优化方法解决无上层约束的线性双层规划问题

In this paper we propose an optimal method for solving the linear bilevel programming problem with no upper-level constraint. The main idea of this method is that the initial point which is in the feasible region goes forward along the optimal direction firstly. When the iterative point reaches the boundary of the feasible region, it can continue to go forward along the suboptimal direction. The iteration is terminated until the iterative point cannot go forward along the suboptimal direction and effective direction, and the new iterative point is the solution of the lower-level programming. An algorithm which bases on the main idea above is presented and the solution obtained via this algorithm is proved to be optimal solution to the bilevel programming problem. This optimal method is effective for solving the linear bilevel programming problem.     

FRAMELET BASED DECONVOLUTION

In this paper, two framelet based deconvolution algorithms are proposed. The basic idea of framelet based approach is to convert the deconvolution problem to the problem of inpainting in a frame domain by constructing a framelet system with one of the masks being the given （discrete） convolution kernel via the unitary extension principle of [26], as introduced in [6-9] . The first algorithm unifies our previous works in high resolution image reconstruction and infra-red chopped and nodded image restoration, and the second one is a combination of our previous frame-based deconvolution algorithm and the iterative thresholding algorithm given by [14, 16]. The strong convergence of the algorithms in infinite dimensional settings is given by employing proximal forward-backward splitting （PFBS） method. Consequently, it unifies iterative algorithms of infinite and finite dimensional setting and simplifies the proof of the convergence of the aluorithms of [6].     

A Robust Algorithm for Blind Total Variation Restoration

Image restoration is a fundamental problem in image processing. Blind image restoration has a great value in its practical application. However, it is not an easy problem to solve due to its complexity and difficulty. In this paper, we combine our robust algorithm for known blur operator with an alternating minimization implicit iterative scheme to deal with blind deconvolution problem, recover the image and identify the point spread function（PSF）. The only assumption needed is satisfy the practical physical sense. Numerical experiments demonstrate that this minimization algorithm is efficient and robust over a wide range of PSF and have almost the same results compared with known PSF algorithm.     

An iterative rounding 2-approximation algorithm for the k-partial vertex cover problem

We study a generalization of the vertex cover problem. For a given graph with weights on the vertices and an integer k, we aim to find a subset of the vertices with minimum total weight, so that at least k edges in the graph are covered. The problem is called the k-partial vertex cover problem. There are some 2-approximation algorithms for the problem. In the paper we do not improve on the approximation ratios of the previous algorithms, but we derive an iterative rounding algorithm. We present our technique in two algorithms. The first is an iterative rounding algorithm and gives a （2 ＋ Q/OPT ）-approximation for the k-partial vertex cover problem where Q is the largest finite weight in the problem definition and OPT is the optimal value for the instance. The second algorithm uses the first as a subroutine and achieves an approximation ratio of 2.     

PRECONDITIONED HSS-LIKE ITERATIVE METHOD FOR SADDLE POINT PROBLEMS

A new HSS-like iterative method is first proposed based on HSS-like splitting of non- Hermitian （1,1） block for solving saddle point problems. The convergence analysis for the new method is given. Meanwhile, we consider the solution of saddle point systems by preconditioned Krylov subspaee method and discuss some spectral properties of the preconditioned saddle point matrices. Numerical experiments are given to validate the performances of the preconditioners.     

AN ITERATIVE METHOD FOR SOLVING THE SINGULAR POINTS

An iterative method for solving a 2-order singular point is proposed in this paper. This method possesses the advantages of very fast convergent rate and higher accuracy.     

AN INFEASIBLE-INTERIOR-POINT PREDICTOR-CORRECTOR ALGORITHM FOR THE SECOND-ORDER CONE PROGRAM

A globally convergent infeasible-interior-point predictor-corrector algorithm is presented for the second-order cone programming （SOCP） by using the Alizadeh- Haeberly-Overton （AHO） search direction. This algorithm does not require the feasibility of the initial points and iteration points. Under suitable assumptions, it is shown that the algorithm can find an -approximate solution of an SOCP in at most O（√n ln（ε0/ε）） iterations. The iteration-complexity bound of our algorithm is almost the same as the best known bound of feasible interior point algorithms for the SOCP.     

