Some Estimation Problems for Linear Model of Time Series,

If{X(n),n≥1}are random variables which satisfy following linear modelwhere α_j,λ_j,l≤j≤n are unknown constants andis real stationary sequence.In thispaper we discuss the estimation of λ_j,α_j and introduce δ-separated periodgram maximum estimatorλ_j for λ_j.Under some moderate conditions we prove the following consistence and asymptotionormality for these estimators,as the size     

Solving ℓ1 Regularization Problems With Piecewise Linear Losses

This article is concerned with the computational aspect of ?₁ regularization problems with a certain class of piecewise linear loss functions. The problem of computing the ?₁ regularization path for a piecewise linear loss can be formalized as a parametric linear programming problem. We propose an efficient implementation method of the parametric simplex algorithm for such a problem. We also conduct a simulation study to investigate the behavior of the number of “breakpoints” of the regularization path when both the number of observations and the number of explanatory variables vary. Our method is also applicable to the computation of the regularization path for a piecewise linear loss and the blockwise ?_∞ penalty. This article has supplementary material online.     

Solving Large Sparse Linear and Quadratic Generalized Eigenvalue Problems

We study the methods for solving the following large order eigenvelue problems occurring in the analysis of structural vibration: (1) (2) and (3) where M and C are both symmetric matrices, while (?) is skew symmetric. Moreover, M is positive definite, and the matrix K in (2) and (3) is also assumed to be symmetric positive definite. 1 The eigenvalue problem for normal matrices in generalized inner prduct Let B be an n×n Hermitian positive definite matrix. Then     

Multisplitting and Schwarz Methods for Solving Linear Complementarity Problems

In this paper we consider some synchronous and asynchronous multisplitting and Schwarz methods for solving the linear complementarity problems. We establish some convergence theorems of the methods by using the concept of M-splitting.     

Solving Linear Boundary Value Problems Via Non-commutative Gröbner Bases

A new approach for symbolically solving linear boundary value problems is presented. Rather than using general-purpose tools for obtaining parametrized solutions of the underlying ODE and fitting them against the specified boundary conditions (which may be quite expensive), the problem is interpreted as an operator inversion problem in a suitable Banach space setting. Using the concept of the oblique Moore—Penrose inverse, it is possible to transform the inversion problem into a system of operator equations that can be attacked by virtue of non-commutative Gröbner bases. The resulting operator solution can be represented as an integral operator having the classical Green’s function as its kernel. Although, at this stage of research, we cannot yet give an algorithmic formulation of the method and its domain of admissible inputs, we do believe that it has promising perspectives of automation and generalization; some of these perspectives are discussed.     

A Mehrotra-Type Predictor-Corrector Algorithm for P*(κ) Linear Complementarity Problems

Mehrotra-type predictor-corrector algorithm,as one of most efficient interior point methods,has become the backbones of most optimization packages.Salahi et al.proposed a cut strategy based algorithm for linear optimization that enjoyed polynomial complexity and maintained its efficiency in practice.We extend their algorithm to P_*（κ）linear complementarity problems.The way of choosing corrector direction for our algorithm is different from theirs. The new algorithm has been proved to have an ο(（1+4κ）（17+19κ） √（1+2κn）^3/2log[（x⁰）^Ts⁰/ε] worst case iteration complexity bound.An numerical experiment verifies the feasibility of the new algorithm.     

The Averaged Robbins – Monro Method for Linear Problems in a Banach Space

We consider a recursive method of Robbins–Monro type to solve the linear problem Ax=V in a Banach space. The bounded linear operator A and the vector V are assumed to be observable with some noise only. According to Polyak and Ruppert we use gains converging to zero slower than 1/n and take the average of the iterates as an estimator for the solution of the linear problem. Under weak conditions on the noise processes almost sure and distributional invariance principles are shown.     

Analysis of Linear Triangular Elements for Convection-diffusion Problems by Streamline Diffusion Finite Element Methods

This paper is devoted to studying the superconvergence of streamline diffusion finite element methods for convection-diffusion problems.In [8],under the condition thatε≤h~2 the optimal finite element error estimate was obtained in L~2-norm.In the present paper,however,the same error estimate result is gained under the weaker condition thatε≤h.     

Obstacle Problems with Linear Growth: Hölder Regularity for the Dual Solution

For a strictly convex integrand f : ℝⁿ → ℝ with linear growth we discuss the variational problem among mappings u : ℝⁿ ⊃ Ω → ℝ of Sobolev class W¹₁ with zero trace satisfying in addition u ≥ ψ for a given function ψ such that ψ|_∂Ω < 0. We introduce a natural dual problem which admits a unique maximizer σ. In further sections the smoothness of σ is investigated using a special J-minimizing sequence with limit u* ∈ C^1,α (Ω) for which the duality relation holds.     

