grants from the São Paulo Research Foundation (FAPESP 2017/27265-5) http://www.fapesp.br/.

The French mycoses study group is composed of: Marine Gosset Woimant, Geneviève blanchard (Centre Hospitaler de Pontoise, Pontoise) ; Souad Silhadi (Centre AA received travel grant and honoraria from Gilead,

was positive in 24/44 (54.5%) specimens and 12/13 (92.3%) patients, and 20/38 (52.6%) and 10/11 (90.9%), respectively (Table 3). In proven acute pulmonary ± mediastinal lymphadenitis, RTqPCR was positive in 12/12 (100%) patients tested with 12/14 (85.7%) positive specimens tested (BAL, biopsies, lymph nodes) (Table 2). In CNS histoplasmosis, RTqPCR in CSF was positive in the 2/2 (100%) cases (4/7, 57.1% of tested specimens) (Table 2). Disseminated Hcd infections had various RTqPCR positive sample types, including skin, bone, brain, urinary and digestive tract specimens (Table 2).

A total of 169 specimens were recovered from these 44 proven cases including 97 PCR-positive specimens (positivity rate = 57.4%). In 94 specimens investigated in parallel to RTqPCR for microscopy and/or culture, PCR was positive in 59/64 (92.2%) specimens associated with a positive microscopy or culture vs. in 10/30 (33.3%) specimens associated with a negative microscopy or culture. In addition, 28/75 (37.3%) specimens not investigated in parallel with microscopy and/or culture were RTqPCR positive.

Most of the possible cases (RTqPCR positive / culture negative) had localized diseases (8/9, 88.9%) [lung (7/8), digestive tract (1/8)] and only 1/9 (11.1%) had a disseminated disease (Patient #16, immunocompetent, with a negative microscopy and negative culture in blood and BAL). RTqPCR positive specimens were respiratory specimens (n=8) and digestive tract (n=1) (Table 2). Parallel culture and/or direct examination/histopathology were negative in the seven patients for whom it was done in parallel to RTqPCR (three patients had no other investigation but the PCR to assess mycological diagnosis in a compatible epidemiological and clinical context). In one case under immunosuppressive therapy (#50), galactomannan detection was positive (index >0.5). Three out of 9 (33.3%) were immunocompromised and all (9/9, 100%) lived or travelled in endemic areas. For all patients, no other diagnosis explained


Mathieu Besançon
email: mathieu.besancon@polymtl.ca

Miguel F Anjos

Luce Brotcorne

Complexity of near-optimal robust versions of multilevel optimization problems

Keywords: near-optimal robustness, multilevel optimization, complexity theory Mathematics Subject Classification (2010) 91A65, 90C26, 90C10, 90C10, 90C11, 90C60

Near-optimality robustness extends multilevel optimization with a limited deviation of a lower level from its optimal solution, anticipated by higher levels. We analyze the complexity of near-optimal robust multilevel problems, where near-optimal robustness is modelled through additional adversarial decisionmakers. Near-optimal robust versions of multilevel problems are shown to remain in the same complexity class as the problem without near-optimality robustness under general conditions.

Introduction

Multilevel optimization is a class of mathematical optimization problems where other problems are embedded in the constraints. They are well suited to model sequential decision-making processes, where a first decisionmaker, the leader intrinsically integrates the reaction of another decision-maker, the follower, into their decision-making problem.

In recent years, most of the research focuses on the study and design of efficient solution methods for the case of two levels, namely bilevel problems [START_REF] Dempe | Bilevel optimization: theory, algorithms and applications[END_REF], which fostered a growing range of applications.

Near-optimal robustness, defined in [START_REF] Besançon | Near-optimal Robust Bilevel Optimization[END_REF], is an extension of bilevel optimization. In this setting, the upper level anticipates limited deviations of the lower level from an optimal solution and aims at a solution that remains feasible for any feasible and near-optimal solution of the lower level. This protection of the upper level against uncertain deviations of the lower-level has led to the characterization of near-optimality robustness as a robust optimization approach for bilevel optimization. The lower-level response corresponds to the uncertain parameter and the maximum deviation of the objective value from an optimal solution to the uncertainty budget. Because the set of near-optimal lower-level solutions potentially has infinite cardinality and depends on the upper-level decision itself, near-optimality robustness adds generalized semi-infinite constraints to the bilevel problem. The additional constraint can also be viewed as a form of robustness under decision-dependent uncertainty.

