A new series of conjectures and open questions in optimization and matrix analysis

ESAIM - Control, Optimisation and Calculus of Variations

Volumn 15, Issue 2, 2009, Pages 454-470

  Hiriart Urruty, Jean Baptiste ^a  

^a INSTITUT DE MATHÉMATIQUES DE TOULOUSE   (France)

Author keywords

Bistochastic matrices; Convex sets; Energy functions; Global optimization; Normal matrices; Permanent function; Positive (semi)definite matrices; Variational problems

Indexed keywords

BISTOCHASTIC MATRICES; CONVEX SETS; ENERGY FUNCTIONS; NORMAL MATRICES; PERMANENT FUNCTION; POSITIVE (SEMI)DEFINITE MATRICES; VARIATIONAL PROBLEMS;

GLOBAL OPTIMIZATION; NONLINEAR ANALYSIS; OPTIMIZATION; SET THEORY;

CONVEX OPTIMIZATION;

EID: 65349132536     PISSN: 12928119     EISSN: 12623377     Source Type: Journal    
DOI: 10.1051/cocv:2008040     Document Type: Article

References (58)

1. T. Andreescu, O. Mushkarov and L. Stoyanov, Geometric problems on maxima and minima. Birkhäuser (2006).
2. M. Atiyah and P. Sutcliffe, The geometry of point particles. Proc. R. Soc. London A 458 (2002) 1089-1115.
3. M. Atiyah and P. Sutcliffe, Polyhedra in physics, chemistry and geometry. Milan J. Math. 71 (2003) 33-58.
4. R. Bapat, Mixed discriminants of positive semidefinite matrices. Linear Algebra Appl. 126 (1989) 107-124.
5. M. Bayart, épreuve de mathématiques générales du concours d'agrégation de mathématiques 1980. Revue de Mathématiques Spéciales (1980-1981) 220-230.
6. A. Ben Tal, A. Nemirovski and C. Roos, Robust solutions of uncertain quadratic and conic-quadratic problems. SIAM J. Optim. 13 (2002) 535-560.
7. E. Bendito, A. Carmona, A.M. Encinas and J.M. Gesto, Estimation of Fekete points. J. Comput. Phys. 225 (2007) 2354-2376.
8. D. Bessis, P. Moussa and M. Villani, Monotonic converging variational approximations to the functional integrals in quantum statistical mechanics. J. Math. Phys. 16 (1975) 2318-2325.
9. R. Bhatia, Matrix analysis. Springer (1997).
10. J. Bochnak and J. Siciak, Polynomials and multilinear mappings in topological vector spaces. Studia Math. 39 (1971) 59-76.
11. G-S. Cheon and I.M. Wanless, An update on Minc's survey of open problems involving permanents. Linear Algebra Appl. 403 (2005) 314-342.
12. H.T. Croft, K.J. Falconer and R.K. Guy, Unsolved problems in geometry. Springer-verlag (1991).
13. K. Derinkuyu and M. Pinar, On the S-procedure and some variants. Math. Meth. Oper. Res. 64 (2006) 55-77.
14. K. Derinkuyu, M. Pinar and A. Camci, An improved probability bound for the approximate S-lemma. Oper. Res. Lett. 35 (2007) 743-746.
15. M. Drmota, W. Schachermayer and J. Teichmann, A hyper-geometric approach to the BMV-conjecture. Monatshefte Math. 146 (2005) 179-201.
16. S.W. Drury, Essentially Hermitian matrices revisited. Electronic J. Linear Algebra 15 (2006) 285-296.
17. G.P. Egorychev, The solution of Van der Waerden's problem for permanents. Dokl. Akad. Sci. SSSR 258 (1981) 1041-1044 (in Russian), Adv. Math. 42 (1981) 299-305.
18. G.P. Egorychev, Proof of the Van der Waerden conjecture. Siberian Math. J. 22 (1982) 854-859.
19. L. Elsner and K.D. Ikramov, Normal matrices: an update. Linear Algebra Appl. 285 (1998) 291-303.
20. D.I. Falikman, A proof of the Van der Waerden conjecture on the permanent of a doubly stochastic matrix. Mat. Zametki 29 (1981) 931-938 (in Russian).
21. M. Fannes and D. Petz, Perturbation of Wigner matrices and a conjecture. Proc. Amer. Math. Soc. 131 (2003) 1981-1988.
22. R. Grone, C.R. Johnson, E.M. Sa and H. Wolkowicz, Normal matrices. Linear Algebra Appl. 87 (1987) 213-225.
23. L. Gurvits, The Van der Waerden conjecture for mixed discriminants. Adv. Math. 200 (2006) 435-454.
24. L. Gurvits, A proof of hyperbolic Van der Waerden conjecture: the right generalization is the ultimate simplification. Preprint (2006).
25. D. Hägele, Proof of the cases p - 7 of the Lieb-Seiringer formulation of the Bessis-Moussa-Villani conjecture. J. Stat. Phys. 127 (2007) 1167-1171.
26. O. Hanner and H. Radstrom, A generalization of a theorem of Fenchel. Proceedings of the American Mathematical Society 2 (1951) 589-593.
