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

Hiriart-Urruty, Jean-Baptiste

doi:10.1051/cocv:2008040

Hiriart-Urruty, Jean-Baptiste

ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 2, pp. 454-470

Résumé

We present below a new series of conjectures and open problems in the fields of (global) Optimization and Matrix analysis, in the same spirit as our recently published paper [J.-B. Hiriart-Urruty, Potpourri of conjectures and open questions in Nonlinear analysis and Optimization. SIAM Review 49 (2007) 255-273]. With each problem come a succinct presentation, a list of specific references, and a view on the state of the art of the subject.

MR

DOI : 10.1051/cocv:2008040

Classification : 15A, 26B, 49K, 65C, 65K, 90C
Keywords: convex sets, positive (semi)definite matrices, variational problems, energy functions, global optimization, permanent function, bistochastic matrices, normal matrices

@article{COCV_2009__15_2_454_0,
     author = {Hiriart-Urruty, Jean-Baptiste},
     title = {A new series of conjectures and open questions in optimization and matrix analysis},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {454--470},
     year = {2009},
     publisher = {EDP Sciences},
     volume = {15},
     number = {2},
     doi = {10.1051/cocv:2008040},
     mrnumber = {2513094},
     language = {en},
     url = {https://www.numdam.org/articles/10.1051/cocv:2008040/}
}

TY  - JOUR
AU  - Hiriart-Urruty, Jean-Baptiste
TI  - A new series of conjectures and open questions in optimization and matrix analysis
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2009
SP  - 454
EP  - 470
VL  - 15
IS  - 2
PB  - EDP Sciences
UR  - https://www.numdam.org/articles/10.1051/cocv:2008040/
DO  - 10.1051/cocv:2008040
LA  - en
ID  - COCV_2009__15_2_454_0
ER  -

%0 Journal Article
%A Hiriart-Urruty, Jean-Baptiste
%T A new series of conjectures and open questions in optimization and matrix analysis
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2009
%P 454-470
%V 15
%N 2
%I EDP Sciences
%U https://www.numdam.org/articles/10.1051/cocv:2008040/
%R 10.1051/cocv:2008040
%G en
%F COCV_2009__15_2_454_0

Hiriart-Urruty, Jean-Baptiste. A new series of conjectures and open questions in optimization and matrix analysis. ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 2, pp. 454-470. doi: 10.1051/cocv:2008040

Bibliographie
Cité par

[1] T. Andreescu, O. Mushkarov and L. Stoyanov, Geometric problems on maxima and minima. Birkhäuser (2006). | Zbl | MR

[2] M. Atiyah and P. Sutcliffe, The geometry of point particles. Proc. R. Soc. London A 458 (2002) 1089-1115. | Zbl | MR

[3] M. Atiyah and P. Sutcliffe, Polyhedra in physics, chemistry and geometry. Milan J. Math. 71 (2003) 33-58. | Zbl | MR

[4] R. Bapat, Mixed discriminants of positive semidefinite matrices. Linear Algebra Appl. 126 (1989) 107-124. | Zbl | MR

[5] M. Bayart, Épreuve de mathématiques générales du concours d'agrégation de mathématiques 1980. Revue de Mathématiques Spéciales (1980-1981) 220-230. | MR

[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. | Zbl | MR

[7] E. Bendito, A. Carmona, A.M. Encinas and J.M. Gesto, Estimation of Fekete points. J. Comput. Phys. 225 (2007) 2354-2376. | Zbl | MR

[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. | Zbl | MR

[9] R. Bhatia, Matrix analysis. Springer (1997). | Zbl | MR

[10] J. Bochnak and J. Siciak, Polynomials and multilinear mappings in topological vector spaces. Studia Math. 39 (1971) 59-76. | Zbl | MR

[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. | Zbl | MR

[12] H.T. Croft, K.J. Falconer and R.K. Guy, Unsolved problems in geometry. Springer-verlag (1991). | Zbl | MR

[13] K. Derinkuyu and M. Pinar, On the S-procedure and some variants. Math. Meth. Oper. Res. 64 (2006) 55-77. | Zbl | MR

[14] K. Derinkuyu, M. Pinar and A. Camci, An improved probability bound for the approximate S-lemma. Oper. Res. Lett. 35 (2007) 743-746. | Zbl | MR

[15] M. Drmota, W. Schachermayer and J. Teichmann, A hyper-geometric approach to the BMV-conjecture. Monatshefte Math. 146 (2005) 179-201. | Zbl | MR

[16] S.W. Drury, Essentially Hermitian matrices revisited. Electronic J. Linear Algebra 15 (2006) 285-296. | Zbl | MR

