On Lagrangian relaxation of quadratic matrix constraints

SIAM Journal on Matrix Analysis and Applications

Volumn 22, Issue 1, 2000, Pages 41-55

  Anstreicher, Kurt ^a   Wolkowicz, Henry ^b  

^a UNIVERSITY OF IOWA   (United States)

^b UNIVERSITY OF WATERLOO   (Canada)

Author keywords

Graph partitioning; Lagrangian relaxations; Max cut problems; Quadratic assignment; Quadratically constrained quadratic programs; Semidefinite programming

Indexed keywords

EID: 0034353656     PISSN: 08954798     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0895479898340299     Document Type: Article

References (50)

1. F. ALIZADEH, Interior point methods in semidefinite programming with applications to combinatorial optimization, SIAM J. Optim., 5 (1995), pp. 13-51.
2. E. BALAS, S. CERIA, AND G. CORNUEJOLS, A lift-and-project cutting plane algorithm for mixed 0-1 programs, Math. Programming, 58 (1993), pp. 295-324.
3. R. BHATIA, Perturbation Bounds for Matrix Eigenvalues, Pitman Res. Notes Math. Ser. 162, Longman Scientific and Technical, Harlow, UK, 1987.
4. M. CELIS, J. DENNIS JR., AND R. TAPIA, A trust region strategy for nonlinear equality constrained optimization, in Proceedings of the SIAM Conference on Numerical Optimization, Boulder, CO, 1984 , pp. 71-82. Also available as Technical Report TR84-1, Rice University, Houston, TX.
5. J.-P. CROUZEIX, J.-E. MARTINEZ-LEGAZ, AND A. SEEGER, An alternative theorem for quadratic forms and extensions, Linear Algebra Appl., 215 (1995), pp. 121-134.
6. W. DONATH AND A. HOFFMAN, Lower bounds for the partitioning of graphs, IBM J. Res. Develop., 17 (1973), pp. 420-425.
7. A. EDELMAN, T. A. ARIAS, AND S. T. SMITH, The geometry of algorithms with orthogonality constraints, SIAM J. Matrix Anal. Appl., 20 (1998), pp. 303-353.
8. G. FINKE, R. BURKHARD, AND F. RENDL, Quadratic assignment problems, Ann. Discrete Math., 31 (1987), pp. 61-82.
9. M. Fu, Z. LUO, AND Y. YE, Approximation algorithms for quadratic programming, J. Combinatorial Optim., 2 (1998), pp. 29-50.
10. T. FUJIE AND M. KOJIMA, Semidefinite programming relaxation for nonconvex quadratic programs, J. Global Optim., 10 (1997), pp. 367-380.
11. M. X. GOEMANS, Semidefinite programming in combinatorial optimization, Math. Programming, 79 (1997), pp. 143-162.
12. M. GOEMANS AND D. WILLIAMSON, Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming, J. Assoc. Comput. Mach., 42 (1995), pp. 1115-1145.
13. S. HADLEY, F. RENDL, AND H. WOLKOWICZ, Bounds for the quadratic assignment problems using continuous optimization, in Integer Programming and Combinatorial Optimization, University of Waterloo Press, Waterloo, Ontario, Canada, 1990, pp. 237-248.
14. S. HADLEY, F. RENDL, AND H. WOLKOWICZ, A new lower bound via projection for the quadratic assignment problem, Math. Oper. Res., 17 (1992), pp. 727-739.
15. C. HELMBERG, An Interior Point Method for Semidefinite Programming and Max-Cut Bounds, Ph.D. thesis, Graz University of Technology, Austria, 1994.
16. C. HELMBERG, Fixing Variables in Semidefinite Relaxations, Lecture Notes in Comput. Sci. 1284, Springer, Berlin, 1997.
17. C. HELMBERG AND F. RENDL, A spectral bundle method for semidefinite programming, SIAM J. Optim, 10 (2000), pp. 673-696.
18. C. HELMBERG, F. RENDL, R. J. VANDERBEI, AND H. WOLKOWICZ, An interior-point method for semidefinite programming, SIAM J. Optim., 6 (1996), pp. 342-361.
19. F. JARRE, On the convergence of the method of analytic centers when applied to convex quadratic programs, Math. Programming, 49 (1990), pp. 341-358.
20. M. KOJIMA AND L. TUNÇEL, Cones of Matrices and Successive Convex Relaxations of Nonconvex Sets, Technical Report B-338, Tokyo Institute of Technology, Tokyo, Japan, 1998.
21. 2P, sequential quadratic constrained quadratic programming, in Advances in Nonlinear Programming, Y. Xiang Yuan, ed., Appl. Optim. 14, Kluwer Academic Publishers, Dordrecht, 1998, pp. 177-204.
22. M. LAURENT, Cuts, matrix completions and graph rigidity, Math. Programming, 79 (1997), pp. 255-284.
