optimization - How to understand this Risk Parity Algorithm?

I am trying to understand an optimization algorithm to achieve risk parity in a portfolio. I need some help figuring out the notation in the following formula:

I found this on THIS paper.

I understand the following, if you could help me by pointing any mistake, would be great!

I understand that this algorithm is suppossed to iterate the allocation for each asset at a time.

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  $x^*_i$ : The iteration n+1 of asset i.

  
  


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  $σ_i$ : The standard deviation of Asset i

  
  


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  $x_j$ : allocation for each asset j

  
  


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  $\sigma_j$ : The standard deviation of asset j

  
  


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  $\rho_{i,j}$ : This is my biggest question. WHAT is this?

  
  
  


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  $b_i$ : The risk budget for the asset, which for risk parity is $\frac{1}{n}$

  
  


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  $\sigma(x)$ : The standard deviation of the portfolio

  
  

What am I missing?

Answer

Your question seems very simple. The $\rho_{ij}$ are the correlations between asset i and asset j, in other words these are the elements of the correlation matrix. This notation is very standard in portfolio optimization problems. The number of securities n, the n-by-n correlation matrix R and the n vector of $\sigma_j$'s are the main inputs of a risk parity problem.