Optimization problems in which the variable is not a vector but a symmetric matrix which is required to be positive semidefinite have been intensely studied in the last ten years. Part of the reason for the interest stems from the applicability of such problems to such diverse areas as designing the strongest column, checking the stability of a differential inclusion, and obtaining tight bounds for hard combinatorial optimization problems. Part also derives from great advances in our ability to solve such problems efficiently in theory and in practice (perhaps “or ” would be more appropriate: the most effective computational methods are not always provably efficient in theory, and vice versa). Here we describe this class of optimization problems, give a number of examples demonstrating its significance, outline its duality theory, and discuss algorithms for solving such problems.

The Goemans-Williamson randomized algorithm guarantees a high-quality approximation to the Max-Cut problem, but the cost associated with such an approximation can be excessively high for large-scale problems due to the need for solving an expensive semidefinite relaxation. In order to achieve better practical performance, we propose an alternative, rank-two relaxation and develop a specialized version of the Goemans-Williamson technique. The proposed approach leads to continuous optimization heuristics applicable to Max-Cut as well as other binary quadratic programs, for example the Max-Bisection problem. A computer code based on the rank-two relaxation heuristics is compared with two state-of-the-art semidefinite programming codes that implement the Goemans-Williamson randomized algorithm, as well as with a purely heuristic code for effectively solving a particular Max-Cut problem arising in physics. Computational results show that the proposed approach is fast and scalable and, more importantly, attains a higher approximation quality in practice than that of the Goemans-Williamson randomized algorithm. An extension to Max-Bisection is also discussed as well as an important difference between the proposed approach and the Goemans-Williamson algorithm, namely that the new approach does not guarantee an upper bound on the Max-Cut optimal value. Key words. Binary quadratic programs, Max-Cut and Max-Bisection, semidefinite relaxation, rank-two relaxation, continuous optimization heuristics. AMS subject classifications. 90C06, 90C27, 90C30 1.

Recently in Burer et al. (Mathematical Programming A, submitted), the authors of this paper introduced a nonlinear transformation to convert the positive definiteness constraint on an n × n matrix-valued function of a certain form into the positivity constraint on n scalar variables while keeping the number of variables unchanged. Based on this transformation, they proposed a first-order interior-point algorithm for solving a special class of linear semidefinite programs. In this paper, we extend this approach and apply the transformation to general linear semidefinite programs, producing nonlinear programs that have not only the n positivity constraints, but also n additional nonlinear inequality constraints. Despite this complication, the transformed problems still retain most of the desirable properties. We propose first-order and second-order interior-point algorithms for this type of nonlinear program and establish their global convergence. Computational results demonstrating the effectiveness of the first-order method are also presented.

– approximations – linear scaling methods – convex program for density matrix 2. Path-following interior point method [Boyd and Vandenberghe (2001)] – formulation – number of iterations – cost of each iteration 3. Automatic differentiation [Griewank (1989), Vavasis (1999)] 4. Automatic CG preconditioning [Morales and Nocedal (2000,2002)]