My questions are originated from the book Convex Optimization written by Stephen P. Boyd, where a proof of a global inequality of perturbation analysis puzzles me. And here are my questions:
- In the proof of the inequality $5.57$, how can the first inequality hold when we might have two different domains for unperturbed and perturbed problems? (I can only get the inequality for all feasible points of unperturbed problem, while having no ideas about the points belonging to the perturbed problem which are not feasible for the unperturbed problem)
- Is this true for a problem of a non-convex objective with a convex inequality constraint (suppose our perturbation is always greater than zero)? Or is there any weaker result for non-convex problems I can resort to?
Those two problems are:
Unperturbed problem \begin{equation} \begin{array} \displaystyle \text{minimize} & f_{0}(x)\\ \textrm{s.t.} & f_{i}(x) \leq 0, i=1,\dots, m \\ & h_{i}(x) = 0, i=1,\dots, p\\ \end{array} \end{equation}
Perturbed problem \begin{equation} \begin{array} \displaystyle \text{minimize} & f_{0}(x)\\ \textrm{s.t.} & f_{i}(x) \leq u_{i}, i=1,\dots, m \\ & h_{i}(x) = v_{i}, i=1,\dots, p\\ \end{array} \end{equation}
Below is the image of the captured content of the inequality and its proof provided by the author.
