Some Complexity Results for Polynomial Ideals

In this paper, we survey some of our new results on the complexity of a number of problems related to polynomial ideals. We consider multivariate polynomials over some ring, like the integers or the rationals. For instance, a polynomial ideal membership problem is a (w+ 1)-tupleP= (f,g₁,g₂, …g_wwherefand theg_iare multivariate polynomials, and the problem is to determine whetherfis in the ideal generated by theg_i. For polynomials over the integers or rationals, this problem is known to be exponential space complete. We discuss further complexity results for problems related to polynomial ideals, like the word and subword problems for commutative semigroups, a quantitative version of Hilbert's Nullstellensatz in a complexity theoretic version, and problems concerning the computation of reduced polynomials and Gröbner bases.     

Complexity Results for Some Global Optimization Problems

We discuss the complexity of a class of highly structured global optimization problems, namely the maximization of separable functions, with each one-dimensional component convex and nondecreasing, over polytopes defined by a 0-1 constraint matrix with at most two variables involved in each constraint. In particular, we prove some inapproximability and approximability results.     

Analyzing the Complexity of Finding Good Neighborhood Functions for Local Search Algorithms

A drawback to using local search algorithms to address NP-hard discrete optimization problems is that many neighborhood functions have an exponential number of local optima that are not global optima (termed L-locals). A neighborhood function η is said to be stable if the number of L-locals is polynomial. A stable neighborhood function ensures that the number of L-locals does not grow too large as the instance size increases and results in improved performance for many local search algorithms. This paper studies the complexity of stable neighborhood functions for NP-hard discrete optimization problems by introducing neighborhood transformations. Neighborhood transformations between discrete optimization problems consist of a transformation of problem instances and a corresponding transformation of solutions that preserves the ordering imposed by the objective function values. In this paper, MAX Weighted Boolean SAT (MWBS), MAX Clause Weighted SAT (MCWS), and Zero-One Integer Programming (ZOIP) are shown to be NPO-complete with respect to neighborhood transformations. Therefore, if MWBS, MCWS, or ZOIP has a stable neighborhood function, then every problem in NPO has a stable neighborhood function. These results demonstrate the difficulty of finding effective neighborhood functions for NP-hard discrete optimization problems.This research is supported in part by the Air Force Office of Scientific Research (F49620-01-1-0007, FA9550-04-1-0110).     

关于p叶解析函数的若干结果

得到了ａ级ｐ叶星象函数的一个判别定理，给出了近于凸函数和单叶性的一些结果。     

关于解析算子值函数的几个结果

关于复值解析函数Riesz—Dunford积分的Ky Fan定理由[1]推广到算子值解析函数,由此函数论中的很多定理得到了推广.本文的目的在于改进[1]中的结果,得到了较弱条件下的Pick定理,从而推广了[2]中的Julia引理,并简化了其证明过程.     

调和单叶函数族的一些结果

借用著名的Ruscheweyh导数,引入了一类单叶保向调和函数族. 通过建立极值理论,得到了关联该族的最优系数边界、最优增长定理和最优偏差定理. 同时,给出了该族与先前已有调和函数族之间的转换半径. 最后,讨论了基于该族的修正哈达玛乘积结果.     

Finding a Cluster of Zeros of Univariate Polynomials

A method to compute an accurate approximation for a zero cluster of a complex univariate polynomial is presented. The theoretical background on which this method is based deals with homotopy, Newton's method, and Rouché's theorem. First the homotopy method provides a point close to the zero cluster. Next the analysis of the behaviour of the Newton method in the neighbourhood of a zero cluster gives the number of zeros in this cluster. In this case, it is sufficient to know three points of the Newton sequence in order to generate an open disk susceptible to contain all the zeros of the cluster. Finally, an inclusion test based on a punctual version of the Rouché theorem validates the previous step. A specific implementation of this algorithm is given. Numerical experiments illustrate how this method works and some figures are displayed.     

Approximation in p-Norm of Univariate Concave Functions

We derive worst-case bounds, with respect to the L ^p norm, on the error achieved by algorithms aimed at approximating a concave function of a single variable, through the evaluation of the function and its subgradient at a fixed number of points to be determined. We prove that, for p larger than 1, adaptive algorithms outperform passive ones. Next, for the uniform norm, we propose an improvement of the Sandwich algorithm, based on a dynamic programming formulation of the problem.     

On the Complexity of Multivalued Logic Functions over Some Infinite Basis

Under study is the complexity of the realization of k-valued logic functions (k ≥ 3) by logic circuits in the infinite basis consisting of the Post negation (i.e., the function (x + 1) mod k) and all monotone functions. The complexity of the circuit is the total number of elements of this circuit. For an arbitrary function f, we find the lower and upper bounds of complexity, which differ from one another at most by 1 and have the form 3 log₃(d(f)+ 1)+O(1), where d(f) is the maximal number of the decrease of the value of f taken over all increasing chains of tuples of values of the variables. We find the exact value of the corresponding Shannon function which characterizes the complexity of the most complex function of a given number of variables.     

