Search

Point load strength test is one of the indirect methods to estimate the uniaxial compressive strength of rocks. Due to its ease of sample preparation, low cost, and simple testing, it has been widely used in rock engineering in the world. The common calculation method of point load strength index is the method of equivalent core diameter of minimum cross-section, followed by the method of actual fracture section and loading distance method. However, because there are different factors being considered in different methods, the point load strength indexes obtained from different formulas varies greatly, which will result in large errors when estimating the compressive strength of rocks. In this study, a large number of point load strength tests and uniaxial compression tests were carried out using four kinds of rock samples. Then, a series of statistical analyses were conducted based on the test data. The results indicated that the distance between two loading points and the width of actual fracture section played an important and non-negligible role for failure of rock specimens, but the effect of width of minimum cross-section of specimens could be ignored. Consequently, by combing the advantages of two traditional methods, a modified calculation method of PLSI was proposed. Finally, uniaxial compressive strength of four kinds of rocks estimated by different methods was compared. The results showed that the modified method had the least estimation error.

In this paper, we introduce the Hermitian completely positive (HCP) matrix, which has a Hermitian completely positive (HCP) decomposition with all real and imaginary parts of the decomposition vectors being nonnegative. Some properties of the Hermitian completely positive matrix are given. A semidefinite algorithm is also proposed for checking whether a Hermitian matrix is HCP or not. If a matrix is not HCP, a certificate for it can be obtained; if it is, an HCP decomposition can be obtained.

In this paper, we introduce the complex completely positive tensor, which has a symmetric complex decomposition with all real and imaginary parts of the decomposition vectors being non-negative. Some properties of the complex completely positive tensor are given. A semidefinite algorithm is also proposed for checking whether a complex tensor is complex completely positive or not. If a tensor is not complex completely positive, a certificate for it can be obtained; if it is complex completely positive, a complex completely positive decomposition can be obtained.

In this paper, we study the higher-degree tensor eigenvalue complementarity problem (HDTEiCP). We give an upper bound for the number of the higher-degree complementarity eigenvalues for the generic HDTEiCP. A semidefinite relaxation algorithm is proposed for computing all the higher-degree complementarity eigenvalues sequentially, as well as the corresponding eigenvectors, and the convergence of the algorithm is discussed. Some numerical results are also given.

In this paper, we study the convergence rate of the Levenberg-Marquardt (LM) method under the Holderian local error bound condition and the Holderian continuity of the Jacobian, which are more gene...

In this paper, we study the completely positive (CP) tensor program, which is a linear optimization problem with the cone of CP tensors and some linear constraints. We reformulate it as a linear program over the cone of moments, then construct a hierarchy of semidefinite relaxations for solving it. We also discuss how to find a best CP approximation of a given tensor. Numerical experiments are presented to show the efficiency of the proposed methods.

A symmetric matrix \begin{document}$A$ \end{document} is completely positive (CP) if there exists an entrywise nonnegative matrix \begin{document}$V$ \end{document} such that \begin{document}$A = VV ^T$ \end{document} . A real symmetric matrix is called completely positive separable (CPS) if it can be written as a sum of rank-1 Kronecker products of completely positive matrices. This paper studies the CPS problem. A criterion is given to determine whether a given matrix is CPS, and a specific CPS decomposition is constructed if the matrix is CPS.

This paper proposes an algorithm for computing the maximum (or minimum) complementarity eigenvalues of tensors. We formulate this as proper polynomial optimization problems that can be efficiently ...

In this paper, we introduce the CP-nuclear value of a completely positive (CP) tensor and study its properties. A semidefinite relaxation algorithm is proposed for solving the minimal CP-nuclear-value tensor recovery. If a partial tensor is CP-recoverable, the algorithm can give a CP tensor recovery with the minimal CP-nuclear value, as well as a CP-nuclear decomposition of the recovered CP tensor. If it is not CP-recoverable, the algorithm can always give a certificate for that, when it is regular. Some numerical experiments are also presented.

In this paper, we study how to compute all real solutions of the tensor complimentary problem, if there are finite many ones. We formulate the problem as a sequence of polynomial optimization probl...

