We present an inexact spectral bundle method for solving convex quadratic semidefinite optimization problems. This method is a first-order method, hence requires much less computational cost each iteration than second-order approaches such as interior-point methods. In each iteration of our method, we solve an eigenvalue minimization problem inexactly, and solve a small convex quadratic semidefinite programming as a subproblem. We give a proof of the global convergence of this method using techniques from the analysis of the standard bundle method, and provide a global error bound under a Slater type condition for the problem in question. Numerical experiments with matrices of order up to 3000 are performed and the computational results establish the effectiveness of this method.

## Citation

33 pages Division of Mathematical Sciences, School of Physical & Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore. School of Mathematics and Computer Science, Fujian Normal University, Fuzhou 350007, PR China.

## Article

View The Inexact Spectral Bundle Method for Convex Quadratic Semidefinite Programming