A Class of Preconditioners for Large Indefinite Linear Systems, as by-product of Krylov subspace Methods: Part II

In this paper we consider the parameter dependent class of preconditioners M(a,d,D) defined in the companion paper The latter was constructed by using information from a Krylov subspace method, adopted to solve the large symmetric linear system Ax = b. We first estimate the condition number of the preconditioned matrix M(a,d,D). Then our preconditioners, which are independent of the choice of the Krylov subspace method adopted, proved to be effective also when solving sequences of slowly changing linear systems, in unconstrained optimization and linear algebra frameworks. A numerical experience is provided to give evidence of the performance of M(a,d,D).preconditioners; large indefinite linear systems; large scale nonconvex optimization; Krylov subspace methods

A Class of Preconditioners for Large Indefinite Linear Systems, as by-product of Krylov subspace Methods: Part I

We propose a class of preconditioners, which are also tailored for symmetric linear systems from linear algebra and nonconvex optimization. Our preconditioners are specifically suited for large linear systems and may be obtained as by-product of Krylov subspace solvers. Each preconditioner in our class is identified by setting the values of a pair of parameters and a scaling matrix, which are user-dependent, and may be chosen according with the structure of the problem in hand. We provide theoretical properties for our preconditioners. In particular, we show that our preconditioners both shift some eigenvalues of the system matrix to controlled values, and they tend to reduce the modulus of most of the other eigenvalues. In a companion paper we study some structural properties of our class of preconditioners, and report the results on a significant numerical experience.preconditioners; large indefinite linear systems; large scale nonconvex optimization; Krylov subspace methods

Bridging the gap between Trust–Region Methods (TRMs) and Linesearch Based Methods (LBMs) for Nonlinear Programming: quadratic sub–problems

We consider the solution of a recurrent sub–problem within both constrained and unconstrained Nonlinear Programming: namely the minimization of a quadratic function subject to linear constraints. This problem appears in a number of LBM frameworks, and to some extent it reveals a close analogy with the solution of trust–region sub–problems. In particular, we refer to a structured quadratic problem where five linear inequality constraints are included. We show that our proposal retains an appreciable versatility, despite its particular structure, so that a number of different real instances may be reformulated following the pattern in our proposal. Moreover, we detail how to compute an exact global solution of our quadratic sub–problem, exploiting first order KKT conditions

On the use of the SYMMBK algorithm for computing negative curvature directions within Newton-Krylov methods

In this paper, we consider the issue of computing negative curvature directions, for nonconvex functions, within Newton-Krylov methods for large scale unconstrained optimization. This issue has been widely investigated in the literature, and different approaches have been proposed. We focus on the well known SYMMBK method proposed for solving large scale symmetric possibly inde finite linear systems [3, 5, 7, 20], and show how to exploit it to yield an effective negative curvature direction. The distinguishing feature of our proposal is that the computation of such negative curvature direction is iteratively carried out, without storing no more than a couple of additional vectors. The results of a preliminary numerical experience are reported showing the reliability of the novel approach we propose

Issues on the use of a modified Bunch and Kaufman decomposition for large scale Newton’s equation

In this work, we deal with Truncated Newton methods for solving large scale (possibly nonconvex) unconstrained optimization problems. In particular, we consider the use of a modified Bunch and Kaufman factorization for solving the Newton equation, at each (outer) iteration of the method. The Bunch and Kaufman factorization of a tridiagonal matrix is an effective and stable matrix decomposition, which is well exploited in the widely adopted SYMMBK [2, 5, 6, 19, 20] routine. It can be used to provide conjugate directions, both in the case of 1 × 1 and 2 × 2 pivoting steps. The main drawback is that the resulting solution of Newton’s equation might not be gradient–related, in the case the objective function is nonconvex. Here we first focus on some theoretical properties, in order to ensure that at each iteration of the Truncated Newton method, the search direction obtained by using an adapted Bunch and Kaufman factorization is gradient–related. This allows to perform a standard Armijo-type linesearch procedure, using a bounded descent direction. Furthermore, the results of an extended numerical experience using large scale CUTEst problems is reported, showing the reliability and the efficiency of the proposed approach, both on convex and nonconvex problems

Preconditioning Strategies for Nonlinear Conjugate Gradient Methods, Based on Quasi-Newton Updates

This paper reports two proposals of possible preconditioners for the Nonlinear Conjugate Gradient (NCG) method, in large scale unconstrained optimization. On one hand, the common idea of our preconditioners is inspired to L-BFGS quasi-Newton updates, on the other hand we aim at explicitly approximating in some sense the inverse of the Hessian matrix. Since we deal with large scale optimization problems, we propose matrix-free approaches where the preconditioners are built using symmetric low-rank updating formulae. Our distinctive new contributions rely on using information on the objective function collected as by-product of the NCG, at previous iterations. Broadly speaking, our first approach exploits the secant equation, in order to impose interpolation conditions on the objective function. In the second proposal we adopt and ad hoc modified-secant approach, in order to possibly guarantee some additional theoretical properties