[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. | Zbl | MR

[18] G.P. Egorychev, Proof of the Van der Waerden conjecture. Siberian Math. J. 22 (1982) 854-859. | Zbl

[19] L. Elsner and K.D. Ikramov, Normal matrices: an update. Linear Algebra Appl. 285 (1998) 291-303. | Zbl | MR

[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). | Zbl | MR

[21] M. Fannes and D. Petz, Perturbation of Wigner matrices and a conjecture. Proc. Amer. Math. Soc. 131 (2003) 1981-1988. | Zbl | MR

[22] R. Grone, C.R. Johnson, E.M. Sa and H. Wolkowicz, Normal matrices. Linear Algebra Appl. 87 (1987) 213-225. | Zbl | MR

[23] L. Gurvits, The Van der Waerden conjecture for mixed discriminants. Adv. Math. 200 (2006) 435-454. | Zbl | MR

[24] L. Gurvits, A proof of hyperbolic Van der Waerden conjecture: the right generalization is the ultimate simplification. Preprint (2006). | MR

[25] D. Hägele, Proof of the cases $p \leq 7$ of the Lieb-Seiringer formulation of the Bessis-Moussa-Villani conjecture. J. Stat. Phys. 127 (2007) 1167-1171. | Zbl | MR

[26] O. Hanner and H. Radstrom, A generalization of a theorem of Fenchel. Proceedings of the American Mathematical Society 2 (1951) 589-593. | Zbl | MR

[27] F. Hansen, Trace functions as Laplace transforms. J. Math. Phys. 47 (2006) 043504. | Zbl | MR

[28] D.P. Hardin and E.B. Saff, Discretizing manifolds via minimum energy points. Notices Amer. Math. Soc. 51 (2004) 1186-1194. | Zbl | MR

[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). | MR

[30] C. Hillar, Advances on the Bessis-Moussa-Villani trace conjecture. Linear Algebra Appl. 426 (2007) 130-142. | Zbl | MR

[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. | Zbl | MR

[32] J.-B. Hiriart-Urruty, Potpourri of conjectures and open questions in Nonlinear analysis and Optimization. SIAM Review 49 (2007) 255-273. | Zbl | MR

[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. | Zbl | MR

[34] R. Holzman and D.J. Kleitman, On the product of sign vectors and unit vectors. Combinatorica 12 (1992) 303-316. | Zbl | MR

[35] R.A. Horn and C.R. Johnson, Matrix analysis. Cambridge University Press (1985). | Zbl | MR

[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. | Zbl | MR

[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. | Zbl | MR

[38] C.R. Johnson, S. Leichenauer, P. Mcnamara and R. Costas, Principal minor sums of ${(A + t B)}^{m}$ . Linear Algebra Appl. 411 (2005) 386-389. | Zbl | MR

[39] H. Joris, Le chasseur perdu dans la forêt : un problème de géométrie plane. Elem. Math. 35 (1980) 1-14. | Zbl | MR

[40] D. Knuth, A permanent inequality. Amer. Math. Monthly 88 (1981) 731-740. | Zbl | MR

[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. | Zbl | MR

[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. | Zbl | MR

[44] M. Marcus and M. Newman, On the minimum of the permanent of a doubly stochastic matrix. Duke Math. J. 26 (1959) 61-72. | Zbl | MR

[45] H. Minc, Permanents, Encyclopedia of Mathematics and its Applications 6. Addison-Wesley, Reading, Mass (1978). | Zbl | MR

[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. | Zbl | MR

[48] P. Moussa, On the representation of $T r [e^{(A - λ B)}]$ as a Laplace transform. Rev. Math. Phy. 12 (2000) 621-655. | Zbl | MR

[49] P.J. Nahin, When least is best. Princeton University Press (2004). | Zbl | MR

[50] Y. Nesterov and A. Nemirovski, Interior-point polynomial algorithms in convex programming. SIAM Studies in Applied Mathematics (1994). | Zbl | MR

[51] D. Niven, Maxima and minima without calculus. Reprinted by the Mathematical Association of America (2006). | Zbl

[52] J.D. Pinter, Globally optimized spherical point arrangements: model variants and illustrative results. Ann. Oper. Res. 104 (2001) 213-230. | Zbl | MR

[53] E.A. Rakhmanov, E.B. Saff and Y. Zhou, Minimal discrete energy on the sphere. Math. Res. Lett. 1 (1994) 647-662. | Zbl | MR

[54] E.B. Saff and A.B.J. Kuijlaars, Distributing many points on the sphere. Math. Intelligencer 19 (1997) 5-11. | Zbl | MR

[55] S. Smale, Mathematical problems for the next century. Math. Intelligencer 20 (1998) 7-15. | Zbl | MR

[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. | Zbl | MR

[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. | Zbl | MR

Cité par Sources :