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.     

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.     

Growth Behavior of Two Classes of Merit Functions for Symmetric Cone Complementarity Problems

In the solution methods of the symmetric cone complementarity problem (SCCP), the squared norm of a complementarity function serves naturally as a merit function for the problem itself or the equivalent system of equations reformulation. In this paper, we study the growth behavior of two classes of such merit functions, which are induced by the smooth EP complementarity functions and the smooth implicit Lagrangian complementarity function, respectively. We show that, for the linear symmetric cone complementarity problem (SCLCP), both the EP merit functions and the implicit Lagrangian merit function are coercive if the underlying linear transformation has the P-property; for the general SCCP, the EP merit functions are coercive only if the underlying mapping has the uniform Jordan P-property, whereas the coerciveness of the implicit Lagrangian merit function requires an additional condition for the mapping, for example, the Lipschitz continuity or the assumption as in (45). The authors would like to thank the two anonymous referees for their helpful comments which improved the presentation of this paper greatly. The research of J.-S. Chen was partially supported by National Science Council of Taiwan.     

Hybrid Proximal-Point Methods for Common Solutions of Equilibrium Problems and Zeros of Maximal Monotone Operators

The purpose of this paper is to introduce and study two hybrid proximal-point algorithms for finding a common element of the set of solutions of an equilibrium problem and the set of solutions to the equation 0∈Tx for a maximal monotone operator T in a uniformly smooth and uniformly convex Banach space X. Strong and weak convergence results of these two hybrid proximal-point algorithms are established, respectively. The research of L.C. Ceng was partially supported by the National Science Foundation of China (10771141), Ph.D. Program Foundation of Ministry of Education of China (20070270004), Science and Technology Commission of Shanghai Municipality Grant (075105118), Innovation Program of Shanghai Municipal Education Commission (09ZZ133), and Shanghai Leading Academic Discipline Project (S30405). The research of J.C. Yao was partially supported by Grant NSC 97-2115-M-110-001. Research was carried on within the agreement between National Sun Yat-Sen University of Kaohsiung, Taiwan and Pisa University, Pisa, Italy, 2008.     

Globally Convergent Interior Point Methods for Variational Inequalities in Unbounded Sets

The finite-dimensional variational inequality problem (VIP) has been studied extensively in the literature because of its successful applications in many fields such as economics, transportation, regional science and operations research. Barker and Pang[1] have given an excellent survey of theories, methods and applications of VIPs.     

Global Saturation of Regularization Methods for Inverse Ill-Posed Problems

In this article the concept of saturation of an arbitrary regularization method is formalized based upon the original idea of saturation for spectral regularization methods introduced by Neubauer (Beiträge zur angewandten Analysis und Informatik, pp. 262–270, 1994). Necessary and sufficient conditions for a regularization method to have global saturation are provided. It is shown that for a method to have global saturation the total error must be optimal in two senses, namely as optimal order of convergence over a certain set which at the same time, must be optimal (in a very precise sense) with respect to the error. Finally, two converse results are proved and the theory is applied to find sufficient conditions which ensure the existence of global saturation for spectral methods with classical qualification of finite positive order and for methods with maximal qualification. Finally, several examples of regularization methods possessing global saturation are shown.     

Asymptotic Behavior of Bifurcation Branch of Positive Solutions for Semilinear Sturm–Liouville Problems

We consider the nonlinear eigenvalue problem

Convergence Properties of Modified and Partially-Augmented Lagrangian Methods for Mathematical Programs with Complementarity Constraints

We present new convergence properties of partially augmented Lagrangian methods for mathematical programs with complementarity constraints (MPCC). Four modified partially augmented Lagrangian methods for MPCC based on different algorithmic strategies are proposed and analyzed. We show that the convergence of the proposed methods to a B-stationary point of MPCC can be ensured without requiring the boundedness of the multipliers.     

