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Exterior point simplex-type algorithms for linear and network optimization problems

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  • Konstantinos Paparrizos
  • Nikolaos Samaras
  • Angelo Sifaleras

Abstract

Two decades of research led to the development of a number of efficient algorithms that can be classified as exterior point simplex-type. This type of algorithms can cross over the infeasible region of the primal (dual) problem and find an optimal solution reducing the number of iterations needed. The main idea of exterior point simplex-type algorithms is to compute two paths/flows. Primal (dual) exterior point simplex-type algorithms compute one path/flow which is basic but not always primal (dual) feasible and the other is primal (dual) feasible but not always basic. The aim of this paper is to explain to the general OR audience, for the first time, the developments in exterior point simplex-type algorithms for linear and network optimization problems, over the recent years. We also present other approaches that, in a similar way, do not preserve primal or dual feasibility at each iteration such as the monotonic build-up Simplex algorithms and the criss-cross methods. Copyright Springer Science+Business Media New York 2015

Suggested Citation

  • Konstantinos Paparrizos & Nikolaos Samaras & Angelo Sifaleras, 2015. "Exterior point simplex-type algorithms for linear and network optimization problems," Annals of Operations Research, Springer, vol. 229(1), pages 607-633, June.
    • Handle: RePEc:spr:annopr:v:229:y:2015:i:1:p:607-633:10.1007/s10479-014-1769-1
    DOI: 10.1007/s10479-014-1769-1
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    1. Kurt M. Anstreicher & Tamás Terlaky, 1994. "A Monotonic Build-Up Simplex Algorithm for Linear Programming," Operations Research, INFORMS, vol. 42(3), pages 556-561, June.
    2. Cochran, James J., 2012. "You want them to remember? Then make it memorable! Means for enhancing operations research education," European Journal of Operational Research, Elsevier, vol. 219(3), pages 659-670.
    3. Pan, Ping-Qi, 2008. "A largest-distance pivot rule for the simplex algorithm," European Journal of Operational Research, Elsevier, vol. 187(2), pages 393-402, June.
    4. Karl-Heinz Borgwardt, 1982. "Some Distribution-Independent Results About the Asymptotic Order of the Average Number of Pivot Steps of the Simplex Method," Mathematics of Operations Research, INFORMS, vol. 7(3), pages 441-462, August.
    5. Illes, Tibor & Szirmai, Akos & Terlaky, Tamas, 1999. "The finite criss-cross method for hyperbolic programming," European Journal of Operational Research, Elsevier, vol. 114(1), pages 198-214, April.
    6. A. Charnes & W. W. Cooper, 1954. "The Stepping Stone Method of Explaining Linear Programming Calculations in Transportation Problems," Management Science, INFORMS, vol. 1(1), pages 49-69, October.
    7. Gregory Dobson & Robert Shumsky, 2006. "Web-based Simulations for Teaching Queueing, Little's Law, and Inventory Management," INFORMS Transactions on Education, INFORMS, vol. 7(1), pages 106-124, September.
    8. Paparrizos, Konstantinos & Samaras, Nikolaos & Stephanides, George, 2003. "An efficient simplex type algorithm for sparse and dense linear programs," European Journal of Operational Research, Elsevier, vol. 148(2), pages 323-334, July.
    9. Csizmadia, Zsolt & Illés, Tibor & Nagy, Adrienn, 2012. "The s-monotone index selection rules for pivot algorithms of linear programming," European Journal of Operational Research, Elsevier, vol. 221(3), pages 491-500.
    10. Jens Vygen, 2002. "On dual minimum cost flow algorithms," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 56(1), pages 101-126, August.
    11. Stanley Zionts, 1969. "The Criss-Cross Method for Solving Linear Programming Problems," Management Science, INFORMS, vol. 15(7), pages 426-445, March.
    12. Zhang, Shuzhong, 1999. "New variants of finite criss-cross pivot algorithms for linear programming," European Journal of Operational Research, Elsevier, vol. 116(3), pages 607-614, August.
