In this paper, we apply the -theory to the function which is the sum of the precomposition of the maximum eigenvalue function  with an affine matrix-valued mapping and a finite-valued convex  twice continuously differentiable function. Three kinds of -space decomposition and the proof of equivalency about the three pair subspaces are given. Moreover, the first-order and second-order approximations of the class of maximum eigenvalue functions are shown later. Finally, we obtain some results about the continuity properties on the set of minimizers  of the -Lagrange function. In addition, the sufficient and necessary condition about the continuity of the set of minimizers at 0 is also obtained.

nonsmooth optimization, -Lagrangian, maximum eigenvalue function.