Conjugate Direction Methods and Polarity for Quadratic Hypersurfaces

Abstract

Abstract:We use some results from polarity theory to recast several geometric properties of Conjugate Gradient-based methods, for the solution of nonsingular symmetric linear systems. This approach allows us to pursue three main theoretical objectives. First, we can provide a novel geometric perspective on the generation of conjugate directions, in the context of positive definite systems. Second, we can extend the above geometric perspective to treat the generation of conjugate directions for handling indefinite linear systems. Third, by exploiting the geometric insight suggested by polarity theory, we can easily study the possible degeneracy (pivot breakdown) of Conjugate Gradientbased methods on indefinite linear systems. In particular, we prove that the degeneracy of the standard Conjugate Gradient on nonsingular indefinite linear systems can occur only once in the execution of the Conjugate Gradient. Once again we strongly appreciated the suggestions by the Editorial Board and the anonymous Referees, which definitely contributed to improve and enhance the paper. Hereafter we reply to the observations raised by each of them. We also remark that the paper has been given to a professional proofreader before resubmission: we hope we were able to comply with all the issues raised by the Reviewers. Powered by Editorial Manager® and ProduXion Manager® from Aries Systems CorporationAnswer to the Editor in Chief and the Associate EditorAuthors: The instructions of the Editor/Associate Editor should have been followed, including both structural modifications (e.g. we splitted the former section of Conclusions) and minor suggestions.Answer to the First Referee Authors: All this Reviewer's comments should have been included, following her/his indications. In addition, an English native speaker has contributed with proofreading.Answer to the Second Referee Authors: All the accurate Reviewer's modifications should have been implemented, following her/his indications. Furthermore, we also did our best to revise the entire paper, on the basis of the specific comments of this Reviewer. This has led to some additional small changes, including punctuation, with respect to the previous version of the paper. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 2 Giovanni Fasano, Raffaele Pesenti by exploiting the geometric insight suggested by polarity theory, we can easily study the possible degeneracy (pivot breakdown) of Conjugate Gradient-based methods on indefinite linear systems. In particular, we prove that the degeneracy of the standard Conjugate Gradient on nonsingular indefinite linear systems can occur only once in the execution of the Conjugate Gradient. Powered by Editorial Manager® and ProduXion Manager® from Aries Systems Corporation

Planar methods and grossone for the Conjugate Gradient breakdown in nonlinear programming

“…The latter techniques are indeed extensions of the CG to non-quadratic functions, and require specific care when computing the steplength along the current search direction. In this regard, on one hand the use of grossone can be the right tool to handle numerical instabilities; on the other hand, the theory of Polarity (see for instance [30]) might suggest useful extensions of the asymptotic cone (see [12]). …”

“…1. Equivalently, the line y k + αp k , α ∈ R, is tangent to another level set (dashed and dotted line), in the point at infinity y k+1 (for a more rigorous justification of the last statement the reader can refer to [12]). Then, in order to formally compute the next finite iterate y k+2 , the search direction p k+1 satisfying p k+1 → + ∞ should be provided, as proved in Proposition 3.1.…”

Iterative Grossone-Based Computation of Negative Curvature Directions in Large-Scale Optimization

“…Proof First observe that by [53], also in case at the iterate x j the Hessian matrix ∇ 2 f (x j ) is indefinite, there exists at most one step k, with 0 ≤ k ≤ n, such that in CG ① we might have p T k ∇ 2 f (x j ) p k = 0. Thus, similarly to the rest of the paper, without loss of generality in this proof we assume that possibly the equality p T k ∇ 2 f (x j ) p k = 0 only holds at step k. Moreover, the matrixD is diagonal, which implies that the unit vector associated with its ith eigenvalue μ i (D) is given by e i .…”