Numerical Experiments with a Primal-Dual Algorithm for Solving Quadratic Problems

Derkaoui Orkia, Lehireche Ahmed

2016

Abstract

This paper provides a new variant of primal-dual interior-point method for solving a SemiDefinite Program (SDP). We use the PDIPA (primal-dual interior-point algorithm) solver entitled SDPA (SemiDefinite Programming Algorithm). This last uses a classical Newton descent method to compute the predictor-corrector search direction. The difficulty is in the computation of this line-search, it induces high computational costs. Here, instead we adopt a new procedure to implement another way to determine the step-size along the direction which is more efficient than classical line searches. This procedure consists in the computation of the step size in order to give a significant decrease along the descent line direction with a minimum cost. With this procedure we obtain à new variant of SDPA. The comparison of the results obtained with the classic SDPA and our new variant is promising.

References

  1. Alizadeh, F., 1995. Interior point methods in semidefinite programming with application to combinatorial optimization, SIAM journal on Optimization, 5:13-51.
  2. Alizadeh, F., Haberly, J.-P., and Overton, M.-L.. 1998. Primal-dual interior-point methods for semidefinite programming, convergence rates, stability and numerical. SIAM J. Optim.8 746-768.
  3. Benterki, D., Crouzeix, J.-P., and Merikhi., B., 2003. A numerical implementation of an interior point method for semi-definite programming. Pesquisa Operacional 23-1, 49-59.
  4. Fujisawa, K. and Kojima, M., 1995. SDPA(Semidefinite Programming Algorithm) Users Manual. Technical Report b-308, Tokyo Institute of Technology.
  5. Jarre. F., 1993. An interior-point method for Programming minimizing the maximum eigenvalue of a linear combination of matrices. SIAM Journal on Control and Optimization, 31:1360-1377.
  6. Nesterov Y. E., and Nemirovskii. A. S., 1990. Optimization over positive semidefinite matrices: Mathematical background and user's manual. Technical report, Central Economic & Mathematical Institute, USSR Acad. Sci. Moscow, USSR.
  7. Nesterov Y. E., and Nemirovskii. A. S., 1993. Interior Point Polynomial Methods in Convex Programming : Theory and Algorithms. SIAM Publications, Philadelphia.
  8. Vandenberghe, L. and Boyd. S., 1995. Primal-dual potential reduction method for problems involving matrix inequalities. Mathematical Programming, 69:205-236.
  9. Nesterov, Y.E., and Nemirovskii, A.S., 1994. InteriorPoint Polynomial Algorithms in Convex Programming. SIAM Studies in Applied Mathematics 13. SIAM, Philadelphia, PA, USA. 185, 461, 564, 584, 602.
  10. Nesterov, Y. E., and Todd, M. J., 1995. Primal-dual interior-point methods for self-scaled cones. Technical Report 1125, School of Operations Research and Industrial Engineering, Cornell University, Ithaca, New York, 14853-3801.
  11. Crouzeix, J.P. Merikhi, B., 2008, A logarithm barrier method for semidefinite programming, R.A.I.R.OOper. Res. 42, pp. 123-139.
  12. Benterki, D., Merikhi, B., 2001. A modified algorithm for the strict feasibility problem, RAIRO Oper. Res. 35, pp. 395-399.
  13. Monteiro, R.D.C., 1997. Primal-dual path- following algorithms for semidefinite programming. SIAM Journal on Optimization, 7, pp. 663-678.
  14. Derkaoui, O., Lehireche, A, 2014. Safe bounds in Semidefinite programming by using interval arithmetic. American Journal of Operations Research, Vol. 4 No. 5, septembre, PP. 293-300. DOI: 10.4236/ajor.2014.45029.
  15. Helmberg, C., Rendl, F., Vanderbei, R.J., and Wolkowicz, H., 1996. An interior point method for semidefinite programming. SIAM Journal on Optimization, 6:342-361.
  16. Monteiro, R. D. C., and Tsuchiya, T., 1996. Polynomial convergence of a new family of primal-dual algorithms for semidefinite programming.Technical report, Georgia Institute of Technology,Atlanta, Georgia, USA.
  17. Mehrotra, S., 1992. On the implementation of a primaldual interior point method, SIAM Journal on Optimization 2 575-601.
  18. Monteiro, R. D. C., 1997. Primal-dual path following algorithms for semidefinite programming, SIAM Journal on Optimization 7 663-678.
  19. Nakata, M., Nakatsuji, H., Ehara, M., Fukuda, M., Nakata, K. and Fujisawa, K., 2001. Variational calculations of fermion second-order deduced density matrices by semidefinite programming algorithm, Journal of Chemical Physics 114 8282-8292.
  20. Nakata, M., Ehara, M., and Nakatsuji, H., 2002. Density matrix variational theory: Application to the potential energy surfaces and strongly correlated systems, Journal of Chemical Physics 116 5432-5439.
  21. Ben-Tal, A., and Nemirovskii, A., 2001. Lectures on Moden Convex Optimizatin Analysis, Algorithms, and Engineering Applications, (SIAM, Philadelphia).
  22. Boyd, S., Ghaoui, L. E., Feron, E., and Balakrishnan, V., 1994. Linear matrix inequalities in system and control theory, Society for Industrial and Applied Mathematics Philadelphia, PA, ISBN 0-89871-334-X.
  23. Goemans, M. X., and Williamson, D. P., 1995. Improved approximation alogrithoms for maxmum cut and satisfiability problems using semidefinite programming, Journal of Association for Computing Machinery 42(6) 1115-1145.
  24. Todd, M. J., 2001. Semidefinite optimization, Acta Numerica 10 515-560.
  25. Vandenberghe, L., Boyd, S., 1994. Positive-Definite Programming, Mathematical Programming: State of the Art J. R. Birge and K. G. Murty ed.s, U. of Michigan.
  26. Wolkowicz, H., Saigal, R., and Vandenberghe, L., 2000. Handbook of Semidefinite Programming, Theory, Algorithms, and Applications, (Kluwer Academic Publishers, Massachusetts.
  27. Kojima, M., Mizuno, S. and Yoshise, A., 1989. A PrimalDual Interior Point Algorithm for Linear Programming, in: N. Megiddo, ed., Progress in Mathematical Programming: Interior Point and Related Methods (Springer-Verlag, New York) 29-47.
  28. Tanabe, K., 1988. Centered Newton Method for Mathematical Programming, in: M. Iri and K. Yajima, eds., System Modeling and Optimization (Springer, New York) 197-206.
  29. Borchers. B., 1999. CSDP, a C library for semidefinite programming. Optimization Methods and Software, 11.
  30. Benson, S. J., Ye, Y., and Zhang, X., 2000. Solving large-scale sparse semidefinite programs for combinatorial optimization. SIAM Journal on Optimization, 10(2):443-461.
  31. Sturm, J. F., 1998. Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones.
  32. Yamashita, M., Fujisawa, K., Nakata, K., Nakata, M., Fukuda, M., Kobayashi, K., and Goto, K.., 2010. A high-performance software package for semidefinite programs: SDPA 7. Technical report, Dept. of Mathematical and Computing Science, Tokyo Institute of Technology.
  33. Burer, S., and Monteiro, R. D. C., 2003. A nonlinear programming algorithm for solving semidefinite programs via low-rank factorization. Math. Program., 95(2):329-357.
  34. Helmberg, C., and Rendl. F., 2000. A spectral bundle method for semidefinite programming. SIAM Journal on Optimization, 10:673-696.
  35. Borchers, B., 1999. SDPLIB 1.2, a library of semidefinte programming test problems, Optimization Methods and Software 11 & 12 683-690.
  36. Chouzenoux, E., Moussaoui, S., Idier, J., 2009. A Majorize-Minimize line search algorithm for barrier function optimization. EURASIP European Signal and Image Processing Conference, Aug, Glasgow, United Kingdom. pp.1379-1383, 2009. <hal-00456013>.
Download


