Feasibility and Structural Feature on Monotone Second-Order Cone Linear Complementarity Problems in Hilbert Space

Given a real finite-dimensional or infinite-dimensional Hilbert space H with a Jordan product, the second-order cone linear complementarity problem(SOCLCP)is considered. Some conditions are investigated, for which the SOCLCP is feasible and solvable for any element q?H. The solution set of a monotone SOCLCP is also characterized. It is shown that the second-order cone and Jordan product are interconnected.

EXTENSION OF SMOOTHING FUNCTIONS TO SYMMETRIC CONE COMPLEMENTARITY PROBLEMS

The paper uses Euclidean Jordan algebras as a basic tool to extend smoothing functions, which include the Chen-Mangasarian class and the Fischer-Burmeister smoothing functions, to symmetric cone complementarity problems. Computable formulas for these functions and their Jacobians are derived. In addition, it is shown that these functions are Lipschitz continuous with respect to parameter # and continuously differentiable on J × J for any μ 〉 0.

A New Generalized FB Complementarity Function for Symmetric Cone Complementarity Problems

We establish that the generalized Fischer-Burmeister（FB） function and penalized Generalized Fischer-Burmeister （FB） function defined on symmetric cones are complementarity functions （C-functions）, in terms of Euclidean Jordan algebras, and the Generalized Fischer-Burmeister complementarity function for the symmetric cone complementarity problem （SCCP）. It provides an affirmative answer to the open question by Kum and Lim （Kum S H, Lim Y. Penalized complementarity functions on symmetric cones. J. Glob. Optim.. 2010, 46： 475-485） for any positive integer.

POSITIVE SOLUTIONS OF BOUNDARY VALUE PROBLEMS FOR SECOND-ORDER SINGULAR NONLINEAR DIFFERENTIAL EQUATIONS

New existence results are presented for the singular second-order nonlinear boundary value problems u ' + g(t)f(u) = 0, 0 < t < 1, au(0) - betau ' (0) = 0, gammau(1) + deltau ' (1) = 0 under the conditions 0 less than or equal to f(0)(+) < M-1, m(1) < f(infinity)(-)less than or equal to infinity or 0 less than or equal to f(infinity)(+)< M-1, m(1) < f (-)(0)less than or equal to infinity where f(0)(+) = lim(u -->0)f(u)/u, f(infinity)(-)= lim(u --> infinity)f(u)/u, f(0)(-)= lim(u -->0)f(u)/u, f(infinity)(+) = lim(u --> infinity)f(u)/u, g may be singular at t = 0 and/or t = 1. The proof uses a fixed point theorem in cone theory.

Generalized Strongly Nonlinear Quasi-Complementarity Problems

Using the algorithm in this paper, we prove the existence of solutions to the gene-ralized strongly nonlinear quasi-complementarity problems and the convergence of theiterative sequences generated by the algorithm. Our results improve and extend thecorresponding results of Noor and Chang-Huang. Moreover, a more general iterativealgorithm for finding the approximate solution of generalized strongly nonlinear quasi-complementarity problems is also given. It is shown that the approximate solution ob-tained by the iterative scheme converges to the exact solution of this quasi-com-plementarity problem.

A New Class of Complementarity Function and the Boundedness of Its Merit Function for Symmetric Cone Complementarity Problem

In this paper, we introduce a new class of two-parametric penalized function,which includes the penalized minimum function and the penalized Fischer-Burmeister function over symmetric cone complementarity problems. We propose that this class of function is a class of complementarity functions(C-function). Moreover, its merit function has bounded level set under a weak condition.

Positive Solutions for a Class of Second-order m-point Boundary Value Problems

Using a fixed point theorem in cones, the paper consider the existence of positive solutions for a class of second-order m-point boundary value problem. Sufficient conditions to ensure the existence of double positive solutions are obtained. The associated Green function of this problem is also given.

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

In this paper,we prove existence results of soutions for the nonlinear implicit complementarity problems NICP(T,S,K) where K is a closed weakly locally compact convex cone in a reflexive Banach space E,T is a nonlinear operator from K into E* (i. e.,the dual space of E) and S is a nonlinear operator from K into E. Our results are the essential improvements and extension of the results obtained previously by several authors including Thera,Ding,and Zeng.

A New Complementarity Function and Applications in Stochastic Second-Order Cone Complementarity Problems

This paper considers the so-called expected residual minimization(ERM)formulation for stochastic second-order cone complementarity problems,which is based on a new complementarity function called termwise residual complementarity function associated with second-order cone.We show that the ERM model has bounded level sets under the stochastic weak R0-property.We further derive some error bound results under either the strong monotonicity or some kind of constraint qualifications.Then,we apply the Monte Carlo approximation techniques to solve the ERM model and establish a comprehensive convergence analysis.Furthermore,we report some numerical results on a stochastic second-order cone model for optimal power flow in radial networks.

A Class of Second-Order Cone Eigenvalue Complementarity Problems for Higher-Order Tensors

In this paper,we consider the second-order cone tensor eigenvalue complementarity problem(SOCTEiCP)and present three different reformulations to the model under consideration.Specifically,for the general SOCTEiCP,we first show its equivalence to a particular variational inequality under reasonable conditions.A notable benefit is that such a reformulation possibly provides an efficient way for the study of properties of the problem.Then,for the symmetric and sub-symmetric SOCTEiCPs,we reformulate them as appropriate nonlinear programming problems,which are extremely beneficial for designing reliable solvers to find solutions of the considered problem.Finally,we report some preliminary numerical results to verify our theoretical results.

