Strong duality in nonconvex quadratic optimization with two quadratic constraints

SIAM Journal on Optimization

Volumn 17, Issue 3, 2006, Pages 844-860

  Beck, Amir ^a   Eldar, Yonina C ^a  

^a TECHNION ISRAEL INSTITUTE OF TECHNOLOGY   (Israel)

Author keywords

Nonconvex optimization; Quadratic mappings; Quadratic programming; Strong duality

Indexed keywords

COMPUTER SIMULATION; CONSTRAINT THEORY; OPTIMIZATION; PROBABILITY; PROBLEM SOLVING;

NONCONVEX OPTIMIZATION; QUADRATIC MAPPINGS; QUADRATIC OPTIMIZATION;

QUADRATIC PROGRAMMING;

EID: 33847613047     PISSN: 10526234     EISSN: None     Source Type: Journal    
DOI: 10.1137/050644471     Document Type: Article

References (37)

1. K. ANSTREICHER AND H. WOLKOWICZ, On Lagrangian relaxation of quadratic matrix constraints, SIAM J. Matrix Anal. Appl., 22 (2000), pp. 41-55.
2. M. S. BAZARAA, H. D. SHERALI, AND C. M. SHETTY, Nonlinear Programming: Theory and Algorithms, 2nd ed., John Wiley and Sons, New York, 1993.
3. A. BEN-TAL AND A. NEMIROVSKI, Robust convex optimization, Math. Oper. Res., 23 (1998), pp. 769-805.
4. A. BEN-TAL AND A. NEMIROVSKI, Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications, MPS-SIAM Ser. Optim. 2, SIAM, Philadelphia, 2001.
5. A. BEN-TAL AND M. TEBOULLE, Hidden convexity in some nonconvex quadratically constrained quadratic programming, Math. Programming, 72 (1996), pp. 51-63.
6. M. R. CELIS, J. E. DENNIS, AND R. A. TAPIA, A trust region strategy for nonlinear equality constrained optimization, in Numerical Optimization, 1984 (Boulder, CO, 1984), SIAM, Philadelphia, 1985, pp. 71-82.
7. S. CHANDRASEKARAN, G. H. GOLUB, M. GU, AND A. H. SAYED, Parameter estimation in the presence of bounded data uncertainties, SIAM J. Matrix Anal. Appl., 19 (1998), pp. 235-252.
8. X. CHEN AND Y.-X. YUAN, On local solutions of the Celis-Dennis-Tapia subproblem, SIAM J. Optim., 10 (1999), pp. 359-383.
9. P. CHEVALIER AND B. PICINBONO, Complex linear-quadratic systems for detection and array processing, IEEE Trans. Signal Process., 44 (1996), pp. 2631-2634.
10. A. R. CONN, N. I. M. GOLD, AND P. L. TOINT, Trust-Region Methods, MPS/SIAM Ser. Optim. 1, SIAM, Philadelphia, 2000.
11. P. FINSLER, Uber das vorkommen definiter und semi-definiter formen in scharen quadratische formen, Commentarii Mathematici Helvetici, 9 (1937), pp. 188-192.
12. C. FORTIN AND H. WOLKOWICZ, The trust region subproblem, and semidefinite programming, Optim. Methods Softw., 19 (2004), pp. 41-67.
13. A. L. FRADKOV AND V. A. YAKUBOVICH, The S-procedure and the duality relation in convex quadratic programming problems, Vestnik Leningrad. Univ., 1 (1973), pp. 81-87, 155.
14. D. M. GAY, Computing optimal locally constrained steps, SIAM J. Sci. Statist. Comput., 2 (1981), pp. 186-197.
15. D. M. GAY, A trust-region approach for linearly constrained optimization, in Proceedings of the Dundee Biennial Conference on Numerical Analysis, Dundee, Austalia, 1983, Lecture Notes in Math. 1066, Springer, Berlin, 1984, pp. 72 105.
16. L. EL GHAOUI AND H. LEBRET, Robust solution to least-squares problems with uncertain data, SIAM J. Matrix Anal. Appl., 18 (1997), pp. 1035-1064.
17. J. B. HIRIART-URRUTY AND M. TORKI, Permanently going back and forth between the "quadratic world" and the "convexity world" in optimization, Appl. Math. Optim., 45 (2002), pp. 169-184.
18. Y. HUANG AND S. ZHANG, Complex Matrix Decomposition and Quadratic Programming, Technical report SEEM2005-02, Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Hong Kong, 2005.
19. J. B. LASSERRE, Semidefinite programming versus LP relaxations for polynomial programming, Math. Oper. Res., 27 (2002), pp. 347-360.
20. J. M. MARTÍNEZ, Local minimizers of quadratic functions on Euclidean balls and spheres, SIAM J. Optim., 4 (1994), pp. 159-176.
21. J. J. MORÉ, Generalizations of the trust region subproblem, Optim. Methods Softw. 2, (1993), pp. 189-209.
22. J. J. MORÉ AND D. C. SORENSEN, Computing a trust region step, SIAM J. Sei. Statist. Comput., 4 (1983), pp. 553-572.
23. J.-M. PENG AND Y. YUAN, Optimality conditions for the minimization of a quadratic with two quadratic constraints, SIAM J. Optim., 7 (1997), pp. 579-594.
24. B. PICINBONO AND P. CHEVALIER, Widely linear estimation with complex data, IEEE Trans. Signal Process., 43 (1995), pp. 2030-2033.
25. B. T. POLYAK, Convexity of quadratic transformations and its use in control and optimization, J. Optim. Theory Appl., 99 (1998), pp. 553-583.
26. M. J. D. POWELL AND Y. YUAN, A trust region algorithm for equality constrained optimization, Math. Programming, 49 (1990/91), pp. 189-211.
27. M. V. RAMANA AND A. J. GOLDMAN, Some geometric results in semidefinite programming, J. Global Optim., 7 (1995), pp. 33-50.
28. P. J. SCHREIER, L. L. SCHARF, AND C. T. MULLIS, Detection and estimation of improper complex random signals, IEEE Trans. Inform. Theory, 51 (2005), pp. 306-312.
29. N. Z. SHOR, Quadratic optimization problems, Izv. Akad. Nauk SSSR Tekhn. Kibernet., 1 (1987), pp. 128-139, 222.
30. D. C. SORENSEN, Newton's method with a model trust region modification, SIAM J. Numer. Anal., 19 (1982), pp. 409-426.
31. R. J. STERN AND H. WOLKOWICZ, Indefinite trust region subproblems and nonsymmetric eigenvalue perturbations, SIAM J. Optim., 5 (1995), pp. 286-313.
32. J. F. STURM, Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones, Optim. Methods Softw., 11/12 (1999), pp. 625-653.
33. J. F. STURM AND S. ZHANG, On cones of nonnegative quadratic functions, Math. Oper. Res., 28 (2003), pp. 246-267.
34. L. VANDENBERGHE AND S. BOYD, Semidefinite programming, SIAM Rev., 38 (1996), pp. 40-95.
35. Y. YE AND S. ZHANG, New results on quadratic minimization, SIAM J. Optim., 14 (2003), pp. 245-267.
36. Y. YUAN, On a subproblem of trust region algorithms for constrained optimization, Math. Programming, 47 (1990), pp. 53-63.
37. Y. ZHANG, Computing a Celis-Dennis-Tapia trust-region step for equality constrained optimization, Math. Programming, 55 (1992), pp. 109-124.