Optimal Value of a Cost Function as a Function of the Constraining variable Consider the optimization problem :
$
\textrm{min } f(\mathbf{x})  
$
$       
\textrm{subject to } \sum_i b_ix_i \leq a
$
Using duality and numerical methods (with subgradient method) i.e.
$d = \textrm{max}_\lambda \{ \textrm{inf}_x ( f(\mathbf{x}) - \lambda(\sum_i b_ix_i - a)) \}$ .
we can obtain the optimal cost.
Now I want to express $d$ as a function of $a$ to calculate the optimal cost as a function of the constraining variable. How should I know if $d = d(a)$ is convex / concave in $a$  , if $f$ is convex and $x$ is in a convex set and the problem fullfils Slater conditions and so on?
 Does anyone know where  can I find some theory about expressing the dual function as a function of the constraining variable and the properties of this function definition?
Kind regards

It could perhaps be useful to know that you are essentially asking about properties of the value function in parametric programming, or multi-parametric programming to be completely general.
https://en.wikipedia.org/wiki/Parametric_programming
And regarding convexity, the answer is yes, see e.g., 
Convexity and Concavity Properties of the Optimal Value Function in Parametric Nonlinear Programming A. V. FIACCO 3 AND J. KYPARISIS 4 
https://apps.dtic.mil/dtic/tr/fulltext/u2/a138202.pdf
