Linearly constrained convex programming of entropy type: Convergence and algorithms.
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Date
1993
Authors
Huang, Wanzhen.
Journal Title
Journal ISSN
Volume Title
Publisher
Dalhousie University
Abstract
Description
The problem of estimating a (non-negative) density function, given a finite number of its moments, arises in numerous practical applications. By introducing an entropy-like objective function, we are able to treat this problem as an infinite-dimensional convex programming problem.
The convergence of our estimate to the underlying density is dependent on the choice of the objective. In this thesis, I studied the most commonly used classes of objectives, which include the Boltzmann-Shannon entropy, the Fermi-Dirac entropy, the truncated $L\sb{p}$-entropy. First, I discussed the duality properties of the convex program ($P\sb{n}$), which involves only n moments, and gave theorems to estimate the bounds of the dual gaps. After proving a general necessary optimality condition and giving rates of norm convergence, I set up a set of uniform convergence theorems for certain choices of entropies, provided that the moment functions are algebraic or trigonometric polynomials.
In Chapter 4, I used Newton's method to solve the dual problem. I compared the computational results of the problem with various choices of entropies. For the problem with the Boltzmann-Shannon entropy, using a special structure among the moments, I have developed a set of very efficient algorithm. By using some additional moments, within much less time, we can find a very good estimate function to the underlying density by solving just a couple of linear systems. The algorithms have been implemented in Fortran. Some 2- and 3-dimensional examples have been tested. Since the algorithm is heuristic instead of iterative, some related error analysis has also been performed.
Thesis (Ph.D.)--Dalhousie University (Canada), 1993.
The convergence of our estimate to the underlying density is dependent on the choice of the objective. In this thesis, I studied the most commonly used classes of objectives, which include the Boltzmann-Shannon entropy, the Fermi-Dirac entropy, the truncated $L\sb{p}$-entropy. First, I discussed the duality properties of the convex program ($P\sb{n}$), which involves only n moments, and gave theorems to estimate the bounds of the dual gaps. After proving a general necessary optimality condition and giving rates of norm convergence, I set up a set of uniform convergence theorems for certain choices of entropies, provided that the moment functions are algebraic or trigonometric polynomials.
In Chapter 4, I used Newton's method to solve the dual problem. I compared the computational results of the problem with various choices of entropies. For the problem with the Boltzmann-Shannon entropy, using a special structure among the moments, I have developed a set of very efficient algorithm. By using some additional moments, within much less time, we can find a very good estimate function to the underlying density by solving just a couple of linear systems. The algorithms have been implemented in Fortran. Some 2- and 3-dimensional examples have been tested. Since the algorithm is heuristic instead of iterative, some related error analysis has also been performed.
Thesis (Ph.D.)--Dalhousie University (Canada), 1993.
Keywords
Mathematics.