Norbert Gaffke

A cyclic projection algorithm via duality

We consider the problem of finding the projection of a given point in a Hilbert space onto the intersection of finitely many closed convex sets. A very simple iterative procedure was established by Dykstra (1983), Boyle & Dykstra (1986), and Han (1988), which employs the projection operators onto the individual sets. When looking at a dual problem, the procedure turns out to be the primal formulation of a classical method. In this way a natural explanation of the projection method and a simpler proof of its convergence are obtained. Further investigations concern the asymptotic behaviour of dual quantities to be computed and convergence rates. Applications are given to convex quadratic programming and restricted least squares estimation, nonnegative MINQUE in covariance components models, and euclidean fit of distance matrices in data analysis.

Divisor methods for proportional representation systems: An optimization approach to vector and matrix apportionment problems

When the seats in a parliamentary body are to be allocated proportionally to some given weights, such as vote counts or population data, divisor methods form a prime class to carry out the apportionment. We present a new characterization of divisor methods, via primal and dual optimization problems. The primal goal function is a cumulative product of the discontinuity points of the rounding rule. The variables of the dual problem are the multipliers used to scale the weights before they get rounded. Our approach embraces pervious and impervious divisor methods, and vector and matrix problems.

On a class of algorithms from experimental design theory

The theory of optimal (approximate) linear regression design has produced several iterative methods to solve a special type of convex minimization problems. The present paper gives a unified and extended theoretical treatment of the methods. The emphasis is on the mathematical structures relevant for the optimization process, rather than on the statistical background of experimental design. So the main body of the paper can be read independently from the experimental design context. Applications are given to a special class of extremum problems arising in statistics. The numerical results obtained indicate that the methods are of practical interest

30 Approximate designs for polynomial regression: Invariance, admissibility, and optimality

Publisher Summary This chapter focuses on approximate linear regression design. It provides an overview of the invariance and admissibility of designs, their interrelations and their implications to design optimality, including numerical algorithms. Invariance structures combined with results on admissibility provide a tool to attack optimal design problems of high dimensions, as occurring for second and third order multiple polynomial models. Invariance with respect to infinite (though compact) transformation and matrix groups usually calls for the Haar probability measures, involving deep measure theoretic results. The Karlin–Studden necessary condition on the support of an admissible design is known to become useless for regression models involving a constant term. Well known numerical algorithms for solving extremum problems in optimal design are pure gradient methods, among them the steepest descent method of Fedorov and Wynn. However, after a quick but rough approximation towards the optimum they become very inefficient. The Quasi–Newton methods of Gaffke and Heiligers provide an efficient way of computing the optimum very accurately.

First and Second Order Rotatability of Experimental Designs, Moment Matrices, and Information Surfaces.

SummaryWe place the well-known notion of rotatable experimental designs into the more general context of invariant design problems. Rotatability is studied as it pertains to the experimental designs themselves, as well as to moment matrices, or to information surfaces. The distinct aspects become visible even in the case of first order rotatability. The case of second order rotatability then is conceptually similar, but technically more involved. Our main result is that second order rotatability may be characterized through a finite subset of the orthogonal group, generated by sign changes, permutations, and a single reflection. This is a great reduction compared to the usual definition of rotatability which refers to the full orthogonal group. Our analysis is based on representing the second order terms in the regression function by a Kronecker power. We show that it is essentially the same as using the Schläflian powers, or the usual minimal set of second order monomials, but it allows a more transparent calculus.

Optimal designs in growth curve models - II Correlated model for quadratic growth : optimal designs for parameter estimation and growth prediction

This is a follow up of a recent paper on the study of the optimality aspects of linear growth models with correlated errors. In this paper, we examine optimality aspects of quadratic growth models with correlated errors and provide optimal designs for parameter estimation and growth prediction. In the former case we examine A-, D- and E-optimal designs and in the latter case we examine A-optimal designs, under two different correlation structures. In the process, we also examine the robustness of optimal designs with respect to the values of the correlation coefficient.

Computing optimal approximate invariant designs for cubic regression on multidimensional balls and cubes

Abstract We consider the problem of computing numerically optimal approximate designs for cubic multiple regression in v ≥ 2 variables under a given convex and differentiable optimality criterion, where the experimental region is either a ball or a symmetric cube, both centered at zero. We restrict to designs which are invariant w.r.t. the group of permutations and sign changes acting on the experimental region. For many optimality criteria this is justified by the fact that within the class of all designs there exists an optimal one which is invariant. In particular, this holds true for any convex and orthogonally invariant criterion, including Kiefers фp-criteria, but also for integrated variance criteria (I-criteria) and for mixtures thereof. Based on the mathematical tools developed by Gaffke and Heiligers (Metrika 42 (1995) 29–48), we show how the algorithms of Gaffke and Mathar (Optimization 24 (1992) 91–126) can be applied to compute (nearly) optimal designs. As examples, we present numerical results for (mixtures of) Kiefers фp-criteria under a fixed regression model, which show that the proposed methods work well. For the case that the experimental region is a ball (centered at zero) we also apply the algorithms to the subclass of (third order) rotatable designs, and evaluate the efficiency loss of optimal rotatable designs. Finally, we examine several strategies for obtaining exact designs from optimal approximate ones by changing the weights to integer multiples of 1 n .

Algorithms for optimal design with application to multiple polynomial regression

The approximate theory of optimal linear regression design leads to specific convex extremum problems for numerical solution. A conceptual algorithm is stated, whose concrete versions lead us from steepest descent type algorithms to improved gradient methods, and finally to second order methods with excellent convergence behaviour. Applications are given to symmetric multiple polynomial models of degree three or less, where invariance structures are utilized. A final section is devoted to the construction of efficientexact designs of sizeN from the optimal approximate designs. For the multifactor cubic model and some of the most popular optimality criteria (D-, A-, andI-criteria) fairly efficient exact designs are obtained, even for small sample sizeN.

Optimal and robust invariant designs for cubic multiple regression

Multiple polynomial regression of third order in v ≥ 2 real variables is considered, where the experimental region is a ball centered at zero or a symmetric cube centered at zero. Besides the full cubic regression we consider also symmetric submodels which are between the full quadratic and the full cubic models. The problem is to find (numerically) an optimal approximate design under a given convex and differentiable optimality criterion. The large dimension of the problem is considerably reduced by restricting to designs which are invariant w.r.t. the group of permutations and sign changes of coordinates. This is justified by the fact that many (convex) optimality criteria are invariant w.r.t. the induced matrix group. We develop the mathematical tools to apply algorittms from Gaffke & Mathar (1992) to compute (nearly) optimal designs. Numerical results are presented for mixtures of Kiefers Φ p -criteria under a fixed regression model, and for mixtures over competing models. These mixture criteria aim at designs which are robust under change of criterion or change of regression model, respectively, and are thus of considerable practical interest

Second order methods for solving extremum problems form optimal linar regression design

Extremum problems arising in optimal design for linear regression models lead to minimizing a convex function over a convex set, which is essentially given as the convex hull of a compact generator set. We assume that the generator is simple enough to allow easy minimization of linear functions. Gradient methods were considered by Gaffke & Mathar in [5]. In the present paper we establish second order methods, one of Newton- and the other of BFGS type. Applications are given firstly to optimal invariant designs for multiple cubic regression, and secondly to Bayes L-optimal design for a first order model which also appears as a dual problem to linear minimax estimation. The former is sketched only briefly, since it has been studied in greater detail in [2]