Exact Solutions of Some Nonconvex Quadratic Optimization Problems via SDP and SOCP Relaxations

We show that SDP (semidefinite programming) and SOCP (second order cone programming) relaxations provide exact optimal solutions for a class of nonconvex quadratic optimization problems. It is a generalization of the results by S.~Zhang for a subclass of quadratic maximization problems that have nonnegative off-diagonal coefficient matrices of objective quadratic functions and diagonal coefficient matrices … Read more

Lagrangian dual interior-point methods for semidefinite programs

This paper proposes a new predictor-corrector interior-point method for a class of semidefinite programs, which numerically traces the central trajectory in a space of Lagrange multipliers. The distinguished features of the method are full use of the BFGS quasi-Newton method in the corrector procedure and an application of the conjugate gradient method with an effective … Read more

Exploiting Sparsity in Semidefinite Programming via Matrix Completion II: Implementation and Numerical Results

In Part I of this series of articles, we introduced a general framework of exploiting the aggregate sparsity pattern over all data matrices of large scale and sparse semidefinite programs (SDPs) when solving them by primal-dual interior-point methods. This framework is based on some results about positive semidefinite matrix completion, and it can be embodied … Read more