Smooth convex approximation and its applications

Abstract

In this thesis, we consider a smooth convex approximation to the sum of the $\kappa$ largest components. To make it applicable to a wide class of applications, the study is conducted on some minmax problems. Based on a special smoothing technique, we give an efficient scheme for nonsmooth convex function. By using the composite property of $g_{\kappa}(\varepsilon; \cdot)$ and eigenvalue function $\Lambda(X)$, we find the smooth approximate function to the sum of the $\kappa$ largest eigenvalue function.