解线性互补问题的区间GAOR方法

1引言设M∈Rn×n,q∈Rn,则线性互补问题LCP(M,q)指的是寻找一个向量x∈Rn,使其满足下面的条件: x≥0 Mx+q≥0 xt(Mx+q)=0由于线性互补问题在工程物理、管理学、经济学、约束最优化等领域的应用非常广泛,所以该问题的研究一直倍受大家的关注,至今已有很多有效的算法.早在20世纪80年代     

Dynamics of a new family of iterative processes for quadratic polynomials

In this work we show the presence of the well-known Catalan numbers in the study of the convergence and the dynamical behavior of a family of iterative methods for solving nonlinear equations. In fact, we introduce a family of methods, depending on a parameter m∈N∪{0}. These methods reach the order of convergence m+2 when they are applied to quadratic polynomials with different roots. Newton’s and Chebyshev’s methods appear as particular choices of the family appear for m=0 and m=1, respectively. We make both analytical and graphical studies of these methods, which give rise to rational functions defined in the extended complex plane. Firstly, we prove that the coefficients of the aforementioned family of iterative processes can be written in terms of the Catalan numbers. Secondly, we make an incursion into its dynamical behavior. In fact, we show that the rational maps related to these methods can be written in terms of the entries of the Catalan triangle. Next we analyze its general convergence, by including some computer plots showing the intricate structure of the Universal Julia sets associated with the methods.     

Newton-Like Dynamics and Forward-Backward Methods for Structured Monotone Inclusions in Hilbert Spaces

In a Hilbert space setting we introduce dynamical systems, which are linked to Newton and Levenberg–Marquardt methods. They are intended to solve, by splitting methods, inclusions governed by structured monotone operators M=A+B, where A is a general maximal monotone operator, and B is monotone and locally Lipschitz continuous. Based on the Minty representation of A as a Lipschitz manifold, we show that these dynamics can be formulated as differential systems, which are relevant to the Cauchy–Lipschitz theorem, and involve separately B and the resolvents of A. In the convex subdifferential case, by using Lyapunov asymptotic analysis, we prove a descent minimizing property and weak convergence to equilibria of the trajectories. Time discretization of these dynamics gives algorithms combining Newton’s method and forward-backward methods for solving structured monotone inclusions.     

Order of convergence for polynomial and analytical nodal methods

In [6] we analyzed the direct analytical nodal methods (ANM) of indexl and show that the corresponding mathematical methods are equivalent to the physical ones when the components of the matrices are calculated by generalized Radau reduced integration. In this article we extend the theorem 8 of [7] to the polynomial nodal methods (PNM) (exact calculation of moments) which are thus the order ofO(h ^l+3?δ _l0. We also show that the analytical nodal methods are only the order ofO(h ^l+2). Forl = 0 our numerical results confirm our theoretical results.     

New efficient numerical procedures for solving stochastic variational problems with a priori maximum pointwise error estimates

In a previous paper we gave a new formulation and derived the Euler equations and other necessary conditions to solve strong, pathwise, stochastic variational problems with trajectories driven by Brownian motion. Thus, unlike current methods which minimize the control over deterministic functionals (the expected value), we find the control which gives the critical point solution of random functionals of a Brownian path and then, if we choose, find the expected value.This increase in information is balanced by the fact that our methods are anticipative while current methods are not. However, our methods are more directly connected to the theory and meaningful examples of deterministic variational theory and provide better means of solution for free and constrained problems. In addition, examples indicate that there are methods to obtain nonanticipative solutions from our equations although the anticipative optimal cost function has smaller expected value.In this paper we give new, efficient numerical methods to find the solution of these problems in the quadratic case. Of interest is that our numerical solution has a maximal, a priori, pointwise error of O(h3/2) where h is the node size. We believe our results are unique for any theory of stochastic control and that our methods of proof involve new and sophisticated ideas for strong solutions which extend previous deterministic results by the first author where the error was O(h²).We note that, although our solutions are given in terms of stochastic differential equations, we are not using the now standard numerical methods for stochastic differential equations. Instead we find an approximation to the critical point solution of the variational problem using relations derived from setting to zero the directional derivative of the cost functional in the direction of simple test functions.Our results are even more significant than they first appear because we can reformulate stochastic control problems or constrained calculus of variations problems in the unconstrained, stochastic calculus of variations formulation of this paper. This will allow us to find efficient and accurate numerical solutions for general constrained, stochastic optimization problems. This is not yet being done, even in the deterministic case, except by the first author.     

Multiple invariants conserving Runge-Kutta type methods for Hamiltonian problems

In a recent series of papers, the class of energy-conserving Runge-Kutta methods named Hamiltonian BVMs (HBVMs) has been defined and studied. Such methods have been further generalized for the efficient solution of general conservative problems, thus providing the class of Line Integral Methods (LIMs). In this paper we derive a further extension, which we name Enhanced Line Integral Methods (ELIMs), more tailored for Hamiltonian problems, allowing for the conservation of multiple invariants of the continuous dynamical system. The analysis of the methods is fully carried out and some numerical tests are reported, in order to confirm the theoretical achievements.     

