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1.
We show a descent method for submodular function minimization based on an oracle for membership in base polyhedra. We assume
that for any submodular function f: ?→R on a distributive lattice ?⊆2
V
with ?,V∈? and f(?)=0 and for any vector x∈R
V
where V is a finite nonempty set, the membership oracle answers whether x belongs to the base polyhedron associated with f and that if the answer is NO, it also gives us a set Z∈? such that x(Z)>f(Z). Given a submodular function f, by invoking the membership oracle O(|V|2) times, the descent method finds a sequence of subsets Z
1,Z
2,···,Z
k
of V such that f(Z
1)>f(Z
2)>···>f(Z
k
)=min{f(Y) | Y∈?}, where k is O(|V|2). The method furnishes an alternative framework for submodular function minimization if combined with possible efficient
membership algorithms.
Received: September 9, 2001 / Accepted: October 15, 2001?Published online December 6, 2001 相似文献
2.
Let G be a graph and W a subset of V(G). Let g,f:V(G)→Z be two integer-valued functions such that g(x)≤f(x) for all x∈V(G) and g(y)≡f(y) (mod 2) for all y∈W. Then a spanning subgraph F of G is called a partial parity (g,f)-factor with respect to W if g(x)≤deg
F
(x)≤f(x) for all x∈V(G) and deg
F
(y)≡f(y) (mod 2) for all y∈W. We obtain a criterion for a graph G to have a partial parity (g,f)-factor with respect to W. Furthermore, by making use of this criterion, we give some necessary and sufficient conditions for a graph G to have a subgraph which covers W and has a certain given property.
Received: June 14, 1999?Final version received: August 21, 2000 相似文献
3.
P. K. SAHO 《数学学报(英文版)》2005,21(5):1159-1166
In this paper, we determine the general solution of the functional equation f1 (2x + y) + f2(2x - y) = f3(x + y) + f4(x - y) + f5(x) without assuming any regularity condition on the unknown functions f1,f2,f3, f4, f5 : R→R. The general solution of this equation is obtained by finding the general solution of the functional equations f(2x + y) + f(2x - y) = g(x + y) + g(x - y) + h(x) and f(2x + y) - f(2x - y) = g(x + y) - g(x - y). The method used for solving these functional equations is elementary but exploits an important result due to Hosszfi. The solution of this functional equation can also be determined in certain type of groups using two important results due to Szekelyhidi. 相似文献
4.
Soon Mo JUNG 《数学学报(英文版)》2006,22(2):583-586
The stability problems of the exponential (functional) equation on a restricted domain will be investigated, and the results will be applied to the study of an asymptotic property of that equation. More precisely, the following asymptotic property is proved: Let X be a real (or complex) normed space. A mapping f : X → C is exponential if and only if f(x + y) - f(x)f(y) → 0 as ||x|| + ||y|| → ∞ under some suitable conditions. 相似文献
5.
In this paper we prove a result on the existence of periodic motions for the periodically forced Liénard differential equation
x
″+f(x)x
′+g(x)=e(t) in a situation where the phase portrait of the associated autonomous equation is similar to that of a centre limited by
an unbounded separatrix. The existence result, which is based on a degree theoretic continuation theorem, enables us to treat
some interesting cases not previously considered in the literature.
Received: April 27, 2000 Published online: December 19, 2001 相似文献
6.
Chun Gil PARK Jin Chuan HOU Sei Qwon OH 《数学学报(英文版)》2005,21(6):1391-1398
It is shown that every almost *-homomorphism h : A→B of a unital JC*-algebra A to a unital JC*-algebra B is a *-homomorphism when h(rx) = rh(x) (r 〉 1) for all x∈A, and that every almost linear mapping h : A→B is a *-homomorphism when h(2^nu o y) - h(2^nu) o h(y), h(3^nu o y) - h(3^nu) o h(y) or h(q^nu o y) = h(q^nu) o h(y) for all unitaries u ∈A, all y ∈A, and n = 0, 1,.... Here the numbers 2, 3, q depend on the functional equations given in the almost linear mappings. We prove that every almost *-homomorphism h : A→B of a unital Lie C*-algebra A to a unital Lie C*-algebra B is a *-homomorphism when h(rx) = rh(x) (r 〉 1) for all x ∈A. 相似文献
7.
