A note on the regularized proximal point algorithm

Recently, Xu (J Glob Optim 36:115–125 (2006)) introduced a regularized proximal point algorithm for approximating a zero of a maximal monotone operator. In this note, we shall prove the strong convergence of this algorithm under some weaker conditions.     

A saddle point variational formulation for projection–regularized parameter identification

Summary. This paper is concerned with the ill-posed problem of identifying a parameter in an elliptic equation and its solution applying regularization by projection. As the theory has shown, the ansatz functions for the parameter have to be sufficiently smooth. In this paper we show that these – for a practical implementation unrealistic – smoothness assumptions can be circumvented by reformulating the problem under consideration as a mixed variational equation. We prove convergence as the discretization gets finer in the noise free case and convergence as the data noise level goes to zero in the case of noisy data, as well as convergence rates under additional smoothness conditions. Received August 4, 2000 / Revised version received March 21, 2001 / Published online October 17, 2001     

Fast Givens rotations for orthogonal similarity transformations

Summary Fast Givens rotations with half as many multiplications are proposed for orthogonal similarity transformations and a matrix notation is introduced to describe them more easily. Applications are proposed and numerical results are examined for the Jacobi method, the reduction to Hessenberg form and the QR-algorithm for Hessenberg matrices. It can be seen that in general fast Givens rotations are competitive with Householder reflexions and offer distinct advantages for sparse matrices.     

Commuting point transformations

Summary The question considered is the following: If two invertible measure preserving point transformations commute, in what sense is one a function of the other? The main theorem is the following: If two invertible measure preserving transformations commute, and if the first admits of approximation by periodic transformations, then the second transformation is a piecewise power of the first, where we say that is a piecewise power of if there exists a sequence [j(n)] of positive integers such that for each measurable set A the limit of the measure of the symmetric difference of (A) and ^j(n) (A) tends to zero.Research supported in part by NSF grant GP-3752.     

Association schemes and quadratic transformations for orthogonal polynomials

Special cases of the Askey-Wilson polynomials are the eigenmatrices of the classical association schemes. Three constructions on the schemes — multiple polynomial structures, bipartite halves, and antipodal quotients — give quadratic transformations for the polynomials. It is shown that these transformations essentially follow from a quadratic transformation for the Askey-Wilson polynomials. Explicit formulas for the eigenmatrices of three related association schemes are given.Partially supported by NSF grant DMS: 8500958 and a fellowship from the Sloan Foundation.     

A partial proximal point algorithm for nuclear norm regularized matrix least squares problems

We introduce a partial proximal point algorithm for solving nuclear norm regularized matrix least squares problems with equality and inequality constraints. The inner subproblems, reformulated as a system of semismooth equations, are solved by an inexact smoothing Newton method, which is proved to be quadratically convergent under a constraint non-degeneracy condition, together with the strong semi-smoothness property of the singular value thresholding operator. Numerical experiments on a variety of problems including those arising from low-rank approximations of transition matrices show that our algorithm is efficient and robust.     

Rational spectral transformations and orthogonal polynomials

We show that generic rational transformations of the Stieltjes function with polynomial coefficients (ST) can be presented as a finite superposition of four fundamental elementary transforms: Christoffel transform (CT), Geronimus transform (GT) and forward and backward associated transformations A⁺T, A⁻T. It is shown that the Laguerre-Hahn polynomials (LHP) on arbitrary nonuniform lattice are covariant with respect to ST (i.e., ST of a LHP yields another LHP), whereas the semi-classical polynomials are covariant with respect to a subclass of linear ST. Some applications of these results to the theory of the semi-classical polynomials are considered.     

A robust multigrid approach for variational image registration models

Variational registration models are non-rigid and deformable imaging techniques for accurate registration of two images. As with other models for inverse problems using the Tikhonov regularization, they must have a suitably chosen regularization term as well as a data fitting term. One distinct feature of registration models is that their fitting term is always highly nonlinear and this nonlinearity restricts the class of numerical methods that are applicable. This paper first reviews the current state-of-the-art numerical methods for such models and observes that the nonlinear fitting term is mostly ‘avoided’ in developing fast multigrid methods. It then proposes a unified approach for designing fixed point type smoothers for multigrid methods. The diffusion registration model (second-order equations) and a curvature model (fourth-order equations) are used to illustrate our robust methodology. Analysis of the proposed smoothers and comparisons to other methods are given. As expected of a multigrid method, being many orders of magnitude faster than the unilevel gradient descent approach, the proposed numerical approach delivers fast and accurate results for a range of synthetic and real test images.     

