The Experts below are selected from a list of 300 Experts worldwide ranked by ideXlab platform

Andries P. Engelbrecht - One of the best experts on this subject based on the ideXlab platform.

Kalyan T. Talluri - One of the best experts on this subject based on the ideXlab platform.

  • On a Piecewise-Linear Approximation for network revenue management
    Mathematics of Operations Research, 2016
    Co-Authors: Sumit Kunnumkal, Kalyan T. Talluri
    Abstract:

    The network revenue management (RM) problem arises in airline, hotel, media, and other industries where the sale products use multiple resources. It can be formulated as a stochastic dynamic program, but the dynamic program is computationally intractable because of an exponentially large state space, and a number of heuristics have been proposed to approximate its value function. In this paper we show that the Piecewise-Linear Approximation to the network RM dynamic program is tractable; specifically we show that the separation problem of the Approximation can be solved as a relatively compact linear program. Moreover, the resulting compact formulation of the approximate dynamic program turns out to be exactly equivalent to the Lagrangian relaxation of the dynamic program, an earlier heuristic method proposed for the same problem. We perform a numerical comparison of solving the problem by generating separating cuts or as our compact linear program. We discuss extensions to versions of the network RM prob...

  • On the tractability of the Piecewise-Linear Approximation for general discrete-choice network revenue management
    2014
    Co-Authors: Sumit Kunnumkal, Kalyan T. Talluri
    Abstract:

    The choice network revenue management (RM) model incorporates customer purchase behavior as customers purchasing products with certain probabilities that are a function of the offered assortment of products, and is the appropriate model for airline and hotel network revenue management, dynamic sales of bundles, and dynamic assortment optimization. The underlying stochastic dynamic program is intractable and even its certainty-equivalence Approximation, in the form of a linear program called Choice Deterministic Linear Program (CDLP) is difficult to solve in most cases. The separation problem for CDLP is NP-complete for MNL with just two segments when their consideration sets overlap; the affine Approximation of the dynamic program is NP-complete for even a single-segment MNL. This is in contrast to the independent-class (perfect-segmentation) case where even the Piecewise-Linear Approximation has been shown to be tractable. In this paper we investigate the Piecewise-Linear Approximation for network RM under a general discrete-choice model of demand. We show that the gap between the CDLP and the Piecewise-Linear bounds is within a factor of at most 2. We then show that the Piecewise-Linear Approximation is polynomially-time solvable for a fixed consideration set size, bringing it into the realm of tractability for small consideration sets; small consideration sets are a reasonable modeling tradeoff in many practical applications. Our solution relies on showing that for any discrete-choice model the separation problem for the linear program of the Piecewise-Linear Approximation can be solved exactly by a Lagrangian relaxation. We give modeling extensions and show by numerical experiments the improvements from using Piecewise-Linear Approximation functions.

Christopher Wesley Cleghorn - One of the best experts on this subject based on the ideXlab platform.

Tomomichi Hagiwara - One of the best experts on this subject based on the ideXlab platform.

  • $L_{1}$ Discretization for Sampled-Data Controller Synthesis via Piecewise Linear Approximation
    IEEE Transactions on Automatic Control, 2016
    Co-Authors: Jung Hoon Kim, Tomomichi Hagiwara
    Abstract:

    This paper develops a new discretization method with piecewise linear Approximation for the $L_{1}$ optimal controller synthesis problem of sampled-data systems, which is the problem of minimizing the $L_{\infty}$ -induced norm of sampled-data systems. We apply fast-lifting on the top of the lifting technique, by which the sampling interval $[0,h)$ is divided into $M$ subintervals with an equal width. The signals on each subinterval are then approximated by linear functions by introducing two types of ‘linearizing operators’ for input and output, which leads to piecewise linear Approximation of sampled-data systems. By using the arguments of preadjoint operators, we provide an important inequality that forms a theoretical basis for tackling the $L_{1}$ optimal controller synthesis problem of sampled-data systems more efficiently than the conventional method. More precisely, a mathematical basis for the piecewise linear Approximation method associated with the convergence rate is shown through this inequality, and this suggests that the piecewise linear Approximation method may drastically outperform the conventional method in the $L_{1}$ optimal controller synthesis problem of sampled-data systems. We then provide a discretization procedure of sampled-data systems by which the $L_{1}$ optimal controller synthesis problem is converted to the discrete-time $l_{1}$ optimal controller synthesis problem. Finally, effectiveness of the proposed method is demonstrated through a numerical example.

  • computing the l 0 h induced norm of a compression operator via fast lifting
    Systems & Control Letters, 2014
    Co-Authors: Jung Hoon Kim, Tomomichi Hagiwara
    Abstract:

    Abstract This paper studies computing the induced norm of a compression operator defined on the Banach space L ∞ [ 0 , h ) , which is a difficult problem since it is an infinite-rank operator. Two methods are provided for this problem, each of which can compute an upper bound and a lower bound of the induced norm by using an idea of staircase or piecewise linear Approximation. Staircase Approximation and piecewise linear Approximation are applied through fast-lifting, by which the interval [ 0 , h ) is divided into M subintervals with equal width, and the Approximation errors in these methods are ensured to be reciprocally proportional to M or M 2 . The effectiveness of the proposed methods is demonstrated through numerical examples.

Fridrich Sloboda - One of the best experts on this subject based on the ideXlab platform.