A complete characterization of the gap between convexity and sos-convexity

SIAM Journal on Optimization

Volumn 23, Issue 2, 2013, Pages 811-833

  Ahmadi, Amir Ali ^a   Parrilo, Pablo A ^b  

^a IBM T J WATSON RESEARCH CENTER   (United States)

^b Department of Electrical Engineering and Computer Science   (United States)

Author keywords

Convexity; Semidefinite programming; Sum of squares

Indexed keywords

ALGEBRAIC CONDITIONS; CONVEXITY; HOMOGENEOUS POLYNOMIALS; NON-NEGATIVE POLYNOMIALS; SEMI-DEFINITE PROGRAMMING; SUFFICIENT CONDITIONS; SUM OF SQUARES; SUMS OF SQUARES;

CHARACTERIZATION; MATHEMATICAL PROGRAMMING; POLYNOMIALS;

SYSTEMS ENGINEERING;

EID: 84880744392     PISSN: 10526234     EISSN: None     Source Type: Journal    
DOI: 10.1137/110856010     Document Type: Article

References (49)

1. A. A. Ahmadi, G. Blekherman, and P. A. Parrilo, Convex ternary quartics are sos-convex, in preparation.
2. A. A. Ahmadi, A. Olshevsky, P. A. Parrilo, and J. N. Tsitsiklis, NP-hardness of deciding convexity of quartic polynomials and related problems, Math. Program., 137 (2013), pp. 453-476.
3. A. A. Ahmadi and P. A. Parrilo, A positive definite polynomial Hessian that does not factor, in Proceedings of the 48th IEEE Conference on Decision and Control, 2009.
4. A. A. Ahmadi and P. A. Parrilo, On the equivalence of algebraic conditions for convexity and quasiconvexity of polynomials, in Proceedings of the 49th IEEE Conference on Decision and Control, 2010.
5. A. A. Ahmadi and P. A. Parrilo, A convex polynomial that is not sos-convex, Math. Program., 135 (2012), pp. 275-292.
6. E. M. Aylward, S. M. Itani, and P. A. Parrilo, Explicit SOS decomposition of univariate polynomial matrices and the Kalman-Yakubovich-Popov lemma, in Proceedings of the 46th IEEE Conference on Decision and Control, 2007.
7. G. Blekherman and C. B. Riener, Symmetric Nonnegative Forms and Sums of Squares, arXiv:1205.3102, 2012.
8. G. Blekherman, Convex forms that are not sums of squares, arXiv:0910.0656, 2009.
9. G. Blekherman, Nonnegative polynomials and sums of squares, J. AMS, 25 (2012), pp. 617-635.
10. S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, London, 2004.
11. G. Chesi and Y. S. Hung, Establishing convexity of polynomial Lyapunov functions and their sublevel sets, IEEE Trans. Automat. Control, 53 (2008), pp. 2431-2436.
12. M. D. Choi, T.-Y. Lam, and B. Reznick, Real zeros of positive semidefinite forms. I, Math. Z., 171 (1980), pp. 1-26.
13. M. D. Choi and T. Y. Lam, Extremal positive semidefinite forms, Math. Ann., 231 (1977), pp. 1-18.
14. M. D. Choi, Positive semidefinite biquadratic forms, Linear Algebra Appl., 12 (1975), pp. 95-100.
15. E. de Klerk and M. Laurent, On the Lasserre hierarchy of semidefinite programming relaxations of convex polynomial optimization problems, SIAM J. Optim., 21 (2011), pp. 824-832.
16. P. H. Diananda, On non-negative forms in real variables some or all of which are non-negative, Proc. Cambridge Philosophical Society, 58 (1962), pp. 17-25.
17. A. C. Doherty, P. A. Parrilo, and F. M. Spedalieri, Distinguishing separable and entangled states, Phys. Rev. Lett., 88 (2002), 187904.
18. K. Gatermann and P. A. Parrilo, Symmetry groups, semidefinite programs, and sums of squares, J. Pure Appl. Algebra, 192 (2004), pp. 95-128.
19. N. Gvozdenović and M. Laurent, Semidefinite bounds for the stability number of a graph via sums of squares of polynomials, Math. Program., 110 (2007), pp. 145-173.
20. J. W. Helton and J. Nie, Semidefinite representation of convex sets, Math. Program., 122 (2010), pp. 21-64.
21. D. Henrion and A. Garulli, eds., Positive Polynomials in Control, Lecture Notes in Control and Inform. Sci. 312, Springer, Berlin, 2005.
