Solving A Minimization Problem For A Class Of Constrained Maximum Eigenvalue Fuction

Nonsmooth convex optimization problem is a class of important problems in operationsresearch．As one of the most important problems in nonsmooth convex optimization, theminimization problem of maximum eigenvalue functions is used in physics, engineering,statistics fields widely. In this paper，the bundle method algorithm with a penalty parameter isgiven to solve the optimization problem which is the sum of a maximum eigenvalue functionand a nonsmooth convex function. The main content of this paper can be summarized asfollows:Based on the fact that some function values are not easy to compute, in chapter2weestablish an approximate model of the objective function. By using the approximate model,we can obtain the candidate point sequences and the stable centers. Then we lay thegroundwork for the implementation of the algorithm. In chapter3, a proximal bundle methodwith a penalty parameter is proposed. We compress the bundle model and solve the thornyproblem which occur in the bundle method. In chapter4, the sequence produced by thealgorithm converge to the optimal solution of the original problem. Applying the knowledgeof front parts, the algorithm is used to solve a class of constrained maximum eigenvaluefunctions. In the chapter6, considering the instability nature of subdifferential of the objectivefunctions, an expanded subdifferential is introduced. Then a new approximate model isobtained. This method is more convenient for application.