In Numerical Optimization book by Nocedal, page 19, The author describes trust region method as follow:
In the second algorithmic strategy, known as trust region, the information gathered about $f$ is used to construct a model function $m_k$ whose behavior near the current point $x_k$ is similar to that of the actual objective function $f$.
we find the candidate step $p$ by approximately solving the following subproblem: $$\min_p ~~m_k(x_k + p)$$ where $x_k + p$ lies inside the trust region.
The model $m_k$ in is usually defined to be a quadratic function of the form $$ m_k(x_k + p) = f_k + p^\top \nabla f_k + \frac 1 2 p^\top B_k p $$
I have a naive question to ask: Specifically, I understand how to approximate $f$ but not why: we can use Taylor expansion to approximate, but I do not understand why we need approximate at fist place? Is that because we have a simpler system (quadratic form) that we can solve? Note, the approximation (quadratic form) may not be convex, as shown in the example in book page 20.