Canonical dual least square method for solving general nonlinear systems of quadratic equations

Computational Optimization and Applications

Volumn 47, Issue 2, 2010, Pages 335-347

  Ruan, N ^a   Gao, David Y ^a   Jiao, Y ^a  

^a VIRGINIA POLYTECHNIC INSTITUTE AND STATE UNIVERSITY   (United States)

Author keywords

Duality theory; Global optimization; Least square method; Nonlinear systems of equations

Indexed keywords

CANONICAL DUALITY THEORIES; CONVEX OPTIMIZATION TECHNIQUES; DUAL APPROACH; DUALITY THEORY; EXISTENCE AND UNIQUENESS; GLOBAL OPTIMAL SOLUTIONS; ILLUSTRATIVE EXAMPLES; LEAST SQUARE METHODS; MAXIMIZATION PROBLEM; N-DIMENSIONAL SPACE; NON-LINEAR ALGEBRAIC SYSTEM; NONCONVEX OPTIMIZATION PROBLEM; NONCONVEX PROBLEM; NONLINEAR SYSTEMS OF EQUATIONS; QUADRATIC EQUATIONS;

CONVEX OPTIMIZATION; GLOBAL OPTIMIZATION; LEAST SQUARES APPROXIMATIONS; NONLINEAR SYSTEMS; OPTIMIZATION;

NONLINEAR EQUATIONS;

EID: 77956790297     PISSN: 09266003     EISSN: 15732894     Source Type: Journal    
DOI: 10.1007/s10589-008-9222-5     Document Type: Article

References (39)