Numerical Approximation of Transcritical Simple Bifurcation Point of the Navier-Stokes Equations

The extended system of nondegenerate simple bifurcation point of the Navier-Stokes equations is constructed in this paper, due to its derivative has a block lower triangular form, we design a Newton-like method, using the extended system and splitting iterative technique to compute transcritical nondegenerate simple bifurcation point, we not only reduces computational complexity, but also obtain quadratic convergence of algorithm.     

A quadratically approximate framework for constrained optimization,global and local convergence

This paper presents a quadratically approximate algorithm framework （QAAF） for solving general constrained optimization problems, which solves, at each iteration, a subproblem with quadratic objective function and quadratic equality together with inequality constraints. The global convergence of the algorithm framework is presented under the Mangasarian-Fromovitz constraint qualification （MFCQ）, and the conditions for superlinear and quadratic convergence of the algorithm framework are given under the MFCQ, the constant rank constraint qualification （CRCQ） as well as the strong second-order sufficiency conditions （SSOSC）. As an incidental result, the definition of an approximate KKT point is brought forward, and the global convergence of a sequence of approximate KKT points is analysed.     

非线性混合似变分不等式的扰动近似点-投影算法

In this paper we introduce a new perturbed proximal-projection algorithm for finding the common element of the set of fixed points of non-expansive mappings and the set of solutions of nonlinear mixed variational-like inequalities. The convergence criteria of the iterative sequences generated by the new iterative algorithm is also given. Our approach and results generalize many known results in this field.     

Some properties for the largest component of random geometric graphs with applications in sensor networks

In this paper we consider the standard Poisson Boolean model of random geometric graphs G（Hλ,s; 1） in Rd and study the properties of the order of the largest component L1 （G（Hλ,s; 1）） . We prove that ElL1 （G（Hλ,s; 1））] is smooth with respect to A, and is derivable with respect to s. Also, we give the expression of these derivatives. These studies provide some new methods for the theory of the largest component of finite random geometric graphs （not asymptotic graphs as s - co） in the high dimensional space （d 〉 2）. Moreover, we investigate the convergence rate of E[L1（G（Hλ,s; 1））]. These results have significance for theory development of random geometric graphs and its practical application. Using our theories, we construct and solve a new optimal energy-efficient topology control model of wireless sensor networks, which has the significance of theoretical foundation and guidance for the design of network layout.     

An Erdös-Ko-Rado theorem for restricted signed sets

A restricted signed r-set is a pair （A, f）, where A lohtain in [n] = {1, 2,…, n} is an r-set and f is a map from A to [n] with f（i） ≠ i for all i ∈ A. For two restricted signed sets （A, f） and （B, g）, we define an order as （A, f） ≤ （B, g） if A C B and g｜A ： f A family .A of restricted signed sets on [n] is an intersecting antiehain if for any （A, f）, （B, g） ∈ A, they are incomparable and there exists x ∈ A ∩ B such that f（x） = g（x）. In this paper, we first give a LYM-type inequality for any intersecting antichain A of restricted signed sets, from which we then obtain ｜A｜≤ （r-1^n-1）（n-1）^r-1 if A. consists of restricted signed r-sets on [n]. Unless r = n = 3, equality holds if and only if A consists of all restricted signed r-sets （A, f） such that x0∈ A and f（x0） =ε0 for some fixed x0 ∈ [n], ε0 ∈ [n] ／ {x0}.     

