Minimization problem

May 2009
301
34
Suppose we have the following minimization problem:

\(\displaystyle \text{minimize}_{z \in \mathbb{R}^{N}} \sum_{i=1}^{N} \frac{|z_i|}{(|z_{n,i}|+ \varepsilon_n)^{1-q}} \) subject to \(\displaystyle A \bold{z} = \bold{y} \). How do we prove the following:

Proposition. For any nonincreasing sequence \(\displaystyle (\varepsilon_n) \) of positive numbers and for any initial vector \(\displaystyle \bold{z}_0 \) satisfying \(\displaystyle A \bold{z}_0 = \bold{y} \), the sequence above admits a convergent subsequence?