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1.
This note studies
A
, a condition number used in the linear programming algorithm of Vavasis and Ye [14] whose running time depends only on the
constraint matrix A∈ℝ
m×n
, and (A), a variant of another condition number due to Ye [17] that also arises in complexity analyses of linear programming problems.
We provide a new characterization of
A
and relate
A
and (A). Furthermore, we show that if A is a standard Gaussian matrix, then E(ln
A
)=O(min{mlnn,n}). Thus, the expected running time of the Vavasis-Ye algorithm for linear programming problems is bounded by a polynomial
in m and n for any right-hand side and objective coefficient vectors when A is randomly generated in this way. As a corollary of the close relation between
A
and (A), we show that the same bound holds for E(ln(A)).
Received: September 1998 / Accepted: September 2000?Published online January 17, 2001 相似文献
2.
Given a linear transformation L:?
n
→?
n
and a matrix Q∈?
n
, where ?
n
is the space of all symmetric real n×n matrices, we consider the semidefinite linear complementarity problem SDLCP(L,?
n
+,Q) over the cone ?
n
+ of symmetric n×n positive semidefinite matrices. For such problems, we introduce the P-property and its variants, Q- and GUS-properties. For a matrix A∈R
n×n
, we consider the linear transformation L
A
:?
n
→?
n
defined by L
A
(X):=AX+XA
T
and show that the P- and Q-properties for L
A
are equivalent to A being positive stable, i.e., real parts of eigenvalues of A are positive. As a special case of this equivalence, we deduce a theorem of Lyapunov.
Received: March 1999 / Accepted: November 1999?Published online April 20, 2000 相似文献
3.
We prove a structure theorem for the connected coassociative magmatic bialgebras. The space of primitive elements is an algebra
over an operad called the primitive operad. We prove that the primitive operad is magmatic generated by n−2 operations of arity n. The dimension of the space of all the n-ary operations of this primitive operad turns out to be the Fine number F
n−1. In short, the triple of operads (As, Mag, MagFine) is good.
The third author work is partially supported by FONDECYT Project 1060224 相似文献
4.
Polynomial convergence of primal-dual algorithms for the second-order cone program based on the MZ-family of directions 总被引:5,自引:0,他引:5
In this paper we study primal-dual path-following algorithms for the second-order cone programming (SOCP) based on a family
of directions that is a natural extension of the Monteiro-Zhang (MZ) family for semidefinite programming. We show that the
polynomial iteration-complexity bounds of two well-known algorithms for linear programming, namely the short-step path-following
algorithm of Kojima et al. and Monteiro and Adler, and the predictor-corrector algorithm of Mizuno et al., carry over to the
context of SOCP, that is they have an O( logε-1) iteration-complexity to reduce the duality gap by a factor of ε, where n is the number of second-order cones. Since the MZ-type family studied in this paper includes an analogue of the Alizadeh,
Haeberly and Overton pure Newton direction, we establish for the first time the polynomial convergence of primal-dual algorithms
for SOCP based on this search direction.
Received: June 5, 1998 / Accepted: September 8, 1999?Published online April 20, 2000 相似文献
5.
A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities 总被引:17,自引:0,他引:17
In this paper we take a new look at smoothing Newton methods for solving the nonlinear complementarity problem (NCP) and the
box constrained variational inequalities (BVI). Instead of using an infinite sequence of smoothing approximation functions,
we use a single smoothing approximation function and Robinson’s normal equation to reformulate NCP and BVI as an equivalent
nonsmooth equation H(u,x)=0, where H:ℜ
2n
→ℜ
2n
, u∈ℜ
n
is a parameter variable and x∈ℜ
n
is the original variable. The central idea of our smoothing Newton methods is that we construct a sequence {z
k
=(u
k
,x
k
)} such that the mapping H(·) is continuously differentiable at each z
k
and may be non-differentiable at the limiting point of {z
k
}. We prove that three most often used Gabriel-Moré smoothing functions can generate strongly semismooth functions, which
play a fundamental role in establishing superlinear and quadratic convergence of our new smoothing Newton methods. We do not
require any function value of F or its derivative value outside the feasible region while at each step we only solve a linear system of equations and if
we choose a certain smoothing function only a reduced form needs to be solved. Preliminary numerical results show that the
proposed methods for particularly chosen smoothing functions are very promising.
Received June 23, 1997 / Revised version received July 29, 1999?Published online December 15, 1999 相似文献
6.
We propose a class of non-interior point algorithms for solving the complementarity problems(CP): Find a nonnegative pair
(x,y)∈ℝ
2n
satisfying y=f(x) and x
i
y
i
=0 for every i∈{1,2,...,n}, where f is a continuous mapping from ℝ
n
to ℝ
n
. The algorithms are based on the Chen-Harker-Kanzow-Smale smoothing functions for the CP, and have the following features;
(a) it traces a trajectory in ℝ
3n
which consists of solutions of a family of systems of equations with a parameter, (b) it can be started from an arbitrary
(not necessarily positive) point in ℝ
2n
in contrast to most of interior-point methods, and (c) its global convergence is ensured for a class of problems including
(not strongly) monotone complementarity problems having a feasible interior point. To construct the algorithms, we give a
homotopy and show the existence of a trajectory leading to a solution under a relatively mild condition, and propose a class
of algorithms involving suitable neighborhoods of the trajectory. We also give a sufficient condition on the neighborhoods
for global convergence and two examples satisfying it.