1. P. Biswas T.C. Liang K.C. Toh T.C. Wang Y. Ye 2006 Semidefinite programming approaches for sensor network localization with noisy distance measurements IEEE Trans. Autom. Sci. Eng. 3 4 360 371 10.1109/TASE.2006.877401 (Pubitemid 44574330)
2. S.-C. Fang D.Y. Gao R.L. Shue S.Y. Wu 2008 Canonical dual approach to solving 0-1 quadratic programming problems J. Ind. Manag. Optim. 4 1 125 142 2379923
3. Floudas, C.A.: Deterministic Optimization. Theory, Methods, and Applications. Kluwer Academic, Dordrecht (2000)
4. C.A. Floudas I.G. Akrotirianakis S. Caratzoulas C.A. Meyer J. Kallrath 2005 Global optimization in the 21th century: advances and challenges Comput. Chem. Eng. 29 1185 1202 10.1016/j.compchemeng.2005.02.006 (Pubitemid 40633581)
5. C.A. Floudas V. Visweswaran 1993 A primal-relaxed dual global optimization approach J. Optim. Theory Appl. 78 2 187 225 0796.90056 10.1007/BF00939667 1236801
6. D.Y. Gao 1998 Duality, triality and complementary extremum principles in nonconvex parametric variational problems with applications IMA J. Appl. Math. 61 199 235 0923.49017 10.1093/imamat/61.3.199 1695251
7. D.Y. Gao 1999 Pure complementary energy principle and triality theory in finite elasticity Mech. Res. Commun. 26 1 31 37 0992.74013 10.1016/S0093- 6413(98)00096-2
8. D.Y. Gao 1999 General analytic solutions and complementary variational principles for large deformation nonsmooth mechanics Meccanica 34 169 198 0970.74011 1732509
9. Gao, D.Y.: Duality Principles in Nonconvex Systems: Theory, Methods and Applications. Kluwer Academic, Dordrecht (2000)
10. D.Y. Gao 2000 Canonical dual transformation method and generalized triality theory in nonsmooth global optimization J. Glob. Optim. 17 1/4 127 160 0983.74053 10.1023/A:1026537630859
11. D.Y. Gao 2000 Analytic solution and triality theory for nonconvex and nonsmooth variational problems with applications Nonlinear Anal. 42 7 1161 1193 10.1016/S0362-546X(99)00129-7 1780722
12. D.Y. Gao 2003 Perfect duality theory and complete solutions to a class of global optimization problems Optimization 52 4-5 467 493 1040.49036 10.1080/02331930310001611501 2039234 (Pubitemid 37468991)
13. Gao, D.Y.: Nonconvex semi-linear problems and canonical dual solutions. In: Gao, D.Y., Ogden, R.W. (eds.) Advances in Mechanics and Mathematics, vol. II, pp. 261-312. Kluwer Academic, Dordrecht (2003)
14. D.Y. Gao 2004 Canonical duality theory and solutions to constrained nonconvex quadratic programming J. Glob. Optim. 29 377 399 10.1023/B:JOGO. 0000048034.94449.e3
15. D.Y. Gao 2007 Solutions and optimality to box constrained nonconvex minimization problems J. Ind. Manag. Optim. 3 2 293 304 1171.90504 2329373
16. Gao, D.Y.: Duality-Mathematics, Wiley Encyclopedia of Electrical and Electronics Engineering, vol. 6, pp. 68-77 (1999). Electronic edition, 2007
17. Gao, D.Y.: Advances in canonical duality theory with applications in global optimization. In: Ierapetriou, M., Bassett, M., Pistikopoulos, S. (eds.) Proceedings of the Fifth International Conference on Foundations of Computer-Aided Process Operations (FOCAPO 2008), pp. 73-82. Omni Press (2008)
18. Gao, D.Y.: Canonical duality theory: theory, method, and applications in global optimization. Comput. Chem. (2009, to appear)
19. D.Y. Gao R.W. Ogden 2008 Closed-form solutions, extremality and nonsmoothness criteria to nonconvex variational problem in finite elasticity Z. Angew. Math. Phys. 59 3 498 517 1143.74018 10.1007/s00033-007-7047-1 2399352
20. D.Y. Gao R.W. Ogden 2008 Multiple solutions to non-convex variational problems with implications for phase transitions and numerical computation Q. J. Mech. Appl. Math. 61 4 497 522 1153.74032 10.1093/qjmam/hbn014 2448642
21. Gao, D.Y., Ruan, N.: On the solutions to quadratic minimization problems with box and integer constraints. J. Glob. Optim. (2008, to appear)
22. Gao, D.Y., Ruan, N., Sherali, H.D.: Solutions and optimality criteria for nonconvex constrained global optimization problems. J. Glob. Optim. (2008, to appear)
23. Gao, D.Y., Ruan, N., Sherali, H.D.: Canonical dual solutions for fixed cost quadratic programs (2009, to be submitted)
24. Gao, D.Y., Sherali, H.D.: Canonical Duality Theory: Connections between Nonconvex Mechanics and Global Optimization. Advances in Mechanics and Mathematics, vol. III. Springer, Berlin (2008)
25. D.Y. Gao G. Strang 1989 Geometric nonlinearity: Potential energy, complementary energy, and the gap function Quart. Appl. Math. 47 3 487 504 0691.73012 1012271
26. D.Y. Gao H.F. Yu 2008 Multi-scale modelling and canonical dual finite element method in phase transitions of solids Int. J. Solids Struct. 45 3660 3673 1169.74518 10.1016/j.ijsolstr.2007.08.027 (Pubitemid 351601771)
27. L. Grippo S. Lucidi 1991 A differentiable exact penalty function for bound constrained quadratic programming problems Optimization 22 4 557 578 0734.90063 10.1080/02331939108843700 1106542
28. C. Grosan A. Abraham 2008 A new approach for solving nonlinear equations systems IEEE Trans. Syst. Man Cybern. 38 3 698 714 10.1109/TSMCA.2008.918599 (Pubitemid 351649898)
29. P.V. Hentenryck D. Mcallester D. Kapur 1997 Solving polynomial systems using a branch and prune approach SIAM J. Numer. Anal. 34 2 797 827 0874.65039 10.1137/S0036142995281504 1442939
30. Y. Jiao Y. Chen 2004 An application of generalized linear model in production model and sequential population analysis Fish. Res. 70 367 376 10.1016/j.fishres.2004.08.027 (Pubitemid 39584283)
31. Jiao, Y., Neves, R., Jones, J.: Models and model selection uncertainty in estimating growth rates of endangered freshwater mussel populations. Can. J. Fish. Aquat. Sci. (2008, in press)
32. Kojima, M., Kim, S., Waki, H.: Sparsity in sums of squares of polynomials. Math. Program. 103(1), 45-62
33. S.F. Li A. Gupta 2006 On dual configuration forces J. Elas. 84 13 31 1103.74013 10.1007/s10659-005-9047-8 2251446 (Pubitemid 43999265)
34. K.G. Murty S.N. Kabadi 1987 Some NP-hard problems in quadratic and nonlinear programming Math. Program. 39 117 129 0637.90078 10.1007/BF02592948 916001
35. Ruan, N., Gao, D.Y., Pardalos, P.: Canonical duality approach for solving sensor network localization problem. In: Pardalos, P.M., Ye, Y., Boginski, V.L., Commander, C.W. (eds.) Sensors: Theory, Algorithms, and Applications. Springer, Berlin (2008)
36. H.D. Sherali C. Tuncbilek 1992 A global optimization for polynomial programming problem using a reformulation-linearization technique J. Glob. Optim. 2 101 112 0787.90088 10.1007/BF00121304 1266898
37. H.D. Sherali C. Tuncbilek 1997 New reformulation-linearization technique based relaxation for univariate and multivariate polynominal programming problems Oper. Res. Lett. 21 1 1 10 0885.90105 10.1016/S0167-6377(97)00013-8 1471663
38. Strongin, R.G., Sergeyev, Ya.D.: Global Optimization with Non-Convex Constraints: Sequential and Parallel Algorithms. Kluwer Academic, Dordrecht (2000)
39. H. Waki S. Kim M. Kojima M. Muramatsu 2006 Sums of squares and semidefinite programming relaxations for polynomial optimization problems with structured sparsity SIAM J. Optim. 17 1 218 242 1109.65058 10.1137/050623802 2219151