An improved Perry conjugate gradient method with adaptive parameter choice

In this paper, we present a conjugate gradient method for solving unconstrained optimization problems. Motivated by Perry conjugate gradient method and Dai-Liao method, an improved Perry update matrix is proposed to overcome the non-symmetric positive definite property of the Perry matrix. The parameter in the update matrix is determined by minimizing the condition number of the iterative matrix which can ensure the positive definite property. The obtained method can also be considered as a modified form of CG-DESCENT method with an adjusted term. Under some mild conditions, the presented method is global convergent. Numerical experiments under CUTEst environment show that the proposed algorithm is promising.     

On the choice of parameters for the weighting method in vector optimization

We present a geometrical interpretation of the weighting method for constrained (finite dimensional) vector optimization. This approach is based on rigid movements which separate the image set from the negative of the ordering cone. We study conditions on the existence of such translations in terms of the boundedness of the scalar problems produced by the weighting method. Finally, using recession cones, we obtain the main result of our work: a sufficient condition under which weighting vectors yield solvable scalar problems. An erratum to this article can be found at     

Bootstrap choice of tuning parameters

Consider the problem of estimating θ=θ(P) based on datax _n from an unknown distributionP. Given a family of estimatorsT _{n, β} of θ(P), the goal is to choose β among β∈I so that the resulting estimator is as good as possible. Typically, β can be regarded as a tuning or smoothing parameter, and proper choice of β is essential for good performance ofT _{n, β} . In this paper, we discuss the theory of β being chosen by the bootstrap. Specifically, the bootstrap estimate of β, , is chosen to minimize an empirical bootstrap estimate of risk. A general theory is presented to establish the consistency and weak convergence properties of these estimators. Confidence intervals for θ(P) based on , are also asymptotically valid. Several applications of the theory are presented, including optimal choice of trimming proportion, bandwidth selection in density estimation and optimal combinations of estimates.     

A characterization of signed graphs with generalized perfect elimination orderings

An important property of chordal graphs is that these graphs are characterized by the existence of perfect elimination orderings on their vertex sets. In this paper, we generalize the notion of perfect elimination orderings to signed graphs, and give a characterization for graphs admitting such orderings, together with characterizations restricted to some subclasses and further properties of those graphs. The definition of our generalized perfect elimination orderings is motivated by a generalization of the classical result that a so-called graphic hyperplane arrangement is free if and only if the corresponding graph is chordal.     

A generalization of simplicial elimination orderings

We consider the class of graphs where every induced subgraph possesses a vertex whose neighborhood has no P₄ and no 2K₂. We prove that Berge's Strong Perfect Graph Conjecture holds for such graphs. The class generalizes several well-known families of perfect graphs, such as triangulated graphs and bipartite graphs. Testing membership in this class and computing the maximum clique size for a graph in this class is not hard, but finding an optimal coloring is NP-hard. We give a polynomial-time algorithm for optimally coloring the vertices of such a graph when it is perfect. © 1996 John Wiley & Sons, Inc.     

Multiple nonlinear regressions derived with choice of free parameters

The common procedure of nonlinear approximation using linear multiple regressions developed in logarithmic space suffers from a number of inherent defects. Nonlinear approximation with choice of free parameters allows us to derive multiple regressions of varied structure, consisting of several members and an intercept. The calculations required for the proposed approximation are simple and can be implemented using standard statistical subprograms. More than 50 equations of the proposed type have been fitted for various purposes. The coefficient of determination is generally higher than 0.90. The application of the proposed method is demonstrated with two numerical examples and a number of equations derived for practical application in hydrology, geomorphology and water pollution studies. The proposed method can be used in any area of study where nonlinear multiple approximations are applicable.     

A choice of forcing terms in inexact Newton method

Inexact Newton method is one of the effective tools for solving systems of nonlinear equations. In each iteration step of the method, a forcing term, which is used to control the accuracy when solving the Newton equations, is required. The choice of the forcing terms is of great importance due to their strong influence on the behavior of the inexact Newton method, including its convergence, efficiency, and even robustness. To improve the efficiency and robustness of the inexact Newton method, a new strategy to determine the forcing terms is given in this paper. With the new forcing terms, the inexact Newton method is locally Q-superlinearly convergent. Numerical results are presented to support the effectiveness of the new forcing terms.     

A practical method for solving large-scale TRS

We present a nearly-exact method for the large scale trust region subproblem (TRS) based on the properties of the minimal-memory BFGS method. Our study is concentrated in the case where the initial BFGS matrix can be any scaled identity matrix. The proposed method is a variant of the Moré–Sorensen method that exploits the eigenstructure of the approximate Hessian B, and incorporates both the standard and the hard case. The eigenvalues of B are expressed analytically, and consequently a direction of negative curvature can be computed immediately by performing a sequence of inner products and vector summations. Thus, the hard case is handled easily while the Cholesky factorization is completely avoided. An extensive numerical study is presented, for covering all the possible cases arising in the TRS with respect to the eigenstructure of B. Our numerical experiments confirm that the method is suitable for very large scale problems.     

