The UV-theory For A Class Of The Maximum Eigenvalue Functions Optimizations

One of the important branch of operations reach is nonsmooth optimization. In nonsmooth optimization, eigenvalue optimization problem has been widely researched in physics, engineering and statistics fields. This paper is concerned with a class of function, which is the sum of the composite function of the maximum eigenvalue function and an affine mapping and a finite-valued convex twice continuously differentiable function, and the unconstrained problem which the object function is the function introduced. In other words, the model problem is as follows: min x∈Rnλ1(A(x))+g(x) where λ1(·) is the maximum eigenvalue function and the mapping A:Rn(?)xâ†'A0+βx is affine:A0 is a given real n×n symmetric matrix and β is linear operator from Rn to the space of n×n symmetric matrices. g(x) is a finite-valued convex twice continuously differentiable functionThe article is mainly discussed from three aspects. The first aspect is researching the object function. We will apply UV-theory which can solve the unconstrained nonsmooth optimization problem. Firstly, three kinds of UV-space decomposition of the object function and the proof of the equivalency about this three kinds of UV-space decomposition will be introduced. Moreover, with the help of U-Lagrange function, the first-order and second-order approximation of the object function will be shown later. By the reason of the relation between the approximations and the set of minimizers, researching the properties of the set of minimizers is the second key point. Finally, based on the UV-theory of the maximum function, we give an algorithm to solve unconstrained problem about the class of maximum function. The conclusions of this paper provide a new method for first-order approximation and second-order approximation of the class of maximum function. Meanw-hile, it also introduced a new way to solve the constrained optimization problem of the maximum eigenvalue function...