Volumetric barrier decomposition algorithms for stochastic quadratic second-order cone programming

Volumetric Barrier Cutting Plane Algorithms for Stochastic Linear Semi-Infinite Optimization

“…Note that the complexity results in Theorems 5 and 6 are the counterparts of those in Theorems 6 and 7 in [27] for two-stage stochastic quadratic linear programs with recourse, those in Theorems 3 and 4 in [29] for two-stage stochastic second-order programs with recourse, and those in Theorems 4 and 5 in [28] for two-stage stochastic semidefinite programs with recourse. Note also that the ''rich flavor'' hidden inside the volumetric barrier can be tasted in Theorems 5 and 6 more than in their counterpart theorems in [27]- [29]. The reason for this is that there are no cuts to be generated in the optimization problems studied in [27]- [29], which in turns makes no big difference by replacing m 1 and m 2 with n 1 and n 2 in case the volumetric barrier is not used in [27]- [29].…”

“…We will see that this significant advantage stems from the use the volumetric barrier instead of using the logarithmic barrier. We will also see that the ''rich flavor'' hidden inside the volumetric barrier can be tasted in the proposed algorithm of this work more than in their counterparts algorithms in [27]- [29].…”

“…Note also that the ''rich flavor'' hidden inside the volumetric barrier can be tasted in Theorems 5 and 6 more than in their counterpart theorems in [27]- [29]. The reason for this is that there are no cuts to be generated in the optimization problems studied in [27]- [29], which in turns makes no big difference by replacing m 1 and m 2 with n 1 and n 2 in case the volumetric barrier is not used in [27]- [29].…”

“…In 2011, Ariyawansa and Zhu [28] generalized their work in [27] to derive a volumetric barrier decomposition interior point algorithm for two-stage stochastic semidefinite programming. In 2015, Alzalg [29] exploited the work of Ariyawansa and Zhu in [27], [28] to derive a volumetric barrier decomposition interior point algorithm for two-stage stochastic second-order cone programming.…”

“…In this appendix, we present proofs for the complexity results stated in Section V that bound the number of iterations. The general scheme of our proofs follows the lines of the proofs from [29] and [28]. The proof of Theorem 5 for the short-step algorithm is given in Sub-appendix VI-A, and the proof of Theorem 6 for the longstep algorithm is given in Sub-appendix VI-B.…”