A Relax Inexact Accelerated Proximal Gradient Method for the Constrained Minimization Problem of Maximum Eigenvalue Functions

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 function $g(X)={\delta }_{\mathrm{\Omega }}(X)\hspace{0.17em}\hspace{0.17em}(\mathrm{\Omega }:=\{X\in S^n:\mathcal{F}(X)=b,X⪰\mathrm{0}\})$ in the problem $\text{min}\{{\lambda }_{\text{max}}(X)+g(X):X\in S^n\}$, where ${\lambda }_{\text{max}}(X)$ is the maximum eigenvalue function, $g(X)$ is a proper lower semicontinuous convex function (possibly nonsmooth) and ${\delta }_{\mathrm{\Omega }}(X)$ denotes the indicator function. But the approximate minimizer generated by AIAPG method must be contained in $\mathrm{\Omega }$ otherwise the method will be invalid. In this paper, we will consider the case where the approximate minimizer cannot be guaranteed in $\mathrm{\Omega }$. 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.