A -Method for a Class of Constrained Minimized Problems of Maximum Eigenvalue Functions

In this paper, we apply the -algorithm to solve the constrained minimization problem of a maximum eigenvalue function which is the composite function of an affine matrix-valued mapping and its maximum eigenvalue. Here, we convert the constrained problem into its equivalent unconstrained problem by the exact penalty function. However, the equivalent problem involves the sum of two nonsmooth functions, which makes it difficult to apply -algorithm to get the solution of the problem. Hence, our strategy first applies the smooth convex approximation of maximum eigenvalue function to get the approximate problem of the equivalent problem. Then the approximate problem, the space decomposition, and the -Lagrangian of the object function at a given point will be addressed particularly. Finally, the -algorithm will be presented to get the approximate solution of the primal problem by solving the approximate problem.

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