Limit Theorems in the Problem of Optimal Linear Transformation

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Limit Theorems in the Problem of Optimal Linear Transformation. / Malozemov, V.N.; Petrov, A.V.

In: Operations Research Forum, Vol. 7, No. 1, 01.03.2026.

Research output: Contribution to journal › Article › peer-review

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Malozemov, V.N. ; Petrov, A.V. / Limit Theorems in the Problem of Optimal Linear Transformation. In: Operations Research Forum. 2026 ; Vol. 7, No. 1.

BibTeX

@article{a0fb011fcfdc4e1fb5828f1bb1bb7693,

title = "Limit Theorems in the Problem of Optimal Linear Transformation",

abstract = "We consider the following extremal problem: Among all matrices that map a given point x of the Euclidean spaceRnto a given point b of the Euclidean spaceRm, find a matrix of minimal norm. The article uses the H{\"o}lder norm of vectors and matrices depending on a parameter p. It is shown that for every p∈(1,+∞) there exists a unique solution of the stated problem and an explicit formula for it is derived. Limits of the optimal matrix are found as p approaches the boundary values p→1 and p→+∞. It is established that the limiting matrices are solutions of the corresponding limiting extremal problems. However, unlike the case p∈(1,+∞), uniqueness of these limiting solutions is not guaranteed. A full description of the entire set of solutions of the nonsmooth limiting problems is given. {\textcopyright} The Author(s), under exclusive licence to Springer Nature Switzerland AG 2025.",

keywords = "H{\"o}lder norms, Inverse linear algebra problem, Limit theorems, minimax problems, Minimization of the sum of moduli",

author = "V.N. Malozemov and A.V. Petrov",

note = "Export Date: 05 February 2026; Cited By: 0; Correspondence Address: V.N. Malozemov; Saint-Petersburg State University, Saint-Petersburg, University Embankment, 199034, Russian Federation; email: v.malozemov@spbu.ru; A.V. Petrov; Saint-Petersburg State University, Saint-Petersburg, University Embankment, 199034, Russian Federation; email: aleksndr19@rambler.ru",

year = "2026",

month = mar,

day = "1",

doi = "10.1007/s43069-025-00585-z",

language = "Английский",

volume = "7",

journal = "Operations Research Forum",

issn = "2662-2556",

publisher = "Springer Nature",

number = "1",

}

RIS

TY - JOUR

T1 - Limit Theorems in the Problem of Optimal Linear Transformation

AU - Malozemov, V.N.

AU - Petrov, A.V.

N1 - Export Date: 05 February 2026; Cited By: 0; Correspondence Address: V.N. Malozemov; Saint-Petersburg State University, Saint-Petersburg, University Embankment, 199034, Russian Federation; email: v.malozemov@spbu.ru; A.V. Petrov; Saint-Petersburg State University, Saint-Petersburg, University Embankment, 199034, Russian Federation; email: aleksndr19@rambler.ru

PY - 2026/3/1

Y1 - 2026/3/1

N2 - We consider the following extremal problem: Among all matrices that map a given point x of the Euclidean spaceRnto a given point b of the Euclidean spaceRm, find a matrix of minimal norm. The article uses the Hölder norm of vectors and matrices depending on a parameter p. It is shown that for every p∈(1,+∞) there exists a unique solution of the stated problem and an explicit formula for it is derived. Limits of the optimal matrix are found as p approaches the boundary values p→1 and p→+∞. It is established that the limiting matrices are solutions of the corresponding limiting extremal problems. However, unlike the case p∈(1,+∞), uniqueness of these limiting solutions is not guaranteed. A full description of the entire set of solutions of the nonsmooth limiting problems is given. © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2025.

AB - We consider the following extremal problem: Among all matrices that map a given point x of the Euclidean spaceRnto a given point b of the Euclidean spaceRm, find a matrix of minimal norm. The article uses the Hölder norm of vectors and matrices depending on a parameter p. It is shown that for every p∈(1,+∞) there exists a unique solution of the stated problem and an explicit formula for it is derived. Limits of the optimal matrix are found as p approaches the boundary values p→1 and p→+∞. It is established that the limiting matrices are solutions of the corresponding limiting extremal problems. However, unlike the case p∈(1,+∞), uniqueness of these limiting solutions is not guaranteed. A full description of the entire set of solutions of the nonsmooth limiting problems is given. © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2025.

KW - Hölder norms

KW - Inverse linear algebra problem

KW - Limit theorems

KW - minimax problems

KW - Minimization of the sum of moduli

UR - https://www.mendeley.com/catalogue/5feebf26-2b71-38e6-97fb-4e5b1ed9ef9c/

U2 - 10.1007/s43069-025-00585-z

DO - 10.1007/s43069-025-00585-z

M3 - статья

VL - 7

JO - Operations Research Forum

JF - Operations Research Forum

SN - 2662-2556

IS - 1

ER -

ID: 148346853