Convex optimization

[magnolias]

Hi, I have to replay the results in this paper, I have some doubts in the section IIIA, which shows the ellipsoid uncertainty model.

I don't understand the equation (at the end of page 5):

(1)

[latex]a(\theta)[/latex] is a 14-element column vector, [latex]\Sigma[/latex] is a 14-by-14 - matrix, [latex]\bar{a}[/latex] is a 14-element column vector, so I can not understand how they performed this equation.

I tried to explain it based on other equations:

• The quadratic form Q shows that P must be a 210-by-210 - matrix:

(2)

 

• But the equation at the end of the page shows that [latex]\bar{P}[/latex] is a 14-by-14 - matrix (P and [latex]\bar{P}[/latex] must have the same size):

(3)

I also read other references and I found this paper, which used the same model (in section III). I believe that they explained the reason for choosing this model is: "using the mean as the center and an inflated covariance" (page 9), so I change (1), (2) and (3):

[latex]\bar{p}_i = vec((a(\theta_i), \Sigma(2\theta_{sur}-\theta_i)) - (\bar{a},\bar{\Sigma}))[/latex]

 

[latex]\alpha = max(vec((a(\theta_i), \Sigma(2\theta_{sur}-\theta_i)) - (\bar{a},\bar{\Sigma}))^* \bar{P}^{-1}(vec((a(\theta_i), \Sigma(2\theta_{sur}-\theta_i)) - (\bar{a},\bar{\Sigma}))))[/latex]

 

But I have another problem: because [latex]\bar{p}_i[/latex] is a column vector, [latex]rank (\bar{p}_i \bar{p}_i^*) =1[/latex], and [latex]rank (\bar{P}) <=N[/latex]. For N=64 (as mentioned in the first paper),[latex] \bar{P}[/latex] is a 210-by-210 - matrix and [latex]rank (\bar{P}) <=64[/latex],[latex] \bar{P}[/latex] is not an invertible matrix.

I try with the pseudoinverse matrix (pinv function in Matlab), but [latex] pinv(\bar{P}) [/latex] is not a positive definite matrix, so the constrain: is invalid (because Q may return a complex value, it can not be compare with 1).
Now I don't have any idea to continue this job. Please help me. (I tried to contact with the authors, but they do not reply )