一类非线性互补问题的信赖域算法

In this paper,an ODE-type trust region algorithm for solving a class of nonlinear complementarity problems is proposed.A feature of this algorithm is that only the solution of linear systems of equations is required at each iteration,thus avoiding the need for solving a quadratic subproblem with a trust region bound.Under some conditions,it is proven that this algorithm is globally and locally superlinear convergent.The limited numerical examples show its efficiency.     

广义Cauchy张量的正定性及互补问题的研究

广义Cauchy张量是由Cauchy张量推广而来,将在Cauchy张量的基础上,围绕广义Cauchy张量的正定性及共正性展开研究,并提出关于广义Cauchy张量互补问题的几个结论.     

Structural Properties of Tensors and Complementarity Problems

In this paper, one of our main purposes is to prove the boundedness of the solution set of tensor complementarity problems such that the specific bounds depend only on the structural properties of such a tensor. To achieve this purpose, firstly, we prove that this class of structured tensors is strictly semi-positive. Subsequently, the strictly lower and upper bounds of operator norms are given for two positively homogeneous operators. Finally, with the help of the above upper bounds, we show that the solution set of tensor complementarity problems has the strictly lower bound. Furthermore, the upper bounds of spectral radius are obtained, which depends only on the principal diagonal entries of tensors.     

The Existence and Uniqueness of Solution for Tensor Complementarity Problem and Related Systems

Journal of Optimization Theory and Applications - In the paper, we are concerned with the existence and uniqueness of solution for tensor complementarity problem (TCP) and tensor absolute value...     

An Iterative Method for Finding the Least Solution to the Tensor Complementarity Problem

In this paper, we are concerned with finding the least solution to the tensor complementarity problem. When the involved tensor is strongly monotone, we present a way to estimate the nonzero elements of the solution in a successive manner. The procedure for identifying the nonzero elements of the solution gives rise to an iterative method of solving the tensor complementarity problem. In each iteration, we obtain an iterate by solving a lower-dimensional tensor equation. After finitely many iterations, the method terminates with a solution to the problem. Moreover, the sequence generated by the method is monotonically convergent to the least solution to the problem. We then extend this idea for general case and propose a sequential mathematical programming method for finding the least solution to the problem. Since the least solution to the tensor complementarity problem is the sparsest solution to the problem, the method can be regarded as an extension of a recent result by Luo et al. (Optim Lett 11:471–482, 2017). Our limited numerical results show that the method can be used to solve the tensor complementarity problem efficiently.     

关于一类自由作业机器排序问题

文章研究文［１］中提出的加工时间依赖于机器的自由作业排序问题。Ｍ．Ｄｏｒｏｒ在［１］中提出了一个算法（算法３．４）。最近，Ａ．Ｊ．Ｖａｋｈａｒｉａ、Ｂ．Ｃａｔａｙ［２］及项思明、唐国春［３］均指出Ｍ．Ｄｏｒｏｒ的算法不是最优的。项思明和唐国春提出对这类问题在机器连续加工情形下的一种求解方法，即将排序问题化成指派问题。本文对这种解法作了简化，并回答文［３］中提出的几个问题。     

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 flmc- tion 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.     

On the Solution of the Inverse Eigenvalue Complementarity Problem

In this paper, we discuss the solution of an Inverse Eigenvalue Complementarity Problem. Two nonlinear formulations are presented for this problem. A necessary and sufficient condition for a stationary point of the first of these formulations to be a solution of the problem is established. On the other hand, to assure global convergence to a solution of this problem when it exists, an enumerative algorithm is designed by exploiting the structure of the second formulation. The use of additional implied constraints for enhancing the efficiency of the algorithm is also discussed. Computational results are provided to highlight the performance of the algorithm.     

一类非单调互补问题的Canonical对偶问题研究

对于一类具有广泛应用背景的非单调互补问题,我们构建了这类问题的Canonical对偶问题。其对偶问题可以写成和原问题类似的互补问题。我们给出了对偶问题和原问题解之间的对偶关系,并且将对偶问题转化成一个一维优化问题,这不但可以方便的求解这类问题,也为研究这类问题性质提供了一个非常直观的研究工具。最后,本文给出了几个算例来演示对偶问题的性质。     

On the Range of the Pseudomonotone Second-Order Cone Linear Complementarity Problem

In this paper, we provide a complete characterization of the range of the pseudomonotone second-order cone linear complementarity problem. In particular, by answering the three questions that under what conditions the range is the whole space, convex and closed, respectively, we explicitly characterize and formulate the range of the pseudomonotone second-order cone linear complementarity problem.     