A robust multigrid approach for variational image registration models

Variational registration models are non-rigid and deformable imaging techniques for accurate registration of two images. As with other models for inverse problems using the Tikhonov regularization, they must have a suitably chosen regularization term as well as a data fitting term. One distinct feature of registration models is that their fitting term is always highly nonlinear and this nonlinearity restricts the class of numerical methods that are applicable. This paper first reviews the current state-of-the-art numerical methods for such models and observes that the nonlinear fitting term is mostly ‘avoided’ in developing fast multigrid methods. It then proposes a unified approach for designing fixed point type smoothers for multigrid methods. The diffusion registration model (second-order equations) and a curvature model (fourth-order equations) are used to illustrate our robust methodology. Analysis of the proposed smoothers and comparisons to other methods are given. As expected of a multigrid method, being many orders of magnitude faster than the unilevel gradient descent approach, the proposed numerical approach delivers fast and accurate results for a range of synthetic and real test images.     

An efficient method for calculating smoothing splines using orthogonal transformations

Summary A procedure for calculating the mean squared residual and the trace of the influence matrix associated with a polynomial smoothing spline of degree 2m–1 using an orthogonal factorization is presented. The procedure substantially overcomes the problem of ill-conditioning encountered by a recently developed method which employs a Cholesky factorization, but still requires only orderm ² n operations and ordermn storage.     

A cutting-plane approach for the two-dimensional orthogonal non-guillotine cutting problem

The two-dimensional orthogonal non-guillotine cutting problem (NGCP) appears in many industries (like wood and steel industries) and consists in cutting a rectangular master surface into a number of rectangular pieces, each with a given size and value. The pieces must be cut with their edges always parallel or orthogonal to the edges of the master surface (orthogonal cuts). The objective is to maximize the total value of the pieces cut.In this paper, we propose a two-level approach for solving the NGCP, where, at the first level, we select the subset of pieces to be packed into the master surface without specifying the layout, while at a second level we check only if a feasible packing layout exists. This approach has been already proposed by Fekete and Schepers [S.P. Fekete, J. Schepers, A new exact algorithm for general orthogonal d-dimensional knapsack problems, ESA 97, Springer Lecture Notes in Computer Science 1284 (1997) 144–156; S.P. Fekete, J. Schepers, On more-dimensional packing III: Exact algorithms, Tech. Rep. 97.290, Universität zu Köln, Germany, 2000; S.P. Fekete, J. Schepers, J.C. van der Veen, An exact algorithm for higher-dimensional orthogonal packing, Tech. Rep. Under revision on Operations Research, Braunschweig University of Technology, Germany, 2004] and Caprara and Monaci [A. Caprara, M. Monaci, On the two-dimensional knapsack problem, Operations Research Letters 32 (2004) 2–14]. We propose improved reduction tests for the NGCP and a cutting-plane approach to be used in the first level of the tree search to compute effective upper bounds.Computational tests on problems derived from the literature show the effectiveness of the proposed approach, that is able to reduce the number of nodes generated at the first level of the tree search and the number of times the existence of a feasible packing layout is tested.     

Classical orthogonal polynomials: A functional approach

We characterize the so-called classical orthogonal polynomials (Hermite, Laguerre, Jacobi, and Bessel) using the distributional differential equation D(u)=u. This result is naturally not new. However, other characterizations of classical orthogonal polynomials can be obtained more easily from this approach. Moreover, three new properties are obtained.     

A method to compute recurrence relation coefficients for bivariate orthogonal polynomials by unitary matrix transformations

We present an algorithm computing recurrence relation coefficients for bivariate polynomials, orthonormal with respect to a discrete inner product. These polynomials make it possible to give the solution of a discrete least squares approximation problem. To compute these polynomials, we pose the inverse eigenvalue problem and solve it efficiently and in a stable way, using a sequence of Givens rotations. We also show how to generalize the algorithm for the case of polynomials in more variables. Several numerical experiments show the validity of the approach.     

A limit field for orthogonal range searches in two-dimensional random point search trees

We consider the cost of general orthogonal range queries in random quadtrees. The cost of a given query is encoded into a (random) function of four variables which characterize the coordinates of two opposite corners of the query rectangle. We prove that, when suitably shifted and rescaled, the random cost function converges uniformly in probability towards a random field that is characterized as the unique solution to a distributional fixed-point equation. We also state similar results for 2-d trees. Our results imply for instance that the worst case query satisfies the same asymptotic estimates as a typical query, and thereby resolve an open question of Chanzy et al. (2001).     

Enumeration of symplectic and orthogonal injective partial transformations

In this paper, we compute the generating function of , where a is a real number with a≥1. We then use this function to determine the generating functions of the symplectic and orthogonal Renner monoids. Furthermore, we show that these functions are closely related to Laguerre polynomials.     

A parallel multiple reference point approach for multi-objective optimization

This paper presents a multiple reference point approach for multi-objective optimization problems of discrete and combinatorial nature. When approximating the Pareto Frontier, multiple reference points can be used instead of traditional techniques. These multiple reference points can easily be implemented in a parallel algorithmic framework. The reference points can be uniformly distributed within a region that covers the Pareto Frontier. An evolutionary algorithm is based on an achievement scalarizing function that does not impose any restrictions with respect to the location of the reference points in the objective space. Computational experiments are performed on a bi-objective flow-shop scheduling problem. Results, quality measures as well as a statistical analysis are reported in the paper.