A note on efficient computation of the gradient in semidefinite programming

In the Goemans-Williamson semidefinite relaxation of MAX-CUT, the gradient of the dual barrier objective function has a term of the form diag(Z), where Z is the slack matrix. The purpose of this note is to show that this term can be computed in time and space proportional to the time and space for computing a sparse Cholesky factor of Z using an algorithm due to Erisman and Tinney. The algorithm for computing the term can also be derived from automatic differentiation in backward mode. 1 Semidefinite relaxation of MAX-CUT The MAX-CUT problem is the following combinatorial optimization problem. Given an undirected n-node graph G, find a partition of the nodes of G into two sets V1, V2 such that the number of edges that cross from V1 to V2 is maximized. This problem is known to be NP-complete. Recently, Goemans and Williamson [5] proposed the following semidefinite relaxation Revised September 23, 1999 Department of Computer Science, 4130 Upson Hall, Cornell University, Ithaca, New York 14853. Email: [email protected]. Part of this work was done while the author was visiting Bell Laboratories, 600 Mountain Avenue, Murray Hill, NJ 07974-0636. Also supported in part by NSF grant CCR-9619489. and in part by NSF through grant DMS-9505155 and ONR through grant N00014-96-1-0050.