Received April 9, 1997 / Revised version received September 2, 1998? Published online May 28, 1999 相似文献
7.
Levent Tunçel 《Mathematical Programming》1999,86(1):219-223
Given an m×n integer matrix A of full row rank, we consider the problem of computing the maximum of ∥B
-1
A∥2 where B varies over all bases of A. This quantity appears in various places in the mathematical programming literature. More recently, logarithm of this number
was the determining factor in the complexity bound of Vavasis and Ye’s primal-dual interior-point algorithm. We prove that
the problem of approximating this maximum norm, even within an exponential (in the dimension of A) factor, is NP-hard. Our proof is based on a closely related result of L. Khachiyan [1].
Received November 13, 1998 / Revised version received January 20, 1999? Published online May 12, 1999 相似文献
8.
K.H. Kwon D.W. Lee F. Marcellán S.B. Park 《Annali di Matematica Pura ed Applicata》2001,180(2):127-146
Given an orthogonal polynomial system {Q
n
(x)}
n=0
∞, define another polynomial system by where α
n
are complex numbers and t is a positive integer. We find conditions for {P
n
(x)}
n=0
∞ to be an orthogonal polynomial system. When t=1 and α1≠0, it turns out that {Q
n
(x)}
n=0
∞ must be kernel polynomials for {P
n
(x)}
n=0
∞ for which we study, in detail, the location of zeros and semi-classical character.
Received: November 25, 1999; in final form: April 6, 2000?Published online: June 22, 2001 相似文献
9.
Giuseppina Vannella 《Annali di Matematica Pura ed Applicata》2002,180(4):429-440
We consider a Neumann problem of the type -εΔu+F
′(u(x))=0 in an open bounded subset Ω of R
n
, where F is a real function which has exactly k maximum points.
Using Morse theory we find that, for ε suitably small, there are at least 2k nontrivial solutions of the problem and we give some qualitative information about them.
Received: October 30, 1999 Published online: December 19, 2001 相似文献
10.
A. I. Badulescu 《manuscripta mathematica》2000,101(1):49-70
We prove the orthogonality relations for characters of GL
n
(F) with F a local field of positive characteristic. Using this, we establish a Jacquet-Langlands correspondence with the group of invertible
elements of a central division algebra of dimension n
2 over F, and the transfer of orbital integrals between these two groups.
Received: 16 February 1999 相似文献
11.
Xia Chen 《Probability Theory and Related Fields》2000,116(1):89-123
Let {X
n
}
n
≥0 be a Harris recurrent Markov chain with state space E and invariant measure π. The law of the iterated logarithm and the law of weak convergence are given for the additive functionals
of the form
where ƒ is a real π-centered function defined on E. Some similar results are also obtained for additive functionals which are martingales associated with {X
n
}
n
≥0.
Received: 15 September 1998 / Revised version: 1 April 1999 相似文献
12.
For an edge-weighted graph G with n vertices and m edges, we present a new deterministic algorithm for computing a minimum k-way cut for k=3,4. The algorithm runs in O(n
k-1
F(n,m))=O(mn
k
log(n
2
/m)) time and O(n
2) space for k=3,4, where F(n,m) denotes the time bound required to solve the maximum flow problem in G. The bound for k=3 matches the current best deterministic bound ?(mn
3) for weighted graphs, but improves the bound ?(mn
3) to O(n
2
F(n,m))=O(min{mn
8/3,m
3/2
n
2}) for unweighted graphs. The bound ?(mn
4) for k=4 improves the previous best randomized bound ?(n
6) (for m=o(n
2)). The algorithm is then generalized to the problem of finding a minimum 3-way cut in a symmetric submodular system.
Received: April 1999 / Accepted: February 2000?Published online August 18, 2000 相似文献
13.
Song Xu 《Mathematical Programming》2000,87(3):501-517
We propose an infeasible non-interior path-following method for nonlinear complementarity problems with uniform P-functions. This method is based on the smoothing techniques introduced by Kanzow. A key to our analysis is the introduction
of a new notion of neighborhood for the central path which is suitable for infeasible non-interior path-following methods.
By restricting the iterates in the neighborhood of the central path, we provide a systematic procedure to update the smoothing
parameter and establish the global linear convergence of this method. Some preliminary computational results are reported.
Received: March 13, 1997 / Accepted: December 17, 1999?Published online February 23, 2000 相似文献
14.
