SpeeDP: A new algorithm to compute the SDP relaxations of Max-Cut for very large graphs

We consider low-rank semidefinite programming (LRSDP) relaxations of unconstrained {−1, 1} quadratic problems (or, equivalently, of Max-Cut problems) that can be formulated as the nonconvex nonlinear programming problem of minimizing a quadratic function subject to separable quadratic equality constraints. We prove the equivalence of the LRSDP problem with the unconstrained minimization of a new merit function and we define an efficient and globally convergent algorithm, called SpeeDP, for finding critical points of the LRSDP problem. We provide evidence of the effectiveness of SpeeDP by comparing it with other existing codes on an extended set of instances of the Max-Cut problem. When the rank of solution matrix is bounded by a given value (independent on the problem size n), SpeeDP is still able to provide a valid upper bound for Max-Cut. This feature makes it possible to design an algorithm, called SpeeDP-MC and based on the Goemans-Williamson heuristic, that has two interesting features: (a) it provides heuristic solutions to Max-Cut along with a guaranteed optimality error; (b) it runs with a O(n + m) memory requirement (where m is the number of edges of the graph), thus overcoming a serious drawback of interior point based methods that demand O(n) memory. Exploiting the latter feature, we could run it on very large graphs with sizes of up to a million nodes, obtaining very small optimality error bounds in reasonable computation times. keywords Semidefinite programming, low rank factorization, unconstrained binary quadratic programming, Max-Cut, nonlinear programming.