Facial Reduction Algorithms for Conic Optimization Problems

Abstract

To obtain a primal-dual pair of conic programming problems having zero duality gap, two methods have been proposed: the facial reduction algorithm due to Borwein and Wolkowicz [1,2] and the conic expansion method due to Luo, Sturm, and Zhang [5]. We establish a clear relationship between them. Our results show that although the two methods can be regarded as dual to each other, the facial reduction algorithm can produce a finer sequence of faces including the feasible region. We illustrate the facial reduction algorithm in LP, SOCP and an example of SDP. A simple proof of the convergence of the facial reduction algorithm for conic programming is also presented.

A Structural Geometrical Analysis of Weakly Infeasible SDPS

“…Waki and Muramatsu [26] considered a FRA for conic programming and showed that FRA can be regarded as a dual version of CEA. See the excellent article by Pataki [15] for an elementary exposition of FRA, where he points out the relation between facial reduction and extended duals.…”

“…We now take a closer look at the connection between FRA/CEA and FP. More details on FRA can be found in [15] and [26]. In [26], it is also explained the connection between FRA and CEA, see section 4 therein.…”

“…More details on FRA can be found in [15] and [26]. In [26], it is also explained the connection between FRA and CEA, see section 4 therein. Roughly speaking, FRA and CEA can be seen as dual approaches to one another.…”

“…Regarding finite certificates, it is worthwhile mentioning that there is a way to obtain a finite and polynomially bounded certificate of weak infeasibility by using Ramana's extended Lagrangian dual [21] or the facial reduction algorithm [3,4,15,26]. This result is based on the fact that (K n , L, c) is weakly infeasible if and only if it is infeasible and not strongly infeasible.…”

The Many Faces of Degeneracy in Conic Optimization

“…Recently, there has been a significant renewed interest in facial reduction, in large part due to the success in applications for graph related problems, such as Euclidean distance matrix completion and molecular conformation [76,75,46,6] and in polynomial optimization [110,111,74,141,140]. In particular, a more modern explanation of the facial reduction procedure can be found in [88,104,107,136,142].…”