### Abstract

In this paper, we present a measure of distance in a second-order cone based on a class of continuously differentiable strictly convex functions on ℝ_{++}. Since the distance function has some favorable properties similar to those of the D-function (Censor and Zenios in J. Optim. Theory Appl. 73:451-464 [1992]), we refer to it as a quasi D-function. Then, a proximal-like algorithm using the quasi D-function is proposed and applied to the second-cone programming problem, which is to minimize a closed proper convex function with general second-order cone constraints. Like the proximal point algorithm using the D-function (Censor and Zenios in J. Optim. Theory Appl. 73:451-464 [1992]; Chen and Teboulle in SIAM J. Optim. 3:538-543 [1993]), under some mild assumptions we establish the global convergence of the algorithm expressed in terms of function values; we show that the sequence generated by the proposed algorithm is bounded and that every accumulation point is a solution to the considered problem.

Original language | English |
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Pages (from-to) | 95-113 |

Number of pages | 19 |

Journal | Journal of Optimization Theory and Applications |

Volume | 138 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2008 Jul 1 |

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### Keywords

- Bregman functions
- Convex second-order cone programming
- Proximal-like methods
- Quasi D-functions

### ASJC Scopus subject areas

- Control and Optimization
- Management Science and Operations Research
- Applied Mathematics