7 research outputs found
Planar methods and grossone for the Conjugate Gradient breakdown in nonlinear programming
This paper deals with an analysis of the Conjugate Gradient (CG) method (Hestenes and Stiefel in J Res Nat Bur Stand 49:409-436, 1952), in the presence of degenerates on indefinite linear systems. Several approaches have been proposed in the literature to issue the latter drawback in optimization frameworks, including reformulating the original linear system or recurring to approximately solving it. All the proposed alternatives seem to rely on algebraic considerations, and basically pursue the idea of improving numerical efficiency. In this regard, here we sketch two separate analyses for the possible CG degeneracy. First, we start detailing a more standard algebraic viewpoint of the problem, suggested by planar methods. Then, another algebraic perspective is detailed, relying on a novel recently proposed theory, which includes an additional number, namely grossone. The use of grossone allows to work numerically with infinities and infinitesimals. The results obtained using the two proposed approaches perfectly match, showing that grossone may represent a fruitful and promising tool to be exploited within Nonlinear Programming
How grossone can be helpful to iteratively compute negative curvature directions
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 a recently introduced numerical 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 Newtonâs equations
Representation of grossone-based arithmetic in simulink for scientific computing
AbstractNumerical computing is a key part of the traditional computer architecture. Almost all traditional computers implement the IEEE 754-1985 binary floating point standard to represent and work with numbers. The architectural limitations of traditional computers make impossible to work with infinite and infinitesimal quantities numerically. This paper is dedicated to the Infinity Computer, a new kind of a supercomputer that allows one to perform numerical computations with finite, infinite, and infinitesimal numbers. The already available software simulator of the Infinity Computer is used in different research domains for solving important real-world problems, where precision represents a key aspect. However, the software simulator is not suitable for solving problems in control theory and dynamics, where visual programming tools like Simulink are used frequently. In this context, the paper presents an innovative solution that allows one to use the Infinity Computer arithmetic within the Simulink environment. It is shown that the proposed solution is user-friendly, general purpose, and domain independent
Issues on the use of a modified Bunch and Kaufman decomposition for large scale Newtonâs equation
In this work, we deal with Truncated Newton methods for solving large scale (possibly
nonconvex) unconstrained optimization problems. In particular, we consider the use of a modified
Bunch and Kaufman factorization for solving the Newton equation, at each (outer) iteration of the
method. The Bunch and Kaufman factorization of a tridiagonal matrix is an effective and stable matrix
decomposition, which is well exploited in the widely adopted SYMMBK [2, 5, 6, 19, 20] routine. It
can be used to provide conjugate directions, both in the case of 1 Ă 1 and 2 Ă 2 pivoting steps. The
main drawback is that the resulting solution of Newtonâs equation might not be gradientârelated, in
the case the objective function is nonconvex. Here we first focus on some theoretical properties, in
order to ensure that at each iteration of the Truncated Newton method, the search direction obtained
by using an adapted Bunch and Kaufman factorization is gradientârelated. This allows to perform
a standard Armijo-type linesearch procedure, using a bounded descent direction. Furthermore, the
results of an extended numerical experience using large scale CUTEst problems is reported, showing
the reliability and the efficiency of the proposed approach, both on convex and nonconvex problems
Bridging the gap between TrustâRegion Methods (TRMs) and Linesearch Based Methods (LBMs) for Nonlinear Programming: quadratic subâproblems
We consider the solution of a recurrent subâproblem within both constrained and unconstrained
Nonlinear Programming: namely the minimization of a quadratic function subject to
linear constraints. This problem appears in a number of LBM frameworks, and to some extent it
reveals a close analogy with the solution of trustâregion subâproblems. In particular, we refer to
a structured quadratic problem where five linear inequality constraints are included. We show that
our proposal retains an appreciable versatility, despite its particular structure, so that a number of
different real instances may be reformulated following the pattern in our proposal. Moreover, we
detail how to compute an exact global solution of our quadratic subâproblem, exploiting first order
KKT conditions
Some paradoxes of infinity revisited
In this article, some classical paradoxes of infinity such as Galileoâs paradox, Hilbertâs paradox of the Grand Hotel, Thomsonâs lamp paradox, and the rectangle paradox of Torricelli are considered. In addition, three paradoxes regarding divergent series and a new paradox dealing with multiplication of elements of an infinite set are also described. It is shown that the surprising counting system of an Amazonian tribe, Pirah Ìa, working with only three numerals (one, two, many) can help us to change our perception of these paradoxes. A recently introduced methodology allowing one to work with finite, infinite, and infinitesimal numbers in a unique computational framework not only theoretically but also numerically is briefly described. This methodology is actively used nowadays in numerous applications in pure and applied mathematics and computer science as well as in teaching. It is shown in the article that this methodology also allows one to consider the paradoxes listed above in a new constructive ligh
Numerical infinities and infinitesimals: Methodology, applications, and repercussions on two Hilbert problems
In this survey, a recent computational methodology paying a special attention to the separation
of mathematical objects from numeral systems involved in their representation is described.
It has been introduced with the intention to allow one to work with infinities and infinitesimals
numerically in a unique computational framework in all the situations requiring these notions. The
methodology does not contradict Cantorâs and non-standard analysis views and is based on the
Euclidâs Common Notion no. 5 âThe whole is greater than the partâ applied to all quantities (finite,
infinite, and infinitesimal) and to all sets and processes (finite and infinite). The methodology uses a
computational device called the Infinity Computer (patented in USA and EU) working numerically
(recall that traditional theories work with infinities and infinitesimals only symbolically) with infinite
and infinitesimal numbers that can be written in a positional numeral system with an infinite radix.
It is argued that numeral systems involved in computations limit our capabilities to compute and lead
to ambiguities in theoretical assertions, as well. The introduced methodology gives the possibility
to use the same numeral system for measuring infinite sets, working with divergent series, probability,
fractals, optimization problems, numerical differentiation, ODEs, etc. (recall that traditionally
different numerals lemniscate; Aleph zero, etc. are used in different situations related to infinity). Numerous numerical examples and theoretical illustrations are given. The accuracy of the achieved results is continuously compared with those obtained by traditional tools used to work with infinities and infinitesimals. In particular, it is shown that the new approach allows one to observe mathematical
objects involved in the Hypotheses of Continuum and the Riemann zeta function with a higher
accuracy than it is done by traditional tools. It is stressed that the hardness of both problems is not
related to their nature but is a consequence of the weakness of traditional numeral systems used to
study them. It is shown that the introduced methodology and numeral system change our perception
of the mathematical objects studied in the two problems