Paper Citation


in Harvard Style

Orkia D. and Ahmed L. (2016). Numerical Experiments with a Primal-Dual Algorithm for Solving Quadratic Problems . In Proceedings of 5th the International Conference on Operations Research and Enterprise Systems - Volume 1: ICORES, ISBN 978-989-758-171-7, pages 204-209. DOI: 10.5220/0005813802040209


in Bibtex Style

@conference{icores16,
author={Derkaoui Orkia and Lehireche Ahmed},
title={Numerical Experiments with a Primal-Dual Algorithm for Solving Quadratic Problems},
booktitle={Proceedings of 5th the International Conference on Operations Research and Enterprise Systems - Volume 1: ICORES,},
year={2016},
pages={204-209},
publisher={SciTePress},
organization={INSTICC},
doi={10.5220/0005813802040209},
isbn={978-989-758-171-7},
}


in EndNote Style

TY - CONF
JO - Proceedings of 5th the International Conference on Operations Research and Enterprise Systems - Volume 1: ICORES,
TI - Numerical Experiments with a Primal-Dual Algorithm for Solving Quadratic Problems
SN - 978-989-758-171-7
AU - Orkia D.
AU - Ahmed L.
PY - 2016
SP - 204
EP - 209
DO - 10.5220/0005813802040209