We propose a new interior point based method to minimize a linear function of a matrix variable subject to linear equality and inequality constraints over the set of positive semidefinite matrices. We show that the approach is very efficient for graph bisection problems, such as max-cut. Other applications include max-min eigenvalue problems and relaxations for the stable set problem.

Semidefinite programming (SDP) is a convex optimization problem in the space of symmetric matrices. Primal-dual interior-point methods for SDP are discussed. These generate primal and dual matrices X and Z which commute only in the limit. A new method is proposed which iterates in the space of commuting matrices. Let S! n\Thetan denote the set of real symmetric n \Theta n matrices. The standard inner product on this space is A ffl B = tr AB = P i;j a ij b ij : By X 0, where X 2 S! n\Thetan , we mean that X is positive semidefinite. Consider the semidefinite programming problem (SDP) min C ffl X (1) s:t: A i ffl X = b i i = 1; . . . ; m; X 0: (2) Here C and A i , i = 1; . . . ; m, are all fixed matrices in S! n\Thetan , and the unknown variable X also lies in S! n\Thetan . The semidefinite constraint on X is said to be nonsmooth, since it is equivalent to a bound constraint on the least eigenvalue of X , which is not a differentiable function of X . The constraint is, how...

. General quadratic matrix minimization problems, with orthogonal constraints, arise in continuous relaxations for the (discrete) quadratic assignment problem (QAP). Currently, bounds for QAP are obtained by treating the quadratic and linear parts of the objective function, of the relaxations, separately. This paper handles general objectives as one function. The objectives can be both nonhomogeneous and nonconvex. The constraints are orthogonal or Loewner partial order (positive semidefinite) constraints. Comparisons are made to standard trust region subproblems. Numerical results are obtained using a parametric eigenvalue technique. Contents 1. Introduction 1 2. Preliminary Notations and Motivation 2 2.1. Notations 2.2. A Survey on Eigenvalue Bounds for the QAP 2.3. Loewner Partial Order 3. Optimality Conditions 6 3.1. First Order Conditions 3.2. Second Order Conditions 1991 Mathematics Subject Classification. Primary 90B80, 90C20, 90C35, 90C27; Secondary 65H20, 65K05. Key words...