On algebraic Riccati equations associated with M-matrices

We consider the algebraic Riccati equation for which the four coefficient matrices form an M-matrix K. When K is a nonsingular M-matrix or an irreducible singular M-matrix, the Riccati equation is known to have a minimal nonnegative solution and several efficient methods are available to find this solution. In this paper we are mainly interested in the case where K is a reducible singular M-matrix. Under a regularity assumption on the M-matrix K, we show that the Riccati equation still has a minimal nonnegative solution. We also study the properties of this particular solution and explain how the solution can be found by existing methods.     

Optimizing fuzzy p-hub center problem with generalized value-at-risk criterion

In this study, we reduce the uncertainty embedded in secondary possibility distribution of a type-2 fuzzy variable by fuzzy integral, and apply the proposed reduction method to p-hub center problem, which is a nonlinear optimization problem due to the existence of integer decision variables. In order to optimize p-hub center problem, this paper develops a robust optimization method to describe travel times by employing parametric possibility distributions. We first derive the parametric possibility distributions of reduced fuzzy variables. After that, we apply the reduction methods to p-hub center problem and develop a new generalized value-at-risk (VaR) p-hub center problem, in which the travel times are characterized by parametric possibility distributions. Under mild assumptions, we turn the original fuzzy p-hub center problem into its equivalent parametric mixed-integer programming problems. So, we can solve the equivalent parametric mixed-integer programming problems by general-purpose optimization software. Finally, some numerical experiments are performed to demonstrate the new modeling idea and the efficiency of the proposed solution methods.     

A further study on inverse linear programming problems

In this paper we continue our previous study (Zhang and Liu, J. Comput. Appl. Math. 72 (1996) 261–273) on inverse linear programming problems which requires us to adjust the cost coefficients of a given LP problem as less as possible so that a known feasible solution becomes the optimal one. In particular, we consider the cases in which the given feasible solution and one optimal solution of the LP problem are 0–1 vectors which often occur in network programming and combinatorial optimization, and give very simple methods for solving this type of inverse LP problems. Besides, instead of the commonly used l₁ measure, we also consider the inverse LP problems under l_∞ measure and propose solution methods.     

The hypoelliptic Laplacian on a compact Lie group

Let G be a compact Lie group, and let g be its Lie algebra. In this paper, we produce a hypoelliptic Laplacian on G×g, which interpolates between the classical Laplacian of G and the geodesic flow. This deformation is obtained by producing a suitable deformation of the Dirac operator of Kostant. We show that various Poisson formulas for the heat kernel can be proved using this interpolation by methods of local index theory. The paper was motivated by papers by Atiyah and Frenkel, in connection with localization formulas in equivariant cohomology and with Kac's character formulas for affine Lie algebras. In a companion paper, we will use similar methods in the context of Selberg's trace formula.     

Error analysis of explicit TSERKN methods for highly oscillatory systems

In this paper, we are concerned with the error analysis for the two-step extended Runge-Kutta-Nyström-type (TSERKN) methods [Comput. Phys. Comm. 182 (2011) 2486–2507] for multi-frequency and multidimensional oscillatory systems y″(t) + My(t) = f(t, y(t)), where high-frequency oscillations in the solutions are generated by the linear part My(t). TSERKN methods extend the two-step hybrid methods [IMA J. Numer. Anal. 23 (2003) 197–220] by reforming both the internal stages and the updates so that they are adapted to the oscillatory properties of the exact solutions. However, the global error analysis for the TSERKN methods has not been investigated. In this paper we construct a new three-stage explicit TSERKN method of order four and present the global error bound for the new method, which is proved to be independent of ∥M∥ under suitable assumptions. This property of our new method is very important for solving highly oscillatory systems (1), where ∥M∥ may be arbitrarily large. We also analyze the stability and phase properties for the new method. Numerical experiments are included and the numerical results show that the new method is very competitive and promising compared with the well-known high quality methods proposed in the scientific literature.     

New constructions for balanced Howell rotations

Howell rotations have been used in bridge tournaments for a long time. But it was not until 1955 that Parker and Mood first gave a rigorous definition of a balanced Howell rotation and began a systematic study of its mathematical properties. Later, Berlekamp and Hwang extended this work to the study of complete balanced Howell rotations (which are special cases of balanced Howell rotations). Surprisingly, even though the concept of balanced Howell rotations precedes that of complete balanced Howell rotations, systematic construction methods have been studied only for the latter. Most of these construction methods use the properties of a Galois field GF(p^γ) where p^γ is a prime power. In this paper, we use the properties of a Galois domain GD(p^γq^s) to construct balanced Howell rotations for n partnerships where n ? 1 is the product of two prime powers satisfying certain conditions. In particular, we construct a balanced Howell rotation for 36 partnerships, this being the smallest number for which the existence of a balanced Howell rotation was not previously known. We also give two composition methods for the constructions of balanced Howell rotations.     

