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EXPLICIT SOLUTIONS FOR A CLASS OF NONLINEAR PDEs THAT ARISE IN ALLOCATION PROBLEMS.

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Title: EXPLICIT SOLUTIONS FOR A CLASS OF NONLINEAR PDEs THAT ARISE IN ALLOCATION PROBLEMS.
Authors: Dupuis, Paul1 dupuis@dam.brown.edu, Zhang, Jim X.2 jimxzhang@ncsu.edu
Source: SIAM Journal on Mathematical Analysis. 2008, Vol. 39 Issue 5, p1627-1667. 41p.
Subjects: Numerical solutions to nonlinear differential equations, Nonlinear assignment problems, Mathematical optimization, Probability theory, Hamilton-Jacobi equations
Abstract: To exploit large deviation approximations for allocation and occupancy problems one must solve a deterministic optimal control problem (or equivalently, a calculus of variations problem). As this paper demonstrates, and in sharp contrast to the great majority of large deviation problems for processes with state dependence, for allocation problems one can construct more or less explicit solutions. Two classes of allocation problems are studied. The first class considers objects of a single type with a parameterized family of placement probabilities. The second class considers only equally likely placement probabilities but allows for more than one type of object. In both cases, we identify the Hamilton--Jacobi--Bellman equation, whose solution characterizes the minimal cost, explicitly construct solutions, and identify the minimizing trajectories. The explicit construction is possible because of the very tractable properties of the relative entropy function with respect to optimization. [ABSTRACT FROM AUTHOR]
Copyright of SIAM Journal on Mathematical Analysis is the property of Society for Industrial & Applied Mathematics and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: EXPLICIT SOLUTIONS FOR A CLASS OF NONLINEAR PDEs THAT ARISE IN ALLOCATION PROBLEMS.
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  Data: <searchLink fieldCode="DE" term="%22Numerical+solutions+to+nonlinear+differential+equations%22">Numerical solutions to nonlinear differential equations</searchLink><br /><searchLink fieldCode="DE" term="%22Nonlinear+assignment+problems%22">Nonlinear assignment problems</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+optimization%22">Mathematical optimization</searchLink><br /><searchLink fieldCode="DE" term="%22Probability+theory%22">Probability theory</searchLink><br /><searchLink fieldCode="DE" term="%22Hamilton-Jacobi+equations%22">Hamilton-Jacobi equations</searchLink>
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  Label: Abstract
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  Data: To exploit large deviation approximations for allocation and occupancy problems one must solve a deterministic optimal control problem (or equivalently, a calculus of variations problem). As this paper demonstrates, and in sharp contrast to the great majority of large deviation problems for processes with state dependence, for allocation problems one can construct more or less explicit solutions. Two classes of allocation problems are studied. The first class considers objects of a single type with a parameterized family of placement probabilities. The second class considers only equally likely placement probabilities but allows for more than one type of object. In both cases, we identify the Hamilton--Jacobi--Bellman equation, whose solution characterizes the minimal cost, explicitly construct solutions, and identify the minimizing trajectories. The explicit construction is possible because of the very tractable properties of the relative entropy function with respect to optimization. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of SIAM Journal on Mathematical Analysis is the property of Society for Industrial & Applied Mathematics and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1137/060659739
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        Text: English
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        PageCount: 41
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      – SubjectFull: Numerical solutions to nonlinear differential equations
        Type: general
      – SubjectFull: Nonlinear assignment problems
        Type: general
      – SubjectFull: Mathematical optimization
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      – SubjectFull: Probability theory
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      – SubjectFull: Hamilton-Jacobi equations
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      – TitleFull: EXPLICIT SOLUTIONS FOR A CLASS OF NONLINEAR PDEs THAT ARISE IN ALLOCATION PROBLEMS.
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