For a polytope in the [0,1]
n
cube, Eisenbrand and Schulz showed recently that the maximum Chvátal rank is bounded above by O(n
2logn) and bounded below by (1+ε)n for some ε>0. Chvátal cuts are equivalent to Gomory fractional cuts, which are themselves dominated by Gomory mixed integer
cuts. What do these upper and lower bounds become when the rank is defined relative to Gomory mixed integer cuts? An upper
bound of n follows from existing results in the literature. In this note, we show that the lower bound is also equal to n. This result still holds for mixed 0,1 polyhedra with n binary variables.
Received: March 15, 2001 / Accepted: July 18, 2001?Published online September 17, 2001 相似文献
15.
On the complexity of a class of combinatorial optimization problems with uncertainty 总被引:2,自引:0,他引:2
Igor Averbakh 《Mathematical Programming》2001,90(2):263-272
We consider a robust (minmax-regret) version of the problem of selecting p elements of minimum total weight out of a set of m elements with uncertainty in weights of the elements. We present a polynomial algorithm with the order of complexity O((min {p,m-p})2
m) for the case where uncertainty is represented by means of interval estimates for the weights. We show that the problem is
NP-hard in the case of an arbitrary finite set of possible scenarios, even if there are only two possible scenarios. This
is the first known example of a robust combinatorial optimization problem that is NP-hard in the case of scenario-represented
uncertainty but is polynomially solvable in the case of the interval representation of uncertainty.
Received: July 1998 / Accepted: May 2000?Published online March 22, 2001 相似文献
16.
J. Huisman 《Annali di Matematica Pura ed Applicata》2003,182(1):21-35
We show that there is a large class of non-special divisors of relatively small degree on a given real algebraic curve. If
the real algebraic curve has many real components, such a divisor gives rise to an embedding (birational embedding, resp.)
of the real algebraic curve into the real projective space ℙ
r
for r≥3 (r=2, resp.). We study these embeddings in quite some detail.
Received: October 17, 2001?Published online: February 20, 2003 相似文献
17.
We consider a class of non-linear mixed integer programs with n integer variables and k continuous variables. Solving instances from this class to optimality is an NP-hard problem. We show that for the cases with
k=1 and k=2, every optimal solution is integral. In contrast to this, for every k≥3 there exist instances where every optimal solution takes non-integral values.
Received: August 2001 / Accepted: January 2002?Published online March 27, 2002 相似文献
18.
Self-regular functions and new search directions for linear and semidefinite optimization 总被引:11,自引:0,他引:11
In this paper, we introduce the notion of a self-regular function. Such a function is strongly convex and smooth coercive on its domain, the positive real axis. We show that any
such function induces a so-called self-regular proximity function and a corresponding search direction for primal-dual path-following
interior-point methods (IPMs) for solving linear optimization (LO) problems. It is proved that the new large-update IPMs enjoy
a polynomial ?(n
log) iteration bound, where q≥1 is the so-called barrier degree of the kernel function underlying the algorithm. The constant hidden in the ?-symbol depends
on q and the growth degree p≥1 of the kernel function. When choosing the kernel function appropriately the new large-update IPMs have a polynomial ?(lognlog) iteration bound, thus improving the currently best known bound for large-update methods by almost a factor . Our unified analysis provides also the ?(log) best known iteration bound of small-update IPMs. At each iteration, we need to solve only one linear system. An extension
of the above results to semidefinite optimization (SDO) is also presented.
Received: March 2000 / Accepted: December 2001?Published online April 12, 2002 相似文献
19.
We describe an O(n
4
hmin{logU,n
2logn}) capacity scaling algorithm for the minimum cost submodular flow problem. Our algorithm modifies and extends the Edmonds–Karp
capacity scaling algorithm for minimum cost flow to solve the minimum cost submodular flow problem. The modification entails
scaling a relaxation parameter δ. Capacities are relaxed by attaching a complete directed graph with uniform arc capacity
δ in each scaling phase. We then modify a feasible submodular flow by relaxing the submodular constraints, so that complementary
slackness is satisfied. This creates discrepancies between the boundary of the flow and the base polyhedron of a relaxed submodular
function. To reduce these discrepancies, we use a variant of the successive shortest path algorithm that augments flow along
minimum cost paths of residual capacity at least δ. The shortest augmenting path subroutine we use is a variant of Dijkstra’s
algorithm modified to handle exchange capacity arcs efficiently. The result is a weakly polynomial time algorithm whose running
time is better than any existing submodular flow algorithm when U is small and C is big. We also show how to use maximum mean cuts to make the algorithm strongly polynomial. The resulting algorithm is the
first capacity scaling algorithm to match the current best strongly polynomial bound for submodular flow.
Received: August 6, 1999 / Accepted: July 2001?Published online October 2, 2001 相似文献
20.
Konrad J. Swanepoel 《Discrete and Computational Geometry》2009,41(1):1-27
We show that the maximum number of unit distances or of diameters in a set of n points in d-dimensional Euclidean space is attained only by specific types of Lenz constructions, for all d≥4 and n sufficiently large depending on d. As a corollary, we determine the exact maximum number of unit distances for all even d≥6 and the exact maximum number of diameters for all d≥4 and all n sufficiently large depending on d.
This material is based upon work supported by the South African National Research Foundation. 相似文献