LetX be a real linear normed space, (G, +) be a topological group, andK be a discrete normal subgroup ofG. We prove that if a continuous at a point or measurable (in the sense specified later) functionf:X →G fulfils the condition:f(x +y) -f(x) -f(y) ∈K whenever ‖x‖ = ‖y‖, then, under some additional assumptions onG,K, andX, there esists a continuous additive functionA :X →G such thatf(x) -A(x) ∈K. 相似文献
8.
Weighted mean convergence of Hakopian interpolation on the disk 总被引:1,自引:0,他引:1
Xuezhang Liang Renzhong Feng Xuenan Sun 《分析论及其应用》2007,23(3):213-227
In this paper, we study weighted mean integral convergence of Hakopian interpolation on the unit disk D. We show that the inner product between Hakopian interpolation polynomial Hn(f;x,y) and a smooth function g(x,y) on D converges to that of f(x,y) and g(x,y) on D when n →∞, provided f(x,y) belongs to C(D) and all first partial derivatives of g(x,y) belong to the space LipαM(0 <α≤ 1). We further show that provided all second partial derivatives of g(x,y) also belong to the space LipαM and f(x,y) belongs to C1 (D), the inner product between the partial derivative of Hakopian interpolation polynomial (6)/(6)xHn(f;x,y) and g(x,y) on D converges to that between (6)/(6)xf(x,y) and g(x,y) on D when n →∞. 相似文献
9.
A comparative study of the functional equationsf(x+y)f(x–y)=f
2(x)–f
2(y),f(y){f(x+y)+f(x–y)}=f(x)f(2y) andf(x+y)+f(x–y)=2f(x){1–2f
2(y/2)} which characterise the sine function has been carried out. The zeros of the functionf satisfying any one of the above equations play a vital role in the investigations. The relation of the equationf(x+y)+f(x–y)=2f(x){1–2f
2(y/2)} with D'Alembert's equation,f(x+y)+f(x–y)=2f(x)f(y) and the sine-cosine equationg(x–y)=g(x)g(y) +f(x)f(y) has also been investigated. 相似文献
10.
Janusz Matkowski 《Aequationes Mathematicae》2010,79(3):203-212
Under some natural assumptions on real functions f and g defined on a real interval I, we show that a two variable function M
f,g
: I
2 → I defined by
Mf,g(x,y)=(f+g)-1(f(x)+g(y))M_{f,g}(x,y)=(f+g)^{-1}(f(x)+g(y)) 相似文献
11.
HaoZHAO GuiZhenLIU XiaoXiaYAN 《数学学报(英文版)》2005,21(2):413-422
Let G be a graph with vertex set V(G) and edge set E(G) and let g and f be two integervalued functions defined on V(G) such that 2k - 2 ≤g(x)≤f(x) for all x∈V(G). Let H be a subgraph of G with mk edges. In this paper, it is proved that every (mg m-1,mf-m 1)-graph G has (g, f)-factorizations randomly k-orthogonal to H under some special conditions. 相似文献
12.
In this paper, we consider functions of the form f(x,y)=f(x)g(y){\phi(x,y)=f(x)g(y)} over a box, where
f(x), x ? \mathbb R{f(x), x\in {\mathbb R}} is a nonnegative monotone convex function with a power or an exponential form, and
g(y), y ? \mathbb Rn{g(y), y\in {\mathbb R}^n} is a component-wise concave function which changes sign over the vertices of its domain. We derive closed-form expressions
for convex envelopes of various functions in this category. We demonstrate via numerical examples that the proposed envelopes
are significantly tighter than popular factorable programming relaxations. 相似文献
13.
We consider the parametric programming problem (Q
p
) of minimizing the quadratic function f(x,p):=x
T
Ax+b
T
x subject to the constraint Cx≤d, where x∈ℝ
n
, A∈ℝ
n×n
, b∈ℝ
n
, C∈ℝ
m×n
, d∈ℝ
m
, and p:=(A,b,C,d) is the parameter. Here, the matrix A is not assumed to be positive semidefinite. The set of the global minimizers and the set of the local minimizers to (Q
p
) are denoted by M(p) and M
loc
(p), respectively. It is proved that if the point-to-set mapping M
loc
(·) is lower semicontinuous at p then M
loc
(p) is a nonempty set which consists of at most ?
m,n
points, where ?
m,n
= is the maximal cardinality of the antichains of distinct subsets of {1,2,...,m} which have at most n elements. It is proved also that the lower semicontinuity of M(·) at p implies that M(p) is a singleton. Under some regularity assumption, these necessary conditions become the sufficient ones.
