## Abstract

In this paper we introduce a conic optimization formulation to solve constrained convex programming, and propose a self-dual embedding model for solving the resulting conic optimization problem. The primal and dual cones in this formulation are characterized by the original constraint functions and their corresponding conjugate functions respectively. Hence they are completely symmetric. This allows for a standard primal-dual path following approach for solving the embedded problem. Moreover, there are two immediate logarithmic barrier functions for the primal and dual cones. We show that these two logarithmic barrier functions are conjugate to each other. The explicit form of the conjugate functions are in fact not required to be known in the algorithm. An advantage of the new approach is that there is no need to assume an initial feasible solution to start with. To guarantee the polynomiality of the path-following procedure, we may apply the self-concordant barrier theory of Nesterov and Nemirovski. For this purpose, as one application, we prove that the barrier functions constructed this way are indeed self-concordant when the original constraint functions are convex and quadratic. We pose as an open question to find general conditions under which the constructed barrier functions are self-concordant.

Original language | English (US) |
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Pages (from-to) | 479-496 |

Number of pages | 18 |

Journal | Journal of Global Optimization |

Volume | 29 |

Issue number | 4 |

DOIs | |

State | Published - Aug 2004 |

### Bibliographical note

Funding Information:∗Research supported by Hong kong RGC Earmarked Grants CUHK4181/00E and CUHK4233/01E.

## Keywords

- Convex cones
- Convex programming
- Self-concordant barrier functions
- Self-dual embedding