This paper studies the tensor maximal correlation problem, which aims at optimizing correlations between sets of variables in many statistical applications. We reformulate the problem as an equivalent polynomial optimization problem, by adding the first order optimality condition to the constraints, then construct a hierarchy of semidefinite relaxations for solving it. The global maximizers of the problem can be detected by solving a finite number of such semidefinite relaxations. Numerical experiments show the efficiency of the proposed method.

This paper studies tensor eigenvalue complementarity problems. Basic properties of standard and complementarity tensor eigenvalues are discussed. We formulate tensor eigenvalue complementarity problems as constrained polynomial optimization. When one tensor is strictly copositive, the complementarity eigenvalues can be computed by solving polynomial optimization with normalization by strict copositivity. When no tensor is strictly copositive, we formulate the tensor eigenvalue complementarity problem equivalently as polynomial optimization by a randomization process. The complementarity eigenvalues can be computed sequentially. The formulated polynomial optimization can be solved by Lasserre's hierarchy of semidefinite relaxations. We show that it has finite convergence for general tensors. Numerical experiments are presented to show the efficiency of proposed methods., Comment: 26 pages

Grapiglia et al. (2013) proved subspace properties for the Celis-Dennis-Tapia (CDT) problem. If a subspace with lower dimension is appropriately chosen to satisfy subspace properties, then one can solve the CDT problem in that subspace so that the computational cost can be reduced. We show how to find subspaces that satisfy subspace properties for the CDT problem, by using the eigendecomposition of the Hessian matrix of the objection function. The dimensions of the subspaces are investigated. We also apply the subspace technologies to the trust region subproblem and the quadratic optimization with two quadratic constraints.

In this paper, we study subspace properties of the quadratically constrained quadratic program (QCQP). We prove that, if an appropriate subspace is chosen to satisfy subspace properties, then the solution of the QCQP lies in that subspace. So, we can solve the QCQP in that subspace rather than solve it in the original space. The computational cost could be reduced significantly if the dimension of the subspace is much smaller. We also show how to construct such subspaces and investigate their dimensions.

In this paper, we study convergence properties of the inexact Levenberg-Marquardt method under the Holderian local error bound condition and the Holderian continuity of the Jacobian. The formula of the convergence rates are given, which are functions with respect to the Levenberg-Marquardt parameter, the perturbation vector, as well as the orders of the Holderian local error bound and Holderian continuity of the Jacobian.

A symmetric tensor, which has a symmetric nonnegative decomposition, is called a completely positive tensor. In this paper, we characterize the completely positive tensor as a truncated moment sequence, and transform the problem of checking whether a tensor is completely positive to checking whether its corresponding truncated moment sequence admits a representing measure, then present a semidefinite algorithm to solve it. If a tensor is not completely positive, a certificate for it can be obtained; if it is completely positive, a nonnegative decomposition can be obtained.

In this paper, we propose a new updating rule of the Levenberg–Marquardt LM parameter for the LM method for nonlinear equations. We show that the global complexity bound of the new LM algorithm is , that is, it requires at most iterations to derive the norm of the gradient of the merit function below the desired accuracy .

In this paper, we consider the problem of computing the distance between the linear matrix pencil and the completely positive cone. We formulate it as a linear optimization problem with the cone of moments and the second order cone. A semidefinite relaxation algorithm is presented and the convergence is studied. We also propose a new model for checking the membership in the completely positive cone.

A symmetric tensor is completely positive (CP) if it is a sum of tensor powers of nonnegative vectors. This paper characterizes completely positive binary tensors. We show that a binary tensor is completely positive if and only if it satisfies two linear matrix inequalities. This result can be used to determine whether a binary tensor is completely positive or not. When it is, we give an algorithm for computing its cp-rank and the decomposition. When the order is odd, we show that the cp-rank decomposition is unique. When the order is even, we completely characterize when the cp-rank decomposition is unique. We also discuss how to compute the nearest cp-approximation when a binary tensor is not completely positive.

A matrix A is monotonically positive (MP) if there exists a matrix U such that A = U U T and each column of U is monotonically nonincreasing or nondecreasing. Following the approach in J. Nie (2014) [15] , we propose a semidefinite algorithm for checking whether or not a matrix is MP. If it is not MP, a certificate for it can be obtained; if it is MP, an MP-decomposition can be obtained. Some computational experiments are presented to show how to do this.

