608 research outputs found
Polarity and conjugacy for quadratic hypersurfaces: A unified framework with recent advances
No full textWe aim at completing the analysis in Fasano and Pesenti (2017) for quadratic hypersurfaces, where the geometric viewpoint suggested by the Polarity theory is considered, in order to recast basic properties of the Conjugate Gradient (CG) method (Hestenes and Stiefel, 1952) [1]. Here, the focus is on possibly exploiting theoretical advances on nonconvex quadratic hypersurfaces, in order to address guidelines for efficient optimization methods converging to second order stationary points, in large scale settings. We first recall some results from Fasano and Pesenti (2017), in order to fully analyze the relationship between the CG and the Polarity theory. Then, we specifically address, from a different perspective, the geometric insight of the pivot breakdown, which might occur when solving a nonsingular indefinite Newton’s equation applying the CG. Furthermore, we fully exploit some novel theoretical advances of the Polarity theory on nonconvex quadratic hypersurfaces not considered in Fasano and Pesenti (2017). Finally, we show that our approach describes a general framework, which also encompasses a class of CG–based methods, namely Planar CG–based methods. The framework we consider intends to emphasize a bridge between the geometry behind stationary points of nonconvex quadratic hypersurfaces and their efficient computation using Krylov–subspace methods.We aim at completing the analysis in Fasano and Pesenti (2017) for quadratic hypersurfaces, where the geometric viewpoint suggested by the Polarity theory is considered, in order to recast basic properties of the Conjugate Gradient (CG) method (Hestenes and Stiefel, 1952) [1]. Here, the focus is on possibly exploiting theoretical advances on nonconvex quadratic hypersurfaces, in order to address guidelines for efficient optimization methods converging to second order stationary points, in large scale settings. We first recall some results from Fasano and Pesenti (2017), in order to fully analyze the relationship between the CG and the Polarity theory. Then, we specifically address, from a different perspective, the geometric insight of the pivot breakdown, which might occur when solving a nonsingular indefinite Newton's equation applying the CG. Furthermore, we fully exploit some novel theoretical advances of the Polarity theory on nonconvex quadratic hypersurfaces not considered in Fasano and Pesenti (2017). Finally, we show that our approach describes a general framework, which also encompasses a class of CG-based methods, namely Planar CG-based methods. The framework we consider intends to emphasize a bridge between the geometry behind stationary points of nonconvex quadratic hypersurfaces and their efficient computation using Krylov-subspace methods
On the iterative computation of a l_2-norm scaling based preconditioner
No full textIn this paper we consider the Krylov subspace based method introduced in [Fasano, 2005a], for
iteratively solving the symmetric and possibly indefinite linear system Ax = b. We emphasize the
application of the latter method to compute a diagonal preconditioner. The approach proposed
is based on the approximate computation of the `2-norm of the rows (columns) of the matrix A
and on its use to equilibrate the matrix A. The distinguishing feature of this approach is that
the computation of the `2-norm is performed without requiring the knowledge of the entries of
the matrix A but only using a routine which provides the product of A times a vector
Marta Catalano, Augusto Fasano, and Giovanni Rebaudo’s contribution to the discussion of ‘Martingale posterior distributions’ by Fong, Holmes and Walker
No full textPolitica, fazioni, istituzioni nell'"Italia spagnola". Dall'incoronazione di Carlo V (1530) alla Pace di Westfalia (1648).
No full textImportanti raccolte di documenti, con relativi apparati critici e introduzioni, originata da un progetto MURST coordinato a livello nazionale da E. Fasano Guarini. La serie di volumi rientra nelle Pubblicazioni degli Archivi di Stato, Fonti, presente in tutte le maggiori biblioteche a livello internazionale e ora accessibile in rete. Il comitato scientifico è formato da E. Fasano Guarini, G. Signorotto e M.A. Visceglia
A novel class of Approximate Inverse Preconditioners for large positive definite systems
No full textWe propose a class of preconditioners for large positive definite linear systems, arising in
nonlinear optimization frameworks. These preconditioners can be computed as by-product of Krylov subspace
solvers. Preconditioners in our class are chosen by setting the values of some user-dependent
parameters. We first provide some basic spectral properties which motivate a theoretical interest for the
proposed class of preconditioners. Then, we report the results of a comparative numerical experience,
among some preconditioners in our class, the unpreconditioned case and the preconditioner in [11].
