125_Linearized Subproblem

Based on what was written in the previous post(124_Linearization), I plan to explain the process of transforming a problem into a linearized subproblem using an example.

 

 

In the given problem, linearize the cost and constraint function based on the point ${x^{(0)}} = \left( {1,1} \right)$. By substituting ${x^{(0)}}$ into the cost and constraint functions and calculating, the results are as follows.

 

 

Also, calculating the gradient at ${x^{(0)}}$ yields the following.

 

 

Using the Taylor expansion to find the linearized cost function at ${x^{(0)}}$ results in the following.

 

 

Similarly, linearizing the constraint function results in the following.

 

 

Therefore, the linearized subproblem is defined as minimizing the cost function in equation $\bar f$, subject to the above constraint conditions. The matrix ${\bf{A}}$, and vectors ${\bf{b}}$ and ${\bf{c}}$ from the equation can be determined as follows.

 

 

Therefore, the linearized subproblem can be written in terms of ${d_1}$ and ${d_2}$, as well as in terms of the original variables ${x_1}$ and ${x_2}$. Replacing ${\bf{d}} = (x - {x^{(0)}})$ in the previous expression formulas, it can be summarized as follows.