23. M. LAURENT, A tour d'horizon on positive semidefinite and Euclidean distance matrix completion problems, in Topics in Semidefinite and Interior-Point Methods, Fields Inst. Commun. 18, AMS, Providence, RI, 1998, pp. 51-76.
24. L. LOVÁSZ AND A. SCHRIJVER, Cones of matrices and set-functions and 0-1 optimization, SIAM J. Optim., 1 (1991), pp. 166-190.
25. Z.-Q. LUO AND J. SUN, An Analytic Center Based Column Generation Algorithm for Convex Quadratic Feasibility Problems, Technical report, McMaster University, Hamilton, Ontario, Canada, 1995; also available from ftp://ftp.nus.sg/pub/NUS/opt/qcut.ps.gz.
26. Z.-Q. LUO AND S. ZHANG, On the Extension of Frank-Wolfe Theorem, Technical report, Erasmus University Rotterdam, The Netherlands, 1997.
27. J. M. MARTINEZ, Local minimizers of quadratic functions on Euclidean balls and spheres, SIAM J. Optim., 4 (1994), pp. 159-176.
28. J. J. MORÉ, Generalizations of the trust region problem, Optim. Methods Softw., 2 (1993), pp. 189-209.
29. J. MORÉ AND D. SORENSEN, Computing a trust region step, SIAM J. Sci. Stat. Comput., 4 (1983), pp. 553-572.
30. Y. E. NESTEROV, Semidefinite relaxation and nonconvex quadratic optimization, Optim. Methods Softw., 9 (1998), pp. 141-160.
31. Y. NESTEROV AND A. NEMIROVSKII, Interior Point Polynomial Algorithms in Convex Programming, SIAM Stud. Appl. Math. 13, SIAM, Philadelphia, PA, 1994.
32. P. PARDALOS, F. RENDL, AND H. WOLKOWICZ, The quadratic assignment problem: A survey and recent developments, in Proceedings of the DIMACS Workshop on Quadratic Assignment Problems, DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 16, AMS, Providence, RI, 1994, pp. 1-41.
33. J.-M. PENG AND Y.-X. YUAN, Optimality conditions for the minimization of a quadratic with two quadratic constraints, SIAM J. Optim., 7 (1997), pp. 579-594.
34. p-approximation. I, in Proceedings of the Princeton Symposium on Mathematical Programming, Princeton Univ., 1967, Princeton University Press, Princeton, NJ, 1970, pp. 445-480.
35. p-approximation. II. Canonical programs, SIAM J. Appl. Math., 17 (1969), pp. 317-340.
36. p-approximation. III. Degenerate programs, J. Math. Anal. Appl., 29 (1970), pp. 365-383.
37. S. POLJAK, F. RENDL, AND H. WOLKOWICZ, A recipe for semidefinite relaxation for (0,1)-quadratic programming, J. Global Optim., 7 (1995), pp. 51-73.
38. S. POLJAK AND H. WOLKOWICZ, Convex relaxations of 0-1 quadratic programming, Math. Oper. Res., 20 (1995), pp. 550-561.
39. A. QUIST, E. D. KLERK, C. ROOS, AND T. TERLAKY, Copositive relaxation for general quadratic programming, Optim. Methods Softw., 9 (1998), pp. 185-208.
40. F. RENDL AND H. WOLKOWICZ, A semidefinite framework for trust region subproblems with applications to large scale minimization, Math. Programming, 77 (1997), pp. 273-299.
41. S. SANTOS AND D. SORENSEN, A New Matrix-Free Algorithm for the Large-Scale Trust-Region Subproblem, Technical Report TR95-20, Rice University, Houston, TX, 1995.
42. N. SHOR, Quadratic optimization problems, Izv. Akad. Nauk SSSR Tekhn. Kibernet., 222 (1987), pp. 128-139.
43. R. J. STERN AND H. WOLKOWICZ, Indefinite trust region subproblems and nonsymmetric eigenvalue perturbations, SIAM J. Optim., 5 (1995), pp. 286-313.
44. p programming, European J. Oper. Res., 22 (1985), pp. 70-100.
45. H. WOLKOWICZ AND Q. ZHAO, Semidefinite relaxations for the graph partitioning problem, Discrete Appl. Math., 96-97 (1999), pp. 461-479.
46. Y. YE, Approximating quadratic programming with bound and quadratic constraints, Math. Programming, 84 (1999), pp. 219-226.
47. Y. YUAN, Some Properties of Trust Region Algorithms for Nonsmooth Optimization, Technical Report DAMTP 1983/NA14, University of Cambridge, Cambridge, UK, 1983.
48. Y. YUAN, On a subproblem of trust region algorithms for constrained optimization, Math. Programming, 47 (1990), pp. 53-63.
49. Y. YUAN, A dual algorithm for minimizing a quadratic function with two quadratic constraints, J. Comput. Math., 9 (1991), pp. 348-359.
50. Q. ZHAO, S. KARISCH, F. RENDL, AND H. WOLKOWICZ, Semidefinite programming relaxations for the quadratic assignment problem, J. Combin. Optim., 2 (1998), pp. 71-109.