Characterizations of strict local minima and necessary conditions for weak sharp minima

In nonlinear programming, sufficient conditions of orderm usually identify a special type of local minimizer, here termed a strict local minimizer of orderm. In this paper, it is demonstrated that, if a constraint qualification is satisfied, standard sufficient conditions often characterize this special sort of minimizer. The first- and second-order cases are treated in detail. Necessary conditions for weak sharp local minima of orderm, a larger class of local minima, are also presented.This paper was completed during a sabbatical leave at the University of Waterloo. The author is grateful for the help and support of the Department of Combinatorics and Optimization. The helpful comments of the referees are also appreciated.     

Existence and Lipschitz regularity for minima

We prove the existence, uniqueness and Lipschitz regularity of the minima of the integral functional

on ( ) for a class of integrands that are convex in and for boundary data satisfying some barrier conditions. We do not impose regularity or growth assumptions on .

     

HOMOGENIZATION PROBLEMS OF PARABOLIC MINIMA

ＨＯＭＯＧＥＮＩＺＡＴＩＯＮＰＲＯＢＬＥＭＳＯＦＰＡＲＡＢＯＬＩＣＭＩＮＩＭＡＪＩＡＮＨＵＡＩＹＵ（简怀玉）（ＤｅｐａｒｔｍｅｎｔｏｆＡｐｐｌｉｅｄＭａｔｈｅｍａｔｉｅｓ，ＴｓｉｎｇｈｕａＵｎｉｖｅｒｓｉｔｙａｎｄＩｎｓｔｉｔｕｔｅｏｆＭａｔｈｅｍａ...     

Non-differentiable functionals and singular sets of minima

We provide bounds for the Hausdorff dimension of the singular set of minima of functionals of the type ${\int }_{\Omega }F(x\text{,}v\text{,}Dv)$, where F is only Hölder continuous with respect to the variables $(x\text{,}v)$. To cite this article: J. Kristensen, G. Mingione, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Klein polyhedra and relative minima of lattices

We prove that in ?³, the relative minima of almost any lattice belong to the surface of the corresponding Klein polyhedron. We also prove, for almost any lattice in ?³, that the set of relative minima with nonnegative coordinates coincides with the union of the set of extremal points of the Klein polyhedron and a set of special points belonging to the triangular faces of the Klein polyhedron.     

Isolated periodic minima are unstable

A classical result, studied, among others, by Carathéodory [C. Carathéodory, Calculus of Variations and Partial Differential Equations of the First Order, Chelsea, New York, 1989], says that, at least generically, periodic minimizers are hyperbolic, and consequently, unstable as solutions of the associated Euler–Lagrange equation. A new version of this fact, also valid in the nonhyperbolic case, is given.     

Operators that attain their minima

In this paper we study the theory of operators on complex Hilbert spaces, which attain theirminima in the unit sphere. We prove someimportant results concerning the characterization of the N*, and also AN* operators, see respectively Definition 1.1 and Definition 1.4. The injective property plays an important role in these operators, and shall be established by these classes.     

Comparison of tests for non-parametric hypotheses

1. Summary The main object of this paper is to find a criterion for comparison of two tests for non-parametric hypotheses, taking advantage of the qualitative information that may exist. After a detailed analysis of the problem and some earlier suggestins for its solution (sections 2–4), a criterion is suggested in section 5. In order to apply it to a concrete case, a location problem is specified in section 6. The rank tests to be compared are analyzed in section 7, and the comparison by way of the criterion is carried out in section 8. It turns out that sign tests, sometimes slightly modified, are very often optimal according to the criterion used.     

On successive minima and the Absolute Siegel's Lemma

We prove a new bound on the product of the nth successive minimum of an automorphism of GL(N,kA) and the (N−n+1)th successive minimum of its dual, extending a classical inequality of K. Mahler for polar lattices to the geometry of numbers over the adèles. As a corollary we derive the Absolute Siegel's Lemma in as stated by D. Roy and J.L. Thunder.     

Principal component decomposition of non-parametric tests

Summary Let denote an arbitrary non-parametric unbiased test for a Gaussian shift given by an infinite dimensional parameter space. Then it is shown that the curvature of its power function has a principal component decomposition based on a Hilbert-Schmidt operator. Thus every test has reasonable curvature only for a finite number of orthogonal directions of alternatives. As application the two-sided Kolmogorov-Smirnov goodnessof-fit test is treated. We obtain lower bounds for their local asymptotic relative efficiency. They converge to one as 0 for the directionh ₀(u)=sign(2u–1) of the gradient of the median test. These results are analogous to earlier results of Hájek and idák for one-sided Kolmogorov-Smirnov tests.     

Local minima of the Epstein zeta-function

Translated from Matematicheskie Zametki, Vol. 45, No. 1, pp. 123–131, January, 1989.     

Asymptotic behaviour of high Gaussian minima

We investigate what happens when an entire sample path of a smooth Gaussian process on a compact interval lies above a high level. Specifically, we determine the precise asymptotic probability of such an event, the extent to which the high level is exceeded, the conditional shape of the process above the high level, and the location of the minimum of the process given that the sample path is above a high level.     

On the convergence of constrained minima

We characterize the convergence of values and Lagrange multipliers for minimum problems of perturbed quadratic functionals in Banach spaces subject to a fixed linear operator constraint with finite-dimensional range. We obtain sufficient conditions for the convergence of the solutions. Examples show the different behaviour of the continuous dependence problem for the analogous unconstrained minimizations, where the gamma-type variational convergences are relevant. Such convergence conditions are extended here to the con strained problems.