    13. Gerhard Reinelt, 1991. "TSPLIB—A Traveling Salesman Problem Library," INFORMS Journal on Computing, INFORMS, vol. 3(4), pages 376-384, November.
    14. Fukuda, Komei & Matsui, Tomomi, 1991. "On the finiteness of the criss-cross method," European Journal of Operational Research, Elsevier, vol. 52(1), pages 119-124, May.
    15. Donald Goldfarb & Jianxiu Hao & Sheng-Roan Kai, 1990. "Efficient Shortest Path Simplex Algorithms," Operations Research, INFORMS, vol. 38(4), pages 624-628, August.
    16. M. L. Balinski, 1985. "Signature Methods for the Assignment Problem," Operations Research, INFORMS, vol. 33(3), pages 527-536, June.
    17. Ming S. Hung, 1983. "Technical Note—A Polynomial Simplex Method for the Assignment Problem," Operations Research, INFORMS, vol. 31(3), pages 595-600, June.
    18. Edward J. Russell, 1969. "Letters to the Editor---Extension of Dantzig's Algorithm to Finding an Initial Near-Optimal Basis for the Transportation Problem," Operations Research, INFORMS, vol. 17(1), pages 187-191, February.
    19. Paparrizos, Konstantinos, 1991. "A relaxation column signature method for assignment problems," European Journal of Operational Research, Elsevier, vol. 50(2), pages 211-219, January.
    20. James B. Orlin, 1993. "A Faster Strongly Polynomial Minimum Cost Flow Algorithm," Operations Research, INFORMS, vol. 41(2), pages 338-350, April.
    21. Fred Glover & D. Klingman & A. Napier, 1972. "Basic Dual Feasible Solutions for a Class of Generalized Networks," Operations Research, INFORMS, vol. 20(1), pages 126-136, February.
    22. Stanley Zionts, 1972. "Some Empirical Tests of the Criss-Cross Method," Management Science, INFORMS, vol. 19(4-Part-1), pages 406-410, December.
    23. Robert G. Bland, 1977. "New Finite Pivoting Rules for the Simplex Method," Mathematics of Operations Research, INFORMS, vol. 2(2), pages 103-107, May.
    24. Akkeles, Arif A. & Balogh, Laszlo & Illes, Tibor, 2004. "New variants of the criss-cross method for linearly constrained convex quadratic programming," European Journal of Operational Research, Elsevier, vol. 157(1), pages 74-86, August.
    25. Jon Lee & John F. Raffensperger, 2006. "Using AMPL for Teaching the TSP," INFORMS Transactions on Education, INFORMS, vol. 7(1), pages 37-69, September.
    26. Fukuda, K. & Terlaky, T., 1999. "On the existence of a short admissible pivot sequence for feasibility and linear optimization problems," Pure Mathematics and Applications, Department of Mathematics, Corvinus University of Budapest, vol. 10(4), pages 431-447.
    27. Ravindra K. Ahuja & James B. Orlin, 1992. "The Scaling Network Simplex Algorithm," Operations Research, INFORMS, vol. 40(1-supplem), pages 5-13, February.
    28. Ploskas, Nikolaos & Samaras, Nikolaos, 2015. "Efficient GPU-based implementations of simplex type algorithms," Applied Mathematics and Computation, Elsevier, vol. 250(C), pages 552-570.
    29. Illes, Tibor & Terlaky, Tamas, 2002. "Pivot versus interior point methods: Pros and cons," European Journal of Operational Research, Elsevier, vol. 140(2), pages 170-190, July.
    30. John Hultz & Darwin Klingman & Robert Russell, 1976. "An Advanced Dual Basic Feasible Solution for a Class of Capacitated Generalized Networks," Operations Research, INFORMS, vol. 24(2), pages 301-313, April.
    31. BLAND, Robert G., 1977. "New finite pivoting rules for the simplex method," LIDAM Reprints CORE 315, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    32. Andreas Alpers & Leslie E. Trotter, 2009. "Teaching Computational Discrete Optimization at the Undergraduate Level," INFORMS Transactions on Education, INFORMS, vol. 9(2), pages 63-69, January.
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    1. Sophia Voulgaropoulou & Nikolaos Samaras & Angelo Sifaleras, 2019. "Computational complexity of the exterior point simplex algorithm," Operational Research, Springer, vol. 19(2), pages 297-316, June.

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