Some geometrical aspects of semidefinite linear complementarity problems

This article studies some geometrical aspects of the semidefinite linear complementarity problem (SDLCP), which can be viewed as a generalization of the well-known linear complementarity problem (LCP). SDLCP is a special case of a complementarity problem over a closed convex cone, where the cone considered is the closed convex cone of positive semidefinite matrices. It arises naturally in the unified formulation of a pair of primal-dual semidefinite programming problems. In this article, we introduce the notion of complementary cones in the semidefinite setting using the faces of the cone of positive semidefinite matrices and show that unlike complementary cones induced by an LCP, semidefinite complementary cones need not be closed. However, under R₀-property of the linear transformation, closedness of all the semidefinite complementary cones induced by L is ensured. We also introduce the notion of a principal subtransformation with respect to a face of the cone of positive semidefinite matrices and show that for a self-adjoint linear transformation, strict copositivity is equivalent to strict semimonotonicity of each principal subtransformation. Besides the above, various other solution properties of SDLCP will be interpreted and studied geometrically.     

柔性稀疏阵的稳健波束形成

本文将导向矢量失配时的稳健波束形成问题归结为二次锥规划问题,利用高效的内点法求解。该波束形成器成功地应用于存在阵元位置误差的柔性稀疏阵,相对于经典的对角线加载法、特征空间法,在不同的输入信噪比下获得了更好的输出信号干扰加噪声比。仿真结果表明对超稀疏分布的柔性阵,阵元位置误差对输出SINR起决定性影响,而阵列稀疏程度对其影响不大。     

Stereoisomeric Tris-BINOL-Menthol Bulky Monophosphites: Synthesis,Characterisation and Application in Rhodium-Catalysed Hydroformylation

Four stereoisomeric monoether derivatives, based on axially chiral (R)- or (S)-BINOL bearing a chiral (+)- or (−)-neomenthyloxy group were synthesised and fully characterised by NMR spectroscopy and X-ray crystallography. The respective tris-monophosphites were thereof prepared and fully characterised. The coordination ability of the new bulky phosphites with Rh(CO)₂(acac), was attested by ³¹P NMR, which presented a doublet in the range of δ = 120 ppm, with a ¹J(¹⁰³Rh-³¹P) coupling constant of 290 Hz. The new tris-binaphthyl phosphite ligands were further characterised by DFT computational methods, which allowed us to calculate an electronic (CEP) parameter of 2083.2 cm⁻¹ and an extremely large cone angle of 345°, decreasing to 265° upon coordination with a metal atom. Furthermore, the monophosphites were applied as ligands in rhodium-catalysed hydroformylation of styrene, leading to complete conversions in 4 h, 100% chemoselectivity for aldehydes and up to 98% iso-regioselectivity. The Rh(I)/phosphite catalytic system was also highly active and selective in the hydroformylation of disubstituted olefins, including (E)-prop-1-en-1-ylbenzene and prop-1-en-2-ylbenzene.     

Analysis of nonsmooth vector-valued functions associated with infinite-dimensional second-order cones

Given a Hilbert space H, the infinite-dimensional Lorentz/second-order cone K is introduced. For any x∈H, a spectral decomposition is introduced, and for any function f:R→R, we define a corresponding vector-valued function f^H(x) on Hilbert space H by applying f to the spectral values of the spectral decomposition of x∈H with respect to K. We show that this vector-valued function inherits from f the properties of continuity, Lipschitz continuity, differentiability, smoothness, as well as s-semismoothness. These results can be helpful for designing and analyzing solution methods for solving infinite-dimensional second-order cone programs and complementarity problems.     

GLOBAL TOPOLOGICAL PROPERTIES OF HOMOGENEOUS VECTOR FIELDS IN {\tf R}$^{^{\hbox{\normal 3}}}$

In this paper, the authors prove that the flows of homogeneous vector field $Q(x)$ at infinity are topologically equivalent to the flows of the tangent vector field $Q_T(u)$ $(u\in S^2)$ on the sphere $S^2$, and show the theorems for the global topological classification of $Q(x)$. They derive the necessary and sufficient conditions for the global asymptotic stability and the boundedness of vector field $Q(x),$ and obtain the criterion for the global topological equivalence of two homogeneous vector fields.     

On Tensor Products of l-Operators and bo-Operators

It is proved that the tensor product of two linear operators is a cone summing operator (respectively, order bounded operator) if and only if both operators are cone summing (respectively, order bounded).     

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.     

二阶积分微分方程的广义拟线性化方法

运用广义拟线性化方法研究了正规锥上的二阶积分微分方程初值问题,获得了逼近解序列一致且平方收敛的结果.     

Banach空间四阶周期边值问题正解的存在性

讨论了Banach空间E中的四阶周期边值问题:( u(4)(t)??u00(t)+′u(t)= f(t; u(t));06 t 61; u(i)(0)= u(i)(1); i =0;1;2;3正解的存在性,其中f :[0;1]￡ P ! P连续, P为E的正元锥,?;′2 R且满足0<′<(?2+2?2)2;?>?2?2;′?4+??2+1>0:通过非紧性测度的估计技巧与凝聚映射的不动点指数理论获得了该问题正解的存在性结果.     

Convergence of the Augmented Lagrangian Method for Nonlinear Optimization Problems over Second-Order Cones

The paper analyzes the rate of local convergence of the augmented Lagrangian method for nonlinear second-order cone optimization problems. Under the constraint nondegeneracy condition and the strong second order sufficient condition, we demonstrate that the sequence of iterate points generated by the augmented Lagrangian method locally converges to a local minimizer at a linear rate, whose ratio constant is proportional to 1/τ with penalty parameter τ not less than a threshold . Importantly and interestingly enough, the analysis does not require the strict complementarity condition. Supported by the National Natural Science Foundation of China under Project 10771026 and by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.