Central European Journal of Operations Research

, Volume 23, Issue 4, pp 913–924 | Cite as

The practical behavior of the homogeneous self-dual formulations in interior point methods

  • Csaba MeszarosEmail author
Original Paper


Interior point methods proved to be efficient and robust tools for solving large-scale optimization problems. The standard infeasible-start implementations scope very well with wide variety of problem classes, their only serious drawback is that they detect primal or dual infeasibility by divergence and not by convergence. As an alternative, approaches based on skew-symmetric and self-dual reformulations were proposed. In our computational study we overview the implementation of interior point methods on the homogeneous self-dual formulation of optimization problems and investigate the effect of the increased dimension from numerical and computational aspects.


Interior point methods Convex quadratically constrained quadratic programming Homogeneous self-dual embedding 


  1. Andersen ED, Ye Y (1998) A computational study of the homogeneous algorithm for large-scale convex optimization. Comput Optim Appl 10:243–289zbMATHMathSciNetCrossRefGoogle Scholar
  2. Andersen ED, Ye Y (1999) On a homogeneous algorithm for the monotone complementarity problem. Math Program 84(2):375–399zbMATHMathSciNetCrossRefGoogle Scholar
  3. Andersen ED, Gondzio J, Meszaros C, Xu X (1996) Implementation of interior point methods for large scale linear programs. In: Terlaky T (ed) Interior point methods of mathematical programming. Kluwer, Dordrecht, pp 189–252Google Scholar
  4. Anstreicher KM, Vial J-P (1994) On the convergence of an infeasible primal-dual interior-point method for convex programming. Optim Methods Softw 3:285–316CrossRefGoogle Scholar
  5. Gay DM (1985) Electronic mail distribution of linear programming test problems. COAL Newsl 13:10–12Google Scholar
  6. Goldman AJ, Tucker AW (1956a) Polyhedral convex cones. In: Kuhn HW, Tucker AW (eds) Linear inequalities and related systems. Princeton University Press, Princeton, NJ, pp 19–40Google Scholar
  7. Goldman AJ, Tucker AW (1956b) Theory of linear programming. In: Kuhn HW, Tucker AW (eds) Linear inequalities and related systems. Princeton University Press, Princeton, NJ, pp 53–97Google Scholar
  8. Gondzio J (1996) Multiple centrality corrections in a primal-dual method for linear programming. Comput Optim Appl 6:137–156zbMATHMathSciNetCrossRefGoogle Scholar
  9. Jansen B, Terlaky T, Roos C (1994) The theory of linear programming: skew symmetric self-dual problems and the central path. Optimization 29:225–233zbMATHMathSciNetCrossRefGoogle Scholar
  10. Kojima M, Megiddo N, Mizuno S (1991) A primal-dual infeasible-interior-point algorithm for linear programming, Technical reportGoogle Scholar
  11. Kojima M, Megiddo N, Mizuno S (1993) A primal-dual infeasible-interior-point algorithm for linear programming. Math Program 61:263–280zbMATHMathSciNetCrossRefGoogle Scholar
  12. Lustig IJ (1990) Feasibility issues in primal-dual interior-point methods for linear programming. Math Program 49:145–162zbMATHMathSciNetCrossRefGoogle Scholar
  13. Lustig IJ, Marsten RE, Shanno DF (1992) On implementing Mehrotra’s predictor-corrector interior-point method for linear programming. SIAM J Optim 2(3):435–449zbMATHMathSciNetCrossRefGoogle Scholar
  14. Lustig IJ, Marsten RE, Shanno DF (1994) Interior point methods for linear programming: computational state of the art. ORSA J Comput 6(1):1–15zbMATHMathSciNetCrossRefGoogle Scholar
  15. Mehrotra S (1991) High order methods and their performance. Technical Report 90–16R1, Department of Industrial Engineering and Managment Sciences Northwestern University, Evanston, USAGoogle Scholar
