Equivalent Lipschitz surrogates for zero-norm and rank optimization problems

This paper proposes a mechanism to produce equivalent Lipschitz surrogates for zero-norm and rank optimization problems by means of the global exact penalty for their equivalent mathematical programs with an equilibrium constraint (MPECs). Specifically, we reformulate these combinatorial problems as equivalent MPECs by the variational characterization of the zero-norm and rank function, show that their penalized problems, yielded by moving the equilibrium constraint into the objective, are the global exact penalization, and obtain the equivalent Lipschitz surrogates by eliminating the dual variable in the global exact penalty. These surrogates, including the popular SCAD function in statistics, are also difference of two convex functions (D.C.) if the function and constraint set involved in zero-norm and rank optimization problems are convex. We illustrate an application by designing a multi-stage convex relaxation approach to the rank plus zero-norm regularized minimization problem.     

关于二个方程同解的条件及其应用

探讨ｆ［ｎ］（ｘ）＝ｘ与ｆ（ｘ）＝ｘ同解的条件，分析一维离散动力系统的周期和混沌问题、并举例说明同解条件在解方程中的应用。     

Smooth and Topological Equivalence of Functions on Surfaces

We obtain conditions under which the Morse functions defined on surfaces are smooth equivalent and functions with isolated critical (singular) points are topologically equivalent.     

An Example of two Cardinals that are Equivalent in the n-Order Logic and not Equivalent in the (n + 1)-Order Logic

It is proved that the property of two models to be equivalent in the nth order logic is definable in the (n + 1)th order logic. Basing on this fact, there is given an (nonconstructive) “example” of two n-order equivalent cardinal numbers that are not (n + 1)-order equivalent.     

辛映射与Hamilton控制系统

给定等价辛流形 ,即辛同态或形变等价的辛流形 ,研究了建立在这些辛流形上的Hamilton控制系统之间的一些性质的联系 ,诸如 (局部 )能观测性 ,强可接近性 ,(拟 )极小性等 .而且 ,利用Cort啨ｓ介绍的 (弱 )外等价系统的概念 ,给出使得两个Hamilton控制系统是辛同态的一个充分条件     

Duality for semi-definite and semi-infinite programming

In this article, we study semi-definite and semi-infinite programming problems (SDSIP), which includes semi-infinite linear programs and semi-definite programs as special cases. We establish that a uniform duality between the homogeneous (SDSIP) and its Lagrangian-type dual problem is equivalent to the closedness condition of certain cone. Moreover, this closedness condition was assured by a generalized canonically closedness condition and a Slater condition. Corresponding results for the nonhomogeneous (SDSIP) problem were obtained by transforming it into an equivalent homogeneous (SDSIP) problem.     

Equivalence of Fourth Order Boundary Value Problems and Matrix Eigenvalue Problems

We show that a class of regular self-adjoint fourth order boundary value problems (BVPs) is equivalent to a certain class of matrix problems. Conversely, for any given matrix problem in this class, there exist fourth order self-adjoint BVPs which are equivalent to the given matrix problem. Equivalent here means that they have exactly the same spectrum.     

A Longitudinal Examination of Middle School Students' Understanding of the Equal Sign and Equivalent Equations

This longitudinal study investigated (a) middle school students' understanding of the equal sign, (b) students' performance solving equivalent equations problems, and (c) changes in students' understanding and performance over time. Written assessment data were collected from 81 students at four time points over a 3-year period. At the group level, understanding and performance improved over the middle school years. However, such improvements were gradual, with many students still showing weak understanding and poor performance at the end of grade 8. More sophisticated understanding of the equal sign was associated with better performance on equivalent equations problems. At the individual level, students displayed a variety of trajectories over the middle school years in their understanding of the equal sign and in their performance on equivalent equations problems. Further, students' performance on the equivalent equations problems varied as a function of when they acquired a sophisticated understanding of the equal sign. Those who acquired a relational understanding earlier were more successful at solving the equivalent equations problems at the end of grade 8.     

Local saddle point and a class of convexification methods for nonconvex optimization problems

A class of general transformation methods are proposed to convert a nonconvex optimization problem to another equivalent problem. It is shown that under certain assumptions the existence of a local saddle point or local convexity of the Lagrangian function of the equivalent problem (EP) can be guaranteed. Numerical experiments are given to demonstrate the main results geometrically.     

