Abstract
For constrained minimization problem of maximum eigenvalue functions, since the objective function is nonsmooth, we can use the approximate inexact accelerated proximal gradient (AIAPG) method (Wang et al., 2013) to solve its smooth approximation minimization problem. When we take the functiong(X)=δΩ(X) (Ω∶={X∈Sn:F(X)=b,X⪰0})in the problemmin{λmax(X)+g(X):X∈Sn}, whereλmax(X)is the maximum eigenvalue function,g(X)is a proper lower semicontinuous convex function (possibly nonsmooth) andδΩ(X)denotes the indicator function. But the approximate minimizer generated by AIAPG method must be contained inΩotherwise the method will be invalid. In this paper, we will consider the case where the approximate minimizer cannot be guaranteed inΩ. Thus we will propose two different strategies, respectively, constructing the feasible solution and designing a new method named relax inexact accelerated proximal gradient (RIAPG) method. It is worth mentioning that one advantage when compared to the former is that the latter strategy can overcome the drawback. The drawback is that the required conditions are too strict. Furthermore, the RIAPG method inherits the global iteration complexity and attractive computational advantage of AIAPG method.
Highlights
The minimization problem of maximum eigenvalue functions in nonsmooth optimization presents a fascinating mathematical challenge
To overcome the drawbacks above we propose relax inexact accelerated proximal gradient (RIAPG) method to solve problem (P2) for which the iterate points Xk generated by method need not be strictly contained in Ω
It should be noted that comparing to the quantities of approximate inexact accelerated proximal gradient (AIAPG) method, ak and Vk here may be negative since the lack of the feasibility of Xk
Summary
The minimization problem of maximum eigenvalue functions in nonsmooth optimization presents a fascinating mathematical challenge. Was introduced in [5] for solving the minimization problem of the sum of maximum eigenvalue function and proper lower semicontinuous convex function. We design the RIAPG method which is based on AIAPG method to solve the smooth approximation problem of constrained minimization problem of maximum eigenvalue functions. We consider the smooth approximation (hε ∘ λ)(X) [6] to the maximum eigenvalue function λmax(X) which is a proper, lower semicontinuous, convex function and ∇(hε∘λ)(X) is Lipschitz continuous This thought for dealing with the problem resembles the technique used in [7]. We use infeasible approximate minimizer to construct feasible solution which satisfies the conditions required by AIAPG method. To overcome the drawbacks above we propose RIAPG method to solve problem (P2) for which the iterate points Xk generated by method need not be strictly contained in Ω
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