Skip to main content
SpringerLink
Log in

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

  • Original Paper
  • Published:
Central European Journal of Operations Research Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Andersen ED, Ye Y (1998) A computational study of the homogeneous algorithm for large-scale convex optimization. Comput Optim Appl 10:243–289

    Article  MATH  MathSciNet  Google Scholar 

  • Andersen ED, Ye Y (1999) On a homogeneous algorithm for the monotone complementarity problem. Math Program 84(2):375–399

    Article  MATH  MathSciNet  Google Scholar 

  • 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 189–252

  • Anstreicher KM, Vial J-P (1994) On the convergence of an infeasible primal-dual interior-point method for convex programming. Optim Methods Softw 3:285–316

    Article  Google Scholar 

  • Gay DM (1985) Electronic mail distribution of linear programming test problems. COAL Newsl 13:10–12

    Google Scholar 

  • 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 19–40

  • 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 53–97

  • Gondzio J (1996) Multiple centrality corrections in a primal-dual method for linear programming. Comput Optim Appl 6:137–156

    Article  MATH  MathSciNet  Google Scholar 

  • Jansen B, Terlaky T, Roos C (1994) The theory of linear programming: skew symmetric self-dual problems and the central path. Optimization 29:225–233

    Article  MATH  MathSciNet  Google Scholar 

  • Kojima M, Megiddo N, Mizuno S (1991) A primal-dual infeasible-interior-point algorithm for linear programming, Technical report

  • Kojima M, Megiddo N, Mizuno S (1993) A primal-dual infeasible-interior-point algorithm for linear programming. Math Program 61:263–280

    Article  MATH  MathSciNet  Google Scholar 

  • Lustig IJ (1990) Feasibility issues in primal-dual interior-point methods for linear programming. Math Program 49:145–162

    Article  MATH  MathSciNet  Google Scholar 

  • Lustig IJ, Marsten RE, Shanno DF (1992) On implementing Mehrotra’s predictor-corrector interior-point method for linear programming. SIAM J Optim 2(3):435–449

    Article  MATH  MathSciNet  Google Scholar 

  • Lustig IJ, Marsten RE, Shanno DF (1994) Interior point methods for linear programming: computational state of the art. ORSA J Comput 6(1):1–15

    Article  MATH  MathSciNet  Google Scholar 

  • Mehrotra S (1991) High order methods and their performance. Technical Report 90–16R1, Department of Industrial Engineering and Managment Sciences Northwestern University, Evanston, USA

  • Mészáros C (1997) On free variables in interior point methods. Optim Methods Softw 9:121–139

    Article  Google Scholar 

  • Mészáros C (1999) The BPMPD interior-point solver for convex quadratic problems. Optim Methods Softw 11 &12:431–449

    Article  Google Scholar 

  • Mészáros C (2005) On the Cholesky factorization in interior point methods. Comput Math Appl 50: 1157–1166

    Google Scholar 

  • Mészáros C (2008) On numerical issues of interior point methods. SIAM J Matrix Anal 30(1):223–235

    Article  MATH  MathSciNet  Google Scholar 

  • Mészáros C (2010) On the implementation of interior point methods for dual-core platforms. Optim Mathods Softw 25(3):449–456

    Article  MATH  Google Scholar 

  • Mészáros C (2011a) On sparse matrix orderings in interior point methods. Working paper, Computer and Automation Institute, Hungarian Academy of Sciences, Budapest

  • Mészáros C (2011b) Solving quadratically constrained convex optimization problems with an interior point method. Optim Methods Softw 26(3):421–429

    Article  MATH  MathSciNet  Google Scholar 

  • Mészáros C (2012) Regularization techniques in interior point methods. J Comput Appl Math 236: 3704–3709

    Google Scholar 

  • Mészáros C, Suhl UH (2004) Advanced preprocessing techniques for linear and quadratic programming. OR Spectrum 25:575–595

    Article  Google Scholar 

  • Mittelmann HD, Spellucci P (1998)Decision tree for optimization software. World Wide Web. http://plato.la.asu.edu/guide.html

  • Mizuno S (1994) Polynomiality of infeasible-interior-point algorithms for linear programming. Math Program 67(1):109–119

    Google Scholar 

  • Mizuno S, Todd MJ (2001) On two homogeneous self-dual approaches to linear programming and its extensions. Math Program 89:517–534

    Article  MATH  MathSciNet  Google Scholar 

  • Nesterov Y, Todd MJ, Ye Y (1999) Infeasible-start primal-dual methods and infeasibility detectors for nonlinear programming problems. Math Program 84:227–267

    MATH  MathSciNet  Google Scholar 

  • 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 3–18

  • Vanderbei RJ (1995) Symmetric quasi-definite matrices. SIAM J Optim 5(1):100–113

    Article  MATH  MathSciNet  Google Scholar 

  • Wolfe P (1961) A duality theorem for non-linear programming. Q Appl Math 19:239–244

    MATH  MathSciNet  Google Scholar 

  • Wright SJ (1999) Modified Cholesky factorizations in interior-point algorithms for linear programming. SIAM J Optim 9(4):1159–1191

    Article  MATH  MathSciNet  Google Scholar 

  • 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 52242

  • Xu X (1996) On the implementation of a homogeneous and self-dual linear programming algorithm. Math Program 76(2):155–181

    Google Scholar 

  • Xu X, Ye Y (1995) A generalized homogeneous and self-dual algorithm for linear programming. Oper Res Lett 17:181–190

    Article  MATH  MathSciNet  Google Scholar 

  • 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):151–171

    Article  MATH  MathSciNet  Google Scholar 

  • Ye Y, Todd MJ, Mizuno S (1994) An \(O(\sqrt{n} L)\)-iteration homogeneous and self-dual linear programming algorithm. Math Oper Res 19:53–67

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Computer and Automation Research Institute, Hungarian Academy of Sciences, Budapest, Hungary

    Csaba Meszaros

Authors
  1. Csaba Meszaros
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Csaba Meszaros.

Additional information

Supported in part by Hungarian Research Fund OTKA K-77420.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Meszaros, C. The practical behavior of the homogeneous self-dual formulations in interior point methods. Cent Eur J Oper Res 23, 913–924 (2015). https://doi.org/10.1007/s10100-013-0336-1

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10100-013-0336-1

Keywords

Search

Navigation