Quasi-bimonads and their representations

In this paper quasi-bimonads on a monoidal category are introduced and investigated. Quasi-bimonads generalize quasi-bialgebras to a non-braided setting. We discuss their representations and investigate the R-matrix of a quasi-bimonad. Such an R-matrix provides a new solution of the version of the Yang-Baxter equation adapted to the situation. We also introduce an equivalent relation on (quasitriangular) quasi-bimonads such that the categories of representations of two (quasitriangular) quasi-bimonads are (braided) monoidal equivalent. Finally, we discuss Drinfeld twists and Hom quasi-bialgebras.     

Comparability Invariance Results for Tolerance Orders

We prove comparability invariance results for three classes of ordered sets: bounded tolerance orders (equivalent to parallelogram orders), unit bitolerance orders (equivalent to point-core bitolerance orders) and unit tolerance orders (equivalent to 50% tolerance orders). Each proof uses a different technique and relies on the alternate characterization.     

H-矩阵和S-双对角占优矩阵

In this paper, the concept of the s-doubly diagonally dominant matrices is introduced and the properties of these matrices are discussed. With the properties of the s-doubly diagonally dominant matrices and the properties of comparison matrices, some equivalent conditions for H-matrices are presented. These conditions generalize and improve existing results about the equivalent conditions for H-matrices. Applications and examples using these new equivalent conditions are also presented, and a new inclusion region of k-multiple eigenvalues of matrices is obtained.     

Fuzzy Principles and Characterization of Trustworthiness

We show that (a) the basic principles of subdifferential calculus (various local and global fuzzy principles, multidirectional mean value theorem, extremal principle) which are shown to be equivalent for -subdifferentials are in fact equivalent for any subdifferential and (b) for -versions of -subdifferentials these principles turn out to be also equivalent to the properties of the space to be a trustworthy or a subdifferentiability space (with respect to a given subdifferential).     

Affinely equivalent Khler-Finsler metrics on a complex manifold

The purpose of the present paper is to investigate affinely equivalent Khler-Finsler metrics on a complex manifold.We give two facts (1) Projectively equivalent Khler-Finsler metrics must be affinely equivalent;(2) a Khler-Finsler metric is a Khler-Berwald metric if and only if it is affinely equivalent to a Khler metric.Furthermore,we give a formula to describe the affine equivalence of two weakly Khler-Finsler metrics.     

Formal and finite order equivalences

We show that two families of germs of real-analytic subsets in ${{\mathbb C}^{n}}$ are formally equivalent if and only if they are equivalent of any finite order. We further apply the same technique to obtain analogous statements for equivalences of real-analytic self-maps and vector fields under conjugations. On the other hand, we provide an example of two sets of germs of smooth curves that are equivalent of any finite order but not formally equivalent.     

An analytic invariant of ordinary curve singularities

It is well known that analytically equivalent ordinary plane curve singularities have projectively equivalent tangent cones. In this note we introduce an analytic invariant in order to show two non analytically equivalent ordinary 5-fold points with projectively equivalent (or equal) tangent cones.

     

Affinely equivalent K?hler-Finsler metrics on a complex manifold

The purpose of the present paper is to investigate affinely equivalent K?hler-Finsler metrics on a complex manifold. We give two facts (1) Projectively equivalent K?hler-Finsler metrics must be affinely equivalent; (2) a K?hler-Finsler metric is a K?hler-Berwald metric if and only if it is affinely equivalent to a K?hler metric. Furthermore, we give a formula to describe the affine equivalence of two weakly K?hler-Finsler metrics.     

On geodesic equivalence of Riemannian metrics and sub-Riemannian metrics on distributions of corank 1

The present paper is devoted to the problem of (local) geodesic equivalence of Riemannian metrics and sub-Riemannian metrics on generic corank 1 distributions. Using the Pontryagin maximum principle, we treat Riemannian and sub-Riemannian cases in a unified way and obtain some algebraic necessary conditions for the geodesic equivalence of (sub-)Riemannian metrics. In this way, first we obtain a new elementary proof of the classical Levi-Civita theorem on the classification of all Riemannian geodesically equivalent metrics in a neighborhood of the so-called regular (stable) point w.r.t. these metrics. Second, we prove that sub-Riemannian metrics on contact distributions are geodesically equivalent iff they are constantly proportional. Then we describe all geodesically equivalent sub-Riemannian metrics on quasi-contact distributions. Finally, we give a classification of all pairs of geodesically equivalent Riemannian metrics on a surface that are proportional at an isolated point. This is the simplest case, which was not covered by Levi-Civita’s theorem. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 21, Geometric Problems in Control Theory, 2004.     

等变序列分歧问题关于右等价群的有限决定性

研究在右等价群R(τ)的作用下,等变序列分歧问题f∈ε_(uλ)×(?)(τ)的有限决定性,得到了有限决定性的充分条件和必要条件.