TY - JOUR

T1 - A Simple Projection Algorithm for Linear Programming Problems

AU - Kitahara, Tomonari

AU - Sukegawa, Noriyoshi

N1 - Funding Information:
Acknowledgements The first author is supported in part by Grant-in-Aid for Young Scientists (B) 15K15941 from the Japan Society for the Promotion of Sciences. The second author is supported in part by Grant-in-Aid for Young Scientists (Start-up) 15H06617 from the Japan Society for the Promotion of Sciences.

PY - 2019/1/15

Y1 - 2019/1/15

N2 - Fujishige et al. propose the LP-Newton method, a new algorithm for linear programming problem (LP). They address LPs which have a lower and an upper bound for each variable, and reformulate the problem by introducing a related zonotope. The LP-Newton method repeats projections onto the zonotope by Wolfe’s algorithm. For the LP-Newton method, Fujishige et al. show that the algorithm terminates in a finite number of iterations. Furthermore, they show that if all the inputs are rational numbers, then the number of projections is bounded by a polynomial in L, where L is the input length of the problem. In this paper, we propose a modification to their algorithm using a binary search. In addition to its finiteness, if all the inputs are rational numbers and the optimal value is an integer, then the number of projections is bounded by L+ 1 , that is, a linear bound.

AB - Fujishige et al. propose the LP-Newton method, a new algorithm for linear programming problem (LP). They address LPs which have a lower and an upper bound for each variable, and reformulate the problem by introducing a related zonotope. The LP-Newton method repeats projections onto the zonotope by Wolfe’s algorithm. For the LP-Newton method, Fujishige et al. show that the algorithm terminates in a finite number of iterations. Furthermore, they show that if all the inputs are rational numbers, then the number of projections is bounded by a polynomial in L, where L is the input length of the problem. In this paper, we propose a modification to their algorithm using a binary search. In addition to its finiteness, if all the inputs are rational numbers and the optimal value is an integer, then the number of projections is bounded by L+ 1 , that is, a linear bound.

UR - http://www.scopus.com/inward/record.url?scp=85044751690&partnerID=8YFLogxK

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U2 - 10.1007/s00453-018-0436-3

DO - 10.1007/s00453-018-0436-3

M3 - Article

AN - SCOPUS:85044751690

SN - 0178-4617

VL - 81

SP - 167

EP - 178

JO - Algorithmica

JF - Algorithmica

IS - 1

ER -