The discrete moment problem with fractional moments

2013 ◽  
Vol 41 (6) ◽  
pp. 715-718 ◽  
Author(s):  
Anh Ninh ◽  
András Prékopa
Keyword(s):  
Moment Problem ◽  
Discrete Moment ◽  
Fractional Moments

Stieltjes moment problem via fractional moments

2005 ◽  
Vol 166 (3) ◽  
pp. 664-677 ◽  
Author(s):  
Pierluigi Novi Inverardi ◽  
Alberto Petri ◽  
Giorgio Pontuale ◽  
Aldo Tagliani
Keyword(s):  
Moment Problem ◽  
Stieltjes Moment Problem ◽  
Fractional Moments

scholarly journals Hausdorff moment problem via fractional moments

2003 ◽  
Vol 144 (1) ◽  
pp. 61-74 ◽  
Author(s):  
Pierluigi Novi Inverardi ◽  
Giorgio Pontuale ◽  
Alberto Petri ◽  
Aldo Tagliani
Keyword(s):  
Moment Problem ◽  
Fractional Moments ◽  
Hausdorff Moment Problem

scholarly journals The Discrete Moment Problem with Nonconvex Shape Constraints

2020 ◽  
Author(s):  
Xi Chen ◽  
Simai He ◽  
Bo Jiang ◽  
Christopher Thomas Ryan ◽  
Teng Zhang
Keyword(s):  
Extreme Point ◽  
Failure Rate ◽  
Moment Problem ◽  
Dimensional Space ◽  
Discrete Distribution ◽  
Exact Algorithm ◽  
Optimal Solutions ◽  
Worst Case ◽  
Shape Constraints ◽  
Discrete Moment

The discrete moment problem aims to find a worst-case discrete distribution that satisfies a given set of moments. This paper studies the discrete moment problems with additional shape constraints that guarantee the worst-case distribution is either log-concave (LC) or has an increasing failure rate (IFR) or increasing generalized failure rate (IGFR). These classes are useful in practice, with applications in revenue management, reliability, and inventory control. The authors characterize the structure of optimal extreme point distributions and show, for example, that an optimal extreme point solution to a moment problem with m moments and LC shape constraints is piecewise geometric with at most m pieces. Using this optimality structure, they design an exact algorithm for computing optimal solutions in a low-dimensional space of parameters. The authors leverage this structure to study a robust newsvendor problem with shape constraints and compute optimal solutions.

Application of the solution of the univariate discrete moment problem for the multivariate case

2010 ◽  
Vol 47 (4) ◽  
pp. 485-504
Author(s):  
Gergely Mádi-Nagy
Keyword(s):  
Probability Distribution ◽  
Objective Function ◽  
Moment Problem ◽  
Random Variable ◽  
Expected Value ◽  
Dual Method ◽  
Finite Support ◽  
Dual Algorithm ◽  
Numerical Examples ◽  
Discrete Moment

The objective of the univariate discrete moment problem (DMP) is to find the minimum and/or maximum of the expected value of a function of a random variable which has a discrete finite support. The probability distribution is unknown, but some of the moments are given. This problem is an ill-conditioned LP, but it can be solved by the dual method developed by Prékopa. The multivariate discrete moment problem (MDMP) is the generalization of the DMP where the objective function is the expected value of a function of a random vector. The MDMP has also been initiated by Prékopa and it can also be considered as an (ill-conditioned) LP. The central results of the MDMP concern the structure of the dual feasible bases and provide us with bounds without any numerical difficulties. Unfortunately, in this case not all the dual feasible bases have been found hence the multivariate counterpart of the dual method of the DMP cannot be developed. However, there exists a method developed by Mádi-Nagy [7] which allows us to get the basis corresponding to the best bound out of the known structures by optimizing independently on each variable. In this paper we present a method using the dual algorithm of the DMP for solving those independent subproblems. The efficiency of this new method is illustrated by numerical examples.

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