Convergence of a smoothing algorithm for symmetric cone complementarity problems with a nonmonotone line search

In this paper, we propose a smoothing algorithm for solving the monotone symmetric cone complementarity problems (SCCP for short) with a nonmonotone line search. We show that the nonmonotone algorithm is globally convergent under an assumption that the solution set of the problem concerned is nonempty. Such an assumption is weaker than those given in most existing algorithms for solving optimization problems over symmetric cones. We also prove that the solution obtained by the algorithm is a maximally complementary solution to the monotone SCCP under some assumptions.

GUS-property for Lorentz cone linear complementarity problems on Hilbert spaces

Given a real(finite-dimensional or infinite-dimensional) Hilbert space H with a Jordan product,we consider the Lorentz cone linear complementarity problem,denoted by LCP(T,Ω,q),where T is a continuous linear operator on H,ΩH is a Lorentz cone,and q ∈ H.We investigate some conditions for which the problem concerned has a unique solution for all q ∈ H(i.e.,T has the GUS-property).Several sufficient conditions and several necessary conditions are given.In particular,we provide two suficient and necessary conditions of T having the GUS-property.Our approach is based on properties of the Jordan product and the technique from functional analysis,which is different from the pioneer works given by Gowda and Sznajder(2007) in the case of finite-dimensional spaces.

Existence of Positive Solutions for Singular Second-order m-Point Boundary Value Problems

The singular second-order m-point boundary value problem , is considered under some conditions concerning the first eigenvalue of the relevant linear operators, where (Lϕ)(x) = (p(x)ϕ′(x))′ + q(x)ϕ(x) and ξ i ∈ (0, 1) with 0 【 ξ1 【 ξ2 【 · · · 【 ξ m−2 【 1, a i ∈ [0, ∞). h(x) is allowed to be singular at x = 0 and x = 1. The existence of positive solutions is obtained by means of fixed point index theory. Similar conclusions hold for some other m-point boundary value conditions.

POSITIVE SOLUTIONS OF BOUNDARY VALUE PROBLEMS FOR SECOND-ORDER DELAY DIFFERENTIAL EQUATION

In this paper, we discuss the existence of positive solutions of the secondorder delay boundary value problems. By applying the fixed-point theorem in a cone, we show the existence of at least one positive solution with singularity and the superlinear semipositone. As an demonstrate our results. application, we also give some examples todemonstrate our results.

Nonsingularity in second-order cone programming via the smoothing metric projector

Based on the differential properties of the smoothing metric projector onto the second-order cone,we prove that,for a locally optimal solution to a nonlinear second-order cone programming problem,the nonsingularity of the Clarke's generalized Jacobian of the smoothing Karush-Kuhn-Tucker system,constructed by the smoothing metric projector,is equivalent to the strong second-order sufficient condition and constraint nondegeneracy,which is in turn equivalent to the strong regularity of the Karush-Kuhn-Tucker point.Moreover,this nonsingularity property guarantees the quadratic convergence of the corresponding smoothing Newton method for solving a Karush-Kuhn-Tucker point.Interestingly,the analysis does not need the strict complementarity condition.

二阶锥线性互补问题的两种新光滑型算法

本文研究二阶锥线性互补问题的两种低阶罚函数光滑型算法.利用核函数卷积积分为正函数和负函数生成光滑函数的方法,提出了两种新的光滑函数,并利用光滑牛顿法进行数值实验,获得了当罚参数趋于无穷大、光滑参数单调下降趋于零时,低阶罚函数方程组解序列在特定条件下收敛于二阶锥线性互补问题解的结果.通过数值实验将新提出的光滑函数与原有的光滑函数进行性能比较,结果表明新光滑函数之一具有更好的数值性能,这推广了投影函数的光滑函数.

基于单参函数求解二阶锥互补问题的光滑牛顿法

结合单参函数,在CHKS函数的框架下定义一种新的二阶锥互补函数,其包含CHKS的两个正则化形式,是一类具有良好性质且应用更加广泛的互补函数.基于该函数提出了一种求解二阶锥互补问题的有效算法,给出了算法的适定性分析以及全局收敛性证明,并进行了数值验证.

Complementarity Properties of the Lyapunov Transformation over Symmetric Cones

The well-known Lyapunov＇s theorem in matrix theory/continuous dynamical systems as- serts that a square matrix A is positive stable if and only if there exists a positive definite matrix X such that AX＋XA^＊ is positive definite. In this paper, we extend this theorem to the setting of any Euclidean Jordan algebra V. Given any element a E V, we consider the corresponding Lyapunov transformation La and show that the P and S-properties are both equivalent to a being positive. Then we characterize the R0-property for La and show that La has the R0-property if and only if a is invertible. Finally, we provide La with some characterizations of the E0-property and the nondegeneracy property.

Banach空间中的相补问题

本文在Ｂａｎａｃｈ空间中研究了三类相补问题解的存在性。所得结果是［４，５，６，９，１１－１４］中相应结果的深入和发展。

对称锥互补问题

对称锥互补问题是一类均衡优化,包括标准互补问题、二阶锥互补问题和半定互补问题等,近几年,人们借助欧几里德若当代数技术,在对称锥互补问题的研究方面获得了突破性进展并使之逐渐受到重视,本文主要从理论和算法两方面总结和评述这些新成果,同时,列出了相应的重要文献。