27. F. Hansen, Trace functions as Laplace transforms. J. Math. Phys. 47 (2006) 043504.
28. D.P. Hardin and E.B. Saff, Discretizing manifolds via minimum energy points. Notices Amer. Math. Soc. 51 (2004) 1186-1194.
29. S. He, Z.-Q. Luo, J. Nie and S. Zhang, Semidefinite relaxation bounds for indefinite homogeneous quadratic optimization. Technical report, Department of systems engineering and engineering management, the Chinese University of Hong-Kong (2007).
30. C. Hillar, Advances on the Bessis-Moussa-Villani trace conjecture. Linear Algebra Appl. 426 (2007) 130-142.
31. C. Hillar and C.R. Johnson, On the positivity of the coefficients of a certain polynomial defined by two positive definite matrices. J. Statist. Phys. 118 (2005) 781-789.
32. J.-B. Hiriart-Urruty, Potpourri of conjectures and open questions in Nonlinear analysis and Optimization. SIAM Review 49 (2007) 255-273.
33. J.-B. Hiriart-Urruty and C. Lemaréchal, Convex Analysis and Minimization Algorithms I, Grundlehren der mathematischen Wissenschaften 305. Springer-Verlag (1993); 2nd edition in 1996.
34. R. Holzman and D.J. Kleitman, On the product of sign vectors and unit vectors. Combinatorica 12 (1992) 303-316.
35. R.A. Horn and C.R. Johnson, Matrix analysis. Cambridge University Press (1985).
36. H.-X. Huang, P. Pardalos and Z.-J. Shen, A point balance algorithm for the spherical code problem. J. Global Optim. 19 (2001) 329-344.
37. C.R. Johnson and C.J. Hillar, Eigenvalues of words in two positive definite letters. SIAM J. Matrix Anal. Appl. 23 (2002) 916-928.
38. C.R. Johnson, S. Leichenauer, P. McNamara and R. Costas, Principal minor sums of (A + tB)m. Linear Algebra Appl. 411 (2005) 386-389.
39. H. Joris, Le chasseur perdu dans la forêt : un problème de géométrie plane. Elem. Math. 35 (1980) 1-14.
40. D. Knuth, A permanent inequality. Amer. Math. Monthly 88 (1981) 731-740.
41. A.B.J. Kuijlaars and E.B. Saff, Asymptotics for minimal discrete energy on the sphere. Trans. Amer. Math. Soc. 350 (1998) 523-538.
42. J.C. Lagarias, The Van der Waerden conjecture: two soviet solutions. Notices Amer. Math. Soc. 29 (1982) 130-133.
43. E.H. Lieb and R. Seiringer, Equivalent forms of the Bessis-Moussa-Villani conjecture. J. Statist. Phys. 115 (2004) 185-190.
44. M. Marcus and M. Newman, On the minimum of the permanent of a doubly stochastic matrix. Duke Math. J. 26 (1959) 61-72.
45. H. Minc, Permanents, Encyclopedia of Mathematics and its Applications 6. Addison-Wesley, Reading, Mass (1978).
46. A. Mouchet, Bounding the ground-sate energy of a many-body system with the differential method. Nuclear Phys. A 765 (2006) 319-341.
47. A. Mouchet, Upper and lower bounds for an eigenvalue associated with a positive eigenvector J. Math. Phys. 47 (2006) 022109.
48. A-λ B)] as a Laplace transform. Rev. Math. Phy. 12 (2000) 621-655.
49. P.J. Nahin, When least is best. Princeton University Press (2004).
50. Y. Nesterov and A. Nemirovski, Interior-point polynomial algorithms in convex programming. SIAM Studies in Applied Mathematics (1994).
51. D. Niven, Maxima and minima without calculus. Reprinted by the Mathematical Association of America (2006).
52. J.D. Pinter, Globally optimized spherical point arrangements: model variants and illustrative results. Ann. Oper. Res. 104 (2001) 213-230.
53. E.A. Rakhmanov, E.B. Saff and Y. Zhou, Minimal discrete energy on the sphere. Math. Res. Lett. 1 (1994) 647-662.
54. E.B. Saff and A.B.J. Kuijlaars, Distributing many points on the sphere. Math. Intelligencer 19 (1997) 5-11.
55. S. Smale, Mathematical problems for the next century. Math. Intelligencer 20 (1998) 7-15.
56. W.J.H. Stortelder, J.J.B. de Swart and J.D. Pinter, Finding elliptic Fekete points sets: two numerical approaches. J. Comput. Appl. Math. 130 (2001) 205-216.
57. P.L. Takouda, Problèmes d'approximation linéaires coniques : Approches par projections et via Optimisation sous contraintes de semidéfinie positivité. Ph.D. thesis, Paul Sabatier University, Toulouse, France (2003).
58. J.H. Van Lint, Notes on Egorychev's proof of the Van der Waerden conjecture. Linear Algebra Appl. 39 (1981) 1-8.