Hostname: page-component-848d4c4894-nr4z6 Total loading time: 0 Render date: 2024-05-16T02:33:27.825Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  24 June 2008

Jean-Baptiste Hiriart-Urruty
Jean-Baptiste Hiriart-Urruty*
Affiliation:
Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex 9, France. jbhu@cict.fr
Get access

Abstract

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 Review49 (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.

Keywords

Convex setspositive (semi)definite matricesvariational problemsenergy functionsglobal optimizationpermanent functionbistochastic matricesnormal matrices
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 15 , Issue 2 , April 2009 , pp. 454 - 470
Copyright
© EDP Sciences, SMAI, 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

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