A convex optimization approach for minimizing the ratio of indefinite quadratic functions over an ellipsoid

We consider the nonconvex problem (RQ) of minimizing the ratio of two nonconvex quadratic functions over a possibly degenerate ellipsoid. This formulation is motivated by the so-called regularized total least squares problem (RTLS), which is a special case of the problem’s class we study. We prove that under a certain mild assumption on the problem’s data, problem (RQ) admits an exact semidefinite programming relaxation. We then study a simple iterative procedure which is proven to converge superlinearly to a global solution of (RQ) and show that the dependency of the number of iterations on the optimality tolerance grows as . This research is partially supported by the Israel Science Foundation, ISF grant #489-06.     

A global minimization algorithm for Lipschitz functions

The global optimization problem with and f(x) satisfying the Lipschitz condition , is considered. To solve it a region-search algorithm is introduced. This combines a local minimum algorithm with a procedure that at the ith iteration finds a region S _i where the global minimum has to be searched for. Specifically, by making use of the Lipschitz condition, S _i , which is a sequence of intervals, is constructed by leaving out from S _i-1 an interval where the global minimum cannot be located. A convergence property of the algorithm is given. Further, the ratio between the measure of the initial feasible region and that of the unexplored region may be used as stop rule. Numerical experiments are carried out; these show that the algorithm works well in finding and reducing the measure of the unexplored region.     

Global behavior for Whitham type equations on a finite interval

We study the initial-boundary value problem for nonlinear nonlocal equations on a finite interval where λ > 0 and pseudodifferential operator is defined by the inverse Laplace transform. The aim of this paper is to prove the global existence of solutions to the inital-boundary value problem (0.1) and to find the main term of the asymptotic representation in the case of the large initial data.     

Number of points of a translated lattice in a domain on a multidimensional ellipsoid

We prove an asymptotic formula for the number of representations of the number m by n-ary quadratic form f which lie in a given residue class (mod a) and in a given domain on the surface . The parameters of the problem are unconstrained. For n=3 the asymptotic formula is conditional (or incomplete).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 151, pp. 176–183, 1986.     

Mathematical Properties of Optimization Problems Defined by Positively Homogeneous Functions

We consider the nonlinear programming problem

Properties of global decaying solution to the Cauchy problem of nonlinear evolution equations

We establish the global existence and decaying results for the Cauchy problem of nonlinear evolution equations,

Bifurcations of Positive and Negative Continua in Quasilinear Elliptic Eigenvalue Problems

The main result of this work is a Dancer-type bifurcation result for the quasilinear elliptic problem

Equivalence and symmetries of first order differential equations

In this article, the equivalence and symmetries of underdetermined differential equations and differential equations with deviations of the first order are considered with respect to the pseudogroup of transformations . That means, the transformed unknown function is obtained by means of the change of the independent variable and subsequent multiplication by a nonvanishing factor. Instead of the common direct calculations, we use some more advanced tools from differential geometry; however, the exposition is self-contained and only the most fundamental properties of differential forms are employed. We refer to analogous achievements in literature. In particular, the generalized higher symmetry problem involving a finite number of invariants of the kind is compared to similar results obtained by means of auxiliary functional equations.     

Stability of properly efficient points and isolated minimizers of constrained vector optimization problems

In this paper the constrained vector optimization problem mic _C f(x), g(x) ∃ − K, is considered, where and are locally Lipschitz functions and and are closed convex cones. Several solution concepts are recalled, among them the concept of a properly efficient point (p-minimizer) and an isolated minimizer (i-minimizer). On the base of certain first-order optimalitty conditions it is shown that there is a close relation between the solutions of the constrained problem and some unconstrained problem. This consideration allows to “double” the solution concepts of the given constrained problem, calling sense II optimality concepts for the constrained problem the respective solutions of the related unconstrained problem, retaining the name of sense I concepts for the originally defined optimality solutions. The paper investigates the stability properties of thep-minimizers andi-minimizers. It is shown, that thep-minimizers are stable under perturbations of the cones, while thei-minimizers are stable under perturbations both of the cones and the functions in the data set. Further, it is shown, that sense I concepts are stable under perturbations of the objective data, while sense II concepts are stable under perturbations both of the objective and the constraints. Finally, the so called structural stability is discused.     

