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.
Mots-clés : 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}, publisher = {EDP-Sciences}, volume = {15}, number = {2}, year = {2009}, doi = {10.1051/cocv:2008040}, mrnumber = {2513094}, language = {en}, url = {http://archive.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 - http://archive.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 http://archive.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. http://archive.numdam.org/articles/10.1051/cocv:2008040/
[1] Geometric problems on maxima and minima. Birkhäuser (2006). | MR | Zbl
, and ,[2] The geometry of point particles. Proc. R. Soc. London A 458 (2002) 1089-1115. | MR | Zbl
and ,[3] Polyhedra in physics, chemistry and geometry. Milan J. Math. 71 (2003) 33-58. | MR | Zbl
and ,[4] Mixed discriminants of positive semidefinite matrices. Linear Algebra Appl. 126 (1989) 107-124. | MR | Zbl
,[5] Épreuve de mathématiques générales du concours d'agrégation de mathématiques 1980. Revue de Mathématiques Spéciales (1980-1981) 220-230. | MR
,[6] Robust solutions of uncertain quadratic and conic-quadratic problems. SIAM J. Optim. 13 (2002) 535-560. | MR | Zbl
, and ,[7] Estimation of Fekete points. J. Comput. Phys. 225 (2007) 2354-2376. | MR | Zbl
, , and ,[8] Monotonic converging variational approximations to the functional integrals in quantum statistical mechanics. J. Math. Phys. 16 (1975) 2318-2325. | MR | Zbl
, and ,[9] Matrix analysis. Springer (1997). | MR | Zbl
,[10] Polynomials and multilinear mappings in topological vector spaces. Studia Math. 39 (1971) 59-76. | MR | Zbl
and ,[11] An update on Minc's survey of open problems involving permanents. Linear Algebra Appl. 403 (2005) 314-342. | MR | Zbl
and ,[12] Unsolved problems in geometry. Springer-verlag (1991). | MR | Zbl
, and ,[13] On the S-procedure and some variants. Math. Meth. Oper. Res. 64 (2006) 55-77. | MR | Zbl
and ,[14] An improved probability bound for the approximate S-lemma. Oper. Res. Lett. 35 (2007) 743-746. | MR | Zbl
, and ,[15] A hyper-geometric approach to the BMV-conjecture. Monatshefte Math. 146 (2005) 179-201. | MR | Zbl
, and ,[16] Essentially Hermitian matrices revisited. Electronic J. Linear Algebra 15 (2006) 285-296. | MR | Zbl
,[17] 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. | MR | Zbl
,[18] Proof of the Van der Waerden conjecture. Siberian Math. J. 22 (1982) 854-859. | Zbl
,[19] Normal matrices: an update. Linear Algebra Appl. 285 (1998) 291-303. | MR | Zbl
and ,[20] A proof of the Van der Waerden conjecture on the permanent of a doubly stochastic matrix. Mat. Zametki 29 (1981) 931-938 (in Russian). | MR | Zbl
,[21] Perturbation of Wigner matrices and a conjecture. Proc. Amer. Math. Soc. 131 (2003) 1981-1988. | MR | Zbl
and ,[22] Normal matrices. Linear Algebra Appl. 87 (1987) 213-225. | MR | Zbl
, , and ,[23] The Van der Waerden conjecture for mixed discriminants. Adv. Math. 200 (2006) 435-454. | MR | Zbl
,[24] A proof of hyperbolic Van der Waerden conjecture: the right generalization is the ultimate simplification. Preprint (2006). | MR
,[25] Proof of the cases of the Lieb-Seiringer formulation of the Bessis-Moussa-Villani conjecture. J. Stat. Phys. 127 (2007) 1167-1171. | MR | Zbl
,[26] A generalization of a theorem of Fenchel. Proceedings of the American Mathematical Society 2 (1951) 589-593. | MR | Zbl
and ,[27] Trace functions as Laplace transforms. J. Math. Phys. 47 (2006) 043504. | MR | Zbl
