Penalty functions,Newton's method,and quadratic programming

In this paper, the search directions computed by two versions of the sequential quadratic programming (SQP) algorithm are compared with that computed by attempting to minimize a quadratic penalty function by Newton's method, and it is shown that the differences are attributable to ignoring certain terms in the equation for the Newton correction. Since the effect of ignoring these terms may be to make the resultant direction a poor descent direction for the quadratic penalty function, it is argued that the latter is an inappropriate merit function for use with SQP. A method is then suggested by which these terms may be included without losing the benefits gained from the use of the orthogonal transformations derived from the constraints Jacobian.The authors wish to thank A. R. Conn and N. I. M. Gould for spirited discussions which took place when the second author spent some time at Waterloo, Ontario, Canada; they also thank L. C. W. Dixon for the clarifications that he suggested to the penultimate draft of this paper.     

Shrinkage estimators of the location parameter for certain spherically symmetric distributions

We consider estimation of a location vector for particular subclasses of spherically symmetric distributions in the presence of a known or unknown scale parameter. Specifically, for these spherically symmetric distributions we obtain slightly more general conditions and larger classes of estimators than Brandwein and Strawderman (1991,Ann. Statist.,19, 1639–1650) under which estimators of the formX +ag(X) dominateX for quadratic loss, concave functions of quadratic loss and general quadratic loss.Research supported by NSF grant DMS-88-22622     

The Probability Content of Cones in Isotropic Random Fields

This paper provides computable representations for the evaluation of the probability content of cones in isotropic random fields. A decomposition of quadratic forms in spherically symmetric random vectors is obtained and a representation of their moments is derived in terms of finite sums. These results are combined to obtain the distribution function of quadratic forms in spherically symmetric or central elliptically contoured random vectors. Some numerical examples involving the sample serial covariance are provided. Ratios of quadratic forms are also discussed.     

THE VARIATION OF INVARIANT LINES AND LIMIT CYCLES FOR A QUADRATIC DIFFERENTIAL SYSTEM

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一个改进的SQP型算法

本文建立非线性等式和不等式约束规划问题的一个序列二次规划(SQP)型算法.算法的每次迭代只需解一个确实可解的二次规划，然后对其解进行简单的显式校正，便可产生关于罚函数是下降的搜索方向，克服Maratos效应.在适当的假设条件下，还论证了算法的全局收敛性和超级收敛性.     

Karl Menger's contributions to social thought

The work of Karl Menger in social science is briefly surveyed, in particular in the areas of marginal utility and diminishing value, utility, and uncertainty, a logic of imperatives based on deterrence, and a theory of voluntary associations in which cohesive groups are studied combinatorially.     

Deterministic optimal control,given only a partial observation of the initial state

The deterministic linear-system, quadratic-cost optimal control problem is considered when the only state information available is a partial linear observation of the initial statex ₀. Thus, it is only known that the initial condition belongs to a particular linear variety. A control function is found which is optimal, in the sense (roughly) that (i) it can be computed using available information aboutx ₀ and (ii) no other control function which can be found using that information gives lower cost than it does for every initial condition that could have given rise to the information. The optimal control can be found easily from the conventional Riccati equation of optimal control. Applications are considered in the presence of unknown exponential disturbances and to the case with a sequence of partial state observations.     

A class of loss functions of catenary form

The catenary form of loss function is considered in the framework of Bayesian decision theory. The mathematical tractability of this form seems to be unrecognized; it contains quadratic loss as a limiting case. For various probability distributions expressions are given for posterior analysis, and limiting properties are investigated.     

A survey of multicriteria optimization or the vector maximum problem,part I: 1776–1960

In this survey, the history of the subject from 1776 until 1960 is presented. A brief biographical sketch of Vilfredo Pareto is given first. Then, the more or less simultaneous development of the concepts of utility, preference, and welfare theory follows, with results which go back to Hausdorff and Cantor. A brief discussion of the work of Borel and von Neumann as initiators of game theory is included. Each of these areas has developed enough to warrant its own survey; hence, they are reviewed here only insofar as they provide necessary foundations. Thereafter, the concepts of efficiency, vector maximum problem, and Pareto optimality are reviewed in connection with production theory, programming, and economics. The survey is presented within a unified mathematical framework, and the emphasis is on mathematical results, rather than psychological or socio-economic discussion. To enable the reader to draw conclusions without having to obtain each article himself, the results have been presented in somewhat more detail than usual.     

On Hsu's theorem in multivariate regression

The paper deals with optimal quadratic unbiased estimation of the unknown dispersion matrix in multivariate regression models without assuming normality of the errors. We show that Hsu's theorem for univariate regression models continues to multivariate models with no additional assumptions. Furthermore optimal quadratic plus linear estimating functions for regression coefficients are considered, and we investigate whether the ordinary linear estimates are the best. This leads to a new theorem which is similar to that of Hsu.