In this paper, we prove complexity results on multilevel problems to which near-optimality robustness constraints are added under various forms. We show that under fairly general conditions, the near-optimal robust version of a multilevel problem remains on the same level of the polynomial hierarchy as the original problem. These results are non-trivial assuming that the polynomial hierarchy does not collapse and open the possibility of solution algorithms for near-optimal robust multilevel problems as efficient as for their canonical counterpart. Even though we focus on near-optimal robust multilevel problems, the complexity results we establish hold for all multilevel problems that present the same hierarchical structure, i.e. the same anticipation and parameterization between levels as the near-optimal formulation with the adversarial problems, as defined in Section 3.

The rest of this paper is organized as follows. Section 2 introduces the notation and the background on near-optimality robustness and existing complexity results in multilevel optimization. Section 3 presents complexity results for the near-optimal robust version of bilevel problems, where the lower level belongs to P and N P. These results are extended in Section 4 to multilevel optimization problems, focusing on integer multilevel linear problems with near-optimal deviations of the topmost intermediate level. Section 5 provides complexity results for a generalized form of near-optimal robustness in integer multilevel problems, where multiple decision-makers anticipate near-optimal reactions of a lower level. Finally, we draw some conclusions in Section 6.

Background on near-optimality robustness and multilevel optimization

In this section, we introduce the notation and terminology for bilevel optimization and near-optimality robustness, and highlight prior complexity results in multilevel optimization. Let us define a bilevel problem as:

min x F (x, v) (1a) s.t. G k (x, v) ≤ 0 ∀k ∈ [[m u ]] (1b) x ∈ X (1c)
where v ∈ arg min

y∈Y {f (x, y) s.t. g i (x, y) ≤ 0 ∀i ∈ [[m l ]]}. ( 1d
)
We denote by X and Y the domain of upper-and lower-level variables respectively. We use the convenience notation [[n]] = {1, . . . , n} for a natural n.

Problem (1) is ill-posed, since multiple solutions to the lower level may exist [START_REF] Dempe | Bilevel programming problems[END_REF]Ch. 1]. Models often rely on additional assumptions to alleviate this ambiguity, the two most common being the optimistic and pessimistic approaches. In the optimistic case (BiP), the lower level selects an optimal decision that most favours the upper level. In this setting, the lower-level decision can be taken by the upper level, as long as it is optimal for the lower-level problem. The upper level can thus optimize over both x and v, leading to:

(BiP): min x,v F (x, v) (2a) s.t. G k (x, v) ≤ 0 ∀k ∈ [[m u ]] (2b) x ∈ X (2c) v ∈ arg min y∈Y {f (x, y) s.t. g i (x, y) ≤ 0 ∀i ∈ [[m l ]]}. ( 2d
)
Constraint (2d) implies that v is feasible for the lower level and that f (x, v) is the optimal value of the lower-level problem, parameterized by x.

The pessimistic approach assumes that the lower level chooses an optimal solution that is the worst for the upper-level objective as in [START_REF] Dempe | Bilevel optimization: theory, algorithms and applications[END_REF] or with respect to the upper-level constraints as in [START_REF] Wiesemann | Pessimistic Bilevel Optimization[END_REF]. . The near-optimal robust version of (BiP) considers that the lower-level solution may not be optimal but near-optimal with respect to the lower-level objective function. The tolerance for near-optimality, denoted by δ is expressed as a maximum deviation of the objective value from optimality. The problem solved at the upper level must integrate this deviation and protects the feasibility of its constraints for any near-optimal lower-level decision. The problem is formulated as:

(NORBiP): min x,v F (x, v) (3a) s.t. (2b) -(2d) (3b) G k (x, z) ≤ 0 ∀z ∈ Z(x; δ) ∀k ∈ [[m u ]] (3c)
where

