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Convergence Analysis of an Inexact Potential Reduction Method for Convex Quadratic Programming

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Abstract

We analyze the convergence of an infeasible inexact potential reduction method for quadratic programming problems. We show that the convergence of this method is achieved if the residual of the KKT system satisfies a bound related to the duality gap. This result suggests stopping criteria for inner iterations that can be used to adapt the accuracy of the computed direction to the quality of the potential reduction iterate in order to achieve computational efficiency.

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References

  1. Benzi, M., Golub, G., Liesen, J.: Numerical solution of saddle point problems. Acta Numer. 14, 1–137 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  2. Cafieri, S., D’Apuzzo, M., De Simone, V., di Serafino, D.: On the iterative solution of KKT systems in potential reduction software for large-scale quadratic problems. Comput. Optim. Appl., Special Issue on High Performance Algorithms and Software for Nonlinear Optimization, Edited by A. Murli and G. Toraldo, doi:10.1007/s10589-007-9035-y (2007)

  3. Baryamureeba, V., Steihaug, T.: On the convergence of an inexact primal-dual interior-point method for linear programming. Report No. 188, Department of Informatics, University of Bergen, Bergen, Norway (2000)

    Google Scholar 

  4. Korzak, J.: Convergence analysis of inexact infeasible-interior-point-algorithms for solving linear programming problems. SIAM J. Optim. 11, 133–148 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  5. Portugal, L., Resende, M., Veiga, G., Judice, J.: A truncated primal-infeasible dual-feasible interior point network flow method. Networks 35, 91–108 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  6. Mizuno, S., Kojima, M., Todd, M.J.: Infeasible interior-point primal-dual potential-reduction algorithms for linear programming. SIAM J. Optim. 5, 52–67 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  7. Cafieri, S., D’Apuzzo, M., De Simone, V., di Serafino, D.: Stopping criteria for inner iterations in inexact potential reduction methods: a computational study. Comput. Optim. Appl., Special Issue on Linear Algebra Issues Arising in Interior Point Methods, Edited by J. Gondzio and G. Toraldo 36(2–3), 165–193 (2007)

  8. Tanabe, K.: Centered newton method for mathematical programming. In: Iri, M., Yajima, K. (eds.) System Modeling and Optimization: Proceedings of the 13th IFIP Conference. Lecture Notes in Control and Information Systems, vol. 113, pp. 197–206. Springer, Berlin (1988)

    Google Scholar 

  9. Todd, M.J., Ye, Y.: A centered projective algorithm for linear programming. Math. Oper. Res. 15, 508–529 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  10. Todd, M.J.: Potential-reduction methods in mathematical programming. Math. Program. 76, 3–45 (1996)

    MathSciNet  Google Scholar 

  11. Kojima, M., Mizuno, S., Yoshise, A.: An \(O(\sqrt{(}n)L)\) iteration potential-reduction algorithm for linear complementarity problems. Math. Program. 50, 331–342 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  12. Cafieri, S., D’Apuzzo, M., Marino, M., Mucherino, A., Toraldo, G.: Interior point solver for large-scale quadratic programming problems with bound constraints. J. Optim. Theory Appl. 129(1), 55–75 (2006)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, 2nd University of Naples, Caserta, Italy

    S. Cafieri, M. D’Apuzzo, V. De Simone & D. di Serafino

  2. Department of Agricultural Engineering and Agronomy, University of Naples Federico II, Naples, Italy

    G. Toraldo

Authors
  1. S. Cafieri
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  2. M. D’Apuzzo
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  3. V. De Simone
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  4. D. di Serafino
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  5. G. Toraldo
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Corresponding author

Correspondence to S. Cafieri.

Additional information

Communicated by P.M. Pardalos.

This research was partially supported by the Italian MIUR, Project FIRB—Large Scale Nonlinear Optimization # RBNE01WBBB and Project PRIN—Innovative Problems and Methods in Nonlinear Optimization # 2005017083.

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Cafieri, S., D’Apuzzo, M., De Simone, V. et al. Convergence Analysis of an Inexact Potential Reduction Method for Convex Quadratic Programming. J Optim Theory Appl 135, 355–366 (2007). https://doi.org/10.1007/s10957-007-9264-3

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  • DOI: https://doi.org/10.1007/s10957-007-9264-3

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