22. D. Hilbert, Über die Darstellung Definiter Formen als Summe von Formenquadraten, Math. Ann., 32 (1888), pp. 342-350.
23. M. Kojima, Sums of Squares Relaxations of Polynomial Semidefinite Programs, Research report B-397, Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, 2003.
24. J. B. Lasserre, Convexity in semialgebraic geometry and polynomial optimization, SIAM J. Optim., 19 (2008), pp. 1995-2014.
25. J. B. Lasserre, Representation of nonnegative convex polynomials, Arch. Math., 91 (2008), pp. 126-130.
26. J. B. Lasserre, Certificates of convexity for basic semi-algebraic sets, Appl. Math. Lett., 23 (2010), pp. 912-916.
27. J. Löfberg, Yalmip: A toolbox for modeling and optimization in MATLAB, in Proceedings of the CACSD Conference, 2004; also available online from http://control.ee.ethz.ch/∼joloef/yalmip.php.
28. A. Magnani, S. Lall, and S. Boyd, Tractable fitting with convex polynomials via sum of squares, in Proceedings of the 44th IEEE Conference on Decision and Control, 2005.
29. T. S. Motzkin, The arithmetic-geometric inequality, in Inequalities, Academic Press, New York, 1967, pp. 205-224.
30. K. G. Murty and S. N. Kabadi, Some NP-complete problems in quadratic and nonlinear programming, Math. Program., 39 (1987), pp. 117-129.
31. P. M. Pardalos and S. A. Vavasis, Open questions in complexity theory for numerical optimization, Math. Program., 57 (1992), pp. 337-339.
32. P. A. Parrilo and B. Sturmfels, Minimizing polynomial functions, in Algorithmic and Quantitative Real Algebraic Geometry, DIMACS Series in Discrete Math. Theoret. Comput. Sci., 60 (2003), pp. 83-99.
33. P. A. Parrilo, Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization, Ph.D. thesis, California Institute of Technology, 2000.
34. P. A. Parrilo, Semidefinite programming relaxations for semialgebraic problems, Math. Program., 96 (2003), pp. 293-320.
35. P. A. Parrilo, Polynomial games and sum of squares optimization, in Proceedings of the 45th IEEE Conference on Decision and Control, 2006.
36. A. Pfister and C. Scheiderer, An Elementary Proof of Hilbert's Theorem on Ternary Quartics, arXiv:1009.3144, 2010.
37. V. Powers, B. Reznick, C. Scheiderer, and F. Sottile, A new approach to Hilbert's theorem on ternary quartics, C. R. Math., 339 (2004), pp. 617-620.
38. S. Prajna, A. Papachristodoulou, and P. A. Parrilo, SOSTOOLS: Sum of Squares Optimization Toolbox for MATLAB, http://www.cds.caltech.edu/sostools and http://www.mit.edu/∼parrilo/sostools (2002).
39. B. Reznick, Uniform denominators in Hilbert's 17th problem, Math Z., 220 (1995), pp. 75-97.
40. B. Reznick, Some Concrete Aspects of Hilbert's 17th Problem, Contemp. Math., 253, AMS, Providence, RI, 2000, pp. 251-272.
41. B. Reznick, On Hilbert's Construction of Positive Polynomials, arXiv:0707.2156, 2007.
42. B. Reznick, Blenders, arXiv:1008.4533, 2010.
43. R. M. Robinson, Some definite polynomials which are not sums of squares of real polynomials, in Selected Questions of Algebra and Logic (Collection Dedicated to the Memory of A. I. Mal-cev), Izdat. "Nauka" Sibirsk. Otdel., Novosibirsk, 1973, pp. 264-282 (in Russian).
44. W. Rudin, Real and Complex Analysis, 3rd ed., McGraw-Hill, New York, 1987.
45. C. W. Scherer and C. W. J. Hol, Matrix sum of squares relaxations for robust semidefinite programs, Math. Program., 107 (2006), pp. 189-211.
46. N. Z. Shor, Class of global minimum bounds of polynomial functions, Cybernetics, 23 (1987), pp. 731-734.
47. J. Sturm, SeDuMi Version 1.05, http://sedumi.ie.lehigh.edu/.
48. A. L. Tits, Lecture Notes on Optimal Control, http://www.isr.umd.edu/ ∼andre/664.pdf, (2008).
49. L. Vandenberghe and S. Boyd, Semidefinite programming, SIAM Rev., 38 (1996), pp. 49-95.