Z(x; δ) = {y ∈ Y | f (x, y) ≤ f (x, v) + δ, g(x, y) ≤ 0}. (3d)
Z(x; δ) denotes the near-optimal set, i.e. the set of near-optimal lower-level solutions, depending on both the upper-level decision x and δ. (NORBiP) is a generalization of the pessimistic bilevel problem since the latter is both a special case and a relaxation of (NORBiP) [START_REF] Besançon | Near-optimal Robust Bilevel Optimization[END_REF]. We refer to (BiP) as the canonical problem for (NORBiP) (or equivalently Problem (4)) and (NORBiP) as the near-optimal robust version of (BiP). In the formulation of (NORBiP), the upper-level objective depends on decision variables of both levels, but is not protected against near-optimal deviations. A more conservative formulation also protecting the objective by moving it to the constraints in an epigraph formulation [START_REF] Besançon | Near-optimal Robust Bilevel Optimization[END_REF] is given by: (NORBiP-Alt): min

x,v,τ τ s.t. (2b) -(2d) G k (x, z) ≤ 0 ∀z ∈ Z(x; δ) ∀k ∈ [[m u ]] F (x, z) ≤ τ ∀z ∈ Z(x; δ),
The optimal values of the three problems are ordered as: opt(BiP) ≤ opt(NORBiP) ≤ opt(NORBiP-Alt).

We next provide a review of complexity results for bilevel and multilevel optimization problems. Bilevel problems are N P-hard in general, even when the objective functions and constraints at both levels are linear [START_REF] Audet | Links between linear bilevel and mixed 0-1 programming problems[END_REF]. When the lower-level problem is convex, a common solution approach consists in replacing it with its KKT conditions [START_REF] Allende | Solving bilevel programs with the KKT-approach[END_REF][START_REF] Dempe | Solution of bilevel optimization problems using the KKT approach[END_REF], which are necessary and sufficient if the problem satisfies certain constraint qualifications. This approach results in a single optimization problem with complementarity constraints, of which the decision problem is N P-complete [START_REF] Chung | NP-completeness of the linear complementarity problem[END_REF]. A specific form of the three-level problem is investigated in [START_REF] Dempe | A special three-level optimization problem[END_REF], where only the objective value of the bottom-level problem appears in the objective functions of the first and second levels. If these conditions hold and all objectives and constraints are linear, the problem can be reduced to a single level one with complementarity constraints of polynomial size. This model is similar to the worst-case reformulation of (NORBiP) presented in Section 3.

Pessimistic bilevel problems for which no upper-level constraint depends on lower-level variables are studied in [START_REF] Zare | On a class of bilevel linear mixed-integer programs in adversarial settings[END_REF]. The problem of finding an optimal solution to the pessimistic case is shown to be N P-hard, even if a solution to the optimistic counterpart of the same problem is provided. A variant is also defined, where the lower level may pick a suboptimal response only impacting the upper-level objective. This variant is comparable to the Objective-Robust Near-Optimal Bilevel Problem defined in [START_REF] Besançon | Near-optimal Robust Bilevel Optimization[END_REF]. In [START_REF] Wiesemann | Pessimistic Bilevel Optimization[END_REF], the independent case of the pessimistic bilevel problem is studied, corresponding to a special case of (NORBiP) with δ = 0 and all lower-level constraints independent of the upper-level variables. It is shown that the linear independent pessimistic bilevel problem, and consequently the linear near-optimal robust bilevel problem, can be solved in polynomial time while it is strongly N P-hard in the non-linear case.

When the lower-level problem cannot be solved in polynomial time, the bilevel problem is in general Σ P 2 -hard. The notion of Σ P 2 -hardness and classes of the polynomial hierarchy are recalled in Section 3. Despite this complexity result, new algorithms and corresponding implementations have been developed to solve these problems and in particular, mixed-integer linear bilevel problems [11,12,?]. Variants of the bilevel knapsack were investigated in [START_REF] Caprara | A study on the computational complexity of the bilevel knapsack problem[END_REF], and proven to be Σ P 2 -hard as the generic mixed-integer bilevel problem.

