Potpourri of Conjectures and Open Questions in Nonlinear Analysis and Optimization

We present a collection of fourteen conjectures and open problems in the fields of nonlinear analysis and optimization. These problems can be classified into three groups: problems of pure mathematical interest, problems motivated by scientific computing and applications, and problems whose solutions are known but for which we would like to know better proofs. For each problem we provide a succinct presentation, a list of appropriate references, and a view of the state of the art of the subject.

[1]  Alberto Seeger,et al.  Second Derivatives of a Convex Function and of Its Legendre-Fenchel Transformate , 1992, SIAM J. Optim..

[2]  J. Hiriart-Urruty New concepts in nondifferentiable programming , 1979 .

[3]  S. Chern,et al.  Exterior Differential Systems , 1990 .

[4]  Eugenio Calabi,et al.  Improper affine hyperspheres of convex type and a generalization of a theorem by K. Jörgens. , 1958 .

[5]  R. Tyrrell Rockafellar,et al.  Variational Analysis , 1998, Grundlehren der mathematischen Wissenschaften.

[6]  J. Hiriart-Urruty,et al.  ENSEMBLES DE TCHEBYCHEV VS ENSEMBLES CONVEXES : L'ETAT DE LA SITUATION VU VIA L'ANALYSE CONVEXE NON LISSE , 1998 .

[7]  J. Hiriart-Urruty,et al.  Fundamentals of Convex Analysis , 2004 .

[8]  W. Greub Linear Algebra , 1981 .

[9]  M. Delfour,et al.  Shape Analysis via Oriented Distance Functions , 1994 .

[10]  P. Cannarsa,et al.  Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control , 2004 .

[11]  J.-B. Hiriart-Urruty,et al.  Permanently Going Back and Forth between the ``Quadratic World'' and the ``Convexity World'' in Optimization , 2002 .

[12]  J. Cheeger A lower bound for the smallest eigenvalue of the Laplacian , 1969 .

[13]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[14]  C. Villani Topics in Optimal Transportation , 2003 .

[15]  Cristian E. Gutiérrez,et al.  The Monge―Ampère Equation , 2001 .

[16]  Jieyong Zhou,et al.  A direct proof and a generalization for a Kantorovich type inequality , 2005 .

[17]  Ilya J. Bakelman,et al.  Convex Analysis and Nonlinear Geometric Elliptic Equations , 1994 .

[18]  Luis A. Caffarelli,et al.  A Liouville theorem for solutions of the Monge–Ampère equation with periodic data , 2004 .

[19]  Giuseppe Buttazzo,et al.  On Newton’s problem of minimal resistance , 1993 .

[20]  Bernd Kawohl,et al.  A symmetry problem in the calculus of variations , 1996 .

[21]  F. Uhlig A recurring theorem about pairs of quadratic forms and extensions: a survey , 1979 .

[22]  Ya-Xiang Yuan,et al.  On a subproblem of trust region algorithms for constrained optimization , 1990, Math. Program..

[23]  B. Kawohl,et al.  CHARACTERIZATION OF CHEEGER SETS FOR CONVEX SUBSETS OF THE PLANE , 2006 .

[24]  S. Kružkov GENERALIZED SOLUTIONS OF THE HAMILTON-JACOBI EQUATIONS OF EIKONAL TYPE. I. FORMULATION OF THE PROBLEMS; EXISTENCE, UNIQUENESS AND STABILITY THEOREMS; SOME PROPERTIES OF THE SOLUTIONS , 1975 .

[25]  Vladimir Tikhomirov,et al.  Stories about maxima and minima , 1990 .

[26]  J. Hiriart-Urruty,et al.  Convex analysis and minimization algorithms , 1993 .

[27]  A. Mennucci,et al.  Hamilton—Jacobi Equations and Distance Functions on Riemannian Manifolds , 2002, math/0201296.

[28]  V. S. Balaganskii,et al.  The problem of convexity of Chebyshev sets , 1996 .

[29]  F. Deutsch Best approximation in inner product spaces , 2001 .

[30]  Ronald L. Graham,et al.  Problem #7 , 1974, SIGS.

[31]  Édouard Oudet,et al.  Minimizing within Convex Bodies Using a Convex Hull Method , 2005, SIAM J. Optim..