### 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 d_{i} 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 language | English |
---|---|

Pages (from-to) | 1-25 |

Number of pages | 25 |

Journal | Optimization Methods and Software |

Volume | 33 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 2 2018 |

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### 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.

Research output: Contribution to journal › Article

*Optimization Methods and Software*, vol. 33, no. 1, pp. 1-25. https://doi.org/10.1080/10556788.2017.1382495

}

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.

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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 -