Multilevel optimization was initially investigated in [START_REF] Jeroslow | The polynomial hierarchy and a simple model for competitive analysis[END_REF] in the case of linear constraints and objectives at all levels. In this setting, the problem is shown to be in Σ P s , with s + 1 being the number of levels. The linear bilevel problem corresponds to s = 1 and is in Σ P 1 ≡ N P. If at least the bottom-level problem involves integrality constraints (or more generally belongs to N P but not P), the multilevel problem with s levels belongs to Σ P s . A model unifying multistage stochastic and multilevel problems is defined in [START_REF] Bolusani | A Unified Framework for Multistage and Multilevel Mixed Integer Linear Optimization[END_REF], based on a risk function capturing the component of the objective function which is unknown to a decision-maker at their stage, either because of a stochastic component or of another decision-maker that acts after the current one.

As highlighted in [START_REF] Bolusani | A Unified Framework for Multistage and Multilevel Mixed Integer Linear Optimization[END_REF][START_REF] Ralphs | Complexity and multi-level optimization[END_REF], most results in the literature on complexity of multilevel optimization use N P-hardness as the sole characterization. This only indicates that a given problem is at least as hard as all problems in N P and that no polynomial-time solution method should be expected unless N P = P.

We characterize near-optimal robust multilevel problems not only on the hardness or "lower bound" on complexity, i.e. being at least as hard as all problems in a given class but through their complexity "upper bound", i.e. the class of the polynomial hierarchy they belong to. The linear optimistic bilevel problem is for instance strongly N P-hard, but belongs to N P and is therefore not Σ P 2 -hard.

Complexity of near-optimal robust bilevel problems

We establish in this section complexity results for near-optimal robust bilevel problems for which the lower level L is a single-level problem parameterized by the upper-level decision. (NORBiP) can be reformulated by replacing each k-th semi-infinite constraint (3c) with the lower-level solution z k in Z(x; δ) that yields the highest value of

G k (x, z k ): min x,v F (x, v) (4a) s.t. (2b) -(2d) (4b) G k (x, z k ) ≤ 0 ∀k ∈ [[m u ]] (4c) z k ∈ arg max y∈Y {G k (x, y) s.t. f (x, y) ≤ f (x, v) + δ, g(x, y) ≤ 0} ∀k ∈ [[m u ]] (4d)
From a game-theoretical perspective, the near-optimal robust version of a bilevel problem can be seen as a three-player hierarchical game. The upper level U and lower level L are identical to the original bilevel problem. The third level is the adversarial problem A and selects the worst near-optimal lower-level solution with respect to upper-level constraints, as represented by the embedded maximization in Constraint (4d). If the upper-level problem has multiple constraints, the adversarial problem can be decomposed into problems

A k , k ∈ [[m u ]]
, where m u is the number of upper-level constraints. The interaction among the three players is depicted in Fig. 1. The blue dashed arcs represent a parameterization of the source vertex by the decisions of the destination vertex, and the solid red arcs represent an anticipation of the destination vertex decisions in the problem of the source vertex. This convention will be used in all figures throughout the paper.

The adversarial problem can be split into m u adversarial problems as done in [START_REF] Besançon | Near-optimal Robust Bilevel Optimization[END_REF], each finding the worst-case with respect to one of the upper-level constraints. The canonical problem refers to the optimistic bilevel problem without near-optimal robustness constraints. We refer to the variable v as the canonical lower-level decision.

The complexity classes of the polynomial hierarchy are only defined for decision problems. We consider that an optimization problem belongs to a given class if that class contains the decision problem of determining if there exists a feasible solution for which the objective value at least as good as a given bound. Example 2 Denoting by H L the set of linear functions from the space of lower-level variables to R, combinatorial optimization problems in P which can be formulated as linear optimization problems with totally unimodular matrices are not in P * [H L ] in general. Indeed, adding a linear constraint may break the integrality of solutions of the linear relaxation of the lower-level problem.