Low regularity solution for a nonlocal perturbation of KdV equation

The Cauchy problem for a nonlocal perturbation of KdV equation is considered by the Fourier restriction norm method. Local well-posedness for initial data in and global results for data in are obtained. The second author was supported by the National Natural Science Foundation of China Grant No.10526011.     

Global regularity for almost minimizers of nonconvex variational problems

We prove some global, up to the boundary of a domain $\Omega \subset {\mathbb{R}}^{n}We prove some global, up to the boundary of a domain , continuity and Lipschitz regularity results for almost minimizers of functionals of the form The main assumption for g is that it be asymptotically convex with respect its third argument. For the continuity results, the integrand is allowed to have some discontinuous behavior with respect to its first and second arguments. For the global Lipschitz regularity result, we require g to be H?lder continuous with respect to its first two arguments.      

Categoricity in ℵ1 of sentences inL_{\omega _1 ,\omega } (Q)

We investigate the categoricity and number of non-isomorphic models in ℵ₁ of sentences in . AssumingV=L we prove that no sentence in has exactly one uncountable model. Thus partially answering problem 24 of a problem list by Friedman.     

Lipschitz regularity of the minimizers of autonomous integral functionals with discontinuous non-convex integrands of slow growth

Let be a Borelian function and let (P) be the problem of minimizing

Some results on the qualitative properties of positive solutions of quasilinear elliptic equations

We consider the Dirichlet problem in Ω with zero Dirichlet boundary conditions. We prove local summability properties of and we exploit these results to give geometric characterizations of the critical set . We extend to the case of changing sign nonlinearities some results known in the case f(s) > 0 for s > 0. Berardino Sciunzi: Supported by MURST, Project “Metodi Variazionali ed Equazioni Differenziali Non Lineari”     

Critical Nonlinear Nonlocal Equations on a Half-Line

We study nonlinear nonlocal equations on a half-line in the critical case

Unconstrained and convex polynomial approximation inC[−1,1]

Some estimates for unconstrained and convex polynomial approximation in the uniform metric are obtained. These results are given in terms of the Ditzian-Totik moduli of smoothness , ≤1 with . The construction of the approximating polynomials does not depend on λ.     

Augmentation and Modification Problems for Hermitian Matrices

We obtain necessary and sufficient conditions for the solvability of the augmentation and modification problems of order for Hermitian matrices. The augmentation problem consists in the construction of a Hermitian -matrix with a given -block in block -representation and with the prescribed eigenvalues. The modification problem consists in the construction of a Hermitian -matrix of rank not greater than so that the obtained matrix, being added to a given Hermitian -matrix , will have the required spectrum. We give an estimate for the minimal number of different eigenvalues of the solutions to these problems.     

Critical Exponents for the Heat-Conduction Equation with Nonlocal,Nonlinear Perturbations

We study the conditions for the existence and nonexistence of global in time (t > 0), nonnegative solutions of the problem

Splitting lemma at infinity and a strongly resonant problem with periodic nonlinearity

In this paper we study the following problem:with periodic nonlinearity g, where and λ₂ is the second eigenvalue of −Δ, on H ¹ ₀(B). We proved that the problem has infinitely many solutions under some additional conditions on g and h. The method we used is a new variational reduction method. Mathematics Subject Classi cation (2000) 35J20, 35J70     

Spectral order and isotonic differential operators of Laguerre–Pólya type

The spectral order on R ⁿ induces a natural partial ordering on the manifold of monic hyperbolic polynomials of degree n. We show that all differential operators of Laguerre–Pólya type preserve the spectral order. We also establish a global monotony property for infinite families of deformations of these operators parametrized by the space ℓ^∞ of real bounded sequences. As a consequence, we deduce that the monoid of linear operators that preserve averages of zero sets and hyperbolicity consists only of differential operators of Laguerre–Pólya type which are both extensive and isotonic. In particular, these results imply that any hyperbolic polynomial is the global minimum of its -orbit and that Appell polynomials are characterized by a global minimum property with respect to the spectral order.