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

Abstract

We consider an iterative computation of negative curvature directions, in large-scale unconstrained optimization frameworks, needed for ensuring the convergence toward stationary points which satisfy second-order necessary optimality conditions. We show that to the latter purpose, we can fruitfully couple the conjugate gradient (CG) method with a recently introduced approach involving the use of the numeral called Grossone. In particular, recalling that in principle the CG method is well posed only when solving positive definite linear systems, our proposal exploits the use of grossone to enhance the performance of the CG, allowing the computation of negative curvature directions in the indefinite case, too. Our overall method could be used to significantly generalize the theory in state-of-the-art literature. Moreover, it straightforwardly allows the solution of Newton's equation in optimization frameworks, even in nonconvex problems. We remark that our iterative procedure to compute a negative curvature direction does not require the storage of any matrix, simply needing to store a couple of vectors. This definitely represents an advance with respect to current results in the literature.

Pure and mixed lexicographic-paretian many-objective optimization: state of the art

“…Grossone Methodology (GM) (Sergeyev 2017) is a numerical framework that makes working with infinite and infinitesimal numbers on a computer possible. It has already been successfully applied to many different optimization problems (Cococcioni et al 2018(Cococcioni et al , 2020De Cosmis and De Leone 2012;De Leone 2018;De Leone et al 2020;Lai et al 2020a;. The GM fundamental element is the infinite unit Grossone, denoted by À, which allows one to build numerical values composed by finite, infinite and infinitesimal components, known as gross-scalar (G-scalar in brief).…”

Non-Archimedean zero-sum games

“…It has been proposed by Sergeyev, and [31] contains an exhaustive discussion of the topic. GM has found several applications in optimization theory, such as regularization [14], conjugate gradient methods [15], and especially in lexicographic multi-objective LP. In [8], indeed, a Grossone-version of the Simplex algorithm (the G-Simplex) has been implemented and theoretically studied.…”

On the Use of Grossone Methodology for Handling Priorities in Multi-objective Evolutionary Optimization

“…A very useful application of Grossone Methodology [44] is a ploy to reformulate lexicographic problems in a way that allows performing actual numerical computations by means of macro-objectives, something that is not possible with standard model. Grossone has been successfully applied in several other optimization problems, such as in linear and non-linear programming [8,9,11], in global optimization [46], in largescale unconstrained optimization [10], in machine learning [2,3,20] and in many other areas. In addition, the Grossone Methodology has been shown to be independent from non-standard analysis [45].…”