The bundle scheme for solving arbitrary eigenvalue optimizations

Abstract

Optimization involving eigenvalues arises in a large spectrum of applications in various domains, such as physics, engineering, statistics and finance. In this paper, we consider the arbitrary eigenvalue minimization problems over an affine family of symmetric matrices, which is a special class of eigenvalue function-D.C. function λ l. An explicit proximal bundle approach for solving this class of nonsmooth, nonconvex (D.C.) optimization problem is developed. We prove the global convergence of our method, in the sense that every accumulation point of the sequence of iterates generated by the proposed algorithm is stationary. As an application, we use the proposed bundle method to solve some particular eigenvalue problems. Some encouraging preliminary numerical results indicate that our method can solve the test problems efficiently.

A proximal-projection partial bundle method for convex constrained minimax problems

“…We also note that our method is very different from that in [16] due to the proximalprojection strategy and the new descent test criterion. Some other recent methods for general-purpose nonsmooth constrained minimization problems can be found in, e.g., [13,32,17,35,36].…”