A quasi-Newton method can be obtained from a method of conjugate directions

In a recent paper McCormick and Ritter consider two classes of algorithms, namely methods of conjugate directions and quasi-Newton methods, for the problem of minimizing a function ofn variablesF(x). They show that the former methods possess ann-step superlinear rate of convergence while the latter are every step superlinear and therefore inherently superior. In this paper a simple and computationally inexpensive modification of a method of conjugate directions is presented. It is shown that the modified method is a quasi-Newton method and is thus every step superlinearly convergent. It is also shown that under certain assumptions on the second derivatives ofF the rate of convergence of the modified method isn-step quadratic.This work was supported by the National Research Council of Canada under Research Grant A8189.     

A nonmonotone truncated Newton–Krylov method exploiting negative curvature directions, for large scale unconstrained optimization

We propose a new truncated Newton method for large scale unconstrained optimization, where a Conjugate Gradient (CG)-based technique is adopted to solve Newton’s equation. In the current iteration, the Krylov method computes a pair of search directions: the first approximates the Newton step of the quadratic convex model, while the second is a suitable negative curvature direction. A test based on the quadratic model of the objective function is used to select the most promising between the two search directions. Both the latter selection rule and the CG stopping criterion for approximately solving Newton’s equation, strongly rely on conjugacy conditions. An appropriate linesearch technique is adopted for each search direction: a nonmonotone stabilization is used with the approximate Newton step, while an Armijo type linesearch is used for the negative curvature direction. The proposed algorithm is both globally and superlinearly convergent to stationary points satisfying second order necessary conditions. We carry out a significant numerical experience in order to test our proposal.     

A note on conjugate distributions for copulas

A family of conjugated distributions for a given type of copulas is defined in this paper. Those copulas can be written as a mixture of d‐dimensional parameter exponential functions. The generalized Farlie–Gumbel–Morgenstern copula is an example of this representation. This family is used to illustrate the estimation technique with real data. Also, the applicability of Bayesian predictive approach is shown in an education policy issue by defining goals for the number of students per class that leads to improve their performance at school. Copyright © 2014 John Wiley & Sons, Ltd.     

The regularized feasible directions method for nonconvex optimization

This paper develops and studies a feasible directions approach for the minimization of a continuous function over linear constraints in which the update directions belong to a predetermined finite set spanning the feasible set. These directions are recurrently investigated in a cyclic semi-random order, where the stepsize of the update is determined via univariate optimization. We establish that any accumulation point of this optimization procedure is a stationary point of the problem, meaning that the directional derivative in any feasible direction is nonnegative. To assess and establish a rate of convergence, we develop a new optimality measure that acts as a proxy for the stationarity condition, and substantiate its role by showing that it is coherent with first-order conditions in specific scenarios. Finally we prove that our method enjoys a sublinear rate of convergence of this optimality measure in expectation.     

A Statistic for Testing the Null Hypothesis of Elliptical Symmetry

We present and study a procedure for testing the null hypothesis of multivariate elliptical symmetry. The procedure is based on the averages of some spherical harmonics over the projections of the scaled residual (1978, N. J. H. Small, Biometrika65, 657–658) of the d-dimensional data on the unit sphere of ^d. We find, under mild hypothesis, the limiting null distribution of the statistic presented, showing that, for an appropriate choice of the spherical harmonics included in the statistic, this distribution does not depend on the parameters that characterize the underlying elliptically symmetric law. We describe a bivariate simulation study that shows that the finite sample quantiles of our statistic converge fairly rapidly, with sample size, to the theoretical limiting quantiles and that our procedure enjoys good power against several alternatives.     

A minimal error conjugate gradient method for ill-posed problems

For the ill-posed operator equationTx=y in Hilbert space, we introduce a modification of the usual conjugate gradient method which minimizes the error, not the residual, at each step. Moreover, the error is minimized over the same finite-dimensional subspace that is associated with the usual method.This work was completed while the author was on leave at the University of Tennessee, Knoxville, Tennessee. Travel support from the Taft Committee and from the University of Tennessee is gratefully acknowledged.     

