An entropic linearized program is defined as follows: \begin{equation*} \begin{array}{ll@{}ll} \text{maximize} & \displaystyle \langle c,x\rangle-\frac{1}{\eta}H(x) \\ \text{subject to}& \displaystyle Ax=b \\ \end{array} \end{equation*} where $H(x)=\sum_{i=1}^d x_i\log x_i$, $\eta$ is a regularization parameter.
Suppose now that we have two different programs with different coefficients $c_1,c_2$, can we bound the distance of solution $x_1,x_2$ in terms of $c_1,c_2$?
Intuitively, when $c_1,c_2$ are close, the solution should not change too much, let alone having the entropic function to regularize it.