An extension of Chubanov's polynomial-time linear programming algorithm to second-order cone programming

Tomonari Kitahara, T. Tsuchiya

Research output: Contribution to journalArticle

4 Citations (Scopus)

Abstract

In this paper, we extend Chubanov's new polynomial-time algorithm for linear programming to second-order cone programming based on the idea of cutting plane method. The algorithm finds an (Formula presented.) -dimensional vector x which satisfies Ax = 0, x ∈ K, where (Formula presented.) and K is a direct product of n second-order cones and half lines. Like Chubanov's algorithm, one iteration of the proposed algorithm consists of two phases: execution of a basic procedure and scaling. Within O(n log ∈−1) calls of the basic procedure, the algorithm either (i) finds an interior feasible solution, (ii) finds a non-zero dual feasible solution, or (iii) verifies that there is no interior feasible solution whose minimum eigenvalue is greater than or equal to ϵ. Each basic procedure requires (Formula presented.) arithmetic operations, where di is the dimension of each second-order cone. If the problem is interior feasible, then the algorithm finds an interior feasible solution in O(n log cond(A,K)) calls of the basic procedure, where cond(A,K) is a condition number associated with the system.

Original languageEnglish
Pages (from-to)1-25
Number of pages25
JournalOptimization Methods and Software
Volume33
Issue number1
DOIs
Publication statusPublished - Jan 2 2018

Fingerprint

Second-order Cone Programming
Linear programming
Cones
Polynomial time
Polynomials
Interior
Second-order Cone
Cutting Plane Method
Direct Product
Condition number
Polynomial-time Algorithm
Half line
Scaling
Verify
Eigenvalue
Iteration

All Science Journal Classification (ASJC) codes

  • Software
  • Control and Optimization
  • Applied Mathematics

Cite this

An extension of Chubanov's polynomial-time linear programming algorithm to second-order cone programming. / Kitahara, Tomonari; Tsuchiya, T.

In: Optimization Methods and Software, Vol. 33, No. 1, 02.01.2018, p. 1-25.

Research output: Contribution to journalArticle

@article{703245733099464eb9b43e7f222475bd,
title = "An extension of Chubanov's polynomial-time linear programming algorithm to second-order cone programming",
abstract = "In this paper, we extend Chubanov's new polynomial-time algorithm for linear programming to second-order cone programming based on the idea of cutting plane method. The algorithm finds an (Formula presented.) -dimensional vector x which satisfies Ax = 0, x ∈ K, where (Formula presented.) and K is a direct product of n second-order cones and half lines. Like Chubanov's algorithm, one iteration of the proposed algorithm consists of two phases: execution of a basic procedure and scaling. Within O(n log ∈−1) calls of the basic procedure, the algorithm either (i) finds an interior feasible solution, (ii) finds a non-zero dual feasible solution, or (iii) verifies that there is no interior feasible solution whose minimum eigenvalue is greater than or equal to ϵ. Each basic procedure requires (Formula presented.) arithmetic operations, where di is the dimension of each second-order cone. If the problem is interior feasible, then the algorithm finds an interior feasible solution in O(n log cond(A,K)) calls of the basic procedure, where cond(A,K) is a condition number associated with the system.",
author = "Tomonari Kitahara and T. Tsuchiya",
year = "2018",
month = "1",
day = "2",
doi = "10.1080/10556788.2017.1382495",
language = "English",
volume = "33",
pages = "1--25",
journal = "Optimization Methods and Software",
issn = "1055-6788",
publisher = "Taylor and Francis Ltd.",
number = "1",

}

TY - JOUR

T1 - An extension of Chubanov's polynomial-time linear programming algorithm to second-order cone programming

AU - Kitahara, Tomonari

AU - Tsuchiya, T.

PY - 2018/1/2

Y1 - 2018/1/2

N2 - In this paper, we extend Chubanov's new polynomial-time algorithm for linear programming to second-order cone programming based on the idea of cutting plane method. The algorithm finds an (Formula presented.) -dimensional vector x which satisfies Ax = 0, x ∈ K, where (Formula presented.) and K is a direct product of n second-order cones and half lines. Like Chubanov's algorithm, one iteration of the proposed algorithm consists of two phases: execution of a basic procedure and scaling. Within O(n log ∈−1) calls of the basic procedure, the algorithm either (i) finds an interior feasible solution, (ii) finds a non-zero dual feasible solution, or (iii) verifies that there is no interior feasible solution whose minimum eigenvalue is greater than or equal to ϵ. Each basic procedure requires (Formula presented.) arithmetic operations, where di is the dimension of each second-order cone. If the problem is interior feasible, then the algorithm finds an interior feasible solution in O(n log cond(A,K)) calls of the basic procedure, where cond(A,K) is a condition number associated with the system.

AB - In this paper, we extend Chubanov's new polynomial-time algorithm for linear programming to second-order cone programming based on the idea of cutting plane method. The algorithm finds an (Formula presented.) -dimensional vector x which satisfies Ax = 0, x ∈ K, where (Formula presented.) and K is a direct product of n second-order cones and half lines. Like Chubanov's algorithm, one iteration of the proposed algorithm consists of two phases: execution of a basic procedure and scaling. Within O(n log ∈−1) calls of the basic procedure, the algorithm either (i) finds an interior feasible solution, (ii) finds a non-zero dual feasible solution, or (iii) verifies that there is no interior feasible solution whose minimum eigenvalue is greater than or equal to ϵ. Each basic procedure requires (Formula presented.) arithmetic operations, where di is the dimension of each second-order cone. If the problem is interior feasible, then the algorithm finds an interior feasible solution in O(n log cond(A,K)) calls of the basic procedure, where cond(A,K) is a condition number associated with the system.

UR - http://www.scopus.com/inward/record.url?scp=85031429865&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85031429865&partnerID=8YFLogxK

U2 - 10.1080/10556788.2017.1382495

DO - 10.1080/10556788.2017.1382495

M3 - Article

AN - SCOPUS:85031429865

VL - 33

SP - 1

EP - 25

JO - Optimization Methods and Software

JF - Optimization Methods and Software

SN - 1055-6788

IS - 1

ER -