Algorithmica

pp 1–12

# A Simple Projection Algorithm for Linear Programming Problems

Article

## Abstract

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.

## Keywords

Linear programming Zonotope Projection Binary search

## Notes

### 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.

## References

1. 1.
Chakrabarty, D., Jain, P., Kothari, P.: Provable submodular minimization using Wolfe’s algorithm. In: Advances in Neural Information Processing Systems, pp. 802–809 (2014)Google Scholar
2. 2.
Fujishige, S.: Submodular functions and optimization, 2nd edn. Annals of Discrete Mathematics, vol. 58. Elsevier, Amsterdam (2005)Google Scholar
3. 3.
Fujishige, S.: Personal reminiscence: combinatorial and discrete optimization problems in which I have been interested. Jpn. J. Ind. Appl. Math. 29, 357–384 (2012)
4. 4.
Fujishige, S., Hayashi, T., Isotani, S.: The minimum-norm-point algorithm applied to submodular function minimization and linear programming. Kyoto University. Research Institute for Mathematical Sciences [RIMS] (2006)Google Scholar
5. 5.
Fujishige, S., Hayashi, T., Yamashita, K., Zimmermann, U.: Zonotopes and the LP-Newton method. Optim. Eng. 10, 193–205 (2009)
6. 6.
Fujishige, S., Liu, X., Zhang, X.: An algorithm for solving the minimum-norm point problem over the intersection of a polytope and an affine set. J. Optim. Theory Appl. 105, 113–147 (2000)
7. 7.
Fujishige, S., Sato, H., Zhan, P.: An algorithm for finding the minimum-norm point in the intersection of a convex polyhedron and a hyperplane. Jpn. J. Ind. Appl. Math. 11, 245–264 (1994)
8. 8.
Fujishige, S., Zhan, P.: A dual algorithm for finding the minimum-norm point in a polytope. J. Oper. Res. Soc. Jpn. 33, 188–195 (1990)
9. 9.
Fujishige, S., Zhan, P.: A dual algorithm for finding a nearest pair of points in two polytopes. J. Oper. Res. Soc. Jpn. 35, 353–365 (1992)
10. 10.
Grunbaum, B.: Convex polytopes, volume 221 of Graduate Texts in Mathematics, 60 (2003)Google Scholar
11. 11.
Kitahara, T., Mizuno, S., Shi, J.: The LP-Newton method for standard form linear programming problems. Oper. Res. Lett. 41, 426–429 (2013)
12. 12.
McMullen, P.: On zonotopes. Trans. Am. Math. Soc. 159, 91–109 (1971)
13. 13.
Sekitani, K., Yamamoto, Y.: A recursive algorithm for finding the minimum norm point in a polytope and a pair of closest points in two polytopes. Math. Program. 61, 233–249 (1993)
14. 14.
Silvestri, F., Reinelt, G.: The LP-Newton Method and Conic Optimization (2016). arXiv preprint arXiv:1611.09260
15. 15.
Wolfe, P.: Finding the nearest point in a polytope. Math. Program. 11, 128–149 (1976)
16. 16.
Ziegler, G.M.: Lectures on Polytopes, vol. 152. Springer, New York (2012)