Abstract
In this paper, we investigate the computational behavior of the exterior point simplex algorithm. Up until now, there has been a major difference observed between the theoretical worst case complexity and practical performance of simplex-type algorithms. Computational tests have been carried out on randomly generated sparse linear problems and on a small set of benchmark problems. Specifically, 6780 linear problems were randomly generated, in order to formulate a respectable amount of experiments. Our study consists of the measurement of the number of iterations that the exterior point simplex algorithm needs for the solution of the above mentioned problems and benchmark dataset. Our purpose is to formulate representative regression models for these measurements, which would play a significant role for the evaluation of an algorithm’s efficiency. For this examination, specific characteristics, such as the number of constraints and variables, the sparsity and bit length, and the condition of matrix A, of each linear problem, were taken into account. What drew our attention was that the formulated model for the randomly generated problems reveal a linear relation among these characteristics.
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References
Berenguer E, Smith L (1986) The expected number of extreme points of a random linear program. Math Program 35:129–134
Borgwardt K (1982a) The average number of pivot steps required by the simplex method is polynomial. Z Oper Res 26(1):157–177
Borgwardt K (1982b) Some distribution independent results about the asymptotic order of the average number of pivot steps in the simplex method. Math Oper Res 7(3):441–462
Dantzig G (1949) Programming of interdependent activities: II, mathematical model. Econometrica 3–4:200–211
Dantzig G (1963) Linear programming and extensions. Princeton University Press, Princeton
Draper N (1998) Smith H applied regression analysis, 3rd edn. Wiley, New York
Durbin J, Watson G (1950) Testing for serial correlation in least squares regression I. Biometrika 37(3,4):409–428
Fisher R (1922) On the interpretation of \(\chi ^2\) from contingency tables and the calculation of P. J R Stat Soc 85(1):87–94
Glavelis T, Samaras N (2013) An experimental investigation of a primal-dual exterior point simplex algorithm. Optim: J Math Program Oper Res 62(8):1143–1152
Hosmer D, Jovanovic B, Lemeshow S (1989) Best subsets logistic regression. Biometrics 45(4):1265–1270
Karmarkar N (1984) A new polynomial-time algorithm for linear programming. Combinatorica 4(4):373–395
Khachiyan L (1979) A polynomial algorithm in linear programming. Sov Math Dokl 20:191–194
Klee V, Minty G (1972) How good is the simplex algorithm? In: Shisha O (ed) Inequalities, vol III. Academic Press, New York, NY, pp 159–175
Kutner M, Neter J, Nachtsheim C, Wasserman W (2004) Applied linear statistical models, 5th edn. McGraw-Hill, New York
Linear Programming Test Problems. http://www.sztaki.hu/~meszaros/public_ftp/lptestset/misc. Last checked on 05 Oct 2016
Maros I, Khaliq M (1999) Advances in design and implementation of optimization software. Technical report
Mathworks MATLAB. http://www.mathworks.com/products/matlab. Last checked on 05 Oct 2016
McGeoch C, Sanders P, Fleischer R, Cohen PR, Precup D (2002) Using finite experiments to study asymptotic performance. In: Experimental algorithmics. Springer, pp 93–126
Minitab. http://www.minitab.com/en-us/academic. Last checked on 05 Oct 2016
NETLIB, netlib benchmark lps. http://www.netlib.org/benchmark. Last checked on 05 Oct 2016
NIST, National Institute of Standards and Technology. http://www.nist.gov. Last checked on 5 Oct 2016
Paparrizos K (1991) An infeasible exterior point simple algorithm for assignment problems. Math Program 51:45–54
Paparrizos K (1993) An exterior point simplex algorithm for (general) linear programming problems. Ann Oper Res 47:497–508
Paparrizos K, Samaras N, Stephanides G (2000) A method for generating random optimal linear problems and a comparative computational study. In: Proceedings of the 13th national conference of the Hellenic operational research society, Piraeus, pp 785–794 (in Greek)
Paparrizos K, Samaras N, Stephanides N (2003a) An efficient simplex type algorithm for sparse and dense linear problems. Eur J Oper Res 148(2):323–334
Paparrizos K, Samaras N, Stephanides N (2003b) A new efficient primal dual simplex algorithm. Comput Oper Res 30(9):1383–1339
Paparrizos K, Samaras N, Sifaleras A (2015) Exterior point simplex-type algorithms for linear and network optimization problems. Ann Oper Res 229(1):607–633
Prišenk J, Turk J, Rozman Čr, Borec A, Zrakić M, Pažek K (2014) Advantages of combining linear programming and weighted goal programming for agriculture application. Oper Res 14(2):253–260
Rao C (1973) Coefficient of determination, linear statistical inference and its applications, 2nd edn. Wiley, New York
Student (1908) The probable error of a mean. Biometrika 6(1):1–25
Triantafyllidis C, Samaras N (2015) Three nearly scaling invariant versions of an exterior point algorithm for linear programming. Optim: J Math Program Oper Res 64(10):2136–2181
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Voulgaropoulou, S., Samaras, N. & Sifaleras, A. Computational complexity of the exterior point simplex algorithm. Oper Res Int J 19, 297–316 (2019). https://doi.org/10.1007/s12351-017-0291-z
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DOI: https://doi.org/10.1007/s12351-017-0291-z