An uncertainty principle for function fields

In a recent paper, Granville and Soundararajan (2007) [5] proved an “uncertainty principle” for arithmetic sequences, which limits the extent to which such sequences can be well-distributed in both short intervals and arithmetic progressions. In the present paper we follow the methods of Granville and Soundararajan (2007) [5] and prove that a similar phenomenon holds in Fq[t].     

Taylor–Galerkin and Taylor-collocation methods for the numerical solutions of Burgers’ equation using B-splines

In this paper, for the numerical solution of Burgers’ equation, we give two B-spline finite element algorithms which involve a collocation method with cubic B-splines and a Galerkin method with quadratic B-splines. In time discretization of the equation, Taylor series expansion is used. In order to verify the stabilities of the purposed methods, von-Neumann stability analysis is employed. To see the accuracy of the methods, L₂ and L_∞ error norms are calculated and obtained results are compared with some earlier studies.     

Correlative sparsity structures and semidefinite relaxations for concave cost transportation problems with change of variables

We present a hierarchy of semidefinite programming (SDP) relaxations for solving the concave cost transportation problem (CCTP), which is known to be NP-hard, with p suppliers and q demanders. In particular, we study cases in which the cost function is quadratic or square-root concave. The key idea of our relaxation methods is in the change of variables to CCTPs, and due to this, we can construct SDP relaxations whose matrix variables are of size O((min {p, q}) ^ω ) in the relaxation order ω. The sequence of optimal values of SDP relaxations converges to the global minimum of the CCTP as the relaxation order ω goes to infinity. Furthermore, the size of the matrix variables can be reduced to O((min {p, q}) ^ω-1 ), ω ≥  2 by using Reznick’s theorem. Numerical experiments were conducted to assess the performance of the relaxation methods.     

Self-Orthogonal and Self-Dual Codes Constructed via Combinatorial Designs and Diophantine Equations

Combinatorial designs have been widely used, in the construction of self-dual codes. Recently, new methods of constructing self-dual codes are established using orthogonal designs (ODs), generalized orthogonal designs (GODs), a set of four sequences and Diophantine equations over GF(p). These methods had led to the construction of many new self-dual codes over small finite fields and rings. In this paper, we used some methods to construct self-orthogonal and self dual codes over GF(p), for some primes p. The construction is achieved by using some special kinds of combinatorial designs like orthogonal designs and GODs. Moreover, we combine eight circulant matrices, a system of Diophantine equations over GF(p), and a recently discovered array to obtain a new construction method. Using this method new self-dual and self-orthogonal codes are obtained. Specifically, we obtain new self-dual codes [32,16,12] over GF(11) and GF(13) which improve the previously known distances.     

New modifications of Potra-Pták’s method with optimal fourth and eighth orders of convergence

In this paper, we present two new iterative methods for solving nonlinear equations by using suitable Taylor and divided difference approximations. Both methods are obtained by modifying Potra-Pták’s method trying to get optimal order. We prove that the new methods reach orders of convergence four and eight with three and four functional evaluations, respectively. So, Kung and Traub’s conjecture Kung and Traub (1974) [2], that establishes for an iterative method based on n evaluations an optimal order p=2ⁿ⁻¹ is fulfilled, getting the highest efficiency indices for orders p=4 and p=8, which are 1.587 and 1.682.We also perform different numerical tests that confirm the theoretical results and allow us to compare these methods with Potra-Pták’s method from which they have been derived, and with other recently published eighth-order methods.     

On the distribution properties of Niederreiter-Halton sequences

We study the distribution properties of sequences which are a generalization of the well-known van der Corput-Halton sequence on the one hand, and digital (T,s)-sequences in the sense of Niederreiter on the other. In this paper we completely answer the question under which conditions such a sequence is uniformly distributed in the s-dimensional unit cube, by using methods based on the q-additive property of the weighted q-ary sum-of-digits function.     

Fourier-Motzkin elimination and its dual

In this paper we investigate block designs with repeated blocks and parameters υ, k; b, r, λ satisfying (b, r, λ) = 1. These designs cannot be constructed by taking a number of identical copies of a smaller design. Several construction methods of such designs are given and we discuss those constructions of Hanani which yield such designs. Tables for υ ? 22 are included.     

A non-cyclic triple-error-correcting BCH-like code and some minimum distance results

In this paper, we give the first example of a non-cyclic triple-error-correcting code which is not equivalent to the primitive BCH code. It has parameters [63, 45, 7]. We also give better bounds on minimum distances of some [2 ⁿ ? 1, 2 ⁿ - 3n - 1] cyclic codes with three small zeroes. Finally, we reprove weight distribution results of Kasami for triple-error-correcting BCH-like codes using direct methods.