Augmented Lagrangians with Adaptive Precision Control for Quadratic Programming with Equality Constraints: Corrigendum and Addendum

In this paper we give corrections to our paper on an augmented Lagrangian type algorithm for strictly convex quadratic programming problems with equality constraints.     

Dynamical Determinants via Dynamical Conjugacies for Postcritically Finite Polynomials

We give an analogue of Levin–Sodin–Yuditskii's study of the dynamical Ruelle determinants of hyperbolic rational maps in the case of subhyperbolic quadratic polynomials. Our main tool is to reduce to an expanding situation. We do so by applying a dynamical change of coordinates on the domains of a Markov partition constructed from the landing ray at the postcritical repelling orbit. We express the dynamical determinants as the product of an (entire) determinant with an explicit expression involving the postcritical repelling orbit, thus explaining the poles in d (z).     

Non-CPMG Fast Spin Echo with full signal

The standard Fast Spin Echo sequence used in MR imaging relies on the CPMG condition. A consequence of this condition is that only one component of the transverse magnetization can be measured. To counter this, some phase modulation schemes (XY, MLEV.) for the pulse train have been proposed, but they are useful only over a very restricted range, close to pi, of the refocusing pulse rotation angle. Some other solutions not relying on phase modulation have also been suggested, but they destroy one half the available signal. Revisiting the phase modulation approach, J. Murdoch ("Second SMR Scientific Meeting," p. 1145, 1994) suggested that a quadratic phase modulation could generate a train of classical echoes. We show here that indeed a quadratic phase modulation has a very suitable property: after an adequate change of frame, the dynamic of the system composed of all the protons situated in one pixel can be seen as stationary. If the parameter of the quadratic phase modulation is well chosen, it is then possible to put the dynamic system in a combination of two suitable states and obtain a signal identical to the signal of a classical spin echo, at least for nutation of the refocusing pulse higher than, approximately, two radians.     

On a Theorem of Dubins and Freedman

Under a notion of splitting the existence of a unique invariant probability, and a geometric rate of convergence to it in an appropriate metric, are established for Markov processes on a general state space S generated by iterations of i.i.d. maps on S. As corollaries we derive extensions of earlier results of Dubins and Freedman;⁽¹⁷⁾ Yahav;⁽³⁰⁾ and Bhattacharya and Lee⁽⁶⁾ for monotone maps. The general theorem applies in other contexts as well. It is also shown that the Dubins–Freedman result on the necessity of splitting in the case of increasing maps does not hold for decreasing maps, although the sufficiency part holds for both. In addition, the asymptotic stationarity of the process generated by i.i.d. nondecreasing maps is established without the requirement of continuity. Finally, the theory is applied to the random iteration of two (nonmonotone) quadratic maps each with two repelling fixed points and an attractive period-two orbit.     

Indivisibility of Class Numbers of Real Quadratic Fields

Let D denote the fundamental discriminant of a real quadratic field, and let h(D) denote its associated class number. If p is prime, then the Cohen and Lenstra Heuristics give a probability that ph(D). If pgt;3 is prime, then subject to a mild condition, we show that

QPECgen, a MATLAB Generator for Mathematical Programs with Quadratic Objectives and Affine Variational Inequality Constraints

We describe a technique for generating a special class, called QPEC, of mathematical programs with equilibrium constraints, MPEC. A QPEC is a quadratic MPEC, that is an optimization problem whose objective function is quadratic, first-level constraints are linear, and second-level (equilibrium) constraints are given by a parametric affine variational inequality or one of its specialisations. The generator, written in MATLAB, allows the user to control different properties of the QPEC and its solution. Options include the proportion of degenerate constraints in both the first and second level, ill-conditioning, convexity of the objective, monotonicity and symmetry of the second-level problem, and so on. We believe these properties may substantially effect efficiency of existing methods for MPEC, and illustrate this numerically by applying several methods to generator test problems. Documentation and relevant codes can be found by visiting http://www.ms.unimelb.edu.au/danny/qpecgendoc.html.     

解线性不等式的神经网络

本文提出两个解线性不等式的Hopfiedl-Tank型的神经网络。第一个网络模拟同时松弛投影方法，第二个网络是二次规划方法。当线性不等式的解集非空时，这两个方法都给出该线性不等式的解。同时我们还给出了这两个网络的数值模拟。     

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