A symmetric matrix A is completely positive (CP) if there exists an entrywise nonnegative matrix B such that $$A = BB^T$$A=BBT. We characterize the interior of the CP cone. A semidefinite algorithm is proposed for checking whether a matrix is in the interior of the CP cone, and its properties are studied. A CP-decomposition of a matrix in Dickinson's form can be obtained if it is an interior of the CP cone. Some computational experiments are also presented.

In this paper, we present a multi-point iterative Levenberg–Marquardt algorithm for singular nonlinear equations. The algorithm converges globally and the convergence order is studied under the local error bound condition, which is weaker than the nonsingularity of the Jacobian at the solution.

A real n × n symmetric matrix P is partially positive (PP) for a given index set I ⊆ {1, ..., n} if there exists a matrix V such that V (I,:) ⩾ 0 and P = V VT. We give a characterization of PP-matrices. A semidefinite algorithm is presented for checking whether a matrix is partially positive or not. Its properties are studied. A PP-decomposition of a matrix can also be obtained if it is partially positive.

In this paper, we present a modified trust region algorithm for nonlinear equations with the trust region radii converging to zero. The algorithm calculates the Jacobian after every two computations of the step. It preserves the global convergence as the traditional trust region algorithms. Moreover, it converges nearly q-cubically under the local error bound condition, which is weaker than the nonsingularity of the Jacobian at a solution. Numerical results show that the algorithm is very efficient for both singular problems and nonsingular problems.

Researchers are often interested in testing for measurement invariance with respect to an ordinal auxiliary variable such as age group, income class, or school grade. In a factor-analytic context, these tests are traditionally carried out via a likelihood ratio test statistic comparing a model where parameters differ across groups to a model where parameters are equal across groups. This test neglects the fact that the auxiliary variable is ordinal, and it is also known to be overly sensitive at large sample sizes. In this paper, we propose test statistics that explicitly account for the ordinality of the auxiliary variable, resulting in higher power against "monotonic" violations of measurement invariance and lower power against "non-monotonic" ones. The statistics are derived from a family of tests based on stochastic processes that have recently received attention in the psychometric literature. The statistics are illustrated via an application involving real data, and their performance is studied via simulation.

In this paper we propose an accelerated version of the modified Levenberg-Marquardt method for nonlinear equations (see Jinyan Fan, Mathematics of Computation 81 (2012), no. 277, 447–466). The original version uses the addition of the LM step and the approximate LM step as the trial step at every iteration, and achieves the cubic convergence under the local error bound condition which is weaker than nonsingularity. The notable differences of the accelerated modified LM method from the modified LM method are that we introduce the line search for the approximate LM step and extend the LM parameter to more general cases. The convergence order of the new method is shown to be a continuous function with respect to the LM parameter. We compare it with both the LM method and the modified LM method; on the benchmark problems we observe competitive performance.

In this paper we propose a new LM algorithm for the singular system of nonlinear equations F(x)=0, where F(x):R^n->R^n is continuously differentiable and F^'(x) is Lipschitz continuous. At every iteration, not only a LM step but also a correction step are computed. The LM parameter is taken as @m"[email protected]?F"[email protected]?^@d, where @[email protected]?(0,2] and @m"k is updated by the trust region technique. Under the local error bound condition which is weaker than nonsingularity, we prove the convergence rate of the algorithm is min{2,[email protected]}. Numerical results show that the new LM algorithm performs very well.

In this paper, we propose a Shamanskii-like Levenberg-Marquardt method for nonlinear equations. At every iteration, not only a LM step but also m?1 approximate LM steps are computed, where m is a positive integer. Under the local error bound condition which is weaker than nonsingularity, we show the Shamanskii-like LM method converges with Q-order m+1. The trust region technique is also introduced to guarantee the global convergence of the method. Since the Jacobian evaluation and matrix factorization are done after every m computations of the step, the overall cost of the Shamanskii-like LM method is usually much less than that of the general LM method (the m=1 case).

