my question is about functions of the following form:
$$ f(t) = \max_{\mathbf{x}}~ \mathbf{c^T x} ~ {\rm s.t. \mathbf{Ax} +t \cdot \mathbf{a} \leq \mathbf{b}}, $$ where $\mathbf{x},\mathbf{b}, $ and $\mathbf{a} $ are vectors and $\mathbf{A} $ is matrix.
Here, the evaluation of $f(t)$ requires to compute the solution of linear optimization problem. I wonder if there is literature available on such functions. In particular I would like to have information about
- the form and properties
- extrema
of this function. Moreover, I would like to know if there is an simple way to obtain function evaluations that do not require to solve an optimization problem.