A Method for Converting a Class of Univariate Functions into d.c. Functions

D.c. functions are functions that can be expressed as the sum of a concave function and a convex function (or as the difference of two convex functions). In this paper, we extend the class of univariate functions that can be represented as d.c. functions. This expanded class is very broad including a large number of nonlinear and/or nonsmooth univariate functions. In addition, the procedure specifies explicitly the functional and numerical forms of the concave and convex functions that comprise the d.c. representation of the univariate functions. The procedure is illustrated using two numerical examples. Extensions of the conversion procedure for discontinuous univariate functions is also discussed.     

Convexity of Products of Univariate Functions and Convexification Transformations for Geometric Programming

We investigate the characteristics that have to be possessed by a functional mapping f:ℝ↦ℝ so that it is suitable to be employed in a variable transformation of the type x→f(y) in the convexification of posynomials. We study first the bilinear product of univariate functions f ₁(y ₁), f ₂(y ₂) and, based on convexity analysis, we derive sufficient conditions for these two functions so that ℱ₂(y ₁,y ₂)=f ₁(y ₁)f ₂(y ₂) is convex for all (y ₁,y ₂) in some box domain. We then prove that these conditions suffice for the general case of products of univariate functions; that is, they are sufficient conditions for every f _i (y _i ), i=1,2,…,n, so as ℱ _n (y ₁,y ₂,…,y _n )=∏ _i=1 ⁿ f _i (y _i ) to be convex. In order to address the transformation of variables that are exponentiated to some power κ≠1, we investigate under which further conditions would the function (f) ^κ be also suitable. The results provide rigorous reasoning on why transformations that have already appeared in the literature, like the exponential or reciprocal, work properly in convexifying posynomial programs. Furthermore, a useful contribution is in devising other transformation schemes that have the potential to work better with a particular formulation. Finally, the results can be used to infer the convexity of multivariate functions that can be expressed as products of univariate factors, through conditions on these factors on an individual basis. The authors gratefully acknowledge support from the National Science Foundation.     

亚纯函数的n阶导数分享1值点的唯一性的一些结果

In this paper, we consider the problem of the uniqueness for meromorphic functions whose n-th derivatives share the same 1-points. The results in this paper are different from all of theorems given by H X Yi and C C Yang and other authors.     

Univariate Global Optimization with Multiextremal Non-Differentiable Constraints Without Penalty Functions

This paper proposes a new algorithm for solving constrained global optimization problems where both the objective function and constraints are one-dimensional non-differentiable multiextremal Lipschitz functions. Multiextremal constraints can lead to complex feasible regions being collections of isolated points and intervals having positive lengths. The case is considered where the order the constraints are evaluated is fixed by the nature of the problem and a constraint i is defined only over the set where the constraint i−1 is satisfied. The objective function is defined only over the set where all the constraints are satisfied. In contrast to traditional approaches, the new algorithm does not use any additional parameter or variable. All the constraints are not evaluated during every iteration of the algorithm providing a significant acceleration of the search. The new algorithm either finds lower and upper bounds for the global optimum or establishes that the problem is infeasible. Convergence properties and numerical experiments showing a nice performance of the new method in comparison with the penalty approach are given. This research was supported by the following grants: FIRB RBNE01WBBB, FIRB RBAU01JYPN, and RFBR 04-01-00455-a. The author thanks Prof. D. Grimaldi for proposing the application discussed in the paper. An erratum to this article is available at .     

关于亚纯函数唯一性象集的一些进一步结果

This paper studies the problem of uniqueness of meromorphic functions which share three common sets with weight, and improves a result of M. L. Fang.     

偶函数在零点处导数为零的六种解法

以一道证明偶函数在零点处导数为零的习题为例,分别使用链式法则、导数定义,带Peano型余项的泰勒公式等知识,给出6种解法,旨在培养学生分析问题和解决问题的能力。     

Efficient Hybrid Algorithms for Finding Zeros of Convex Functions

We consider two hybrid algorithms for finding an ϵ-approximation of a root of a convex real function that is twice differentiable and satisfies a certain growth condition on the intervial [0, R]. The first algorithm combines a binary search procedure with Newton′s method. The binary search produces an interval contained in the region of quadratic convergence of Newton′s method. The computational cost of the binary search, as well as the computational cost of Newton′s method, is of order O(log log(R/ϵ)). The second algorithm combines a binary search with the secant method in a similar fashion. This results in a lower overall computational cost when the cost of evaluating the derivative is more than .44042 of the cost of evaluating the function. Our results generalize same recent results of Ye.