Existence of Solutions for Nonlinear Implicit Complementarity Problems in Reflexive Banach Spaces

51.IntroductionNonlinearcomp1ementaritytheoryhasemergedasaninterestingandfascinatingbranchofapplicablemathematics.Thistheoryhasbecomearichsourceofinspirationandmotivationforscientistsandengineerstoalargenumberofproblemsarisingincontactproblemsinelasticity,fluidflowthroughporousmedia,generalequilibriumoftransportationandeconomics,optimiza-tionandcontrolproblems,etc.IthasbeenshownbyKaramardian[8jthatiftheconvexsetin-volvedinavariationalinequalityproblemandacomplmentarityproblemisaconvexcone,then…     

The Local Asymptotic Behavior of Solutions of Second Order Linear Differential Equations in a Nutshell

1.IntroductionGiven a second order linear differential equation with meromorphic coefficienta(x)y” b(x)y’ c(x)y=0 (1.1)in,a domain D the local analytic theory for such equations is well developed,see e.g.[8].If x_0 is a regular point for(1.1)then the theory advocates to utilize Taylor seriesexpansions of two linearly indepentdent solutions of(1.1).If x_0 is a regular singular pointfor(1.1),Frobenius’method tells us to follow sometimes a laborious procedure which     

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.     

Asymptotic Behavior of Solutions of Boundary-Value Problems for the Equation Δu − ku = f in a Layer

For the solutions of boundary-value problems for the equation Δu???ku?=?f in the layer $$ \varPi =\left\{ {\left( {x^{\prime},{x_n}} \right)\in {{\mathbb{R}}^n}|{x}^{\prime}\in {{\mathbb{R}}^{n-1 }},{x_n}\in \left( {a,b} \right)} \right\},\quad -\infty <a<b<+\infty, \quad n\geq 3, $$ one obtains the first term of their asymptotics at infinity.     

On the Asymptotic Behavior of Faber Polynomials for Domains With Piecewise Analytic Boundary

Let φ(z) be an analytic function on a punctured neighborhood of ∞, where it has a simple pole. The nth Faber polynomial F _n (z) (n=0,1,2,…) associated with φ is the polynomial part of the Laurent expansion at ∞ of [φ(z)] ⁿ . Assuming that ψ (the inverse of φ) conformally maps |w|>1 onto a domain Ω bounded by a piecewise analytic curve without cusps pointing out of Ω, and under an additional assumption concerning the “Lehman expansion” of ψ about those points of |w|=1 mapped onto corners of ∂ Ω, we obtain asymptotic formulas for F _n that yield fine results on the limiting distribution of the zeros of Faber polynomials.      

Modified Extragradient Methods for a System of Variational Inequalities in Banach Spaces

In this paper, we introduce a new system of general variational inequalities in Banach spaces. We establish the equivalence between this system of variational inequalities and fixed point problems involving the nonexpansive mapping. This alternative equivalent formulation is used to suggest and analyze a modified extragradient method for solving the system of general variational inequalities. Using the demi-closedness principle for nonexpansive mappings, we prove the strong convergence of the proposed iterative method under some suitable conditions.     

Asymptotic behavior for non-oscillatory solutions of difference equations with several delays in the neutral term

In this paper, we first consider difference equations with several delays in the neutral term of the form * $$\Delta\left(y_{n}+\sum_{i=1}^{L}p_{i}y_{n-{k_{i}}}-\sum_{j=1}^{M}r_{j}y_{n-{\rho_{j}}}\right)+q_{n}y_{n-\tau}=0\quad \mbox{for}\ n\in\mathbb{Z}^{+}(0),$$ study various cases of coefficients in the neutral term and obtain the asymptotic behavior for non-oscillatory solution of (*) under some hypotheses. Moreover, we consider reaction-diffusion difference equations with several delays in the neutral term of the form $$\begin{array}{l}\Delta_{1}\left(u_{n,m}+\displaystyle \sum_{i=1}^{L}p_{i}u_{n-{k_{i}},m}-\displaystyle \sum_{j=1}^{M}r_{j}u_{n-{\rho_{j}},m}\right)+q_{n,m}u_{n-\tau,m}\\[18pt]\quad {}=a^{2}\Delta_{2}^{2}u_{n+1,m-1}\end{array}$$ for (n,m)∈?⁺(0)×Ω, study various cases of coefficients in the neutral term and obtain the asymptotic behavior for non-oscillatory solution under some hypotheses.     

