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One-sided heavy tailed distributions have been used in many engineering applications, ranging from teletraffic modelling to financial engineering. In practice, the most interesting heavy tailed distributions are those having a finite mean and a diverging variance. The LogNormal distribution is sometimes discarded from modelling heavy tailed phenomena because it has a finite variance, even when it seems the most appropriate one to fit the data. In this work we provide for the first time a LogNormal distribution having a finite mean and a variance which converges to a well-defined infinite value. This is possible thanks to the use of Non-Standard Analysis. In particular, we have been able to obtain a Non-Standard LogNormal distribution, for which it is possible to numerically and experimentally verify whether the expected mean and variance of a set of generated pseudo-random numbers agree with the theoretical ones. Moreover, such a check would be much more cumbersome (and sometimes even impossible) when considering heavy tailed distributions in the traditional framework of standard analysis. [ABSTRACT FROM AUTHOR]

This work presents a generalized implementation of the infeasible primal-dual interior point method (IPM) achieved by the use of non-Archimedean values, i.e., infinite and infinitesimal numbers. The extended version, called here the non-Archimedean IPM (NA-IPM), is proved to converge in polynomial time to a global optimum and to be able to manage infeasibility and unboundedness transparently, i.e., without considering them as corner cases: by means of a mild embedding (addition of two variables and one constraint), the NA-IPM implicitly and transparently manages their possible presence. Moreover, the new algorithm is able to solve a wider variety of linear and quadratic optimization problems than its standard counterpart. Among them, the lexicographic multi-objective one deserves particular attention, since the NA-IPM overcomes the issues that standard techniques (such as scalarization or preemptive approach) have. To support the theoretical properties of the NA-IPM, the manuscript also shows four linear and quadratic non-Archimedean programming test cases where the effectiveness of the algorithm is verified. This also stresses that the NA-IPM is not just a mere symbolic or theoretical algorithm but actually a concrete numerical tool, paving the way for its use in real-world problems in the near future. [ABSTRACT FROM AUTHOR]