Exploiting sparsity in primal-dual interior-point methods for semidefinite programming

Implementation of interior point methods for mixed semidefinite and second order cone optimization problems

“…Instead of dealing with (14)- (16) directly, one often concentrates on the so-called reduced system (21). The reduced system is derived by pre-multiplying (16) with AΠ yielding 0 = AΠ(A T ∆y + ∆z) (14) = AΠA T ∆y + A(r − ∆x) (15) = AΠA T ∆y + Ar.…”

Conic Optimization Software

“…These methods [34,[36][37][38][39] exploit the fact that the dual slack variable S inherits the sparsity structure of the coefficient matrices. By enforcing positive semidefiniteness only on a small number of carefully chosen submatrices of a large sparse matrix, one can make sure that the original matrix can be completed to be positive semidefinite.…”

Parallel implementation of a semidefinite programming solver based on CSDP on a distributed memory cluster

“…, m, the worst case complexity in computing M is O(mn 3 + m 2 n 2 ) [19]. In practice the constraint matrices are very sparse and we exploit sparsity in the construction of the Schur complement matrix in the same way as in [10]. For sparse A ′ i s with O(1) entries, the matrix Z −1 A i X can be computed in O(n 2 ) operations (i = 1, .…”

“…When we have dense constraints, construction of M , called Elements from now on, can be the most time consuming computation in each iteration when m >> n. To reduce the computational time of our code takes advantage of the of the approach by Fujisawa, Kojima, and Nakata [10]. As test results with SDPLIB suggest [5], [26] the most computational time in general large scale problems using primal-dual IPM algorithms is occupied by constructing Elements and solving the linear system which involves Cholesky.…”