Lexicographic multi-objective linear programming using grossone methodology: Theory and algorithm

Cococcioni, Marco; Pappalardo, M.; Sergeyev, Yaroslav D.

doi:10.1016/j.amc.2017.05.058

Cited by 66 publications

(91 citation statements)

References 31 publications

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“…Following an idea of the x-based lexicographic ordering proposed in [100] it has been shown in [20] how to transform LMOLP into a single-objective LP problem. This is done by multiplying the most important objective by 1, the second by x −1 , the third by x −2 etc., where x −i+1 , 1 ≤ i ≤ r, are infinitesimals and r is a finite number of objectives.…”

Section: 3mentioning

confidence: 99%

“…In particular, metamathematical investigations on the new theory and its consistency can be found in [66]. The methodology described here has been successfully applied in several areas of Mathematics and Computer Science: single and multiple criteria optimization (see [20,32,33,34,43,116]), cellular automata (see [29,30,31]), Euclidean and hyperbolic geometry (see [68,69]), percolation (see [56,57,110]), fractals (see [15,87,89,97,102,110]), infinite series and the Riemann zeta function (see [91,96,99,101,114]), the first Hilbert problem, Turing machines, and supertasks (see [81,93,103,104]), numerical differentiation and numerical solution of ordinary differential equations (see [2,74,95,98,105]), etc. Some of these applications will be discussed in the following pages.…”

Section: Introductionmentioning

confidence: 99%

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Numerical infinities and infinitesimals: Methodology, applications, and repercussions on two Hilbert problems

Sergeyev¹

2017

EMS Surv. Math. Sci.

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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 Cantor's and non-standard analysis views and is based on the Euclid's Common Notion no. 5 'The whole is greater than the part' 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 ∞, ℵ 0 , ω, 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.

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Section: 3mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Numerical infinities and infinitesimals: Methodology, applications, and repercussions on two Hilbert problems

Sergeyev¹

2017

EMS Surv. Math. Sci.

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“…The problem involves multi-objective optimization, and two optimization targets are considered in the proposed model. Although some algorithms, such as the Pareto solution set method [24] and the lexicographic method [25], are efficient for solving multi-objective optimization, the linear weighting-sum method [26] is used here for its better practicability and convenience. The two optimization goals are first normalized to the same range and then processed into a single objective function.…”

Section: Objective Functionmentioning

confidence: 99%

Spatial and Temporal Optimization Strategy for Plug-In Electric Vehicle Charging to Mitigate Impacts on Distribution Network

et al. 2018

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The large deployment of plug-in electric vehicles (PEVs) challenges the operation of the distribution network. Uncoordinated charging of PEVs will cause a heavy load burden at rush hour and lead to increased power loss and voltage fluctuation. To overcome these problems, a novel coordinated charging strategy which considers the moving characteristics of PEVs is proposed in this paper. Firstly, the concept of trip chain is introduced to analyze the spatial and temporal distribution of PEVs. Then, a stochastic optimization model for PEV charging is established to minimize the distribution network power loss (DNPL) and maximal voltage deviation (MVD). After that, the particle swarm optimization (PSO) algorithm with an embedded power flow program is adopted to solve the model, due to its simplicity and practicality. Last, the feasibility and efficiency of the proposed strategy is tested on the IEEE 33 distribution system. Simulation results show that the proposed charging strategy not only reduces power loss and the peak valley difference, but also improves voltage profile greatly.

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“…In particular, metamathematical investigations on the new theory and its non-contradictory can be found in [17]. The x-based methodology has been successfully applied in several areas of Mathematics and Computer Science: single and multiple criteria optimization (see [21,22,23,24,25]), cellular automata (see [26,27]), Euclidean and hyperbolic geometry (see [28,29]), percolation (see [30]), fractals (see [31,32,33,34,35]), infinite series and the Riemann zeta function (see [36,37,38,39,40]), the first Hilbert problem, Turing machines, and supertasks (see [41,42,20,43]), numerical differentiation and numerical solution of ordinary differential equations (see [44,45,46,47,48]), etc. In this paper, divergent series and Ramanujan summation are studied.…”

Section: Introductionmentioning

confidence: 99%

Numerical infinities applied for studying Riemann series theorem and Ramanujan summation

Sergeyev¹

2018

AIP Conference Proceedings

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A computational methodology called Grossone Infinity Computing introduced with the intention to allow one to work with infinities and infinitesimals numerically has been applied recently to a number of problems in numerical mathematics (optimization, numerical differentiation, numerical algorithms for solving ODEs, etc.). The possibility to use a specially developed computational device called the Infinity Computer (patented in USA and EU) for working with infinite and infinitesimal numbers numerically gives an additional advantage to this approach in comparison with traditional methodologies studying infinities and infinitesimals only symbolically. The grossone methodology uses the Euclid's Common Notion no. 5 'The whole is greater than the part' and applies it to finite, infinite, and infinitesimal quantities and to finite and infinite sets and processes. It does not contradict Cantor's and non-standard analysis views on infinity and can be considered as an applied development of their ideas. In this paper we consider infinite series and a particular attention is dedicated to divergent series with alternate signs. The Riemann series theorem states that conditionally convergent series can be rearranged in such a way that they either diverge or converge to an arbitrary real number. It is shown here that Riemann's result is a consequence of the fact that symbol ∞ used traditionally does not allow us to express quantitatively the number of addends in the series, in other words, it just shows that the number of summands is infinite and does not allows us to count them. The usage of the grossone methodology allows us to see that (as it happens in the case where the number of addends is finite) rearrangements do not change the result for any sum with a fixed infinite number of summands. There are considered some traditional summation techniques such as Ramanujan summation producing results where to divergent series containing infinitely many positive integers negative results are assigned. It is shown that the careful counting of the number of addends in infinite series allows us to avoid this kind of results if grossone-based numerals are used.

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Lexicographic multi-objective linear programming using grossone methodology: Theory and algorithm

Cited by 66 publications

References 31 publications

Numerical infinities and infinitesimals: Methodology, applications, and repercussions on two Hilbert problems

Numerical infinities and infinitesimals: Methodology, applications, and repercussions on two Hilbert problems

Spatial and Temporal Optimization Strategy for Plug-In Electric Vehicle Charging to Mitigate Impacts on Distribution Network

Numerical infinities applied for studying Riemann series theorem and Ramanujan summation

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