  16. Mészáros C (1997) On free variables in interior point methods. Optim Methods Softw 9:121–139CrossRefGoogle Scholar
  17. Mészáros C (1999) The BPMPD interior-point solver for convex quadratic problems. Optim Methods Softw 11 &12:431–449CrossRefGoogle Scholar
  18. Mészáros C (2005) On the Cholesky factorization in interior point methods. Comput Math Appl 50: 1157–1166Google Scholar
  19. Mészáros C (2008) On numerical issues of interior point methods. SIAM J Matrix Anal 30(1):223–235zbMATHMathSciNetCrossRefGoogle Scholar
  20. Mészáros C (2010) On the implementation of interior point methods for dual-core platforms. Optim Mathods Softw 25(3):449–456zbMATHCrossRefGoogle Scholar
  21. Mészáros C (2011a) On sparse matrix orderings in interior point methods. Working paper, Computer and Automation Institute, Hungarian Academy of Sciences, BudapestGoogle Scholar
  22. Mészáros C (2011b) Solving quadratically constrained convex optimization problems with an interior point method. Optim Methods Softw 26(3):421–429zbMATHMathSciNetCrossRefGoogle Scholar
  23. Mészáros C (2012) Regularization techniques in interior point methods. J Comput Appl Math 236: 3704–3709Google Scholar
  24. Mészáros C, Suhl UH (2004) Advanced preprocessing techniques for linear and quadratic programming. OR Spectrum 25:575–595CrossRefGoogle Scholar
  25. Mittelmann HD, Spellucci P (1998)Decision tree for optimization software. World Wide Web.
  26. Mizuno S (1994) Polynomiality of infeasible-interior-point algorithms for linear programming. Math Program 67(1):109–119Google Scholar
  27. Mizuno S, Todd MJ (2001) On two homogeneous self-dual approaches to linear programming and its extensions. Math Program 89:517–534zbMATHMathSciNetCrossRefGoogle Scholar
  28. Nesterov Y, Todd MJ, Ye Y (1999) Infeasible-start primal-dual methods and infeasibility detectors for nonlinear programming problems. Math Program 84:227–267zbMATHMathSciNetGoogle Scholar
  29. Tucker AW (1956) Dual systems of homogeneous linear relations. In: Kuhn HW, Tucker AW (eds) Linear inequalities and related systems. Princeton University Press, Princeton, NJ, pp 3–18Google Scholar
  30. Vanderbei RJ (1995) Symmetric quasi-definite matrices. SIAM J Optim 5(1):100–113zbMATHMathSciNetCrossRefGoogle Scholar
  31. Wolfe P (1961) A duality theorem for non-linear programming. Q Appl Math 19:239–244zbMATHMathSciNetGoogle Scholar
  32. Wright SJ (1999) Modified Cholesky factorizations in interior-point algorithms for linear programming. SIAM J Optim 9(4):1159–1191zbMATHMathSciNetCrossRefGoogle Scholar
  33. Xu X (August 1994) An \({\cal O}(\sqrt{n} L)\)-iteration large-step infeasible path-following algorithm for linear programming. Technical report, College of Business Administration, The University of Iowa, Iowa City, IA 52242Google Scholar
  34. Xu X (1996) On the implementation of a homogeneous and self-dual linear programming algorithm. Math Program 76(2):155–181Google Scholar
  35. Xu X, Ye Y (1995) A generalized homogeneous and self-dual algorithm for linear programming. Oper Res Lett 17:181–190zbMATHMathSciNetCrossRefGoogle Scholar
  36. Xu X, Hung P-F, Ye Y (1996) A simplified homogeneous and self-dual linear programming algorithm and its implementation. Ann Oper Res 62(1):151–171zbMATHMathSciNetCrossRefGoogle Scholar
  37. Ye Y, Todd MJ, Mizuno S (1994) An \(O(\sqrt{n} L)\)-iteration homogeneous and self-dual linear programming algorithm. Math Oper Res 19:53–67zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Computer and Automation Research InstituteHungarian Academy of SciencesBudapestHungary

Personalised recommendations