Tight Bound for Sum of Heterogeneous Random Variables: Application to Chance Constrained Programming

We study a tight Bennett-type concentration inequality for sums of heterogeneous and independent variables, defined as a one-dimensional minimization. We show that this refinement, which outperforms the standard known bounds, remains computationally tractable: we develop a polynomial-time algorithm to compute confidence bounds, proved to terminate with an epsilon-solution. From the proposed inequality, we deduce tight distributionally robust bounds to Chance-Constrained Programming problems. To illustrate the efficiency of our approach, we consider two use cases. First, we study the chance-constrained binary knapsack problem and highlight the efficiency of our cutting-plane approach by obtaining stronger solution than classical inequalities (such as Chebyshev-Cantelli or Hoeffding). Second, we deal with the Support Vector Machine problem, where the convex conservative approximation we obtain improves the robustness of the separation hyperplane, while staying computationally tractable

A Rank-Based Reward between a Principal and a Field of Agents: Application to Energy Savings

We consider a problem where a Principal aims to design a reward function to a field of heterogeneous agents. In our setting, the agents compete with each other through their rank within the population in order to obtain the best reward. We first explicit the equilibrium for the mean-field game played by the agents, and then characterize the optimal reward in the homogeneous setting. For the general case of a heterogeneous population, we develop a numerical approach, which is then applied to the specific case study of the market of Energy Saving Certificates

Quadratic Regularization of Unit-Demand Envy-Free Pricing Problems and Application to Electricity Markets

We consider a profit-maximizing model for pricing contracts as an extension of the unit-demand envy-free pricing problem: customers aim to choose a contract maximizing their utility based on a reservation bill and multiple price coefficients (attributes). A classical approach supposes that the customers have deterministic utilities; then, the response of each customer is highly sensitive to price since it concentrates on the best offer. A second approach is to consider logit model to add a probabilistic behavior in the customers' choices. To circumvent the intrinsic instability of the former and the resolution difficulties of the latter, we introduce a quadratically regularized model of customer's response, which leads to a quadratic program under complementarity constraints (QPCC). This allows to robustify the deterministic model, while keeping a strong geometrical structure. In particular, we show that the customer's response is governed by a polyhedral complex, in which every polyhedral cell determines a set of contracts which is effectively chosen. Moreover, the deterministic model is recovered as a limit case of the regularized one. We exploit these geometrical properties to develop an efficient pivoting heuristic, which we compare with implicit or non-linear methods from bilevel programming. These results are illustrated by an application to the optimal pricing of electricity contracts on the French market.Comment: 37 pages, 9 figures; adding a section on the pricing of electricity contract

Ergodic control of a heterogeneous population and application to electricity pricing

We consider a control problem for a heterogeneous population composed of customers able to switch at any time between different contracts, depending not only on the tariff conditions but also on the characteristics of each individual. A provider aims to maximize an average gain per time unit, supposing that the population is of infinite size. This leads to an ergodic control problem for a "mean-field" MDP in which the state space is a product of simplices, and the population evolves according to a controlled linear dynamics. By exploiting contraction properties of the dynamics in Hilbert's projective metric, we show that the ergodic eigenproblem admits a solution. This allows us to obtain optimal strategies, and to quantify the gap between steady-state strategies and optimal ones. We illustrate this approach on examples from electricity pricing, and show in particular that the optimal policies may be cyclic-alternating between discount and profit taking stages

Spectroscopic measurement of the excitation spectrum on effectively curved spacetimes in a polaritonic fluid of light

Quantum fields in regions of extreme spacetime curvature give rise to a wealth of effects, like Hawking radiation at the horizon of black holes. While quantum field theory can only be studied theoretically in black holes, it can be tested in controlled laboratory experiments. Typically, a fluid accelerating from sub- to supersonic speed will create an effectively curved spacetime for the acoustic field, with an apparent horizon where the speed of the fluid equals the speed of sound. Here we create effective curved spacetimes with a quantum fluid of light, with smooth and steep acoustic horizons and various supersonic fluid speeds. We use a recently developed spectroscopy method to measure the spectrum of acoustic excitations on these spacetimes, thus observing negative energy modes in the supersonic regions. This demonstrates the potential of quantum fluids of light for the study of field theories on curved spacetimes.Comment: 5 pages, 3 figure