In this paper, we propose a regularized Newton method for the system of monotone nonlinear equations. The regularization parameter is taken as the norm of the residual, and a correction step with little additional calculations is also computed to compensate for the shorter trial step due to the introduction of the regularization parameter. Under the local error bound condition which is weaker than nonsingularity, we show that the new regularized Newton method with correction has quadratic convergence. We also apply the new method to the unconstrained convex optimization problems which may have singular Hessian at the solutions and develop a globally convergent regularized Newton algorithm by using trust region technique. Numerical results show that the algorithm is very efficient and robust.

In this paper, both the constrained Levenberg-Marquardt method and the projected Levenberg-Marquardt method are presented for nonlinear equations $F(x)=0$ subject to $x\in X$, where $X$ is a nonempty closed convex set. The Levenberg-Marquardt parameter is taken as $\| F(x_k) \|_2^\delta$ with $\delta\in (0, 2]$. Under the local error bound condition which is weaker than nonsingularity, the methods are shown to have the same convergence rate, which includes not only the convergence results obtained in [12] for $\delta=2$ but also the results given in [7] for unconstrained nonlinear equations.

In this paper we study the convergence rate of the inexact Levenberg-Marquardt method for nonlinear equations. Under the local error bound condition which is weaker than nonsingularity, we derive an explicit formula of the convergence order of the inexact LM method, which is a continuous function with respect to not only the LM parameter but also the perturbation vector. The new formula includes many convergence rate results in the literature as its special cases.

In this article, we present a trust region algorithm for the nonlinear equations with a new updating rule of the trust region radius, which takes some function of the residual. We show that under the local error bound condition which is weaker than the non-singularity, the new algorithm converges quadratically to some solution of the nonlinear equations. Numerical results show that the new algorithm performs very well for some singular nonlinear equations.

In this paper, we present a new trust region algorithm for the system of singular nonlinear equations with the regularized trust region subproblem. The new algorithm preserves the global convergence of the traditional trust region algorithm, and has the quadratic convergence under some suitable conditions. Finally, some numerical results are given.

In [C. Ma, L. Jiang, Some research on Levenberg–Marquardt method for the nonlinear equations, Appl. Math. Comput. 184 (2007) 1032–1040], the LM parameter at the kth iteration is chosen as λ k = θ ‖ F k ‖ 2 + ( 1 - θ ) ‖ J k T F k ‖ 2 where F is the residual function, J is the Jacobi of F, and θ ∈ [ 0 , 1 ] is a constant. In this note, we point out that the LM parameter can be any combination of ‖ F k ‖ 2 and ‖ J k T F k ‖ 2 provided it is positive. Furthermore, we give a more general choice of the LM parameter, and show that the LM method still preserves the quadratic convergence under the local error condition which is weaker than nonsingularity.

In this paper, we present the new trust region method for nonlinear equations with the trust region converging to zero. The new method preserves the global convergence of the traditional trust region methods in which the trust region radius will be larger than a positive constant. We study the convergence rate of the new method under the local error bound condition which is weaker than the nonsingularity. An example given by Y.X. Yuan shows that the convergence rate can not be quadratic. Finally, some numerical results are given.

We characterize the optimal solution of a quadratic program over the Stiefel manifold with an objective function in trace formulation. The result is applied to relaxations of HQAP and MTLS. Finally, we show that strong duality holds for the Lagrangian dual, provided some redundant constraints are added to the primal program.

We propose a new self-adaptive Levenberg-Marquardt algorithm for the system of nonlinear equations F(x) = 0. The Levenberg-Marquardt parameter is chosen as the product of ?Fk?? with ? being a positive constant, and some function of the ratio between the actual reduction and predicted reduction of the merit function. Under the local error bound condition which is weaker than the nonsingularity, we show that the Levenberg-Marquardt method converges superlinearly to the solution for ?? (0, 1), while quadratically for ?? [1, 2]. Numerical results show that the new algorithm performs very well for the nonlinear equations with high rank deficiency.

We consider the duality theories in nonlinear semidefinite programming. Some duality theorems are established to show the important relations among the optimal solutions and optimal values of the primal, the dual and the saddle point problems of nonlinear semidefinite programming.

