Variational Analysis of the Ky Fan $k$-norm

TL;DR

This paper investigates the variational properties of the Ky Fan k-norm of matrices, introducing concepts like nondegeneracy and strict complementarity, and deriving formulas essential for stability analysis in nonlinear optimization.

Contribution

It provides explicit formulas for the conjugate of the second order directional derivative of the Ky Fan k-norm, advancing theoretical understanding for optimization involving this norm.

Findings

Explicit formulas for conjugate functions derived

Concepts of nondegeneracy and strict complementarity introduced

Foundations laid for sensitivity and stability analysis

Abstract

In this paper, we will study some variational properties of the Ky Fan $k$-norm $\theta=\|\cdot\|_{(k)}$ of matrices, which are closed related to a class of basic nonlinear optimization problems involving the Ky Fan $k$-norm. In particular, for the basic nonlinear optimization problems, we will introduce the concept of nondegeneracy, strict complementarity and the critical cones associated with the generalized equations. Finally, we present the explicit formulas of the conjugate function of the parabolic second order directional derivative of $\theta$, which will be referred to as the sigma term of the second order optimality conditions. The results obtain in this paper provide the necessary theoretical foundations for future work on sensitivity and stability analysis of the nonlinear optimization problems involving the Ky Fan $k$-norm.