Spectrum of collective excitations of a quantum fluid of polaritons

We use a recently developed high-resolution coherent probe spectroscopy method to investigate the dispersion of collective excitations of a polaritonic quantum fluid. We measure the dispersion relation with high energy and wavenumber resolution, which allows us to determine the speed of sound in the fluid and to evidence the contribution of an excitonic reservoir. We report on the generation of collective excitations at negative energies, on the ghost branch of the dispersion curve. Precursors of dynamical instabilities are also identified. Our methods open the way to the precise study of quantum hydrodynamics of quantum fluids of light

Observation of the diffusive Nambu-Goldstone mode of a non-equilibrium phase transition

Second-order phase transitions are governed by spontaneous symmetry breaking, which yield collective excitations with a gapless spectrum called Nambu-Goldstone (NG) modes. While NG modes in conservative systems are propagating excitations, non-equilibrium phase transitions have been predicted to feature a diffusive NG mode. We present the first experimental evidence of a diffusive NG mode in a non-equilibrium Bose-Einstein condensate of microcavity polaritons. The NG mode is observed as a spectral narrowing in the spectroscopic response of the condensate. Additionally, explicitly breaking the symmetry causes the opening of a gap in the spectrum and the disappearance of the NG mode. Our observations confirm the diffusive dynamics of the NG mode of non-equilibrium phase transitions and establish a promising framework to investigate fundamental questions in statistical mechanics.Comment: 8 pages + refs + appendix, 6 figure

Sensitivity and specifi city of HAT Sero-K-SeT, a rapid diagnostic test for serodiagnosis of sleeping sickness caused by Trypanosoma brucei gambiense: a case-control study

Background Human African trypanosomiasis (HAT) is a life-threatening infection aff ecting rural populations in sub- Saharan Africa. Large-scale population screening by antibody detection with the Card Agglutination Test for Trypanosomiasis (CATT)/Trypanosoma brucei (T b) gambiense helped reduce the number of reported cases of gambiense HAT to fewer than 10 000 in 2011. Because low case numbers lead to decreased cost-eff ectiveness of such active screening, we aimed to assess diagnostic accuracy of a rapid serodiagnostic test (HAT Sero-K-SeT) applicable in primary health-care centres. Methods In our case-control study, we assessed participants older than 11 years who presented for HAT Sero-K-SeT and CATT/T b gambiense at primary care centres or to mobile teams (and existing patients with confi rmed disease status at these centres) in Bandundu Province, DR Congo. We defi ned cases as patients with trypanosomes that had been identifi ed in lymph node aspirate, blood, or cerebrospinal fl uid. During screening, we recruited controls without previous history of HAT or detectable trypanosomes in blood or lymph who resided in the same area as the cases. We assessed diagnostic accuracy of three antibody detection tests for gambiense HAT: HAT Sero-K-SeT and CATT/T b gambiense (done with venous blood at the primary care centres) and immune trypanolysis (done with plasma at the Institute of Tropical Medicine, Antwerp, Belgium). Findings Between June 6, 2012, and Feb 25, 2013, we included 134 cases and 356 controls. HAT Sero-K-SeT had a sensitivity of 0·985 (132 true positives, 95% CI 0·947–0·996) and a specifi city of 0·986 (351 true negatives, 0·968–0·994), which did not diff er signifi cantly from CATT/T b gambiense (sensitivity 95% CI 0·955, 95% CI 0·906–0·979 [128 true positives] and specifi city 0·972, 0·949–0·985 [346 true negatives]) or immune trypanolysis (sensitivity 0·985, 0·947–0·996 [132 true positives] and specifi city 0·980, 0·960–0·990 [349 true negatives]). Interpretation The diagnostic accuracy of HAT Sero-K-SeT is adequate for T b gambiense antibody detection in local health centres and could be used for active screening whenever a cold chain and electricity supply are unavailable and CATT/T b gambiense cannot be done

Exploration of steam explosion treatment for the recovery of phenolic compounds

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