Solving Semide nite Programs via Nonlinear Programming

In a semidefinite programming (SDP) problem, a linear function of a symmetric matrix variable X is minimized subject to linear equality constraints on X and the essential constraint that X be positive semidefinite. We show how a special class of semidefinite programming (SDP) problems can be transformed into well-behaved nonlinear programs over ”orthants” <++×< , where n is the size of the matrices involved in the problem and N is a nonnegative integer dependent upon the problem. The class of transformable problems includes most, if not all, instances of SDP relaxations of combinatorial optimization problems with binary variables as well as other important SDP problems. We also develop interior point methods for solving this class of SDP problems. These new interior point methods have the advantage of working entirely within the space of the transformed problem while still maintaining close ties with the original SDP. Under very mild and reasonable assumptions, global convergence of these methods is proved. Strong computational results are presented to show that the method is very promising. (This is joint work with Sam Burer and Yin Zhang.)