Received: November 5, 1997 / Accepted: September 12, 2000?Published online November 17, 2000 相似文献
14.
Convexity of Products of Univariate Functions and Convexification Transformations for Geometric Programming 总被引:1,自引:0,他引:1
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. 相似文献
15.
David E. Dobbs 《International Journal of Mathematical Education in Science & Technology》2013,44(3):358-362
It is proved that if the differential equations y ( n )=f(x, y, y′, …, y ( n ?1 )) and y ( m )=g(x, y, y′, …, y ( n ?1 )) have the same particular solutions in a suitable region where f and g are continuous real-valued functions with continuous partial derivatives (alternatively, continuous functions satisfying the classical Lipschitz condition), then n?=?m and the functions f and g are equal. This note could find classroom use in a course on differential equations as enrichment material for the unit on the existence and uniqueness theorems for solutions of initial value problems. 相似文献
16.
H. Stetkær 《Aequationes Mathematicae》1997,54(1-2):144-172
Summary We produce complete solution formulas of selected functional equations of the formf(x +y) ±f(x + σ (ν)) = Σ
I
2
=1
g
l
(x)h
l
(y),x, y∈G, where the functionsf,g
1,h
1 to be determined are complex valued functions on an abelian groupG and where σ:G→G is an involution ofG. The special case of σ=−I encompasses classical functional equations like d’Alembert’s, Wilson’s first generalization of it, Jensen’s equation and
the quadratic equation. We solve these equations, the equation for symmetric second differences in product form and similar
functional equations for a general involution σ. 相似文献
17.
We determine the general solution of the functional equation f(x + ky) + f(x-ky) = g(x + y) + g(x-y) + h(x) + h(y) for fixed integers with k ≠ 0; ±1 without assuming any regularity conditions for the unknown functions f, g, h, and0020[(h)\tilde] \tilde{h} . The method used for solving these functional equations is elementary but it exploits an important result due to Hosszú.
The solution of this functional equation can also be obtained in groups of certain type by using two important results due
to Székelyhidi. 相似文献
18.
T. Noiri 《Acta Mathematica Hungarica》2003,99(4):315-328
A subset A of a topological space X is said to be β-open [1] if A ⊂ Cl (Int (Cl (A))). A function f : X → Y is said to be β-irresolute [4] if for every β-open set V of Y, f
-1(V) is β-open in X. In this paper we introduce weak and strong forms of β-irresolute functions and obtain several basic properties of such functions.
This revised version was published online in August 2006 with corrections to the Cover Date. 相似文献
19.
Raymond Mortini 《Monatshefte für Mathematik》2009,158(1):81-95
We characterize those sequences (x
n
) in the spectrum of H
∞ whose Nevanlinna–Pick interpolation problems admit thin Blaschke products as solutions. We also study under which conditions
there is a Blaschke product B with prescribed zero-set distribution and solving problems of the form B(x) = f
n
(x) for every x ∈ P(x
n
), where P(x
n
) is the Gleason part associated with the point x
n
and where (f
n
) is an arbitrary sequence of functions in the unit ball of H
∞. As a corollary we get a new characterization of Carleson–Newman Blaschke products in terms of bounded universal functions,
a result first proved by Gallardo and Gorkin.
相似文献
20.
Jochen Gorski Frank Pfeuffer Kathrin Klamroth 《Mathematical Methods of Operations Research》2007,66(3):373-407
The problem of optimizing a biconvex function over a given (bi)convex or compact set frequently occurs in theory as well as
in industrial applications, for example, in the field of multifacility location or medical image registration. Thereby, a
function is called biconvex, if f(x,y) is convex in y for fixed x∈X, and f(x,y) is convex in x for fixed y∈Y. This paper presents a survey of existing results concerning the theory of biconvex sets and biconvex functions and gives
some extensions. In particular, we focus on biconvex minimization problems and survey methods and algorithms for the constrained
as well as for the unconstrained case. Furthermore, we state new theoretical results for the maximum of a biconvex function
over biconvex sets.
J. Gorski and K. Klamroth were partially supported by a grant of the German Research Foundation (DFG). 相似文献
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