BFGS-like updates of constraint preconditioners for sequences of KKT linear systems

We focus on efficient preconditioning techniques for sequences of KKT linear systems arising from the interior point solution of large convex quadratic programming problems. Constraint Preconditioners (CPs), though very effective in accelerating Krylov methods in the solution of KKT systems, have a very high computational cost in some instances, because their factorization may be the … Read more

On the application of the spectral projected gradient method in image segmentation

We investigate the application of the nonmonotone spectral projected gradient (SPG) method to a region-based variational model for image segmentation. We consider a “discretize-then-optimize” approach and solve the resulting nonlinear optimization problem by an alternating minimization procedure that exploits the SPG2 algorithm by Birgin, Martì­nez and Raydan (SIAM J. Optim., 10(4), 2000). We provide a … Read more

On the regularizing behavior of recent gradient methods in the solution of linear ill-posed problems

We analyze the regularization properties of two recently proposed gradient methods applied to discrete linear inverse problems. By studying their filter factors, we show that the tendency of these methods to eliminate first the eigencomponents of the gradient corresponding to large singular values allows to reconstruct the most significant part of the solution, thus yielding … Read more

On the update of constraint preconditioners for regularized KKT systems

We address the problem of preconditioning sequences of regularized KKT systems, such as those arising in Interior Point methods for convex quadratic programming. In this case, Constraint Preconditioners (CPs) are very effective and widely used; however, when solving large-scale problems, the computational cost for their factorization may be high, and techniques for approximating them appear … Read more

Updating constraint preconditioners for KKT systems in quadratic programming via low-rank corrections

This work focuses on the iterative solution of sequences of KKT linear systems arising in interior point methods applied to large convex quadratic programming problems. This task is the computational core of the interior point procedure and an efficient preconditioning strategy is crucial for the efficiency of the overall method. Constraint preconditioners are very effective … Read more

An efficient gradient method using the Yuan steplength

We propose a new gradient method for quadratic programming, named SDC, which alternates some SD iterates with some gradient iterates that use a constant steplength computed through the Yuan formula. The SDC method exploits the asymptotic spectral behaviour of the Yuan steplength to foster a selective elimination of the components of the gradient along the … Read more

A matrix-free approach to build band preconditioners for large-scale bound-constrained optimization

We propose a procedure for building symmetric positive definite band preconditioners for large-scale symmetric, possibly indefinite, linear systems, when the coefficient matrix is not explicitly available, but matrix-vector products involving it can be computed. We focus on linear systems arising in Newton-type iterations within matrix-free versions of projected methods for bound-constrained nonlinear optimization. In this … Read more

On spectral properties of steepest descent methods

In recent years it has been made more and more clear that the critical issue in gradient methods is the choice of the step length, whereas using the gradient as search direction may lead to very effective algorithms, whose surprising behaviour has been only partially explained, mostly in terms of the spectrum of the Hessian … Read more

Parallel algebraic multilevel Schwarz preconditioners for a class of elliptic PDE systems

We present algebraic multilevel preconditioners for linear systems arising from the discretization of systems of coupled elliptic partial differential equations (PDEs). These preconditioners are based on modifications of Schwarz methods and of the smoothed aggregation technique, where the coarsening strategy and the restriction and prolongation operators are defined using a point-based approach with a primary … Read more

A preconditioning framework for sequences of diagonally modified linear systems arising in optimization

We propose a framework for building preconditioners for sequences of linear systems of the form $(A+\Delta_k) x_k=b_k$, where $A$ is symmetric positive semidefinite and $\Delta_k$ is diagonal positive semidefinite. Such sequences arise in several optimization methods, e.g., in affine-scaling methods for bound-constrained convex quadratic programming and bound-constrained linear least squares, as well as in trust-region … Read more