An inexact spectral bundle method for convex quadratic semidefinite programming

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 in 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 program 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.