An unconstrained minimization method for solving low-rank SDP relaxations of the maxcut problem

Abstract

In this paper we consider low-rank semidefinite programming (LRSDP) relaxations of combinatorial quadratic problems that are equivalent to the maxcut problem. Using the Gramian representation of a positive semidefinite matrix, the LRSDP problem can be formulated as the nonconvex nonlinear programming problem of minimizing a quadratic function with quadratic equality constraints. For the solution of this problem we propose a continuously differentiable exact merit function that exploits the special structure of the constraints and we use this function to define an efficient and globally convergent algorithm. Finally, we test our code on an extended set of instances of the maxcut problem and we report comparisons with other existing codes. © 2009 Springer and Mathematical Programming Society

Necessary and sufficient global optimality conditions for NLP reformulations of linear SDP problems

“…This has been already observed in [5] for the special LRSDP deriving from the maxcut problem. Following the same reasoning we show that the condition of Proposition 4 is also necessary.…”

“…Similar conditions have already been proved to be sufficient in [3]. These conditions extend the necessary and sufficient ones proved in [5] for the special case of the semidefinite relaxation of the max cut problem and can be related to the necessary and sufficient global optimality conditions established for some classes of nonconvex quadratic problems (see e.g. [6,11,9]).…”

“…It has been proved in [2,5] that if Problem (SDP) admits a solution X * or rank r, this can be found by solving Problem (1). Actually Problem (1) can be rewritten as a standard nonlinear programming problem, by…”

A feasible method for optimization with orthogonality constraints

“…Relaxing the rank-1 matrix xx to a positive semidefinite matrix X and ignoring the rank-1 requirement, we obtain the SDP relaxation problem (4.7),where C is the Laplace matrix of the graph divided by 4, i.e., [25], which forms an unconstrained differentiable exact penalty function and minimizes it using nonmontone Barzilai-Borwein gradient iterations. As opposed to our algorithms, these two algorithms do not preserve the spherical constraints during the iterations.…”

Computing the nearest low-rank correlation matrix by a simplified SQP algorithm

“…However, our method does not belong to the category of optimization on manifolds. For more background on using the Gramian representation to solve the low-rank semidefinite programming problem, the reader is referred to Burer and Monteiro [5,6] and Grippo, Palagi and Piccialli [12,13]. The following proposition reveals that the local minima of (1) and the local minima of (3) are equivalent.…”