$${\cal U}{\cal V}$$ U V -theory of a Class of Semidefinite Programming and Its Applications

In this paper we study optimization problems involving convex nonlinear semidefinite programming (CSDP). Here we convert CSDP into eigenvalue problem by exact penalty function, and apply the $${\cal U}$$ -Lagrangian theory to the function of the largest eigenvalues, with matrix-convex valued mappings. We give the first-and second-order derivatives of $${\cal U}$$ -Lagrangian in the space of decision variables Rm when transversality condition holds. Moreover, an algorithm frame with superlinear convergence is presented. Finally, we give one application: bilinear matrix inequality (BMI) optimization; meanwhile, list their $${\cal U}{\cal V}$$ decomposition results.