Limited memory restarted l(p)-l(q) minimization methods using generalized Krylov subspaces

Regularization of certain linear discrete ill-posed problems, as well as of certain regression problems, can be formulated as large-scale, possibly nonconvex, minimization problems, whose objective function is the sum of the p(th) power of the l(p)-norm of a fidelity term and the qth power of the lq-norm of a regularization term, with 0 < p,q = 2. We describe new restarted iterative solution methods that require less computer storage and execution time than the methods described by Huang et al. (BIT Numer. Math. 57,351-378, 14). The reduction in computer storage and execution time is achieved by periodic restarts of the method. Computed examples illustrate that restarting does not reduce the quality of the computed solutions

Generalized cross validation for ℓ p-ℓ q minimization

Discrete ill-posed inverse problems arise in various areas of science and engineering. The presence of noise in the data often makes it difficult to compute an accurate approximate solution. To reduce the sensitivity of the computed solution to the noise, one replaces the original problem by a nearby well-posed minimization problem, whose solution is less sensitive to the noise in the data than the solution of the original problem. This replacement is known as regularization. We consider the situation when the minimization problem consists of a fidelity term, that is defined in terms of a p-norm, and a regularization term, that is defined in terms of a q-norm. We allow 0 < p,q ≤ 2. The relative importance of the fidelity and regularization terms is determined by a regularization parameter. This paper develops an automatic strategy for determining the regularization parameter for these minimization problems. The proposed approach is based on a new application of generalized cross validation. Computed examples illustrate the performance of the method proposed

Quality issues impacting production planning

Among the various problems affecting production processes, the unpredictability of quality factors is one of the main issues which concern manufacturing enterprises. In make-to-order or in perishable good production systems, the gap between expected and real output quality increases product cost mainly in two different ways: through the costs of extra production or reworks due to the presence of non-compliant items and through the costs originating from inefficient planning and the need of unscheduled machine changeovers. While the first are relatively easy to compute, even ex-ante, the latter are much more difficult to estimate because they depend on several planning variables such as lot size, sequencing, deliveries due dates, etc. This paper specifically addresses this problem in a make-to-order multi-product customized production system; here, the enterprise diversifies each production lot due to the fact that each order is based on the customer specific requirements and it is unique (in example, packaging or textiles and apparel industry). In these contexts, using a rule-of-thumb in overestimating the input size may cause high costs because all the excess production will generate little or no revenues on top of contributing to increasing wastes in general. On the other hand, the underestimation of the lots size is associated to the eventual need of launching a new, typically very small production order, thus a single product will bear twice the changeover costs. With little markups, it may happen that these extra costs can reduce profit to zero. Aim of this paper is to provide a critical analysis of the literature state-of-art while introducing some elements that can help the definition of lot-sizing policies considering poor quality costs

Theoretical and numerical aspects of a non-stationary preconditioned iterative method for linear discrete ill-posed problems

This work considers some theoretical and computational aspects of the recent paper (Buccini et al., 2021), whose aim was to relax the convergence conditions in a previous work by Donatelli and Hanke, and thereby make the iterative method discussed in the latter work applicable to a larger class of problems. This aim was achieved in the sense that the iterative method presented convergences for a larger class of problems. However, while the analysis presented is correct, it does not establish the superior behavior of the iterative method described. The present note describes a slight modification of the analysis that establishes the superiority of the iterative method. The new analysis allows to discuss the behavior of the algorithm when varying the involved parameters, which is also useful for their empirical estimation

An Arnoldi-based preconditioner for iterated Tikhonov regularization

Many problems in science and engineering give rise to linear systems of equations that are commonly referred to as large-scale linear discrete ill-posed problems. These problems arise, for instance, from the discretization of Fredholm integral equations of the first kind. The matrices that define these problems are typically severely ill-conditioned and may be rank-deficient. Because of this, the solution of linear discrete ill-posed problems may not exist or be very sensitive to perturbations caused by errors in the available data. These difficulties can be reduced by applying Tikhonov regularization. We describe a novel "approximate Tikhonov regularization method" based on constructing a low-rank approximation of the matrix in the linear discrete ill-posed problem by carrying out a few steps of the Arnoldi process. The iterative method so defined is transpose-free. Our work is inspired by a scheme by Donatelli and Hanke, whose approximate Tikhonov regularization method seeks to approximate a severely ill-conditioned block-Toeplitz matrix with Toeplitz-blocks by a block-circulant matrix with circulant-blocks. Computed examples illustrate the performance of our proposed iterative regularization method