On the Analyticity of Underlying HKM Paths for Monotone Semidefinite Linear Complementarity Problems

An interior point method (IPM) defines a search direction at an interior point of the feasible region. These search directions form a direction field, which in turn defines a system of ordinary differential equations (ODEs). The solutions of the system of ODEs are called off-central paths, underlying paths lying in the interior of the feasible region. It is known that not all off-central paths are analytic, whether w.r.t. μ or , where μ represents the duality gap, at a solution of a given semidefinite linear complementarity problem, SDLCP (Sim and Zhao, Math. Program. 110:475–499, 2007). In Sim and Zhao (J. Optim. Theory Appl. 137:11–25, 2008), we give a necessary and sufficient condition for when an off-central path is analytic as a function of at a solution of a general SDLCP. It is then natural to ask about the analyticity of a SDLCP off-central path at a solution, as a function of μ. We investigate this in the current paper. Again, we work under the assumption that the given SDLCP satisfies strict complementarity condition.     

Admissible Linear Estimation in Linear Models

ＡｄｍｉｓｓｉｂｌｅＬｉｎｅａｒＥｓｔｉｍａｔｉｏｎｉｎＬｉｎｅａｒＭｏｄｅｌｓＬｕＣｈａｎｇｙｕ（鹿长余）ａｎｄＺｈｕＸｉａｎｈａｉ（朱显海）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＮｏｒｔｈｅａｓｔＮｏｒｍａｌＵｎｉｖｅｒｓｉｔｙ，Ｃｈ...     

Existence of Multiple Positive Solutions of Inhomogeneous Semi–Linear Elliptic Problems Involving Critical Exponents ∗

We consider the obstacle problem for the degenerate Monge-Ampére equation. We prove the existence of the greatest viscosity sub-solution u(x) below a given obstacle φ(x), and its C ^{1, 1}-regularity which is optimal. Then the solution satisfies the concave uniformly elliptic equation if it doesn't touch the obstacle. We use the author's previous work to show the C ^{1, α}-regularity of the free boundary, ?{u(x) = φ(x)}. Finally, we discuss the stability of this free boundary.     

Existence’s Results for Parabolic Problems Related to Fully Non Linear Operators Degenerate or Singular

In this paper we prove some existence and regularity results concerning parabolic equations

$ u_t = F(x, \nabla u, D^2 u) + f(x,t) $

with some boundary conditions, on Ω×]0, T[, where Ω is some bounded domain which possesses the exterior cone property and F is some fully nonlinear elliptic operator, singular or degenerate.

     

A Note on the Paper “A Simplified Novel Technique for Solving Fully Fuzzy Linear Programming Problems”

In this note, it is shown that there is error in the existing technique (Khan, I.U., Ahmad, T., Maan, N: A simplified novel technique for solving fully fuzzy linear programming problems. J. Optim. Theory Appl. 159:536–546, 2013).     

A Petrov－Galerkin Method with Linear Trial and Quadratic Test Spaces for Parabolic Convection－Diffusion Problems

ＡＰｅｔｒｏｖ－ＧａｌｅｒｋｉｎＭｅｔｈｏｄｗｉｔｈＬｉｎｅａｒＴｒｉａｌａｎｄＱｕａｄｒａｔｉｃＴｅｓｔＳｐａｃｅｓｆｏｒＰａｒａｂｏｌｉｃＣｏｎｖｅｃｔｉｏｎ－ＤｉｆｆｕｓｉｏｎＰｒｏｂｌｅｍｓＺｏｕＹｏｎｇｋｕｉ（邹永魁）；ＷｕＷｅｉ（吴微）；...     

Asymptotic Behavior of Underlying NT Paths in Interior Point Methods for Monotone Semidefinite Linear Complementarity Problems

An interior point method (IPM) defines a search direction at each interior point of the feasible region. These search directions form a direction field, which in turn gives rise to a system of ordinary differential equations (ODEs). Thus, it is natural to define the underlying paths of the IPM as solutions of the system of ODEs. In Sim and Zhao (Math. Program. Ser. A 110:475–499, 2007), these off-central paths are shown to be well-defined analytic curves and any of their accumulation points is a solution to the given monotone semidefinite linear complementarity problem (SDLCP). In Sim and Zhao (Math. Program. Ser. A 110:475–499, 2007; J. Optim. Theory Appl. 137:11–25, 2008) and Sim (J. Optim. Theory Appl. 141:193–215, 2009), the asymptotic behavior of off-central paths corresponding to the HKM direction is studied. In particular, in Sim and Zhao (Math. Program. Ser. A 110:475–499, 2007), the authors study the asymptotic behavior of these paths for a simple example, while, in Sim and Zhao (J. Optim. Theory Appl. 137:11–25, 2008) and Sim (J. Optim. Theory Appl. 141:193–215, 2009), the asymptotic behavior of these paths for a general SDLCP is studied. In this paper, we study off-central paths corresponding to another well-known direction, the Nesterov-Todd (NT) direction. Again, we give necessary and sufficient conditions for these off-central paths to be analytic w.r.t. \(\sqrt{\mu}\) and then w.r.t. μ, at solutions of a general SDLCP. Also, as in Sim and Zhao (Math. Program. Ser. A 110:475–499, 2007), we present off-central path examples using the same SDP, whose first derivatives are likely to be unbounded as they approach the solution of the SDP. We work under the assumption that the given SDLCP satisfies a strict complementarity condition.     

IMO 1996 Problems

~~IMO 1996 Problems@无边~~     

IMO 2000 Problems

~~IMO 2000 Problems@无边~~