A Non-Archimedean Interior Point Method and Its Application to the Lexicographic Multi-Objective Quadratic Programming

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 t...

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Vydáno v:Mathematics (Basel) Ročník 10; číslo 23; s. 4536
Hlavní autoři: Fiaschi, Lorenzo, Cococcioni, Marco
Médium: Journal Article
Jazyk:angličtina
Vydáno: Basel MDPI AG 01.12.2022
Témata:
Algorithms
Fixed point theory
Food science
Game theory
infinite/infinitesimal numbers
interior point methods
lexicographic multi-objective optimization
Lexicography
Linear algebra
Linear programming
Mathematical research
Methods
non-Archimedean scientific computing
non-standard analysis
Optimization
Polynomials
Preempting
Quadratic programming
Italy
ISSN:2227-7390, 2227-7390
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Shrnutí: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.
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ISSN:2227-7390
2227-7390
DOI:10.3390/math10234536