Quality issues impacting production planning

Among the various problems affecting production processes, the unpredictability of quality factors is one of the main issues which concern manufacturing enterprises. In make-to-order or in perishable good production systems, the gap between expected and real output quality increases product cost mainly in two different ways: through the costs of extra production or reworks due to the presence of non-compliant items and through the costs originating from inefficient planning and the need of unscheduled machine changeovers. While the first are relatively easy to compute, even ex-ante, the latter are much more difficult to estimate because they depend on several planning variables such as lot size, sequencing, deliveries due dates, etc. This paper specifically addresses this problem in a make-to-order multi-product customized production system; here, the enterprise diversifies each production lot due to the fact that each order is based on the customer specific requirements and it is unique (in example, packaging or textiles and apparel industry). In these contexts, using a rule-of-thumb in overestimating the input size may cause high costs because all the excess production will generate little or no revenues on top of contributing to increasing wastes in general. On the other hand, the underestimation of the lots size is associated to the eventual need of launching a new, typically very small production order, thus a single product will bear twice the changeover costs. With little markups, it may happen that these extra costs can reduce profit to zero. Aim of this paper is to provide a critical analysis of the literature state-of-art while introducing some elements that can help the definition of lot-sizing policies considering poor quality costs

Range restricted iterative methods for linear discrete ill-posed problems

Linear systems of equations with a matrix whose singular values decay to zero with increasing index number, and without a significant gap, are commonly referred to as linear discrete ill-posed problems. Such systems arise, e.g., when discretizing a Fredholm integral equation of the first kind. The right-hand side vectors of linear discrete ill-posed problems that arise in science and engineering often represent an experimental measurement that is contaminated by measurement error. The solution to these problems typically is very sensitive to this error. Previous works have shown that error propagation into the computed solution may be reduced by using specially designed iterative methods that allow the user to select the subspace in which the approximate solution is computed. Since the dimension of this subspace often is quite small, its choice is important for the quality of the computed solution. This work describes algorithms for three iterative methods that modify the GMRES, block GMRES, and global GMRES methods for the solution of appropriate linear systems of equations. We contribute to the work already available on this topic by introducing two block variants for the solution of linear systems of equations with multiple right-hand side vectors. The dominant computational aspects are discussed, and software for each method is provided. Additionally, we illustrate the utility of these iterative subspace methods through numerical examples focusing on image reconstruction. This paper is accompanied by software

Fractional graph Laplacian for image reconstruction

Image reconstruction problems, like image deblurring and computer tomography, are usually ill-posed and require regularization. A popular approach to regularization is to substitute the original problem with an optimization problem that minimizes the sum of two terms, an term and an term with . The first penalizes the distance between the measured data and the reconstructed one, the latter imposes sparsity on some features of the computed solution. In this work, we propose to use the fractional Laplacian of a properly constructed graph in the term to compute extremely accurate reconstructions of the desired images. A simple model with a fully automatic method, i.e., that does not require the tuning of any parameter, is used to construct the graph and enhanced diffusion on the graph is achieved with the use of a fractional exponent in the Laplacian operator. Since the fractional Laplacian is a global operator, i.e., its matrix representation is completely full, it cannot be formed and stored. We propose to replace it with an approximation in an appropriate Krylov subspace. We show that the algorithm is a regularization method under some reasonable assumptions. Some selected numerical examples in image deblurring and computer tomography show the performance of our proposal

Simplifying the Virtual Safety Stock formula

The paper deepen the analysis into the Virtual Safety Stock theory, which is an approach intended to drastically reduce safety inventory levels exploiting the eventual time lag between the moment when a product is ordered and the time the product needs to be available, while preserving the same performance as a production system that operates with physical safety stock. The original virtual safety stock definition embeds two major problems: a double Gaussian integral in the formulation together with the heritage of the unrealistic assumptions already included in the earliest Hadley and Whitin’s safety stock conception. This paper describes an alternative approach in which the virtual safety stock is defined with a closed-form expression much easier to compute and use in operations management practice

A comparison of parameter choice rules for ℓp - ℓq minimization

Images that have been contaminated by various kinds of blur and noise can be restored by the minimization of an ℓp-ℓq functional. The quality of the reconstruction depends on the choice of a regularization parameter. Several approaches to determine this parameter have been described in the literature. This work presents a numerical comparison of known approaches as well as of a new one