Geometry of 2×2 Hermitian matrices over any division ring

Let D be a division ring with an involution-,H2(D) be the set of 2 × 2 Hermitian matrices over D. Let ad(A,B) = rank(A-B) be the arithmetic distance between A,B ∈ H2(D) . In this paper,the fundamental theorem of the geometry of 2 × 2 Hermitian matrices over D(char(D) = 2) is proved:if  :H2(D) → H2(D) is the adjacency preserving bijective map,then  is of the form (X) = tP XσP +(0) ,where P ∈ GL2(D) ,σ is a quasi-automorphism of D. The quasi-automorphism of D is studied,and further results are obtained.     

The fundamental theory of abstract majorization inequalities

Using the axiomatic method,abstract concepts such as abstract mean, abstract convex function and abstract majorization are proposed. They are the generalizations of concepts of mean, convex function and majorization, respectively. Through the logical deduction, the fundamental theorems about abstract majorization inequalities are established as follows: for arbitrary abstract mean Σ and Σ , and abstract Σ → Σ strict convex function f(x) on the interval I, if xi, yi ∈ I (i = 1, 2, . . . , n) satisfy that (x1...     

Invariant Metrics and Laplacians on Siegel-Jacobi Disk

Let Dn be the generalized unit disk of degree n. In this paper, Riemannian metrics on the Siegel-Jacobi disk Dn × C（^m,n） which are invariant under the natural action of the Jacobi group are found explicitly and the Laplacians of these invariant metrics are computed explicitly. These are expressed in terms of the trace form.     

Numerical Approximation of a Reaction-Diffusion System with Fast Reversible Reaction

The authors consider the finite volume approximation of a reaction-diffusion system with fast reversible reaction.It is deduced from a priori estimates that the approximate solution converges to the weak solution of the reaction-diffusion problem and satisfies estimates which do not depend on the kinetic rate.It follows that the solution converges to the solution of a nonlinear diffusion problem,as the size of the volume elements and the time steps converge to zero while the kinetic rate tends to infinity.     

Generalization of an elementary inequality in Fourier analysis

The inequality plays an important role in Fourier analysis and approximation theory. It has recently been generalized by Telyakovskii and Leindler. This paper further generalizes and improves their results by introducing a new class of sequences called γ-piecewise bounded variation sequence (γ-PBVS).     

<Emphasis Type="Italic">H</Emphasis><Superscript>1</Superscript>-estimates of Littlewood-Paley and Lusin functions for Jacobi analysis

For α≥β≥ -1/2 let Δ(x) = (2shx)2α+1(2chx)2β+1 denote the weight function on R+ and L1(Δ) the space of integrable functions on R+ with respect to Δ(x)dx, equipped with a convolution structure. For a suitable φ∈ L1(Δ), we put φt(x) = t-1Δ(x)-1Δ(x/t)φ(x/t) for t > 0 and define the radial maximal operator Mφ as usual manner. We introduce a real Hardy space H1(Δ) as the set of all locally integrable functions f on R+ whose radial maximal function Mφ(f) belongs to L1(Δ). In this paper we obtain a relation between...     

Strong Convergence for Weighted Sums of Negatively Associated Arrays

Let {Xni} be an array of rowwise negatively associated random variables and Tnk=k∑i=1 i^a Xni for a ≥ -1, Snk =∑｜i｜≤k Ф（i/nη）1/nη Xni for η∈（0,1],where Ф is some function. The author studies necessary and sufficient conditions of ∞∑n=1 AnP（max 1≤k≤n｜Tnk｜〉εBn）〈∞ and ∞∑n=1 CnP（max 0≤k≤mn｜Snk｜〉εDn）〈∞ for all ε 〉 0, where An, Bn, Cn and Dn are some positive constants, mn ∈ N with mn /nη →∞. The results of Lanzinger and Stadtmfiller in 2003 are extended from the i.i.d, case to the case of the negatively associated, not necessarily identically distributed random variables. Also, the result of Pruss in 2003 on independent variables reduces to a special case of the present paper; furthermore, the necessity part of his result is complemented.     

Homogenization for Degenerate Quasilinear Parabolic Equations of Second Order

In this paper we study the homogenization of degenerate quasilinear parabolic equations: where a(t, y, a, λ) is periodic in (t, y).