Products of positive forms, linear matrix inequalities, and Hilbert 17th problem for ternary forms

European Journal of Operational Research

Volumn 157, Issue 1, 2004, Pages 39-45

  De Klerk, Etienne ^a   Pasechnik, Dmitrii V ^b  

^a UNIVERSITY OF WATERLOO   (Canada)

^b GOETHE UNIVERSITY   (Germany)

Author keywords

Global optimization; Hilbert's 17th problem; Semi definite programming; Ternary forms

Indexed keywords

ALGORITHMS; COMPUTATIONAL COMPLEXITY; GLOBAL OPTIMIZATION; LINEAR PROGRAMMING; MATHEMATICAL TECHNIQUES; PROBLEM SOLVING; THEOREM PROVING;

HILBERT'S 17TH PROBLEM; SEMI-DEFINITE PROGRAMMING; TERNARY FORMS;

POLYNOMIALS;

EID: 1842530967     PISSN: 03772217     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.ejor.2003.08.014     Document Type: Conference Paper

References (27)

1. Artin E. The Collected Papers of Emil Artin. 1965;Addison-Wesley Publishing Co., Inc. Reading, MA.
2. Blum L., Shub M., Smale S. On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions and universal machines. Bulletin of the American Mathematical Society (N.S.). 21(1):1989;1-46.
3. Choi M.D., Lam T.Y. Extremal positive semidefinite forms. Mathematische Annalen. 231(1):1977/78;1-18.
4. Choi M.D., Lam T.Y., Reznick B. Sums of squares of real polynomials. K -theory and algebraic geometry: Connections with quadratic forms and division algebras (Santa Barbara, CA, 1992):1995;103-126 American Mathematical Society, Providence, RI.
5. de Klerk E., Roos C., Terlaky T. Initialization in semidefinite programming via a self-dual, skew-symmetric embedding. OR Letters. 20:1997;213-221.
6. de Klerk E., Roos C., Terlaky T. Infeasible-start semidefinite programming algorithms via self-dual embeddings. Pardalos P.M., Wolkowicz H. Topics in Semidefinite and Interior-Point Methods. Fields Institute Communication Series. vol. 18:1998;215-236 American Mathematical Society.
7. Hilbert D. Über die Darstellung definiter Formen als Summe von Formenquadraten. Mathematische Annalen. 32:1888;342-350.
8. Hilbert D. Über ternäre definite Formen. Acta Mathematica. 17:1893;169-197.
9. Hilbert D. Mathematical problems. Bulletin of the American Mathematical Society (N.S.). 37(4):2000;407-436. (electronic), reprinted from Bulletin of the American Mathematical Society 8 (1902) 437-479.
10. Hilbert D. Mathematical problems. Bulletin of the American Mathematical Society (N.S.). 37(4):2000;407-436. (electronic), reprinted from Bulletin of the American Mathematical Society 8 (1902) 437-479.
11. Lasserre J.B. Semidefinite programming vs. LP relaxations for polynomial programming. Mathematical Methods of Operations Research. 27:2002;347-360.
12. Lasserre J.B. An explicit equivalent positive semidefinite program for nonliner 0-1 programs. SIAM Journal on Optimization. 12:2002;756-769.
13. Motzkin T.S. The arithmetic-geometric inequality. Inequalities (Proc. Sympos. Wright-Patterson Air Force Base, Ohio, 1965). 1967;205-224 Academic Press, New York.
14. Parrilo P.A. Semidefinite programming relaxations for semialgebraic problems. Mathematical Programming, Series B. 96:2003;293-320.
15. G. Pataki, A simple derivation of a facial reduction algorithm, and extended dual systems, Technical Report, Columbia University, New York, USA, 2000.
16. Porkolab L., Khachiyan L. On the complexity of semidefinite programs. Journal of Global Optimization. 10(4):1997;351-365.
17. Powers V., Reznick B. A new bound for Pólya's Theorem with applications to polynomials positive on polyhedra. Journal of Pure and Applied Algebra. 164:2001;221-229.
18. Prestel A., Delzell C.N. Positive Polynomials. 2001;Springer-Verlag, Berlin. From Hilbert's 17th problem to real algebra.
19. Rajwade A.R. Squares. 1993;Cambridge University Press, Cambridge.
20. M.V. Ramana, An algorithmic analysis of multiquadratic and semidefinite programming problems, PhD Thesis, Johns Hopkins University, 1993.
21. Ramana M.V. An exact duality theory for semidefinite programming and its complexity implications. Mathematical Programming, Series B. 77(2):1997;129-162.
22. Ramana M.V., Pardalos P.M. Semidefinite programming. Interior Point Methods of Mathematical Programming. 1996;369-398 Kluwer Academic Publishers, Dordrecht.
23. Reznick B. Sums of even powers of real linear forms. Memoirs of the American Mathematical Society. 96(463):1992;viii+155.
24. Reznick B. Uniform denominators in Hilbert's seventeenth problem. Mathematische Zeitschrift. 220(1):1995;75-97.
25. Reznick B. Some concrete aspects of Hilbert's 17th problem. Real algebraic geometry and ordered structures (Baton Rouge, LA, 1996). 2000;American Mathematical Society, Providence, RI.
26. Rockafellar R. Convex Analysis. 1970;Princeton University Press, Princeton, NJ.
27. Wolkowicz H., Saigal R., Vandenberghe L. Handbook of Semidefinite Programming. 2000;Kluwer Academic Publishers, Norwell, MA.