Σ P s is the complexity class at the s-th level of the polynomial hierarchy [START_REF] Stockmeyer | The polynomial-time hierarchy[END_REF][START_REF] Jeroslow | The polynomial hierarchy and a simple model for competitive analysis[END_REF], defined recursively as Σ P 0 = P, Σ P 1 = N P, and problems of the class Σ P s , s > 1 being solvable in non-deterministic polynomial time, provided an oracle for problems of class Σ P s-1 . In particular, a positive answer to a decision problem in N P can be verified, given a certificate, in polynomial time. If the decision problem associated with an optimization problem is in N P, and given a potential solution, the objective value of the solution can be compared to a given bound and the feasibility can be verified in polynomial time. We reformulate these statements in the following proposition:

Proposition 1 [START_REF] Jeroslow | The polynomial hierarchy and a simple model for competitive analysis[END_REF] An optimization problem is in Σ P s+1 if verifying that a given solution is feasible and attains a given bound can be done in polynomial time, when equipped with an oracle instantaneously solving problems in Σ P s . Proposition 1 is the main property of the classes of the polynomial hierarchy used to determine the complexity of near-optimal robust bilevel problems in various settings throughout this paper.

Lemma 1 Given a bilevel problem in the form of Problem (2), if the lower-level problem is in P * [H], and

-G k (x, •) ∈ H ∀x, ∀k ∈ [[m u ]],
then the adversarial problem (4d) is in P.

Proof The lower-level problem can equivalently be written in an epigraph form:

(v, w) ∈ arg min y,u u s.t. f (x, y) -u ≤ 0 g(x, y) ≤ 0.
Given a solution of the lower-level problem (v, w) and an upper-level constraint G k (x, y) ≤ 0, the adversarial problem is defined by: min y,u -G(x, y)

s.t. f (x, y) -u ≤ 0 g(x, y) ≤ 0 u ≤ w.
Compared to the lower-level problem, the adversarial problem contains an additional linear constraint u ≤ w and an objective function updated to -G(x, •).

⊓ ⊔ Theorem 1 Given a bilevel problem (P ), if there exists H such that the lower-level problem is in N P * [H] and

-G k (x, •) ∈ H ∀k ∈ [[m u ]], ∀x ∈ X ,
then the near-optimal robust version of the bilevel problem is in Σ P 2 like the canonical bilevel problem.

Proof The proof relies on the ability to verify that a given solution (x, v) results in an objective value at least as low as a bound Γ according to Proposition 1. This verification can be carried out with the following steps:

1. Compute the upper-level objective value F (x, v) and verify that F (x, v) ≤ Γ ; 2. Verify that upper-level constraints are satisfied; 3. Verify that lower-level constraints are satisfied; 4. Compute the optimum value L(x) of the lower-level problem parameterized by x and check if:

f (x, v) ≤ min y L(x);
5. Compute the worst case: Find

z k ∈ arg max y∈Y A k (x, v) ∀k ∈ [[m u ]];
where A k (x, v) is the k-th adversarial problem parameterized by (x, v); 6. Verify near-optimal robustness: ∀k ∈ [[m u ]], verify that the k-th upper-level constraint is feasible for the worst-case z k .

Steps 1 and 2 can be carried out in polynomial time by assumption.

Step 3 requires to check the feasibility of a solution to a problem in N P. This can be done in polynomial time. Step

f (x, y) ≤ f (x, v) + δ (5c)
is convex. Furthermore, by definition of P * [H], the adversarial problem is in P.

Applying the same reasoning as in the proof of Theorem 1, Steps 1-3 are identical and can be carried out in polynomial time. Step 4 can be performed in polynomial time since L is in P. Step 5 is also performed in polynomial time since ∀k ∈ [[m u ]], each k-th adversarial problem ( 5) is a convex problem that can be solved in polynomial time since L is in

P * [H].
Step 6 simply is a simple comparison of two quantities.