A note on the choice of control parameters in full-step iterative processes

Translated from Issledovaniya po Prikladnoi Matematike, No. 10, pp. 90–94, 1984.     

A DIRECT METHOD IN OPTIMAL PORTFOLIO AND CONSUMPTION CHOICE

In this paper, we use a direct method to solve the optimal portfolio and consumption choice problem in the security market for a specific case, in which the utility function is of a given homogenous form, i.e. the so-called CRRA case. The idea comes from the completion technique ever used in LQ optimal control.     

A family of kurtosis orderings for multivariate distributions

In this paper, a family of kurtosis orderings for multivariate distributions is proposed and studied. Each ordering characterizes in an affine invariant sense the movement of probability mass from the “shoulders” of a distribution to either the center or the tails or both. All even moments of the Mahalanobis distance of a random vector from its mean (if exists) preserve a subfamily of the orderings. For elliptically symmetric distributions, each ordering determines the distributions up to affine equivalence. As applications, the orderings are used to study elliptically symmetric distributions. Ordering results are established for three important families of elliptically symmetric distributions: Kotz type distributions, Pearson Type VII distributions, and Pearson Type II distributions.     

A practical method of determining project risk contingency budgets

Given a collection of accepted risks with corresponding impacts and probabilities over the life of a project (or a relevant portion), a method to estimate the total potential impact at a given certainty is presented. Project sponsors and managers can decide their risk tolerance and set aside corresponding contingency funds. This analytic method uses a binomial distribution with a probability equal to the risks’ average probability. At a desired statistical confidence level, the number of risks to consider is determined, and risks are ranked in descending order of either impact or expected value of impact. The ranked risks’ impacts are then summed over the specified number of risks, leading to an appropriate contingency budget. The budget found with this method compares extremely well with the one ascertained from numerical simulations of the risk occurrences.     

A note on geometric constructions of bi-invariant orderings

We construct bi-invariant total orderings of residually torsion-free nilpotent groups by using Chen's iterated integrals. This construction can be seen as a generalization of the Magnus ordering of the free groups, and equivalent to the classical construction which uses an iteration of central extensions. Our geometric construction provides a connection between bi-orderings and the rational homotopy theory.     

GEM: A novel evolutionary optimization method with improved neighborhood search

A new optimization technique called Grenade Explosion Method (GEM) is introduced and its underlying ideas, including the concept of Optimal Search Direction (OSD), are elaborated. The applicability and efficiency of the technique is demonstrated using standard benchmark functions. Comparison of the results with those of other, widely-used, evolutionary algorithms shows that the proposed algorithm outperforms its rivals both in the success rate and rate of convergence. The method is also shown to be capable of finding most, or even all, optima of functions having multiple global optima. Moreover, it is shown that the performance of GEM is invariant against shifting and scaling of the search space and objective function.     

A new method for solving differential equations with vague parameters

In the literature, it is pointed out that it is better to use vague sets instead of fuzzy sets. Several authors have proposed different methods for solving such differential equations in which all the parameters are represented by fuzzy numbers but to the best of our knowledge till now no one have represented the same as vague sets. In this paper, a new representation of (α, β)-cut, named as JMD (α, β)-cut, is proposed and with the help of JMD (α, β)-cut a new method is proposed to find the analytical solution of vague differential equations. To show the application of proposed method in real life problems the vague Kolmogorov’s differential equations, obtained by using vague Markov model of piston manufacturing system, are solved by proposed method. Also, to show the advantage of JMD (α, β)-cut over existing (α, β)-cut the same vague Kolmogorov’s differential equations are solved by using the proposed method with the help of existing (α, β)-cut and it is shown that the obtained results are not necessarily vague sets while the results, obtained by using JMD (α, β)-cut, are always vague sets.     

A practical termination criterion for the conjugate gradient method

The conjugate gradient method for the iterative solution of a set of linear equationsAx=b is essentially equivalent to the Lanczos method, which implies that approximations to certain eigen-values ofA can be obtained at low cost. In this paper it is shown how the smallest active eigenvalue ofA can be cheaply approximated, and the usefulness of this approximation for a practical termination criterion for the conjugate gradient method is studied. It is proved that this termination criterion is reliable in many relevant situations.     

A note on number of orderings of higher level

The main aim of the paper is to find an explicit formula for the number of orderings of higher level. Among others, we discuss the relationship between the number of orderings of higher level and the number of orderings of level 1. We also construct a field with a given possible number of orderings of higher level. Received: 10 December 2004