A Determinant Representation for the Distribution of Quadratic Forms in Complex Normal Vectors

Let the column vectors of X:: M×N, M<N, be distributed as independent complex normal vectors with the same covariance matrix Σ. Then the usual quadratic form in the complex normal vectors is denoted by Z=XLX^H where L: N×N is a positive definite hermitian matrix. This paper deals with a representation for the density function of Z in terms of a ratio of determinants. This representation also yields a compact form for the distribution of the generalized variance |Z|.     

A conjugate gradient method for the unconstrained minimization of strictly convex quadratic splines

In this paper, we show that an analogue of the classical conjugate gradient method converges linearly when applied to solving the problem of unconstrained minimization of a strictly convex quadratic spline. Since a strictly convex quadratic program with simple bound constraints can be reformulated as unconstrained minimization of a strictly convex quadratic spline, the conjugate gradient method is used to solve the unconstrained reformulation and find the solution of the original quadratic program. In particular, if the solution of the original quadratic program is nondegenerate, then the conjugate gradient method finds the solution in a finite number of iterations. This author's research is partially supported by the NASA/Langley Research Center under grant NCC-1-68 Supplement-15.     

A conjugate gradient method with descent direction for unconstrained optimization

A modified conjugate gradient method is presented for solving unconstrained optimization problems, which possesses the following properties: (i) The sufficient descent property is satisfied without any line search; (ii) The search direction will be in a trust region automatically; (iii) The Zoutendijk condition holds for the Wolfe–Powell line search technique; (iv) This method inherits an important property of the well-known Polak–Ribière–Polyak (PRP) method: the tendency to turn towards the steepest descent direction if a small step is generated away from the solution, preventing a sequence of tiny steps from happening. The global convergence and the linearly convergent rate of the given method are established. Numerical results show that this method is interesting.     

Admissible minimax estimators of a mean vector of scale mixtures of multivariate normal distributions

The problem of estimating a mean vector of scale mixtures of multivariate normal distributions with the quadratic loss function is considered. For a certain class of these distributions, which includes at least multivariate-t distributions, admissible minimax estimators are given.     

A note on characterizations of multivariate stable distributions

Several characterizations of multivariate stable distributions together with a characterization of multivariate normal distributions and multivariate stable distributions with Cauchy marginals are given. These are related to some standard characterizations of marcinkiewicz.Research supported, in part, by the Air Force Office of Scientific Research under Contract AFOSR 84-0113. Reproduction in whole or part is permitted for any purpose of the United States Government.     

Merging experts’ opinions: A Bayesian hierarchical model with mixture of prior distributions

In this paper, a general approach is proposed to address a full Bayesian analysis for the class of quadratic natural exponential families in the presence of several expert sources of prior information. By expressing the opinion of each expert as a conjugate prior distribution, a mixture model is used by the decision maker to arrive at a consensus of the sources. A hyperprior distribution on the mixing parameters is considered and a procedure based on the expected Kullback–Leibler divergence is proposed to analytically calculate the hyperparameter values. Next, the experts’ prior beliefs are calibrated with respect to the combined posterior belief over the quantity of interest by using expected Kullback–Leibler divergences, which are estimated with a computationally low-cost method. Finally, it is remarkable that the proposed approach can be easily applied in practice, as it is shown with an application.     

Interval estimates for posterior probabilities in a multivariate normal classification model

This paper is devoted to the asymptotic distribution of estimators for the posterior probability that a p-dimensional observation vector originates from one of k normal distributions with identical covariance matrices. The estimators are based on training samples for the k distributions involved. Observation vector and prior probabilities are regarded as given constants. The validity of various estimators and approximate confidence intervals is investigated by simulation experiments.     

