`SBmethod`, Version 1.1.1, is an implementation of the spectral bundle
method [15,13] for eigenvalue optimization
problems of the form

The design variables may be sign constrained, and the are given real symmetric matrices, allows to specify a linear cost term, and is a constant multiplier for the maximum eigenvalue function . The code is intended for large scale problems and allows to exploit structural properties of the matrices such as sparsity and low rank structure (see the manual [12]). Problem (1) is equivalent to the dual of semidefinite programs for with constant trace, .

The code is a subgradient method in the style of the proximal bundle method of
[16]. Function values and subgradients of the nonsmooth convex
function
are
determined via computing the maximum eigenvalue and a corresponding eigenvector
by the Lanczos method. Starting from a given point
, the algorithm
generates the next *candidate*
, where is an accumulated * semidefinite cutting model* minorizing and the dynamically updated * weight* keeps near . A *descent step*
occurs if
, where
. Otherwise a *null step*
is made but
the next model is improved with the new eigenvector computed at .

The algorithm stops, when for a given termination parameter which is by default.

- The code never factorizes the matrix but uses matrix vector multiplications exclusively. Thus, there is no danger of increasing memory requirements or computational work by additional fill-in.
- Experimentally, the method seems to exhibit fast convergence if the semidefinite subcone that is used to generate the semidefinite cutting surface model, spans the optimal solution of the corresponding primal problem.
- If the subcone, however, is too small, the typical tailing off effect of subgradient methods is observed.