,[28] Discretizing manifolds via minimum energy points. Notices Amer. Math. Soc. 51 (2004) 1186-1194. | MR | Zbl
and ,[29] 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
, , and ,[30] Advances on the Bessis-Moussa-Villani trace conjecture. Linear Algebra Appl. 426 (2007) 130-142. | MR | Zbl
,[31] On the positivity of the coefficients of a certain polynomial defined by two positive definite matrices. J. Statist. Phys. 118 (2005) 781-789. | MR | Zbl
and ,[32] Potpourri of conjectures and open questions in Nonlinear analysis and Optimization. SIAM Review 49 (2007) 255-273. | MR | Zbl
,[33] Convex Analysis and Minimization Algorithms I, Grundlehren der mathematischen Wissenschaften 305. Springer-Verlag (1993); 2nd edition in 1996. | MR | Zbl
and ,[34] On the product of sign vectors and unit vectors. Combinatorica 12 (1992) 303-316. | MR | Zbl
and ,[35] Matrix analysis. Cambridge University Press (1985). | MR | Zbl
and ,[36] A point balance algorithm for the spherical code problem. J. Global Optim. 19 (2001) 329-344. | MR | Zbl
, and ,[37] Eigenvalues of words in two positive definite letters. SIAM J. Matrix Anal. Appl. 23 (2002) 916-928. | MR | Zbl
and ,[38] Principal minor sums of . Linear Algebra Appl. 411 (2005) 386-389. | MR | Zbl
, , and ,[39] Le chasseur perdu dans la forêt : un problème de géométrie plane. Elem. Math. 35 (1980) 1-14. | MR | Zbl
,[40] A permanent inequality. Amer. Math. Monthly 88 (1981) 731-740. | MR | Zbl
,[41] Asymptotics for minimal discrete energy on the sphere. Trans. Amer. Math. Soc. 350 (1998) 523-538. | MR | Zbl
and ,[42] The Van der Waerden conjecture: two soviet solutions. Notices Amer. Math. Soc. 29 (1982) 130-133.
,[43] Equivalent forms of the Bessis-Moussa-Villani conjecture. J. Statist. Phys. 115 (2004) 185-190. | MR | Zbl
and ,[44] On the minimum of the permanent of a doubly stochastic matrix. Duke Math. J. 26 (1959) 61-72. | MR | Zbl
and ,[45] Permanents, Encyclopedia of Mathematics and its Applications 6. Addison-Wesley, Reading, Mass (1978). | MR | Zbl
,[46] Bounding the ground-sate energy of a many-body system with the differential method. Nuclear Phys. A 765 (2006) 319-341.
,[47] Upper and lower bounds for an eigenvalue associated with a positive eigenvector J. Math. Phys. 47 (2006) 022109. | MR | Zbl
,[48] On the representation of as a Laplace transform. Rev. Math. Phy. 12 (2000) 621-655. | MR | Zbl
,[49] When least is best. Princeton University Press (2004). | MR | Zbl
,[50] Interior-point polynomial algorithms in convex programming. SIAM Studies in Applied Mathematics (1994). | MR | Zbl
and ,[51] Maxima and minima without calculus. Reprinted by the Mathematical Association of America (2006). | Zbl
,[52] Globally optimized spherical point arrangements: model variants and illustrative results. Ann. Oper. Res. 104 (2001) 213-230. | MR | Zbl
,[53] Minimal discrete energy on the sphere. Math. Res. Lett. 1 (1994) 647-662. | MR | Zbl
, and ,[54] Distributing many points on the sphere. Math. Intelligencer 19 (1997) 5-11. | MR | Zbl
and ,[55] Mathematical problems for the next century. Math. Intelligencer 20 (1998) 7-15. | MR | Zbl
,[56] Finding elliptic Fekete points sets: two numerical approaches. J. Comput. Appl. Math. 130 (2001) 205-216. | MR | Zbl
, and ,[57] 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] Notes on Egorychev's proof of the Van der Waerden conjecture. Linear Algebra Appl. 39 (1981) 1-8. | MR | Zbl
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