I have an otherwise standard LP or PSD QP problem as below:

$\min\limits_x {c}' x$ subject to $Ax\leq b$ or $\min\limits_x \frac{1}{2}{x}' Qx + {c}' x$ subject to $Ax\leq b$

the only exception is that the coefficients, $[a_{i,j}]$, in $A$ are not so constant – they can be changed slightly via other less convenient means (not through x).

My questions: (1) How to effectively find out which $[a_{i,j}]$’s are more impactive for further reducing the objective function? (There will be some cost to investigate if any specific $a_{i,j}$ can be modified.)

(2) Once I know which $a_{i,j}$'s are easily changeable and by how much, how to append some selected $a_{i,j}$'s into the decision variables without having to deal with the quadratic (possibly non-convex) constraints (Online solution time is important to me). Is there any iterative LP/QP scheme that can solve the new optimization problem? Sub-optimal solution (wrt $a_{i,j}$) is acceptable.