The experience was carried on first considering some relevant linear systems proposed in the literature.
Then, we embedded our preconditioners within a linesearch-based Truncated Newton method, where
sequences of linear systems (namely Newton’s equations), are required to be solved. We performed
an extensive numerical testing over the entire medium-large scale convex unconstrained optimization
test set of CUTEst collection [15], confirming the efficiency of our proposal and the improvement with
respect to the preconditioner in [11]
Stochastic model of agent interaction with opinion leaders
No full textWe analyze the problem of agents' interactions in a given population. The purpose of this paper is twofold. Starting from a scheme proposed by Galam [ Physica A 320 571 (2003)], which is based on a majority rule to treat the individuals’ interactions, we first study some of its relevant properties. Then, we introduce special individuals, called opinion leaders, who play a key role in information spreading in several practical applications. Opinion leaders have the special feature of strongly interfering with the process based on the majority rule, speeding up the diffusion. We consider a model describing agents’ interactions, which encompasses Galam's proposal, where opinion leaders are included as special agents. Then we study its specific properties which significantly recast and extend some conclusions drawn for the models given by Galam and Ellero, Fasano, and Sorato [ Physica A 388 3901 (2009)]. Finally, we provide theoretical and numerical results concerning the dynamics of our model, showing that a small percentage of opinion leaders may both accelerate and/or even reverse the overall consensus among all the agents
Planar-Conjugate Gradient algorithm for Large Scale Unconstrained Optimization, Part 2: Application
No full textIn this paper, we present a new conjugate gradient (CG) based algorithm in the class of planar conjugate gradient methods. These methods aim at solving systems of linear equations whose coefficient matrix is indefinite and nonsingular. This is the case where the application of the standard CG algorithm by Hestenes and Stiefel (Ref. 1) may fail, due to a possible division by zero. We give a complete proof of global convergence for a new planar method endowed with a general structure; furthermore, we describe some important features of our planar algorithm, which will be used within the optimization framework of the companion paper (Part 2, Ref. 2). Here, preliminary numerical results are reported
Lanczos-Conjugate Gradient method and pseudoinverse computation, in unconstrained optimization
No full textThis paper extends some theoretical properties of the Conjugate Gradient-type method FLR
[Fas05], for iteratively solving indefinite linear systems of equations. The latter algorithm is a
generalization of the Conjugate Gradient (CG) by Hestenes and Stiefel [HS52].
On one hand, here we carry out a complete relationship between algorithm FLR and the Lanczos
process, in case of indefinite and possibly singular matrices. On the other hand we develop
simple theoretical results for algorithm FLR, in order to construct an approximation of the Moore-Penrose pseudoinverse of an indefinite matrix. Our approach supplies theory for applications within nonconvex optimization
Uso delle Direzioni Coniugate negli algoritmi per l'Ottimizzazione Non Vincolata a grande dimensione
No full textConjugate Gradient (CG)-type Method for the Solution of Newton's equation within Optimization Frameworks
No full textA conjugate gradient (CG)-type algorithm CG Plan is introduced for calculating an approximate solution of Newton’s
equation within large-scale optimization frameworks. The approximate solution must satisfy suitable properties to
ensure global convergence. In practice, the CG algorithm is widely used, but it is not suitable when the Hessian matrix
is indefinite, as it can stop prematurely. CG Plan is a symmetric variant of the composite step Bi-CG method of Bank
and Chan, suitably adapted for optimization problems. It is an alternative to CG that copes with the indefinite case.
We showconvergence for CG Plan, then prove that the practical implementation always provides a gradient related
direction within a truncated Newton method (algorithm TN_Plan). Some preliminary numerical results support the
theory
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