If the upper-level constraints are convex non-affine with respect to the lower-level variables, Problem (5) maximizes a convex non-affine function over a convex set. Such a problem is N P-hard in general. Therefore, the verification that a given solution is feasible and satisfies a predefined bound on the objective value requires solving the m u N P-hard adversarial problems. If L is in N P * [H], then these adversarial problems are in N P by Eq. ( 3), and the near-optimal robust problem is in Σ P 2 according to Proposition 1. ⊓ ⊔

Complexity of near-optimal robust mixed-integer multilevel problems

In this section, we study the complexity of a near-optimal robust version of mixed-integer multilevel linear problems (MIMLP), where the lower level itself is a s-level problem and is Σ P s -hard. The canonical multilevel problem is, therefore, Σ P s+1 -hard [START_REF] Jeroslow | The polynomial hierarchy and a simple model for competitive analysis[END_REF]. For some instances of mixed-integer bilevel, the optimal value can be approached arbitrarily but not reached [START_REF] Moore | The mixed integer linear bilevel programming problem[END_REF]. To avoid such pathological cases, we restrict our attention to multilevel problems satisfying the criterion for mixed-integer bilevel problems from [START_REF] Fischetti | A new general-purpose algorithm for mixed-integer bilevel linear programs[END_REF]:

Property 1
The continuous variables at any level s do not appear in the problems at levels that are lower than s (the levels deciding after s).

More specifically, we will focus on mixed-integer multilevel linear problems where the upper-most lower level L 1 may pick a solution deviating from the optimal value, while we ignore deviations of the levels L i>1 . This problem is noted (NOMIMLP s ) and depicted in Fig. 1.

U L 1 L 2 A

Fig. 2 Near-optimality robustness for multilevel problems

The adversarial problem corresponds to a decision of the level L 1 different from the canonical decision. This decision induces a different reaction from the subsequent levels L 2 , L 3 . Since the top-level constraints depend on the joint reaction of all following levels, we will note z ki = (z k1 , z k2 , z k3 ) the worst-case joint near-optimal solution of all lower levels with respect to the top-level constraint k.

Theorem 3 If L 1 is in Σ P * s [H L ],
the decision problem associated with (NOMIMLP s ) is in Σ P s+1 as the canonical multilevel problem.

Proof Given a solution to all levels (x U , v 1 , v 2 , ...v s ) and a bound Γ , verifying that this solution is (i) feasible, (ii) near-optimal robust of parameter δ, and (iii) has an objective value at least as good as the bound Γ can be done through the following steps: In this section, we study the complexity of a variant of the problem presented in Section 4 with s + 1 decision-makers at multiple top levels U 1 , U 2 , ...U s and a single bottom level L. We denote by U 1 the topmost level. We assume that the bottom-level entity may choose a solution deviating from optimality. This requires that the entities at all U i ∀i ∈ {1..s} levels anticipate this deviation, thus solving a near-optimal robust problem to protect their feasibility from it. The variant, coined GNORMP s , is illustrated in Fig. 3. We assume throughout this section that ?? 1 holds in order to avoid the unreachability problem previously mentioned. The decision variables of all upper levels are denoted by x (i) , and the objective functions by F (i) (x (1) , x (2) ...x (s) ). The lower-level canonical decision is denoted v as in previous sections.

If the lowest level L belongs to N P, U s belongs to Σ P 2 and the original problem is in Σ P s+1 . In a more general multilevel case, if the lowest level L solves a problem in Σ P r , U s solves a problem in Σ P r+1 and U 1 in Σ P r+s .

We note that for all fixed decisions x (i) ∀i ∈ {1..s -1}, U s is a near-optimal robust bilevel problem. This differs from the model presented in Section 4 where, for a fixed upper-level decision, the top-most lower level L 1 is the same parameterized problem as in the canonical setting. Furthermore, as all levels U i anticipate deviations of the lower-level decision in the near-optimal set, the worst case can be formulated with respect to the constraints of each of these levels. In conclusion, distinct adversarial problems A i ∀i ∈ {1..s} can be formulated. Each upper level U i integrates the reaction of the corresponding adversarial problem in its near-optimality robustness constraint. This formulation of GNORMP s is depicted Fig. 4.

