Delta-rational chord: Difference between revisions
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<math> \beta_i : \beta_i + \delta_{i,1} : \cdots : \beta_i + \sum_{l_i=1}^{n_i} \delta_{i,l_i}, </math> | <math> \beta_i : \beta_i + \delta_{i,1} : \cdots : \beta_i + \sum_{l_i=1}^{n_i} \delta_{i,l_i}, </math> | ||
each chord being approximated by | |||
<math> \beta_i : \beta_ir_{i,1} : \cdots : \beta_i r_{i,n_i}. </math> | |||
This implies the constraint | |||
<math> 0 < \beta_1 < \beta_2 < \cdots < \beta_m </math> | <math> 0 < \beta_1 < \beta_2 < \cdots < \beta_m </math> | ||
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and turning the objective function into a sum of univariate objective functions | and turning the objective function into a sum of univariate objective functions | ||
<math> \displaystyle{ \sum_{k_i=1}^{n_i} \Bigg( \beta_i r_{k_i} - \beta_i - \sum_{{l_i}=1}^i \delta_{i,l_i} \Bigg)^2. } </math> | <math> \displaystyle{ \sum_{k_i=1}^{n_i} \Bigg( \beta_i r_{i,k_i} - \beta_i - \sum_{{l_i}=1}^i \delta_{i,l_i} \Bigg)^2. } </math> | ||
The Hessian of the resulting objective function is positive-definite, thus the global optimum can be found by setting all individual partial derivatives to zero. | The Hessian of the resulting objective function is positive-definite, thus the global optimum can be found by setting all individual partial derivatives to zero. | ||