Re-Solving Stochastic Programming Models for Airline Revenue Management

We study some mathematical programming formulations for the origin-destination model in airline revenue management. In particular, we focus on the traditional probabilistic model proposed in the literature. The approach we study consists of solving a sequence of two-stage stochastic programs with simple recourse, which can be viewed as an approximation to a multi-stage stochastic programming formulation to the seat allocation problem. Our theoretical results show that the proposed approximation is robust, in the sense that solving more successive two-stage programs can never worsen the expected revenue obtained with the corresponding allocation policy. Although intuitive, such a property is known not to hold for the traditional deterministic linear programming model found in the literature. We also show that this property does not hold for some bid-price policies. In addition, we propose a heuristic method to choose the re-solving points, rather than re-solving at equally spaced times as customary. Numerical results are presented to illustrate the effectiveness of the proposed approach

Inner approximations of stochastic programs for data-driven stochastic barrier function design

This paper studies finite-horizon safety guarantees for discrete-time piece-wise affine systems with stochastic noise of unknown distributions. Our approach is based on a novel approach to synthesise a stochastic barrier function from noise data. In particular, we first build a chance-constraint tightening to obtain an inner approximation of a stochastic program. Then, we apply this methodology for stochastic barrier function design, yielding a robust linear program to which the scenario approach theory applies. In contrast to existing approaches, our method is data efficient as it only requires the number of data to be proportional to the logarithm in the negative inverse of the confidence level and is computationally efficient due to its reduction to linear programming. Furthermore, while state-of-the-art methods assume known statistics on the noise distribution, our approach does not require any information about it. We empirically evaluate the efficacy of our method on various verification benchmarks. Experiments show a significant improvement with respect to state-of-the-art, obtaining tighter certificates with a confidence that is several orders of magnitude higher

Theory and Applications of Robust Optimization

In this paper we survey the primary research, both theoretical and applied, in the area of Robust Optimization (RO). Our focus is on the computational attractiveness of RO approaches, as well as the modeling power and broad applicability of the methodology. In addition to surveying prominent theoretical results of RO, we also present some recent results linking RO to adaptable models for multi-stage decision-making problems. Finally, we highlight applications of RO across a wide spectrum of domains, including finance, statistics, learning, and various areas of engineering.Comment: 50 page