My Math Forum replace indicator functions with convex conjugates?

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 February 27th, 2014, 05:56 AM #1 Newbie   Joined: Feb 2014 Posts: 1 Thanks: 0 replace indicator functions with convex conjugates? I'm trying to get rid of the indicator functions in the following convex optimization problem. Let ${\mathcal {B}}:=\{B_i:i \in N \}$ be a countable set of convex subsets of $R^n, n \in N$, and assume that the vectors $y_j \in R^n,~j=1..m$ are given for some $m \in N$. $\min_{x_1,\dots,x_m} \sum_{i \in N}|\sum_{j=1}^{m} I_{B_i}(x_j)-\sum_{j=1}^{m} I_{B_i}(y_j)|$ where $x_j \in R^n,~j=1..m$ and $I_B(x)= 1$ if $x \in B$ and $I_B(x)=0$ if $x \notin B.$ So here's my approach so far: 1. Getting rid of $|\cdot|$: $\min_{x_1,\dots,x_m} \sum_{i \in N} \mu_i$ subject to: $\sum_{j=1}^{m} I_{B_i}(x_j)-\sum_{j=1}^{m} I_{B_i}(y_j) \leq \mu_i, i \in N$ $-\sum_{j=1}^{m} I_{B_i}(x_j)+\sum_{j=1}^{m} I_{B_i}(y_j) \leq \mu_i, i \in N$ 2. Dual: $\inf_{x_1,\dots,x_m}\sum_{i \in N} \mu_i+\sum_{i \in N}\lambda_i(\sum_{j=1}^{m} I_{B_i}(x_j)-\sum_{j=1}^{m} I_{B_i}(y_j) - \mu_i)- \sum_{i \in N}\nu_i(\sum_{j=1}^{m} I_{B_i}(x_j)-\sum_{j=1}^{m} I_{B_i}(y_j) - \mu_i)$ subject to $\nu_i\geq 0, ~\lambda_i\geq 0$ 3. At this point I'd like to get rid of the indicator functions, perhaps using their convex conjugates. But I'm a bit stuck. Any ideas? Thanks!

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