Theorem 4 Given a s + 1-level problem GNORMP s , if the bottom-level problem parameterized by all upperlevel decisions

L(x (1) , x (2) ...x (s) ) is in Σ P * r , then GNORMP s is in Σ P r+s like the corresponding canonical bilevel problem.
Proof We denote by x U = (x (1) , x (2) , ...x (s) ) and m Ui the number of constraints of problem U i . As for Theorem 1, this proof is based on the complexity of verifying that a given solution (x U , v) is feasible and results in an objective value below a given bound. The verification requires the following steps:

1. Compute the top-level objective value and assert that it is below the bound; 2. Verify feasibility of (x U , v) with respect to the constraints at all levels; 3. Verify optimality of v for L parameterized by x U ; 4. Verify optimality of x (i) for the near-optimal robust problem solved by the i-th level U i (x (1) , x (2) ...x (i-1) ; δ) parameterized by all the decisions at levels above and the near-optimality tolerance δ; Let us assume that Step 4 requires to solve a problem outside Σ P r+s-1 . Then GNORMP s-1 is not in Σ P r+s-1 as the associated canonical problem, and that Step 4 requires to solve a problem not in Σ P r+s-2 . By recurrence, GNORMP 1 is not in Σ P r+1 . However, GNORMP 1 is a near-optimal robust bilevel problem where the lower level itself in Σ P r ; this corresponds to the setting of Section 4. This contradicts Theorem 3, GNORMP s-1 is therefore in Σ P r+s-1 . Verifying the feasibility of a given solution to GNORMP s requires solving a problem at most in Σ P r+s-1 . Based on Proposition 1, GNORMP s is in Σ P r+s as its canonical multilevel problem.

U 1 U 2 L A Fig. 3 Generalized NOR multilevel U 1 U 2 L A 2 A 1

⊓ ⊔

In conclusion, Theorem 4 shows that adding near-optimality robustness at an arbitrary level of the multilevel problem does not increase its complexity in the polynomial hierarchy. By combining this property with the possibility to add near-optimal deviation at an intermediate level as in Theorem 3, near-optimality robustness can be added at multiple levels of a multilevel model without changing its complexity class in the polynomial hierarchy.

Conclusion

In this paper, we have shown that for many configurations of bilevel and multilevel optimization problems, adding near-optimality robustness to the canonical problem does increase its complexity in the polynomial hierarchy. This result is obtained even though near-optimality robustness constraints add another level to the multilevel problem, which in general would change the complexity class. We defined the class Σ P * s [H] that is general enough to capture many non-linear multilevel problems but avoids corner cases where the modified objective or additional linear constraint makes the problem harder to solve. Future work will consider specialized solution algorithms for some classes of near-optimal robust bilevel and multilevel problems.

Fig. 1

 1 Fig.1Near-optimal robust bilevel problem

Fig. 4

 4 Fig. 4 Decoupling of the two adversarial problems

  4 consists in solving the lower-level problem, while Step 5 corresponds to solving m u problems similar to the lower level, with the objective function modified and an additional linear constraint ensuring near-optimality.Proof If the upper-level constraints are concave with respect to the lower-level variables, the adversarial problem defined as:

	y∈Y Theorem 2 Given a bilevel problem (P ), if the lower-level problem is convex and in P max G k (x, y) s.t. g(x, y) ≤ 0	⊓ ⊔ (5a) (5b)

*

[H] with H a set of convex functions, and if the upper-level constraints are such that -G k (x, •) ∈ H, then the near-optimal robust version of the bilevel problem is in N P. If the upper-level constraints are convex non-affine with respect to the lower-level constraints, the near-optimal robust version is in general not in N P.

Complexity of a generalized near-optimal robust multilevel problem

  1. Compute the objective value and verify that it is lower than Γ ; 2. Verify variable integrality; 3. Solve the problem L 1 , parameterized by x U , and verify that the solution (v 1 , v 2 , ...) is optimal; 4. ∀k ∈ [[m u ]], solve the k-th adversarial problem. Let z k = (z k1 , z k2 ...z ks ) be the solution; 5. ∀k ∈ [[m u ]], verify that the k-th upper-level constraint is feasible for the adversarial solution z k . Steps 1, 2, and 5 can be performed in polynomial time. Step 3 requires solving a problem in Σ P s , while step 4 consists in solving m u problems in Σ P s , since L 1 is in Σ P * s [H L ]. Checking the validity of a solution thus requires solving problems in Σ P s and is itself in Σ P s+1 , like the original problem.

	⊓ ⊔
	5

Acknowledgements

The work of Mathieu Besançon was supported by the Mermoz scholarship and the GdR RO.