Linear Approximation and Asymptotic Expansion of Solutions in Many Small Parameters for a Nonlinear Kirchhoff Wave Equation with Mixed Nonhomogeneous Conditions

In this paper, we consider the following nonlinear Kirchhoff wave equation

$\left\{\begin{array}{l}u_{tt}-\frac{\partial }{\partial x}(\mu (u,\Vert u_{x}\Vert ^{2})u_{x})=f(x,t,u,u_{x},u_{t}),\quad 0

(1)

where \(\widetilde{u}_{0}\), \(\widetilde{u}_{1}\), μ, f, g are given functions and \(\Vert u_{x}\Vert ^{2}=\int_{0}^{1}u_{x}^{2}(x,t)dx.\) To the problem (1), we associate a linear recursive scheme for which the existence of a local and unique weak solution is proved by applying the Faedo–Galerkin method and the weak compact method. In particular, motivated by the asymptotic expansion of a weak solution in only one, two or three small parameters in the researches before now, an asymptotic expansion of a weak solution in many small parameters appeared on both sides of (1)₁ is studied.

     

Equations for the Missing Boundary Values in the Hamiltonian Formulation of Optimal Control Problems

Partial differential equations for the unknown final state and initial costate arising in the Hamiltonian formulation of regular optimal control problems with a quadratic final penalty are found. It is shown that the missing boundary conditions for Hamilton’s canonical ordinary differential equations satisfy a system of first-order quasilinear vector partial differential equations (PDEs), when the functional dependence of the H-optimal control in phase-space variables is explicitly known. Their solutions are computed in the context of nonlinear systems with ℝ ⁿ -valued states. No special restrictions are imposed on the form of the Lagrangian cost term. Having calculated the initial values of the costates, the optimal control can then be constructed from on-line integration of the corresponding 2n-dimensional Hamilton ordinary differential equations (ODEs). The off-line procedure requires finding two auxiliary n×n matrices that generalize those appearing in the solution of the differential Riccati equation (DRE) associated with the linear-quadratic regulator (LQR) problem. In all equations, the independent variables are the finite time-horizon duration T and the final-penalty matrix coefficient S, so their solutions give information on a whole two-parameter family of control problems, which can be used for design purposes. The mathematical treatment takes advantage from the symplectic structure of the Hamiltonian formalism, which allows one to reformulate Bellman’s conjectures concerning the “invariant-embedding” methodology for two-point boundary-value problems. Results for LQR problems are tested against solutions of the associated differential Riccati equation, and the attributes of the two approaches are illustrated and discussed. Also, nonlinear problems are numerically solved and compared against those obtained by using shooting techniques.     

Absolute Continuity and Convergence in Variation for Distributions of Functionals of Poisson Point Measure

General sufficient conditions are given for absolute continuity and convergence in variation of the distributions of the functionals on the probability space generated by a Poisson point measure. The phase space of the Poisson point measure is supposed to be of the form \mathbbR⁺×\mathbbU{\mathbb{R}}^{+}\times{\mathbb{U}}, and its intensity measure to equal dt Π(du). We introduce the family of time stretching transformations of the configurations of the point measure. Sufficient conditions for absolute continuity and convergence in variation are given in terms of the time stretching transformations and the relative differential operators. These conditions are applied to solutions of SDEs driven by Poisson point measures, including SDEs with non-constant jump rate.     

Asymptotic Behavior of Renormalization Constants in Higher Orders of the Perturbation Expansion for the (4−∈)-Dimensionally Regularized O(n)-Symmetric φ4 Theory

Higher-order asymptotic expansions for renormalization constants and critical exponents of the O(n)-symmetric ⁴ theory are found based on the instanton approach in the minimal subtraction scheme for the (4–)-dimensional regularization. The exactly known expansion terms differ substantially from their asymptotic values. We find expressions that improve the asymptotic expansions for unknown expansion terms of the renormalization constants.