Linear programming with nonparametric penalty programs and iterated thresholding

It is known  [Mangasarian, A Newton method for linear programming, J. Optim. Theory Appl. 121 (2004), pp. 1-18] that every linear program can be solved exactly by minimizing an unconstrained quadratic penalty program. The penalty program is parameterized by a scalar t>0, and one is able to solve...

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Bibliographic Details
Published in:Optimization methods & software Vol. 38; no. 1; pp. 107 - 127
Main Authors: Kline, Jeffery, Fung, Glenn Martin
Format: Journal Article
Language:English
Published: Abingdon Taylor & Francis 02.01.2023
Taylor & Francis Ltd
Subjects:
Algorithms
duality
fixed point
Iterative methods
Linear programming
Newton methods
nonexpansive map
Nonparametric statistics
Parameterization
penalty method
Quadratic programming
ISSN:1055-6788, 1029-4937
Online Access:Get full text
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Summary:It is known  [Mangasarian, A Newton method for linear programming, J. Optim. Theory Appl. 121 (2004), pp. 1-18] that every linear program can be solved exactly by minimizing an unconstrained quadratic penalty program. The penalty program is parameterized by a scalar t>0, and one is able to solve the original linear program in this manner when t is selected larger than a finite, but unknown  . In this paper, we show that every linear program can be solved using the solution to a parameter-free penalty program. We also characterize the solutions to the quadratic penalty programs using fixed points of certain nonexpansive maps. This leads to an iterative thresholding algorithm that converges to a desired limit point. We show in numerical experiments that this iterative method can outperform a variety of standard quadratic program solvers. Finally, we show that for every , the solution one obtains by solving a parameterized penalty program is guaranteed to lie in the feasible set of the original linear program.
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ISSN:1055-6788
1029-4937
DOI:10.1080/10556788.2022.2117356