The Space Decomposition Theory for a Class of Semi‐Infinite Maximum Eigenvalue Optimizations

We study optimization problems involving eigenvalues of symmetric matrices. We present a nonsmooth optimization technique for a class of nonsmooth functions which are semi‐infinite maxima of eigenvalue functions. Our strategy uses generalized gradients and 𝒰𝒱 space decomposition techniques suited for the norm and other nonsmooth performance criteria. For the class of max‐functions, which possesses the so‐called primal‐dual gradient structure, we compute smooth trajectories along which certain second‐order expansions can be obtained. We also give the first‐ and second‐order derivatives of primal‐dual function in the space of decision variables Rm under some assumptions.