Tensor Networks and Hierarchical Tensors for the Solution of High-Dimensional Partial Differential Equations

Hierarchical tensors can be regarded as a generalisation, preserving many crucial features, of the singular value decomposition to higher-order tensors. For a given tensor product space, a recursive decomposition of the set of coordinates into a dimension tree gives a hierarchy of nested subspaces and corresponding nested bases. The dimensions of these subspaces yield a notion of multilinear rank. This rank tuple, as well as quasi-optimal low-rank approximations by rank truncation, can be obtained by a hierarchical singular value decomposition. For fixed multilinear ranks, the storage and operation complexity of these hierarchical representations scale only linearly in the order of the tensor. As in the matrix case, the set of hierarchical tensors of a given multilinear rank is not a convex set, but forms an open smooth manifold. A number of techniques for the computation of hierarchical low-rank approximations have been developed, including local optimisation techniques on Riemannian manifolds as well as truncated iteration methods, which can be applied for solving high-dimensional partial differential equations. This article gives a survey of these developments. We also discuss applications to problems in uncertainty quantification, to the solution of the electronic Schrödinger equation in the strongly correlated regime, and to the computation of metastable states in molecular dynamics.     

非光滑非线性互补问题的牛顿法

研究了非光滑的非线性互补问题. 首先将非光滑的非线性互补问题转化为一个非光滑方程组,然后用牛顿法求解这个非光滑方程组. 在该牛顿法中,每次迭代只需一个原始函数B-微分中的一个元素. 最后证明了该牛顿法的超线性收敛性.     

On Minimizing and Stationary Sequences of a New Class of Merit Functions for Nonlinear Complementarity Problems

Motivated by the work of Fukushima and Pang (Ref. 1), we study the equivalent relationship between minimizing and stationary sequences of a new class of merit functions for nonlinear complementarity problems (NCP). These merit functions generalize that obtained via the squared Fischer–Burmeister NCP function, which was used in Ref. 1. We show that a stationary sequence {x^k} /Reⁿ is a minimizing sequence under the condition that the function value sequence {F(x ^k)} is bounded above or the Jacobian matrix sequence {F(x ^k)} is bounded, where F is the function involved in NCP. The latter condition is also assumed by Fukushima and Pang. The converse is true under the assumption of {F(x ^k)} bounded. As an example shows, even for a bounded function F, the boundedness of the sequence {F(x ^k)} is necessary for a minimizing sequence to be a stationary sequence.     

A power penalty approach to a Nonlinear Complementarity Problem

We propose a novel power penalty approach to a Nonlinear Complementarity Problem (NCP) in which the NCP is approximated by a nonlinear equation containing a power penalty term. We show that the solution to the penalty equation converges to that of the NCP at an exponential rate when the function involved is continuous and ξ-monotone. A higher convergence rate is also obtained when the function becomes Lipschitz continuous. Numerical results are presented to confirm the theoretical findings.     

Copositivity for a Class of Fourth-Order Symmetric Tensors Given by Scalar Dark Matter

Journal of Optimization Theory and Applications - The mathematical model of general scalar potentials may be written as a fourth-order symmetric tensor with a particular structure in particle...     

一类非线性渗流方程的Cauchy问题

讨论具有强非线性源和对流项的一般渗流方程以ＲＮ中某有界连续函数ｕ０（ｘ）或某一Ｒａｄｏｎ测度为初值的Ｃａｕｃｈｙ问题弱解的存在性，得到关于解的一系列重要估计．     

On a Class of Population Dynamics Problem with Mixed Boundary Conditions

In this paper, we consider a linear population dynamics model with age dependence and spatial structure in which birth process is described by a nonlocal term. Carleman estimate for the backward nonlocal adjoint problem providing the observability results is fundamental to the analysis and by duality it provides the controllability result. More precisely, we observe that one can expect to compute the initial distribution using observations on a small part of the boundary.