Numerical infinitesimals in a variable metric method for convex nonsmooth optimization

Gaudioso, Manlio; Giallombardo, Giovanni; Mukhametzhanov, Marat S.

doi:10.1016/j.amc.2017.07.057

Cited by 30 publications

(32 citation statements)

References 26 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…where {a i }, {β i }, e k+1 are suitable scalars, and we assume (for the sake of simplicity) that the method performed all CG iterations, with the exception of only one planar iteration (namely the k-th iteration -see [1]and [2]). Then, our novel approach proposes to introduce the numeral grossone, as in [5][6][7][8], and follow some guidelines from [4], so that we can compute the lower block triangular matrix…”

Section: Our Proposalmentioning

confidence: 99%

How Grossone Can Be Helpful to Iteratively Compute Negative Curvature Directions

Leone

Fasano

Roma

et al. 2018

Lecture Notes in Computer Science

View full text Add to dashboard Cite

We consider an iterative computation of negative curvature directions, in large scale optimization frameworks. We show that to the latter purpose, borrowing the ideas in [1], [3] and [4], we can fruitfully pair the Conjugate Gradient (CG) method with an algebraic approach involving the use of grossone [5]. In particular, though in principle the CG method is well-posed only on positive definite linear systems, the use of grossone can enhance the performance of the CG, allowing the computation of negative curvature directions, too. The overall method in our proposal significantly generalizes the theory proposed for [1] and [3], and straightforwardly allows the use of a CG-based method on indefinite Newton's equations.

show abstract

Section: Our Proposalmentioning

confidence: 99%

How Grossone Can Be Helpful to Iteratively Compute Negative Curvature Directions

Leone

Fasano

Roma

et al. 2018

Lecture Notes in Computer Science

View full text Add to dashboard Cite

show abstract

“…In their turn, the values α and β can be finite, infinite, and infinitesimal numbers representable in the numeral system (9). The finiteness of the original Lipschitz constant L from (3) is the essence of the Lipschitz condition allowing people to construct optimization methods for traditional computers.…”

Section: Functions With Infinite/infinitesimal Lipschitz Constantsmentioning

confidence: 99%

“…Notice that in the introduced class of functions infinities and infinitesimals are expressed in numerals (9), and Lemma 1 describes the first property of this class. Notice also that symbol ∞ representing a generic infinity cannot be used together with numerals (9) allowing us to distinguish a variety of infinite (and infinitesimal) numbers.…”

Section: Functions With Infinite/infinitesimal Lipschitz Constantsmentioning

confidence: 99%

“…This computational methodology has already been successfully applied in a number of applications mentioned above. In particular, it has been successfully used for studying strong homogeneity of the P-algorithm and the one-step Bayesian algorithm (see [41]) and to handle ill-conditioning in local optimization (see [9]). …”

Section: Infinity Computing Brieflymentioning

confidence: 99%

“…This computational methodology has already been successfully applied in optimization and numerical differentiation [3,5,6,26] and in a number of other theoretical and applied research areas such as, e.g., cellular automata [4], hyperbolic geometry [16], percolation [13], fractals [2,27], infinite series [25,38], Turing machines [22], numerical solution of ordinary differential equations [1,33], etc. In particular, in the recent paper [9], numerical infinities and infinitesimals from [24,28] have been successfully used to handle ill-conditioning in a multidimensional optimization problem.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On strong homogeneity of a class of global optimization algorithms working with infinite and infinitesimal scales

Sergeyev

Kvasov

Mukhametzhanov

2018

Communications in Nonlinear Science and Numerical Simulation

View full text Add to dashboard Cite

The necessity to find the global optimum of multiextremal functions arises in many applied problems where finding local solutions is insufficient. One of the desirable properties of global optimization methods is strong homogeneity meaning that a method produces the same sequences of points where the objective function is evaluated independently both of multiplication of the function by a scaling constant and of adding a shifting constant. In this paper, several aspects of global optimization using strongly homogeneous methods are considered. First, it is shown that even if a method possesses this property theoretically, numerically very small and large scaling constants can lead to illconditioning of the scaled problem. Second, a new class of global optimization problems where the objective function can have not only finite but also infinite or infinitesimal Lipschitz constants is introduced. Third, the strong homogeneity of several Lipschitz global optimization algorithms is studied in the framework of the Infinity Computing paradigm allowing one to work numerically with a variety of infinities and infinitesimals. Fourth, it is proved that a class of efficient univariate methods enjoys this property for finite, infinite and infinitesimal scaling and shifting constants. Finally, it is shown that in certain cases the usage of numerical infinities and infinitesimals can avoid ill-conditioning produced by scaling. Numerical experiments illustrating theoretical results are described.

show abstract

A Dynamic Precision Floating-Point Arithmetic Based on the Infinity Computer Framework

Amodio

Brugnano

Iavernaro

et al. 2020

Lecture Notes in Computer Science

View full text Add to dashboard Cite

show abstract

Numerical infinitesimals in a variable metric method for convex nonsmooth optimization

Cited by 30 publications

References 26 publications

How Grossone Can Be Helpful to Iteratively Compute Negative Curvature Directions

How Grossone Can Be Helpful to Iteratively Compute Negative Curvature Directions

On strong homogeneity of a class of global optimization algorithms working with infinite and infinitesimal scales

A Dynamic Precision Floating-Point Arithmetic Based on the Infinity Computer Framework

Contact Info

Product

Resources

About