The conjugate gradient method for computing all the extremal stationary probability vectors of a stochastic matrix

Summary The conjugate gradient method is developed for computing stationary probability vectors of a large sparse stochastic matrixP, which often arises in the analysis of queueing system. When unit vectors are chosen as the initial vectors, the iterative method generates all the extremal probability vectors of the convex set formed by all the stationary probability vectors ofP, which are expressed in terms of the Moore-Penrose inverse of the matrix (P−I). A numerical method is given also for classifying the states of the Markov chain defined byP. One particular advantage of this method is to handle a very large scale problem without resorting to any special form ofP. The Institute of Statistical Mathematics     

A sufficient descent LS conjugate gradient method for unconstrained optimization problems

In this paper, we make a modification to the Liu-Storey (LS) conjugate gradient method and propose a descent LS method. The method can generate sufficient descent directions for the objective function. This property is independent of the line search used. We prove that the modified LS method is globally convergent with the strong Wolfe line search. The numerical results show that the proposed descent LS method is efficient for the unconstrained problems in the CUTEr library.     

Unified steerable phase I-phase II method of feasible directions for semi-infinite optimization

In this paper, we complete a cycle in the construction of methods of feasible directions for solving semi-infinite constrained optimization problems. Earlier phase I-phase II methods of feasible directions used one search direction rule in all of ⁿ with two stepsize rules, one for feasible points and one for infeasible points. The algorithm presented in this paper uses both a single search direction rule and a single stepsize rule in all of ⁿ . In addition, the new algorithm incorporates a steering parameter which can be used to control the speed with which feasibility is achieved. The new algorithm is simpler to analyze and performs somewhat better than existing, first order, phase I-phase II methods. The new algorithm is globally convergent, with linear rate.The research reported herein was sponsored in part by the National Science Foundation Grant ECS-8713334, the Air Force Office of Scientific Research Contract AFOSR-86-0116, and the State of California MICRO Program Grant 532410-19900.The authors would like to thank Dr. J. Higgins for providing the C-code of Algorithm 3.1.     

A globally convergent version of the Polak-Ribière conjugate gradient method

In this paper we propose a new line search algorithm that ensures global convergence of the Polak-Ribière conjugate gradient method for the unconstrained minimization of nonconvex differentiable functions. In particular, we show that with this line search every limit point produced by the Polak-Ribière iteration is a stationary point of the objective function. Moreover, we define adaptive rules for the choice of the parameters in a way that the first stationary point along a search direction can be eventually accepted when the algorithm is converging to a minimum point with positive definite Hessian matrix. Under strong convexity assumptions, the known global convergence results can be reobtained as a special case. From a computational point of view, we may expect that an algorithm incorporating the step-size acceptance rules proposed here will retain the same good features of the Polak-Ribière method, while avoiding pathological situations. This research was supported by Agenzia Spaziale Italiana, Rome, Italy.     

二次型之比服从F—分布的充分必要条件

考察F－分布的密度和矩，本文给出了正态随机向量二次型之比服从F－分布的充分必要条件，进而给出了椭球等高随机向量二次型之比服从F分布的充分必要条件．作为应用，我们减弱了传统F－检验中对两简单子样独立性的要求．     

Asymptotic distributions of the likelihood ratio test statistics for covariance structures of the complex multivariate normal distributions

In this paper, the authors derived asymptotic expressions for the null distributions of the likelihood ratio test statistics for multiple independence and multiple homogeneity of the covariance matrices when the underlying distributions are complex multivariate normal. Also, asymptotic expressions are obtained in the non-null cases for the likelihood ratio test statistics for independence of two sets of variables and the equality of two covariance matrices. The expressions obtained in this paper are in terms of beta series. In the null cases, the accuracy of the first terms alone is sufficient for many practical purposes.     

A variational conjugate gradient method for determining the fluid velocity of a slow viscous flow

The problem considered is that of determining the fluid velocity for linear hydrostatics Stokes flow of slow viscous fluids from measured velocity and fluid stress force on a part of the boundary of a bounded domain. A variational conjugate gradient iterative procedure is proposed based on solving a series of mixed well-posed boundary value problems for the Stokes operator and its adjoint. In order to stabilize the Cauchy problem, the iterations are ceased according to an optimal order discrepancy principle stopping criterion. Numerical results obtained using the boundary element method confirm that the procedure produces a convergent and stable numerical solution.