Preconditioned Nonlinear Conjugate Gradient methods based on a modified secant equation

This paper includes a twofold result for the Nonlinear Conjugate Gradient (NCG) method, in large scale unconstrained optimization. First we consider a theoretical analysis, where preconditioning is embedded in a strong convergence framework of an NCG method from the literature. Mild conditions to be satisfied by the preconditioners are defined, in order to preserve NCG convergence. As a second task, we also detail the use of novel matrix-free preconditioners for NCG. Our proposals are based on quasi-Newton updates, and either satisfy the secant equation or a secant-like condition at some of the previous iterates. We show that, in some sense, the preconditioners we propose also approximate the inverse of the Hessian matrix. In particular, the structures of our preconditioners depend on low-rank updates used, along with different choices of specific parameters. The low-rank updates are obtained as by-product of NCG iterations. The results of an extended numerical experience using large scale CUTEst problems is reported, showing that our preconditioners can considerably improve the performance of NCG methods

Cost-effectiveness of tenofovir in the treatment of patients with chronic hepatitis B: data from literature

Chronic hepatitis B (CHB) is a complex disease with significant social impact both on the patients' quality of life of and the economic resources involved. Its chronicity affects considerably not only the clinical management of the disease (for the need for drugs with proven long-term safety and low rate of resistance), but also the economic impact (for the high costs of treatment, the management of complications, and the constant monitoring of therapy).Since, as is well known, the main problem of modern health care systems is the general scarcity of available resources in the face of growing demand for health, the issue of economic evaluation of therapies for the treatment of chronic hepatitis B has been addressed in numerous national and international studies. In fact, clinicians find a strong support for the choice of the most suitable therapeutic pathway in the major scientific societies' guidelines (European Association for the Study of The Liver – EASL, American Association for the Study of Liver Diseases – AASLD, Associazione Italiana per lo Studio del Fegato – AISF), while the analysis of the economic implications is rather more difficult, even for the methodological differences and peculiarities of the different countries.The aim of this paper is to present a brief summary of some of the recently conducted cost-effectiveness analyses and extrapolate some data to support the economic evidence related to the treatment of CHB with nucleos(t)ide analogs. In particular, the article focuses on the comparison between entecavir (ETV) and tenofovir (TDF), the two oral antiviral therapies recommended for first-line treatment. In the selected studies, the comparison between the different treatment options was conducted in order to assess the incremental cost-effectiveness ratio (ICER) and the results were expressed in terms of QALYs (Quality Adjusted Life Years) gained.Despite the methodological differences among the selected studies, tenofovir is found to be, in the context of first-line oral antiviral therapies, the most cost-effective treatment for patients with chronic hepatitis B

Cost-effectiveness of tenofovir in the treatment of patients with chronic hepatitis B: data from literature

Chronic hepatitis B (CHB) is a complex disease with significant social impact both on the patients’ quality of life of and the economic resources involved. Its chronicity affects considerably not only the clinical management of the disease (for the need for drugs with proven long-term safety and low rate of resistance), but also the economic impact (for the high costs of treatment, the management of complications, and the constant monitoring of therapy).Since, as is well known, the main problem of modern health care systems is the general scarcity of available resources in the face of growing demand for health, the issue of economic evaluation of therapies for the treatment of chronic hepatitis B has been addressed in numerous national and international studies. In fact, clinicians find a strong support for the choice of the most suitable therapeutic pathway in the major scientific societies’ guidelines (European Association for the Study of The Liver – EASL, American Association for the Study of Liver Diseases – AASLD, Associazione Italiana per lo Studio del Fegato – AISF), while the analysis of the economic implications is rather more difficult, even for the methodological differences and peculiarities of the different countries.The aim of this paper is to present a brief summary of some of the recently conducted cost-effectiveness analyses and extrapolate some data to support the economic evidence related to the treatment of CHB with nucleos(t)ide analogs. In particular, the article focuses on the comparison between entecavir (ETV) and tenofovir (TDF), the two oral antiviral therapies recommended for first-line treatment. In the selected studies, the comparison between the different treatment options was conducted in order to assess the incremental cost-effectiveness ratio (ICER) and the results were expressed in terms of QALYs (Quality Adjusted Life Years) gained.Despite the methodological differences among the selected studies, tenofovir is found to be, in the context of first-line oral antiviral therapies, the most cost-effective treatment for patients with chronic hepatitis B