In this work, nonlinear complementarity problems (NCPs) are considered. Some NCPs, which come from practice, have no derivatives or it is very difficult to obtain their derivatives. Derivative-free methods are hence considered. Generalized pattern search (GPS) method, as a kind of derivative-free method, plays a very significant role. We combine generalized pattern search method with filter method to attack NCPs because one can analyze the theory of generalized pattern search filter method without sufficient reduction conditions and avoid using of derivatives. Some interesting results are obtained in this work without sufficient reduction conditions. Furthermore, generalized pattern search filter method is efficient to some special problems.

Recently Yamashita and Fukushima [11] established an interesting quadratic convergence result for the Levenberg-Marquardt method without the nonsingularity assumption. This paper extends the result of Yamashita and Fukushima by using µk = |F(xk|2-where δ ∈(1,2) instead of µk = F(xk)2 as the Levenberg-Marquardt parameter. If |F(x)| provides a local error bound for the system of nonlinear equations F(x) = 0, it is shown that the sequence {xk} generated by the new method converges to a solution quadratically, which is stronger than dist(xkċX∞) -- 0 given by Yamashita and Fukushima. Numerical results show that the method performs well for singular problems.

In this paper, we present an inexact Levenberg-Marquardt (LM) method for singular system of nonlinear equations, where the LM parameter is chosen as the norm of the function and the trial step is computed approximately. Under the local error bound condition which is weaker than the non- singularity, we show that the new inexact LM method preserves the quadratic convergence of the traditional LM method where the parameter is chosen to be larger than a positive constant and the Jacobi at the solution is nonsingular.

A symmetric matrix $A$ is completely positive (CP) if there exists an entrywise nonnegative matrix $V$ such that $A = V V ^T$. In this paper, we study the CP-matrix approximation problem of projecting a matrix onto the intersection of a set of linear constraints and the cone of CP matrices. We formulate the problem as the linear optimization with the norm cone and the cone of moments. A semidefinite algorithm is presented for the problem. A CP-decomposition of the projection matrix can also be obtained if the problem is feasible.

Although the least squares (LS) problem and the total least squares (TLS) problem have received much attention, the mixed LS-TLS problem still remains to be investigated. In this paper, we study the necessary and sufficient solvability conditions for the mixed LS-TLS problem in Frobenius norm. The solution set, the minimum Frobenius norm solution and an algorithm to obtain the solution are also given. Further we propose the mixed LS-TLS problem in spectral norm and present the solvability condition.

A symmetric matrix $C$ is completely positive (CP) if there exists an entrywise nonnegative matrix $B$ such that $C=BB^T$. The CP-completion problem is to study whether we can assign values to the missing entries of a partial matrix (i.e., a matrix having unknown entries) such that the completed matrix is completely positive. We propose a semidefinite algorithm for solving general CP-completion problems, and study its properties. When all the diagonal entries are given, the algorithm can give a certificate if a partial matrix is not CP-completable, and it almost always gives a CP-completion if it is CP-completable. When diagonal entries are partially given, similar properties hold. Computational experiments are also presented to show how CP-completion problems can be solved., Comment: arXiv admin note: text overlap with arXiv:1210.6930 by other authors

We associate a binary neighborhood system with a given binary relation, and then re-examine the structure of binary relational rough set model from the viewpoint of binary neighborhood systems. Based on the associated binary neighborhood system, we define an equivalence relation on the universe of discourse by regarding objects of the universe as equivalent if their associated binary neighborhood systems are identical. The induced equivalence relation is interpreted as the indiscernibility relation on the universe. From this view, we propose equivalent form of lower and upper approximations in binary relational rough set model in ways paralleling the corresponding definitions for Pawlak's lower and upper approximations. In addition, we extend variable precision rough set model to variable precision binary relational rough set model.

The main purpose of this paper is to extend the conventional separation theorems concerning the convex subset of R m to generalized separation theorems concerning the convex subset of R m × S n . This is accomplished by the introduction of the generalized inner product in R m × S n . Then we derive the famous Farkas’ lemma in nonlinear semidefinite programming, which may be very important for the analysis of the dual theory and optimality conditions of nonlinear semidefinite programming.